Research ArticleOn Positive Solutions and Mann Iterative Schemes of a ThirdOrder Difference Equation
Zeqing Liu1 Heng Wu1 Shin Min Kang2 and Young Chel Kwun3
1 Department of Mathematics Liaoning Normal University Dalian Liaoning 116029 China2Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea3 Department of Mathematics Dong-A University Pusan 614-714 Republic of Korea
Correspondence should be addressed to Young Chel Kwun yckwundauackr
Received 14 October 2013 Accepted 16 December 2013 Published 28 January 2014
Academic Editor Zhi-Bo Huang
Copyright copy 2014 Zeqing Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinearneutral delay difference equation are proved Six examples are given to illustrate the results presented in this paper
1 Introduction and Preliminaries
Recently many researchers studied the oscillation nonoscil-lation and existence of solutions for linear and nonlinearsecond and third order difference equations and systemssee for example [1ndash23] and the references cited therein Bymeans of the Reccati transformation techniques Saker [18]discussed the third order difference equation
Δ3119909119899+ 119901119899119909119899+1
= 0 forall119899 ge 1198990 (1)
and presented some sufficient conditions which ensure thatall solutions are to be oscillatory or tend to zero Utilizing theSchauder fixed point theorem Yan and Liu [22] proved theexistence of a bounded nonoscillatory solution for the thirdorder difference equation
Δ3119909119899+ 119891 (119899 119909
119899 119909119899minus120591) = 0 forall119899 ge 119899
0 (2)
Agarwal [2] established the oscillatory and asymptotic prop-erties for the third order nonlinear difference equation
Δ3119909119899+ 119902119899119891 (119909119899+1) = 0 forall119899 ge 1 (3)
Andruch-Sobiło andMigda [4] studied the third order lineardifference equation of neutral type
Δ3(119909119899minus 119901119899119909120590119899) plusmn 119902119899119909120591119899= 0 forall119899 ge 119899
0 (4)
and obtained sufficient conditions which ensure that allsolutions of the equation are oscillatory Grace andHamedani[6] discussed the difference equation
Δ3(119909119899minus 119909119899minus120591) plusmn 119902119899
1003816100381610038161003816119909119899minus1205901003816100381610038161003816
3 sgn119909119899minus120590
= 0 forall119899 ge 0 (5)
and gave some new criteria for the oscillation of all solutionsand all bounded solutions
Our goal is to discuss solvability and convergence ofthe Mann iterative schemes for the following third ordernonlinear neutral delay difference equation
Δ3(119909119899+ 119887119899119909119899minus120591) + Δℎ (119899 119909
ℎ1119899
119909ℎ2119899
119909ℎ119896119899
)
+119891 (119899 1199091198911119899
1199091198912119899
119909119891119896119899
) = 119888119899 forall119899 ge 119899
0
(6)
where 120591 119896 1198990isin N 119887
119899119899isinN1198990
119888119899119899isinN1198990
sub R ℎ 119891 isin 119862(N1198990
times
R119896R) ℎ119897119899119899isinN1198990
119891119897119899119899isinN1198990
sube N and
lim119899rarrinfin
ℎ119897119899= lim119899rarrinfin
119891119897119899= +infin 119897 isin 1 2 119896 (7)
By employing the Banach fixed point theorem and somenew techniques we establish the existence of uncountablymany positive solutions of (6) conceive a few Mann iter-ative schemes for approximating these positive solutionsand prove their convergence and the error estimates Sixnontrivial examples are included
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 470181 16 pageshttpdxdoiorg1011552014470181
2 Abstract and Applied Analysis
Throughout this paper we assume that Δ is the forwarddifference operator defined by Δ119909
119899= 119909119899+1
minus 119909119899 R =
(minusinfin +infin) R+ = [0 +infin) N0and N denote the sets of
nonnegative integers and positive integers respectively
N119905= 119899 119899 isin N with 119899 ge 119905 forall119905 isin N
120573 = min 1198990minus 120591 inf ℎ
119897119899 119891119897119899 1 le 119897 le 119896 119899 isin N
1198990
isin N
119867119899= max ℎ2
119897119899 119897 isin 1 2 119896 forall119899 isin N
1198990
119865119899= max 1198912
119897119899 119897 isin 1 2 119896 forall119899 isin N
1198990
(8)
and 119897infin
120573represents the Banach space of all real sequences
on N120573with norm
119909 = sup119899isinN120573
1003816100381610038161003816100381610038161003816
119909119899
1198992
1003816100381610038161003816100381610038161003816lt +infin for each 119909 = 119909
119899119899isinN120573
isin 119897infin
120573
119860 (119873119872) = 119909 = 119909119899119899isinN120573
isin 119897infin
120573 119873 le
119909119899
1198992le 119872 119899 isin N
120573
for any 119872 gt 119873 gt 0
(9)
It is easy to see that 119860(119873119872) is a closed and convex subsetof 119897infin120573 By a solution of (6) wemean a sequence 119909
119899119899isinN120573
witha positive integer 119879 ge 119899
0+120591+120573 such that (6) holds for all 119899 ge
119879
Lemma 1 Let 119901119905119905isinN be a nonnegative sequence and 120591 isin N
(i) If lim119899rarrinfin
(11198992) suminfin
119905=119899+1205911199052119901119905
= 0
then lim119899rarrinfin
(11198992) suminfin
119894=1suminfin
119904=119899+119894120591suminfin
119905=119904119901119905= 0
(ii) If lim119899rarrinfin
(11198992) suminfin
119905=119899+1205911199053119901119905
= 0
then lim119899rarrinfin
(11198992) suminfin
119894=1suminfin
119906=119899+119894120591suminfin
119904=119906suminfin
119905=119904119901119905= 0
Proof Note that
0 le1
1198992
infin
sum
119894=1
infin
sum
119904=119899+119894120591
infin
sum
119905=119904
119901119905
=1
1198992
infin
sum
119894=1
(
infin
sum
119905=119899+119894120591
119901119905+
infin
sum
119905=119899+1+119894120591
119901119905+
infin
sum
119905=119899+2+119894120591
119901119905+ sdot sdot sdot )
=1
1198992
infin
sum
119894=1
infin
sum
119905=119899+119894120591
(1 + 119905 minus 119899 minus 119894120591) 119901119905le1
1198992
infin
sum
119894=1
infin
sum
119905=119899+119894120591
119905119901119905
=1
1198992(
infin
sum
119905=119899+120591
119905119901119905+
infin
sum
119905=119899+2120591
119905119901119905+
infin
sum
119905=119899+3120591
119905119901119905+ sdot sdot sdot )
le1
1198992
infin
sum
119905=119899+120591
(1 +119905 minus 119899 minus 120591
120591) 119905119901119905=1
1198992
infin
sum
119905=119899+120591
119905 minus 119899
120591119905119901119905
le1
1198992120591
infin
sum
119905=119899+120591
1199052119901119905997888rarr 0 as 119899 997888rarr infin
(10)
that is
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119904=119899+119894120591
infin
sum
119905=119904
119901119905= 0 (11)
As in the proof of (10) we infer that
0 le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
infin
sum
119905=119904
119901119905
=1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119905=119906
(1 + 119905 minus 119906) 119901119905
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119905=119906
119905119901119905le
1
1198992120591
infin
sum
119905=119899+120591
1199053119901119905997888rarr 0 as 119899 997888rarr infin
(12)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
infin
sum
119905=119904
119901119905= 0 (13)
This completes the proof
2 Uncountably Many Positive Solutions andMann Iterative Schemes
In this section using the Banach fixed point theoremand Mann iterative schemes we establish the existence ofuncountably many positive solutions of (6) prove conver-gence of the Mann iterative schemes relative to these positivesolutions and compute the error estimates between theManniterative schemes and the positive solutions
Theorem 2 Assume that there exist twoconstants 119872 and 119873 with 119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying1003816100381610038161003816119891 (119899 1199061 1199062 119906119896) minus 119891 (119899 1199061 1199062 119906119896)
1003816100381610038161003816
le 119875119899max 1003816100381610038161003816119906119897 minus 119906119897
1003816100381610038161003816 1 le 119897 le 119896
1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896) minus ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816
le 119877119899max 1003816100381610038161003816119906119897 minus 119906119897
1003816100381610038161003816 1 le 119897 le 119896
forall (119899 119906119897 119906119897) isin N1198990
times (R+ 0)2
1 le 119897 le 119896
(14)
1003816100381610038161003816119891 (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119876119899
1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119882119899
forall (119899 119906119897) isin N1198990
times (R+ 0) 1 le 119897 le 119896
(15)
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (16)
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (17)
119887119899= minus1 eventually (18)
Abstract and Applied Analysis 3
Then one has the following(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
+
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(19)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin 119860(119873119872) of (6)with lim
119899rarrinfin119911119899= +infin and has the following error estimate
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] such thatinfin
sum
119898=0
120572119898= +infin (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Firstly we show that (a) holds Put 119871 isin (119873119872) Itfollows from (16)sim(18) that there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (22)
1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119873
(23)
119887119899= minus1 forall119899 ge 119879 (24)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879 119878119871119909119879 120573 le 119899 lt 119879
(25)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In light of (14) (15) (22)(23) and (25) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872)
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816100381610038161003816
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
4 Abstract and Applied Analysis
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
lt min 119872 minus 119871 119871 minus 119873
(26)which yield that
119878119871 (119860 (119873119872)) sube 119860 (119873119872)
1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119860 (119873119872)
(27)
which implies that 119878119871is a contraction in 119860(119873119872) The
Banach fixed point theorem and (27) ensure that 119878119871has a
unique fixed point 119911 = 119911119899119899isinN120573
isin 119860(119873119872) that is
119911119899= 1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879
119911119899minus120591
= (119899 minus 120591)2119871
+
infin
sum
119894=1
infin
sum
119906=119899+(119894minus1)120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
)
minus 119888119905] forall119899 ge 119879 + 120591
(28)which mean that119911119899minus 119911119899minus120591
= (2119899120591 minus 1205912) 119871
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
(29)which yields thatΔ (119911119899minus 119911119899minus120591)
= 2120591119871 +
infin
sum
119904=119899
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
Δ2(119911119899minus 119911119899minus120591)
= minusℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
+
infin
sum
119905=119899
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905] forall119899 ge 119879 + 120591
(30)
which gives that
Δ3(119911119899minus 119911119899minus120591)
= minusΔℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
minus 119891 (119905 1199111198911119899
1199111198912119899
119911119891119896119899
) + 119888119899 forall119899 ge 119879 + 120591
(31)
which together with (24) implies that 119911 = 119911119899119899isinN120573
is apositive solution of (6) in 119860(119873119872) Note that
119873 le119911119899
1198992le 119872 forall119899 isin N
120573 (32)
which guarantees that lim119899rarrinfin
119911119899= +infin It follows from (19)
(22) (24) (25) and (27) that for any 119898 isin N0and 119899 ge 119879
1003816100381610038161003816100381610038161003816
119909119898+1119899
1198992minus119911119899
1198992
1003816100381610038161003816100381610038161003816
=1
1198992
1003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)] minus 119911
119899
1003816100381610038161003816100381610038161003816100381610038161003816
le (1 minus 120572119898)
1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816
1198992+ 120572119898
1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816
1198992
le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817
le [1 minus (1 minus 120579) 120572119898]1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 119899 ge 119879
(33)
which implies that
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (34)
That is (20) holds Thus Lemma 1 (20) and (21) guaranteethat lim
119898rarrinfin119909119898= 119911
Next we show that (b) holds Let 1198711 1198712
isin
(119873119872) and 1198711= 1198712 As in the proof of (a) we deduce
similarly that for each 119888 isin 1 2 there exist constants 120579119888isin
(0 1) and 119879119888ge 1198990+ 120591 + 120573 and a mapping 119878
119871119888
satisfying
Abstract and Applied Analysis 5
(22)sim(27) where 120579 119871 and 119879 are replaced by 120579119888 119871119888 and 119879
119888
respectively and the mapping 119878119871119888
has a fixed point 119911119888 =
119911119888
119899119899isinN120573
isin 119860(119873119872) which is a positive solution of (6) in119860(119873119872) with lim
119899rarrinfin119911119888
119899= +infin It follows that
119911119888
119899= 1198992119871119888
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911119888
ℎ1119904
119911119888
ℎ2119904
119911119888
ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 119911119888
1198911119905
119911119888
1198912119905
119911119888
119891119896119905
) minus 119888119905]
forall119899 ge 119879119888
(35)
which together with (14) and (20) means that for 119899 ge
max1198791 1198792
100381610038161003816100381610038161003816100381610038161003816
1199111
119899
1198992minus1199112
119899
1198992
100381610038161003816100381610038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119911
1
ℎ1119904
1199111
ℎ2119904
1199111
ℎ119896119904
)
minus ℎ (119904 1199112
ℎ1119904
1199112
ℎ2119904
1199112
ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119911
1
1198911119905
1199111
1198912119905
1199111
119891119896119905
) minus 119891 (119905 1199112
1198911119905
1199112
1198912119905
1199112
119891119896119905
)10038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119911
1
ℎ119897119904
minus 1199112
ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119911
1
119891119897119905
minus 1199112
119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
max 11987921 1198792
2
infin
sum
119894=1
infin
sum
119906=max1198791 1198792+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
(36)
which yields that
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge
10038161003816100381610038161198711 minus 11987121003816100381610038161003816
1 +max 1205791 1205792gt 0 (37)
that is 1199111 = 1199112 This completes the proof
Theorem 3 Assume that there exist two constants119872 and 119873with 119872 gt 119873 gt 0 and four nonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) and
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (38)
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)
119887119899= 1 eventually (40)
Then one has the following
(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
minus
infin
sum
119894=1
119879+2119894120591minus1
sum
119906=119879+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(41)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
6 Abstract and Applied Analysis
Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (42)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873
(43)
119887119899= 1 forall119899 ge 119879 (44)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus119888119905] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(45)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909
119899119899isinN120573
119910 = 119910119899119899isinN120573
isin
119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])
lt min 119872 minus 119871 119871 minus 119873
(46)
which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof
Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)
Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any
1199090
= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated bythe scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871 minus 119887119899119909119898119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus 119887119879119909119898119879minus120591
minus
infin
sum
119906=119879
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(48)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Throughout this paper we assume that Δ is the forwarddifference operator defined by Δ119909
119899= 119909119899+1
minus 119909119899 R =
(minusinfin +infin) R+ = [0 +infin) N0and N denote the sets of
nonnegative integers and positive integers respectively
N119905= 119899 119899 isin N with 119899 ge 119905 forall119905 isin N
120573 = min 1198990minus 120591 inf ℎ
119897119899 119891119897119899 1 le 119897 le 119896 119899 isin N
1198990
isin N
119867119899= max ℎ2
119897119899 119897 isin 1 2 119896 forall119899 isin N
1198990
119865119899= max 1198912
119897119899 119897 isin 1 2 119896 forall119899 isin N
1198990
(8)
and 119897infin
120573represents the Banach space of all real sequences
on N120573with norm
119909 = sup119899isinN120573
1003816100381610038161003816100381610038161003816
119909119899
1198992
1003816100381610038161003816100381610038161003816lt +infin for each 119909 = 119909
119899119899isinN120573
isin 119897infin
120573
119860 (119873119872) = 119909 = 119909119899119899isinN120573
isin 119897infin
120573 119873 le
119909119899
1198992le 119872 119899 isin N
120573
for any 119872 gt 119873 gt 0
(9)
It is easy to see that 119860(119873119872) is a closed and convex subsetof 119897infin120573 By a solution of (6) wemean a sequence 119909
119899119899isinN120573
witha positive integer 119879 ge 119899
0+120591+120573 such that (6) holds for all 119899 ge
119879
Lemma 1 Let 119901119905119905isinN be a nonnegative sequence and 120591 isin N
(i) If lim119899rarrinfin
(11198992) suminfin
119905=119899+1205911199052119901119905
= 0
then lim119899rarrinfin
(11198992) suminfin
119894=1suminfin
119904=119899+119894120591suminfin
119905=119904119901119905= 0
(ii) If lim119899rarrinfin
(11198992) suminfin
119905=119899+1205911199053119901119905
= 0
then lim119899rarrinfin
(11198992) suminfin
119894=1suminfin
119906=119899+119894120591suminfin
119904=119906suminfin
119905=119904119901119905= 0
Proof Note that
0 le1
1198992
infin
sum
119894=1
infin
sum
119904=119899+119894120591
infin
sum
119905=119904
119901119905
=1
1198992
infin
sum
119894=1
(
infin
sum
119905=119899+119894120591
119901119905+
infin
sum
119905=119899+1+119894120591
119901119905+
infin
sum
119905=119899+2+119894120591
119901119905+ sdot sdot sdot )
=1
1198992
infin
sum
119894=1
infin
sum
119905=119899+119894120591
(1 + 119905 minus 119899 minus 119894120591) 119901119905le1
1198992
infin
sum
119894=1
infin
sum
119905=119899+119894120591
119905119901119905
=1
1198992(
infin
sum
119905=119899+120591
119905119901119905+
infin
sum
119905=119899+2120591
119905119901119905+
infin
sum
119905=119899+3120591
119905119901119905+ sdot sdot sdot )
le1
1198992
infin
sum
119905=119899+120591
(1 +119905 minus 119899 minus 120591
120591) 119905119901119905=1
1198992
infin
sum
119905=119899+120591
119905 minus 119899
120591119905119901119905
le1
1198992120591
infin
sum
119905=119899+120591
1199052119901119905997888rarr 0 as 119899 997888rarr infin
(10)
that is
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119904=119899+119894120591
infin
sum
119905=119904
119901119905= 0 (11)
As in the proof of (10) we infer that
0 le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
infin
sum
119905=119904
119901119905
=1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119905=119906
(1 + 119905 minus 119906) 119901119905
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119905=119906
119905119901119905le
1
1198992120591
infin
sum
119905=119899+120591
1199053119901119905997888rarr 0 as 119899 997888rarr infin
(12)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
infin
sum
119905=119904
119901119905= 0 (13)
This completes the proof
2 Uncountably Many Positive Solutions andMann Iterative Schemes
In this section using the Banach fixed point theoremand Mann iterative schemes we establish the existence ofuncountably many positive solutions of (6) prove conver-gence of the Mann iterative schemes relative to these positivesolutions and compute the error estimates between theManniterative schemes and the positive solutions
Theorem 2 Assume that there exist twoconstants 119872 and 119873 with 119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying1003816100381610038161003816119891 (119899 1199061 1199062 119906119896) minus 119891 (119899 1199061 1199062 119906119896)
1003816100381610038161003816
le 119875119899max 1003816100381610038161003816119906119897 minus 119906119897
1003816100381610038161003816 1 le 119897 le 119896
1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896) minus ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816
le 119877119899max 1003816100381610038161003816119906119897 minus 119906119897
1003816100381610038161003816 1 le 119897 le 119896
forall (119899 119906119897 119906119897) isin N1198990
times (R+ 0)2
1 le 119897 le 119896
(14)
1003816100381610038161003816119891 (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119876119899
1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119882119899
forall (119899 119906119897) isin N1198990
times (R+ 0) 1 le 119897 le 119896
(15)
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (16)
lim119899rarrinfin
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (17)
119887119899= minus1 eventually (18)
Abstract and Applied Analysis 3
Then one has the following(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
+
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(19)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin 119860(119873119872) of (6)with lim
119899rarrinfin119911119899= +infin and has the following error estimate
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] such thatinfin
sum
119898=0
120572119898= +infin (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Firstly we show that (a) holds Put 119871 isin (119873119872) Itfollows from (16)sim(18) that there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (22)
1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119873
(23)
119887119899= minus1 forall119899 ge 119879 (24)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879 119878119871119909119879 120573 le 119899 lt 119879
(25)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In light of (14) (15) (22)(23) and (25) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872)
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816100381610038161003816
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
4 Abstract and Applied Analysis
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
lt min 119872 minus 119871 119871 minus 119873
(26)which yield that
119878119871 (119860 (119873119872)) sube 119860 (119873119872)
1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119860 (119873119872)
(27)
which implies that 119878119871is a contraction in 119860(119873119872) The
Banach fixed point theorem and (27) ensure that 119878119871has a
unique fixed point 119911 = 119911119899119899isinN120573
isin 119860(119873119872) that is
119911119899= 1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879
119911119899minus120591
= (119899 minus 120591)2119871
+
infin
sum
119894=1
infin
sum
119906=119899+(119894minus1)120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
)
minus 119888119905] forall119899 ge 119879 + 120591
(28)which mean that119911119899minus 119911119899minus120591
= (2119899120591 minus 1205912) 119871
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
(29)which yields thatΔ (119911119899minus 119911119899minus120591)
= 2120591119871 +
infin
sum
119904=119899
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
Δ2(119911119899minus 119911119899minus120591)
= minusℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
+
infin
sum
119905=119899
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905] forall119899 ge 119879 + 120591
(30)
which gives that
Δ3(119911119899minus 119911119899minus120591)
= minusΔℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
minus 119891 (119905 1199111198911119899
1199111198912119899
119911119891119896119899
) + 119888119899 forall119899 ge 119879 + 120591
(31)
which together with (24) implies that 119911 = 119911119899119899isinN120573
is apositive solution of (6) in 119860(119873119872) Note that
119873 le119911119899
1198992le 119872 forall119899 isin N
120573 (32)
which guarantees that lim119899rarrinfin
119911119899= +infin It follows from (19)
(22) (24) (25) and (27) that for any 119898 isin N0and 119899 ge 119879
1003816100381610038161003816100381610038161003816
119909119898+1119899
1198992minus119911119899
1198992
1003816100381610038161003816100381610038161003816
=1
1198992
1003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)] minus 119911
119899
1003816100381610038161003816100381610038161003816100381610038161003816
le (1 minus 120572119898)
1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816
1198992+ 120572119898
1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816
1198992
le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817
le [1 minus (1 minus 120579) 120572119898]1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 119899 ge 119879
(33)
which implies that
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (34)
That is (20) holds Thus Lemma 1 (20) and (21) guaranteethat lim
119898rarrinfin119909119898= 119911
Next we show that (b) holds Let 1198711 1198712
isin
(119873119872) and 1198711= 1198712 As in the proof of (a) we deduce
similarly that for each 119888 isin 1 2 there exist constants 120579119888isin
(0 1) and 119879119888ge 1198990+ 120591 + 120573 and a mapping 119878
119871119888
satisfying
Abstract and Applied Analysis 5
(22)sim(27) where 120579 119871 and 119879 are replaced by 120579119888 119871119888 and 119879
119888
respectively and the mapping 119878119871119888
has a fixed point 119911119888 =
119911119888
119899119899isinN120573
isin 119860(119873119872) which is a positive solution of (6) in119860(119873119872) with lim
119899rarrinfin119911119888
119899= +infin It follows that
119911119888
119899= 1198992119871119888
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911119888
ℎ1119904
119911119888
ℎ2119904
119911119888
ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 119911119888
1198911119905
119911119888
1198912119905
119911119888
119891119896119905
) minus 119888119905]
forall119899 ge 119879119888
(35)
which together with (14) and (20) means that for 119899 ge
max1198791 1198792
100381610038161003816100381610038161003816100381610038161003816
1199111
119899
1198992minus1199112
119899
1198992
100381610038161003816100381610038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119911
1
ℎ1119904
1199111
ℎ2119904
1199111
ℎ119896119904
)
minus ℎ (119904 1199112
ℎ1119904
1199112
ℎ2119904
1199112
ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119911
1
1198911119905
1199111
1198912119905
1199111
119891119896119905
) minus 119891 (119905 1199112
1198911119905
1199112
1198912119905
1199112
119891119896119905
)10038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119911
1
ℎ119897119904
minus 1199112
ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119911
1
119891119897119905
minus 1199112
119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
max 11987921 1198792
2
infin
sum
119894=1
infin
sum
119906=max1198791 1198792+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
(36)
which yields that
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge
10038161003816100381610038161198711 minus 11987121003816100381610038161003816
1 +max 1205791 1205792gt 0 (37)
that is 1199111 = 1199112 This completes the proof
Theorem 3 Assume that there exist two constants119872 and 119873with 119872 gt 119873 gt 0 and four nonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) and
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (38)
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)
119887119899= 1 eventually (40)
Then one has the following
(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
minus
infin
sum
119894=1
119879+2119894120591minus1
sum
119906=119879+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(41)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
6 Abstract and Applied Analysis
Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (42)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873
(43)
119887119899= 1 forall119899 ge 119879 (44)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus119888119905] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(45)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909
119899119899isinN120573
119910 = 119910119899119899isinN120573
isin
119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])
lt min 119872 minus 119871 119871 minus 119873
(46)
which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof
Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)
Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any
1199090
= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated bythe scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871 minus 119887119899119909119898119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus 119887119879119909119898119879minus120591
minus
infin
sum
119906=119879
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(48)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied Analysis 3
Then one has the following(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
+
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(19)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin 119860(119873119872) of (6)with lim
119899rarrinfin119911119899= +infin and has the following error estimate
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] such thatinfin
sum
119898=0
120572119898= +infin (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Firstly we show that (a) holds Put 119871 isin (119873119872) Itfollows from (16)sim(18) that there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (22)
1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119873
(23)
119887119899= minus1 forall119899 ge 119879 (24)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879 119878119871119909119879 120573 le 119899 lt 119879
(25)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In light of (14) (15) (22)(23) and (25) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872)
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816100381610038161003816
1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
4 Abstract and Applied Analysis
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
lt min 119872 minus 119871 119871 minus 119873
(26)which yield that
119878119871 (119860 (119873119872)) sube 119860 (119873119872)
1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119860 (119873119872)
(27)
which implies that 119878119871is a contraction in 119860(119873119872) The
Banach fixed point theorem and (27) ensure that 119878119871has a
unique fixed point 119911 = 119911119899119899isinN120573
isin 119860(119873119872) that is
119911119899= 1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879
119911119899minus120591
= (119899 minus 120591)2119871
+
infin
sum
119894=1
infin
sum
119906=119899+(119894minus1)120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
)
minus 119888119905] forall119899 ge 119879 + 120591
(28)which mean that119911119899minus 119911119899minus120591
= (2119899120591 minus 1205912) 119871
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
(29)which yields thatΔ (119911119899minus 119911119899minus120591)
= 2120591119871 +
infin
sum
119904=119899
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
Δ2(119911119899minus 119911119899minus120591)
= minusℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
+
infin
sum
119905=119899
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905] forall119899 ge 119879 + 120591
(30)
which gives that
Δ3(119911119899minus 119911119899minus120591)
= minusΔℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
minus 119891 (119905 1199111198911119899
1199111198912119899
119911119891119896119899
) + 119888119899 forall119899 ge 119879 + 120591
(31)
which together with (24) implies that 119911 = 119911119899119899isinN120573
is apositive solution of (6) in 119860(119873119872) Note that
119873 le119911119899
1198992le 119872 forall119899 isin N
120573 (32)
which guarantees that lim119899rarrinfin
119911119899= +infin It follows from (19)
(22) (24) (25) and (27) that for any 119898 isin N0and 119899 ge 119879
1003816100381610038161003816100381610038161003816
119909119898+1119899
1198992minus119911119899
1198992
1003816100381610038161003816100381610038161003816
=1
1198992
1003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)] minus 119911
119899
1003816100381610038161003816100381610038161003816100381610038161003816
le (1 minus 120572119898)
1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816
1198992+ 120572119898
1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816
1198992
le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817
le [1 minus (1 minus 120579) 120572119898]1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 119899 ge 119879
(33)
which implies that
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (34)
That is (20) holds Thus Lemma 1 (20) and (21) guaranteethat lim
119898rarrinfin119909119898= 119911
Next we show that (b) holds Let 1198711 1198712
isin
(119873119872) and 1198711= 1198712 As in the proof of (a) we deduce
similarly that for each 119888 isin 1 2 there exist constants 120579119888isin
(0 1) and 119879119888ge 1198990+ 120591 + 120573 and a mapping 119878
119871119888
satisfying
Abstract and Applied Analysis 5
(22)sim(27) where 120579 119871 and 119879 are replaced by 120579119888 119871119888 and 119879
119888
respectively and the mapping 119878119871119888
has a fixed point 119911119888 =
119911119888
119899119899isinN120573
isin 119860(119873119872) which is a positive solution of (6) in119860(119873119872) with lim
119899rarrinfin119911119888
119899= +infin It follows that
119911119888
119899= 1198992119871119888
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911119888
ℎ1119904
119911119888
ℎ2119904
119911119888
ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 119911119888
1198911119905
119911119888
1198912119905
119911119888
119891119896119905
) minus 119888119905]
forall119899 ge 119879119888
(35)
which together with (14) and (20) means that for 119899 ge
max1198791 1198792
100381610038161003816100381610038161003816100381610038161003816
1199111
119899
1198992minus1199112
119899
1198992
100381610038161003816100381610038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119911
1
ℎ1119904
1199111
ℎ2119904
1199111
ℎ119896119904
)
minus ℎ (119904 1199112
ℎ1119904
1199112
ℎ2119904
1199112
ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119911
1
1198911119905
1199111
1198912119905
1199111
119891119896119905
) minus 119891 (119905 1199112
1198911119905
1199112
1198912119905
1199112
119891119896119905
)10038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119911
1
ℎ119897119904
minus 1199112
ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119911
1
119891119897119905
minus 1199112
119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
max 11987921 1198792
2
infin
sum
119894=1
infin
sum
119906=max1198791 1198792+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
(36)
which yields that
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge
10038161003816100381610038161198711 minus 11987121003816100381610038161003816
1 +max 1205791 1205792gt 0 (37)
that is 1199111 = 1199112 This completes the proof
Theorem 3 Assume that there exist two constants119872 and 119873with 119872 gt 119873 gt 0 and four nonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) and
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (38)
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)
119887119899= 1 eventually (40)
Then one has the following
(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
minus
infin
sum
119894=1
119879+2119894120591minus1
sum
119906=119879+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(41)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
6 Abstract and Applied Analysis
Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (42)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873
(43)
119887119899= 1 forall119899 ge 119879 (44)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus119888119905] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(45)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909
119899119899isinN120573
119910 = 119910119899119899isinN120573
isin
119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])
lt min 119872 minus 119871 119871 minus 119873
(46)
which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof
Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)
Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any
1199090
= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated bythe scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871 minus 119887119899119909119898119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus 119887119879119909119898119879minus120591
minus
infin
sum
119906=119879
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(48)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119894=1
infin
sum
119906=119879+119894120591
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
lt min 119872 minus 119871 119871 minus 119873
(26)which yield that
119878119871 (119860 (119873119872)) sube 119860 (119873119872)
1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119860 (119873119872)
(27)
which implies that 119878119871is a contraction in 119860(119873119872) The
Banach fixed point theorem and (27) ensure that 119878119871has a
unique fixed point 119911 = 119911119899119899isinN120573
isin 119860(119873119872) that is
119911119899= 1198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879
119911119899minus120591
= (119899 minus 120591)2119871
+
infin
sum
119894=1
infin
sum
119906=119899+(119894minus1)120591
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
)
minus 119888119905] forall119899 ge 119879 + 120591
(28)which mean that119911119899minus 119911119899minus120591
= (2119899120591 minus 1205912) 119871
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
(29)which yields thatΔ (119911119899minus 119911119899minus120591)
= 2120591119871 +
infin
sum
119904=119899
ℎ (119904 119911ℎ1119904
119911ℎ2119904
119911ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905]
forall119899 ge 119879 + 120591
Δ2(119911119899minus 119911119899minus120591)
= minusℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
+
infin
sum
119905=119899
[119891 (119905 1199111198911119905
1199111198912119905
119911119891119896119905
) minus 119888119905] forall119899 ge 119879 + 120591
(30)
which gives that
Δ3(119911119899minus 119911119899minus120591)
= minusΔℎ (119899 119911ℎ1119899
119911ℎ2119899
119911ℎ119896119899
)
minus 119891 (119905 1199111198911119899
1199111198912119899
119911119891119896119899
) + 119888119899 forall119899 ge 119879 + 120591
(31)
which together with (24) implies that 119911 = 119911119899119899isinN120573
is apositive solution of (6) in 119860(119873119872) Note that
119873 le119911119899
1198992le 119872 forall119899 isin N
120573 (32)
which guarantees that lim119899rarrinfin
119911119899= +infin It follows from (19)
(22) (24) (25) and (27) that for any 119898 isin N0and 119899 ge 119879
1003816100381610038161003816100381610038161003816
119909119898+1119899
1198992minus119911119899
1198992
1003816100381610038161003816100381610038161003816
=1
1198992
1003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)] minus 119911
119899
1003816100381610038161003816100381610038161003816100381610038161003816
le (1 minus 120572119898)
1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816
1198992+ 120572119898
1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816
1198992
le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817
le [1 minus (1 minus 120579) 120572119898]1003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 119899 ge 119879
(33)
which implies that
1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum
119898
119894=01205721198941003817100381710038171003817119909119898 minus 119911
1003817100381710038171003817 forall119898 isin N0 (34)
That is (20) holds Thus Lemma 1 (20) and (21) guaranteethat lim
119898rarrinfin119909119898= 119911
Next we show that (b) holds Let 1198711 1198712
isin
(119873119872) and 1198711= 1198712 As in the proof of (a) we deduce
similarly that for each 119888 isin 1 2 there exist constants 120579119888isin
(0 1) and 119879119888ge 1198990+ 120591 + 120573 and a mapping 119878
119871119888
satisfying
Abstract and Applied Analysis 5
(22)sim(27) where 120579 119871 and 119879 are replaced by 120579119888 119871119888 and 119879
119888
respectively and the mapping 119878119871119888
has a fixed point 119911119888 =
119911119888
119899119899isinN120573
isin 119860(119873119872) which is a positive solution of (6) in119860(119873119872) with lim
119899rarrinfin119911119888
119899= +infin It follows that
119911119888
119899= 1198992119871119888
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911119888
ℎ1119904
119911119888
ℎ2119904
119911119888
ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 119911119888
1198911119905
119911119888
1198912119905
119911119888
119891119896119905
) minus 119888119905]
forall119899 ge 119879119888
(35)
which together with (14) and (20) means that for 119899 ge
max1198791 1198792
100381610038161003816100381610038161003816100381610038161003816
1199111
119899
1198992minus1199112
119899
1198992
100381610038161003816100381610038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119911
1
ℎ1119904
1199111
ℎ2119904
1199111
ℎ119896119904
)
minus ℎ (119904 1199112
ℎ1119904
1199112
ℎ2119904
1199112
ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119911
1
1198911119905
1199111
1198912119905
1199111
119891119896119905
) minus 119891 (119905 1199112
1198911119905
1199112
1198912119905
1199112
119891119896119905
)10038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119911
1
ℎ119897119904
minus 1199112
ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119911
1
119891119897119905
minus 1199112
119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
max 11987921 1198792
2
infin
sum
119894=1
infin
sum
119906=max1198791 1198792+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
(36)
which yields that
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge
10038161003816100381610038161198711 minus 11987121003816100381610038161003816
1 +max 1205791 1205792gt 0 (37)
that is 1199111 = 1199112 This completes the proof
Theorem 3 Assume that there exist two constants119872 and 119873with 119872 gt 119873 gt 0 and four nonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) and
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (38)
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)
119887119899= 1 eventually (40)
Then one has the following
(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
minus
infin
sum
119894=1
119879+2119894120591minus1
sum
119906=119879+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(41)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
6 Abstract and Applied Analysis
Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (42)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873
(43)
119887119899= 1 forall119899 ge 119879 (44)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus119888119905] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(45)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909
119899119899isinN120573
119910 = 119910119899119899isinN120573
isin
119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])
lt min 119872 minus 119871 119871 minus 119873
(46)
which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof
Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)
Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any
1199090
= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated bythe scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871 minus 119887119899119909119898119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus 119887119879119909119898119879minus120591
minus
infin
sum
119906=119879
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(48)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
(22)sim(27) where 120579 119871 and 119879 are replaced by 120579119888 119871119888 and 119879
119888
respectively and the mapping 119878119871119888
has a fixed point 119911119888 =
119911119888
119899119899isinN120573
isin 119860(119873119872) which is a positive solution of (6) in119860(119873119872) with lim
119899rarrinfin119911119888
119899= +infin It follows that
119911119888
119899= 1198992119871119888
+
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
ℎ (119904 119911119888
ℎ1119904
119911119888
ℎ2119904
119911119888
ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 119911119888
1198911119905
119911119888
1198912119905
119911119888
119891119896119905
) minus 119888119905]
forall119899 ge 119879119888
(35)
which together with (14) and (20) means that for 119899 ge
max1198791 1198792
100381610038161003816100381610038161003816100381610038161003816
1199111
119899
1198992minus1199112
119899
1198992
100381610038161003816100381610038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119911
1
ℎ1119904
1199111
ℎ2119904
1199111
ℎ119896119904
)
minus ℎ (119904 1199112
ℎ1119904
1199112
ℎ2119904
1199112
ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119911
1
1198911119905
1199111
1198912119905
1199111
119891119896119905
) minus 119891 (119905 1199112
1198911119905
1199112
1198912119905
1199112
119891119896119905
)10038161003816100381610038161003816
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus1
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119911
1
ℎ119897119904
minus 1199112
ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119911
1
119891119897119905
minus 1199112
119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
1198992
infin
sum
119894=1
infin
sum
119906=119899+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816
minus
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
max 11987921 1198792
2
infin
sum
119894=1
infin
sum
119906=max1198791 1198792+119894120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
ge10038161003816100381610038161198711 minus 1198712
1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817
(36)
which yields that
100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge
10038161003816100381610038161198711 minus 11987121003816100381610038161003816
1 +max 1205791 1205792gt 0 (37)
that is 1199111 = 1199112 This completes the proof
Theorem 3 Assume that there exist two constants119872 and 119873with 119872 gt 119873 gt 0 and four nonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) and
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (38)
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)
119887119899= 1 eventually (40)
Then one has the following
(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge
1198990+ 120591 + 120573 such that for each 119909
0= 119909
0119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871
minus
infin
sum
119894=1
119879+2119894120591minus1
sum
119906=119879+(2119894minus1)120591
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
) minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(41)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
6 Abstract and Applied Analysis
Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (42)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873
(43)
119887119899= 1 forall119899 ge 119879 (44)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus119888119905] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(45)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909
119899119899isinN120573
119910 = 119910119899119899isinN120573
isin
119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])
lt min 119872 minus 119871 119871 minus 119873
(46)
which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof
Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)
Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any
1199090
= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated bythe scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871 minus 119887119899119909119898119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus 119887119879119909119898119879minus120591
minus
infin
sum
119906=119879
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(48)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 =1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (42)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873
(43)
119887119899= 1 forall119899 ge 119879 (44)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871
minus
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus119888119905] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(45)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909
119899119899isinN120573
119910 = 119910119899119899isinN120573
isin
119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus 119871
10038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119894=1
119899+2119894120591minus1
sum
119906=119899+(2119894minus1)120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])
lt min 119872 minus 119871 119871 minus 119873
(46)
which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof
Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)
Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any
1199090
= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated bythe scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+1205721198981198992119871 minus 119887119899119909119898119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus 119887119879119909119898119879minus120591
minus
infin
sum
119906=119879
infin
sum
119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum
119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(48)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = 119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus 119887119872 minus119873
0 le 119887119899le 119887 lt 1 forall119899 ge 119879
(49)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus 119887119899119909119899minus120591
minus
infin
sum
119906=119899
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(50)for each 119909 = 119909
119899119899isinN120573
isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le 119887119899
100381610038161003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
(119899 minus 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 minus 120591)2
1198992
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le 1198871003817100381710038171003817119909 minus 119910
1003817100381710038171003817
+
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max ℎ2
119897119904 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 1198912
119897119905 1 le 119897 le 119896]
le [119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus 119887119872
minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus 119887119872 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873
(51)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14)(15) (38) (39) and
minus1 lt 119887 le 119887119899le 0 eventually (52)
Then one has the following
(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for
any 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by(48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 satisfying
120579 = minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905) (53)
1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 119887)119872 minus 119871 119871 minus 119873
(54)
minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by (50) By virtue of
(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573
119910 = 119910
119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le 119887119899
1003816100381610038161003816100381610038161003816
119909119899minus120591
minus 119910119899minus120591
1198992
1003816100381610038161003816100381610038161003816
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le [minus119887 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 minus 119887119872
+1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
le 119871 minus 119887119872 +1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873
(56)
That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882
119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899ge 119887 gt 1 eventually (57)
Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin
(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909
0=
1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 1205721198981198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+1205721198981198792119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(58)
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 =1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (59)
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min 119872 minus 119871 119871 minus1
119887119872 minus119873
(60)
119887119899ge 119887 gt 1 forall119899 ge 119879 (61)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[(119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
)
minus 119888119905)] 119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(62)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909
119899119899isinN120573
119910 =
119910119899119899isinN120573
isin 119860(119873119872) and 119899 ge 119879
10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
+1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
+1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le 119871 +1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816
+10038161003816100381610038161198881199051003816100381610038161003816 ]
le 119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt 119871 +min 119872 minus 119871 119871 minus1
119887119872 minus119873 le 119872
119878119871119909119899
1198992
ge 119871 minus1
119887119872
minus1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge 119871 minus1
119887119872 minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
gt 119871 minus1
119887119872 minusmin 119872 minus 119871 119871 minus
1
119887119872 minus119873 ge 119873
(63)
which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875
119899119899isinN1198990
119876119899119899isinN1198990
119877119899119899isinN1198990
and 119882119899119899isinN1198990
satisfying (14) (15) (38) (39) and
119887119899le 119887 lt minus1 eventually (64)
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
Then one has the following
(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899
0+ 120591 + 120573 such that for any 119909
0= 1199090119899119899isinN120573
isin
119860(119873119872) the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by the scheme
119909119898+1119899
=
(1 minus 120572119898) 119909119898119899
+ 120572119898 minus 1198992119871 minus
119909119898119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum119906=119899+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
119899 ge 119879 119898 ge 0
(1 minus 120572119898) 119909119898119879
+120572119898 minus 119879
2119871 minus
119909119898119879+120591
119887119879+120591
minus
infin
sum
119906=119879+120591
infin
sum119904=119906
[ℎ (119904 119909119898ℎ1119904
119909119898ℎ2119904
119909119898ℎ119896119904
)
minus
infin
sum119905=119904
(119891 (119905 1199091198981198911119905
1199091198981198912119905
119909119898119891119896119905
)
minus 119888119905)]
120573 le 119899 lt 119879 119898 ge 0
(65)
converges to a positive solution 119911 = 119911119899119899isinN120573
isin
119860(119873119872) of (6) with lim119899rarrinfin
119911119899= +infin and has the
error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in [0 1] satisfying (21)
(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)
Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 +
120573 satisfying
120579 = minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)] (66)
minus1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt min (1 + 1
119887)119872 minus 119871 119871 minus
1
119887119872 minus119873
(67)
119887119899ge 119887 gt 1 forall119899 ge 119879 (68)
Define a mapping 119878119871 119860(119873119872) rarr 119897
infin
120573by
119878119871119909119899
=
minus1198992119871 minus
119909119899+120591
119887119899+120591
minus1
119887119899+120591
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
ℎ (119904 119909ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus
infin
sum
119905=119904
[119891 (119905 1199091198911119905
1199091198912119905
119909119891119896119905
) minus 119888119905]
119899 ge 119879
119878119871119909119879 120573 le 119899 lt 119879
(69)
for each 119909 = 119909119899119899isinN120573
isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816
119878119871119909119899
1198992minus119878119871119910119899
1198992
10038161003816100381610038161003816100381610038161003816
le minus1
119887119899+120591
1003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
1198992
1003816100381610038161003816100381610038161003816
minus1
119887119899+1205911198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
[10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)
minus ℎ (119904 119910ℎ1119904
119910ℎ2119904
119910ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)
minus 119891 (119905 1199101198911119905
1199101198912119905
119910119891119896119905
)10038161003816100381610038161003816]
le minus1
119887119899+120591
100381610038161003816100381610038161003816100381610038161003816
119909119899+120591
minus 119910119899+120591
(119899 + 120591)2
100381610038161003816100381610038161003816100381610038161003816
(119899 + 120591)2
1198992
minus1
119887119899+1205911198992
infin
sum
119906=119899
infin
sum
119904=119906
[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904
10038161003816100381610038161003816 1 le 119897 le 119896
+
infin
sum
119905=119904
119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905
10038161003816100381610038161003816 1 le 119897 le 119896]
le minus1
119887[(1 +
120591
119879)
2
+1
1198792
infin
sum
119906=119879
infin
sum
119904=119906
(119877119904119867119904+
infin
sum
119905=119904
119875119905119865119905)]
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
= 1205791003817100381710038171003817119909 minus 119910
1003817100381710038171003817
119878119871119909119899
1198992
le minus119871 minus119872
119887minus
1
1198871198992
times
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
le minus119871 minus119872
119887minus
1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
(119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))
lt minus119871 minus119872
119887+min (1 + 1
119887)119872 + 119871 minus119871 minus 119873 le 119872
119878119871119909119899
1198992
ge minus119871
+1
1198871198992
infin
sum
119906=119899+120591
infin
sum
119904=119906
10038161003816100381610038161003816ℎ (119904 119909
ℎ1119904
119909ℎ2119904
119909ℎ119896119904
)10038161003816100381610038161003816
+
infin
sum
119905=119904
[10038161003816100381610038161003816119891 (119905 119909
1198911119905
1199091198912119905
119909119891119896119905
)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]
ge minus119871 +1
1198871198792
infin
sum
119906=119879
infin
sum
119904=119906
[119882119904+
infin
sum
119905=119904
(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]
gt minus119871 minusmin (1 + 1
119887)119872 + 119871 minus119871 minus 119873 ge 119873
(70)
which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof
3 Examples
In this section we suggest six examples to explain the resultspresented in Section 2
Example 1 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus 119909119899minus120591) + Δ(
sin2119909119899minus3
1198997) +
1
(1198999 + 21198995 + 1) (1 + 1199092
1198992)
=1198992minus 2119899
1198998 + 1198993 + 1 forall119899 ge 4
(71)
where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1
and let119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= minus1 119888
119899=
1198992minus 2119899
1198998 + 1198993 + 1
119891 (119899 119906) =1
(1198999 + 21198995 + 1) (1 + 1199062)
ℎ (119899 119906) =sin21199061198997
1198911119899= 1198992 119865
119899= 1198994
ℎ1119899= 119899 minus 3 119867
119899= (119899 minus 3)
2 119875
119899=2119872
1198999
119876119899=1
1198999 119877
119899=2
1198997 119882
119899=1
1198997
forall (119899 119906) isin N1198990
timesR
(72)
It is easy to see that (14) (15) and (18) are satisfied Note that
1
1198992
infin
sum
119905=119899+120591
1199052max 119877
119905119867119905119882119905
=1
1198992
infin
sum
119905=119899+120591
1199052max2(119905 minus 3)
2
11990571
1199057
=
infin
sum
119905=119899+120591
2(119905 minus 3)2+ 1
1199055
le2
1198992
infin
sum
119905=119899+120591
1
1199053997888rarr 0 as 119899 997888rarr infin
1
1198992
infin
sum
119905=119899+120591
1199053max 119875
119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119905=119899+120591
1199053max2119872
11990551
1199059
100381610038161003816100381610038161199052minus 2119905
10038161003816100381610038161003816
1199058 + 1199053 + 1
lemax 1 2119872
1198992
infin
sum
119905=119899+120591
1
1199052997888rarr 0 as 119899 997888rarr infin
(73)
which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899
0+120591+120573 such
that for each 1199090= 1199090119899119899isinN120573
isin 119860(119873119872) the Mann iterativesequence 119909
119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (19)converges to a positive solution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of(71) with lim
119899rarrinfin119911119899= +infin and has the error estimate (20)
where 120572119898119898isinN0
is an arbitrary sequence in [0 1] satisfying(21)
Example 2 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ 119909119899minus120591) + Δ(
sin211990931198993+1
1198993 (1198996 + 2) (1 + 1199094
21198992minus3)
)
+(minus1)1198991198993(1199091198992minus119899minus1
+ 119909(119899+1)(119899+2)
)
(11989913 + 1198995 + 1) (1 + 1199092
1198992minus119899minus1
+ 1199092
(119899+1)(119899+2))
=1198992minus ln 119899
1198996 + 1198995 + 1 forall119899 ge 5
(74)
where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let
119872 and 119873 be two positive constants with 119872 gt 119873 and
119887119899= 1 119888
119899=
1198992minus ln 119899
1198996 + 1198995 + 1
119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)
(11989913 + 1198995 + 1) (1 + 1199062 + V2)
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Abstract and Applied Analysis
ℎ (119899 119906 V) =sin2V
1198993 (1198996 + 2) (1 + 1199064) 119891
1119899= 1198992minus 119899 minus 1
1198912119899= (119899 + 1) (119899 + 2)
119865119899= (119899 + 1)
2(119899 + 2)
2 ℎ
1119899= 21198992minus 3
ℎ2119899= 31198993+ 1
119867119899= (3119899
3+ 1)2
119875119899= 119876119899=
4
11989910 119877
119899= 119882119899=10
1198999
forall (119899 119906 V) isin N1198990
timesR2
(75)
It is clear that (14) (15) and (40) are fulfilled Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
10(31199043+ 1)2
119904910
1199049
le160
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le160
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(76)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (77)
Observe that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max4(119905 + 1)2(119905 + 2)
2
119905104
119905101199052minus ln 119905
1199056 + 1199055 + 1
le196
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054=196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
119905 minus 119906 + 1
1199054
le196
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le196
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(78)
which yields that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)
Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +
1198990+ 120573 such that the Mann iterative sequence 119909
119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (74) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 3 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899
119909119899minus120591)
+ Δ(
(minus1)119899 sin (119890minus119899
2|11990951198992minus3|)
11989915 minus radic119899 + 3
+1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|11990921198993+1|)
+(minus1)119899
1198996 (1 + 1199092
119899minus3)minus
1
(1198997 + 21198994 minus 1) (1 + 1199092
119899+4)
=3(minus1)1198991198992
911989910ln3119899 forall119899 ge 7
(80)
where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and
120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and
119887119899=1 + 3 ln 1198992 + 4 ln 119899
119888119899=3(minus1)1198991198992
911989910ln3119899
119891 (119899 119906 V) =(minus1)119899
1198996 (1 + 1199062)minus
1
(1198997 + 21198994 minus 1) (1 + V2)
ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899
2|119906|)
11989915 minus radic119899 + 3+
1198992+ (minus1)
119899(119899+1)2
(11989912 + 611989910 + 7) 119890|V|
1198911119899= 119899 minus 3 119891
2119899= 119899 + 4
119865119899= (119899 + 4)
2 ℎ
1119899= 51198992minus 3
ℎ2119899= 21198993+ 1 119867
119899= (2119899
3+ 1)2
119875119899= 119876119899=3
1198996 119877
119899= 119882119899=
2
11989910
forall (119899 119906 V) isin N1198990
timesR2
(81)
It is not difficult to verify that (14) (15) and (47) are fulfilledNote that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max
2(21199043+ 1)2
119904102
11990410
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le18
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(82)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (83)
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 13
Observe that1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max3(119905 + 4)2
11990563
1199056
100381610038161003816100381610038161003816100381610038161003816
3(minus1)1199051199052
911990510ln3119905
100381610038161003816100381610038161003816100381610038161003816
le12
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199054
le12
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199053le12
1198992
infin
sum
119905=119899
1
1199052997888rarr 0 as 119899 997888rarr infin
(84)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)
That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (41) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (80) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 4 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+1 minus 5119899
3
2 + 61198993119909119899minus120591) + Δ(
21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199092
3119899minus7))
+
sin (119899211990931198992minus2)
(radic119899 + 14)22
=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9
(86)
where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and
120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and
119887119899=1 minus 5119899
3
2 + 61198993 119888
119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2
1198999 + 1198998 + 21198995 + 1198993 + 7
119891 (119899 119906) =
sin (1198992119906)
(radic119899 + 14)22
ℎ (119899 119906) =21198992+ 119899 minus 1
(1198998 + 31198996 + 2) (1 + 1199062)
1198911119899= 31198992minus 2 119865
119899= (3119899
2minus 2)2
ℎ1119899= 3119899 minus 7
119867119899= (3119899 minus 7)
2 119875
119899= 119876119899=3
1198999 119877119899= 119882119899=1
1198995
forall (119899 119906) isin N1198990
timesR
(87)
Obviously (14) (15) and (52) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(3119904 minus 7)2
11990451
1199045
le9
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199043le9
1198992
infin
sum
119904=119899
1
1199042997888rarr 0 as 119899 997888rarr infin
(88)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (89)
Notice that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max
3(31199052minus 2)2
11990593
1199059
100381610038161003816100381610038161003816100381610038161003816
(minus1)1199051199053+ 51199052+ 4119905 minus 2
1199059 + 1199058 + 21199055 + 1199053 + 7
100381610038161003816100381610038161003816100381610038161003816
le27
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le27
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le27
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(90)
which gives that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)
That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that the Mann
iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generatedby (48) converges to a positive solution 119911 = 119911
119899119899isinN120573
isin
119860(119873119872) of (86) with lim119899rarrinfin
119911119899= +infin and has the error
estimate (20) where 120572119898119898isinN0
is an arbitrary sequencein [0 1] satisfying (21)
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Abstract and Applied Analysis
Example 5 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899+ (
120587
2+ 119899 sin 1
119899) 119909119899minus120591)
+ Δ((minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119909
2119899+1)))
+119899 sin (119899119909
119899minus2)
2 + (119899 + 5)16
=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
forall119899 ge 3
(92)
where 120591 isin N is fixed Let 1198990= 3 119896 = 1 119887 = 1205872 and
120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and
119887119899=120587
2+ 119899 sin 1
119899 119888
119899=
(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899
119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)
16
ℎ (119899 119906) =(minus1)119899(119899+1)2
(119899 + 4)8(119899 + 5)
3(1 + cos (1198992119906))
1198911119899= 119899 minus 2 119865
119899= (119899 minus 2)
2
ℎ1119899= 2119899 + 1 119867
119899= (2119899 + 1)
2
119875119899= 119876119899=
1
11989914 119877119899= 119882119899=2
1198999 forall (119899 119906) isin N
1198990
timesR
(93)
Clearly (14) (15) and (61) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max2(2119904 + 1)2
11990492
1199049
le18
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199047le18
1198992
infin
sum
119904=119899
1
1199046997888rarr 0 as 119899 997888rarr infin
(94)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 2)2
119905141
11990514
10038161003816100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus1cos3 (1199052 + 1)11990516 + ln 119905
10038161003816100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
11990514le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
11990513
le1
1198992
infin
sum
119905=119899
1
11990512997888rarr 0 as 119899 997888rarr infin
(95)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)
That is (38) and (39) hold Consequently Theorem 6 impliesthat (92) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((2120587)119872 +
119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +
120573 such that the Mann iterative sequence 119909119898119898isinN0
=
119909119898119899119899isinN120573
119898isinN0
generated by (58) converges to a positivesolution 119911 = 119911
119899119899isinN120573
isin 119860(119873119872) of (92) with lim119899rarrinfin
119911119899=
+infin and has the error estimate (20) where 120572119898119898isinN0
is anarbitrary sequence in [0 1] satisfying (21)
Example 6 Consider the third order nonlinear neutral delaydifference equation
Δ3(119909119899minus21198995+ 91198992minus 1
1198995 + 31198992 + 2119909119899minus120591) + Δ(
cos ((minus1)119899119890119899)
(119899 + 7)6radic1 +
1003816100381610038161003816119909119899minus21003816100381610038161003816
)
+
sin (1198992119909119899minus1)
1198999 + 31198995 + 21198992 + 1
=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2 forall119899 ge 6
(97)
where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =
min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and
119887119899= minus
21198995+ 91198992minus 1
1198995 + 31198992 + 2
119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1
11989911 + 61198993 + 7119899 + 2
119891 (119899 119906) =
sin (1198992119906)1198999 + 31198995 + 21198992 + 1
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 15
ℎ (119899 119906) =cos ((minus1)119899119890119899)
(119899 + 7)6radic1 + |119906|
1198911119899= 119899 minus 1 119865
119899= (119899 minus 1)
2
ℎ1119899= 119899 minus 2 119867
119899= (119899 minus 2)
2
119875119899= 119876119899=1
1198997 119877119899= 119882119899=1
1198996
forall (119899 119906) isin N1198990
timesR
(98)
Obviously (14) (15) and (64) are satisfied Note that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max(119904 minus 2)2
11990461
1199046
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
1
1199044le1
1198992
infin
sum
119904=119899
1
1199043997888rarr 0 as 119899 997888rarr infin
(99)
which means that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
max 119877119904119867119904119882119904 = 0 (100)
It is clear that
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816
=1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max(119905 minus 1)2
11990571
1199057
100381610038161003816100381610038161003816100381610038161003816
(minus1)119905minus11199054+ 41199052+ 119905 minus 1
11990511 + 61199053 + 7119905 + 2
100381610038161003816100381610038161003816100381610038161003816
le1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
1
1199055le1
1198992
infin
sum
119906=119899
infin
sum
119905=119906
1
1199054
le1
1198992
infin
sum
119905=119899
1
1199053997888rarr 0 as 119899 997888rarr infin
(101)
which implies that
lim119899rarrinfin
1
1198992
infin
sum
119906=119899
infin
sum
119904=119906
infin
sum
119905=119904
max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)
That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899
0+ 120591 + 120573 such that
the Mann iterative sequence 119909119898119898isinN0
= 119909119898119899119899isinN120573
119898isinN0
generated by (65) converges to a positive solution 119911 =
119911119899119899isinN120573
isin 119860(119873119872) of (97) with lim119899rarrinfin
119911119899= +infin and
has the error estimate (20) where 120572119898119898isinN0
is an arbitrarysequence in[0 1] satisfying (21)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)
References
[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000
[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000
[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998
[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005
[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004
[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000
[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007
[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002
[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007
[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004
[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004
[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012
[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009
[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011
[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Abstract and Applied Analysis
Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005
[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011
[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007
[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003
[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004
[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012
[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002
[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995
[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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