Research Article Cyclic ( )-Contractions in Uniform Spaces and...
8
Research Article Cyclic ()-Contractions in Uniform Spaces and Related Fixed Point Results N. Hussain, 1 E. KarapJnar, 2 S. Sedghi, 3 N. Shobkolaei, 4 and S. Firouzian 5 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Atilim University, ˙ Incek, 06836 Ankara, Turkey 3 Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran 4 Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran 5 Payame Noor University, Babol, Iran Correspondence should be addressed to N. Hussain; [email protected] Received 25 December 2013; Accepted 30 January 2014; Published 11 March 2014 Academic Editor: Chi-Ming Chen Copyright © 2014 N. Hussain et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. First, we define cyclic ()-contractions of different types in a uniform space. en, we apply these concepts of cyclic ()-contractions to establish certain fixed and common point theorems on a Hausdorff uniform space. Some more general results are obtained as corollaries. Moreover, some examples are provided to demonstrate the usability of the proved theorems. 1. Introduction Let be a nonempty set. A nonempty family, , of subsets of × is called the uniform structure of if it satisfies the following properties: (i) if is in , then contains the diagonal {(, ) | ∈ }; (ii) if is in and is a subset of × which contains , then is in ; (iii) if and are in , then ∩ is in ; (iv) if is in , then there exists in , such that, whenever (, ) and (, ) are in , then (, ) is in ; (v) if is in , then {(, ) | (, ) ∈ } is also in . e pair (, ) is called a uniform space and the element of is called entourage or neighbourhood or surrounding. e pair (, ) is called a quasi-uniform space (see, e.g., [1, 2]) if property (v) is omitted. Existence and uniqueness of fixed points for various contractive mappings in the setting of uniform spaces have been investigated by several authors; see, for example, [3–12] and the references therein. Recently, an interesting and remarkable notion of cyclic mapping was introduced and studied by Kirk et al. [13]. Following this paper, a number of authors introduced con- tractive mapping via the cyclic mappings and reported certain fixed point results in the setting of different type of spaces; see, for example, [13–17]. In this paper, we will give the characterization of cyclic mapping in the context of uniform spaces and, further, prove the existence and uniqueness of fixed and common fixed points of such mappings via -distance and -distance, introduced by Aamri and El Moutawakil [18]. For the sake of completeness, we recollect some basic definitions and fundamental results. Let Δ = {(, ) | ∈ } be the diagonal of a nonempty set . For , ∈ × , we will use the following setting in the sequel: ∘ = {(, ) | there exists ∈ : (,) ∈ , (, ) ∈ } , −1 = {(, ) | (, ) ∈ } . (1) For subset ∈, a pair of points and are said to be - close if (, ) ∈ and (, ) ∈ . Moreover, a sequence { } Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 976859, 7 pages http://dx.doi.org/10.1155/2014/976859
Transcript of Research Article Cyclic ( )-Contractions in Uniform Spaces and...
Research ArticleCyclic (120601)-Contractions in Uniform Spaces and RelatedFixed Point Results
N Hussain1 E KarapJnar2 S Sedghi3 N Shobkolaei4 and S Firouzian5
1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics Atilim University Incek 06836 Ankara Turkey3 Department of Mathematics Qaemshahr Branch Islamic Azad University Qaemshahr Iran4Department of Mathematics Babol Branch Islamic Azad University Babol Iran5 Payame Noor University Babol Iran
Correspondence should be addressed to N Hussain nhusainkauedusa
Received 25 December 2013 Accepted 30 January 2014 Published 11 March 2014
Academic Editor Chi-Ming Chen
Copyright copy 2014 N Hussain 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
First we define cyclic (120601)-contractions of different types in a uniform spaceThen we apply these concepts of cyclic (120601)-contractionsto establish certain fixed and common point theorems on a Hausdorff uniform space Some more general results are obtained ascorollaries Moreover some examples are provided to demonstrate the usability of the proved theorems
1 Introduction
Let119883 be a nonempty set A nonempty family 120599 of subsets of119883 times 119883 is called the uniform structure of 119883 if it satisfies thefollowing properties
(i) if 119866 is in 120599 then 119866 contains the diagonal (119909 119909) | 119909 isin119883
(ii) if 119866 is in 120599 and119867 is a subset of119883times119883 which contains119866 then119867 is in 120599
(iii) if 119866 and119867 are in 120599 then 119866 cap 119867 is in 120599(iv) if 119866 is in 120599 then there exists 119867 in 120599 such that
whenever (119909 119910) and (119910 119911) are in 119867 then (119909 119911) is in119866
(v) if 119866 is in 120599 then (119910 119909) | (119909 119910) isin 119866 is also in 120599
The pair (119883 120599) is called a uniform space and the element of120599 is called entourage or neighbourhood or surrounding Thepair (119883 120599) is called a quasi-uniform space (see eg [1 2]) ifproperty (v) is omitted
Existence and uniqueness of fixed points for variouscontractive mappings in the setting of uniform spaces havebeen investigated by several authors see for example [3ndash12]and the references therein
Recently an interesting and remarkable notion of cyclicmapping was introduced and studied by Kirk et al [13]Following this paper a number of authors introduced con-tractivemapping via the cyclicmappings and reported certainfixed point results in the setting of different type of spaces seefor example [13ndash17]
In this paper we will give the characterization of cyclicmapping in the context of uniform spaces and further provethe existence and uniqueness of fixed and common fixedpoints of such mappings via 119860-distance and 119864-distanceintroduced by Aamri and El Moutawakil [18]
For the sake of completeness we recollect some basicdefinitions and fundamental results Let Δ = (119909 119909) | 119909 isin 119883be the diagonal of a nonempty set 119883 For 119881119882 isin 119883 times 119883 wewill use the following setting in the sequel
119881 ∘119882
= (119909 119910) | there exists 119911 isin 119883 (119909 119911) isin 119882 (119911 119910) isin 119881
119881minus1
= (119909 119910) | (119910 119909) isin 119881
(1)
For subset 119881 isin 120599 a pair of points 119909 and 119910 are said to be 119881-close if (119909 119910) isin 119881 and (119910 119909) isin 119881 Moreover a sequence 119909
119899
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 976859 7 pageshttpdxdoiorg1011552014976859
2 Abstract and Applied Analysis
in 119883 is called a Cauchy sequence for 120599 if for any 119881 isin 120599 thereexists 119873 ge 1 such that 119909
119899
and 119909119898
are 119881-close for 119899119898 ge 119873For (119883 120599) there is a unique topology 120591(120599) on119883 generated by119881(119909) = 119910 isin 119883 | (119909 119910) isin 119881 where 119881 isin 120599
A sequence 119909119899
in119883 is convergent to 119909 for 120599 denoted bylim119899rarrinfin
119909119899
= 119909 if for any 119881 isin 120599 there exists 1198990
isin N such that119909119899
isin 119881(119909) for every 119899 ge 1198990
A uniform space (119883 120599) is calledHausdorff if the intersection of all the 119881 isin 120599 is equal to Δ of119883 that is if (119909 119910) isin 119881 for all119881 isin 120599 implies 119909 = 119910 If119881 = 119881minus1then we say that a subset 119881 isin 120599 is symmetrical Throughoutthe paper we assume that each 119881 isin 120599 is symmetrical Formore details see for example [1 18ndash21]
Now we recall the notions of 119860-distance and 119864-distance
Definition 1 (see eg [18 19]) Let (119883 120599) be a uniform spaceA function 119901 119883 times 119883 rarr [0infin) is said to be an 119860-distanceif for any 119881 isin 120599 there exists 120575 gt 0 such that if 119901(119911 119909) le 120575 and119901(119911 119910) le 120575 for some 119911 isin 119883 then (119909 119910) isin 119881
Definition 2 (see eg [18 19]) Let (119883 120599) be a uniform spaceA function 119901 119883times119883 rarr [0infin) is said to be an 119864-distance if
(1199011
) 119901 is an 119860-distance(1199012
) 119901(119909 119910) le 119901(119909 119911) + 119901(119911 119910) forall119909 119910 119911 isin 119883
Example 3 (see eg [18 19]) Let (119883 120599) be a uniform spaceand let 119889 be a metric on 119883 It is evident that (119883 120599
119889
) is auniform space where 120599
119889
is the set of all subsets of 119883 times 119883containing a ldquobandrdquo 119861
120598
= (119909 119910) isin 1198832 | 119889(119909 119910) lt 120598 forsome 120598 gt 0 Moreover if 120599 sube 120599
119889
then 119889 is an 119864-distance on(119883 120599)
Lemma 4 (see eg [18 19]) Let (119883 120599) be a Hausdorffuniform space and let 119901 be an 119860-distance on X Let 119909
119899
and119910119899
be sequences in 119883 and 120572119899
and let 120573119899
be sequences in[0infin) converging to 0 Then for 119909 119910 119911 isin 119883 the followinghold
(a) If 119901(119909119899
119910) le 120572119899
and 119901(119909119899
119911) le 120573119899
for all 119899 isin N then119910 = 119911 In particular if 119901(119909 119910) = 0 and 119901(119909 119911) = 0then 119910 = 119911
(b) If 119901(119909119899
119910119899
) le 120572119899
and 119901(119909119899
119911) le 120573119899
for all 119899 isin N then119910119899
converges to 119911(c) If 119901(119909
119899
119909119898
) le 120572119899
for all 119899119898 isin N with 119898 gt 119899 then119909119899
is a Cauchy sequence in (119883 120599)
Let 119901 be an 119860-distance A sequence in a uniform space(119883 120599)with an119860-distance is said to be a 119901-Cauchy if for every120598 gt 0 there exists 119899
0
isin N such that 119901(119909119899
119909119898
) lt 120598 for all119899119898 ge 119899
0
Definition 5 (see eg [18 19]) Let (119883 120599) be a uniform spaceand let 119901 be an 119860-distance on119883
(1) 119883 is 119878-complete if for every 119901-Cauchy sequence 119909119899
there exists 119909 in119883 with lim
119899rarrinfin
119901(119909119899
119909) = 0(2) 119883 is 119901-Cauchy complete if for every 119901-Cauchy
sequence 119909119899
there exists 119909 in119883with lim119899rarrinfin
119909119899
= 119909with respect to 120591(120599)
Remark 6 Let (119883 120599) be a Hausdorff uniform space which is119878-complete If a sequence 119909
119899
is a 119901-Cauchy sequence thenwe have lim
119899rarrinfin
119901(119909119899
119909) = 0 Regarding Lemma 4(b) wederive that lim
119899rarrinfin
119909119899
= 119909 with respect to the topology 120591(120599)and hence 119878-completeness implies 119901-Cauchy completeness
Definition 7 Let (119883 120599) be a Hausdorff uniform space andlet 119901 be an 119860-distance on 119883 Two self-mappings 119891 and 119892 of119883 are said to be weak compatible if they commute at theircoincidence points that is 119891119909 = 119892119909 implies that 119891119892119909 = 119892119891119909
We denote by F the class of functions 120601 [0infin) rarr[0infin) nondecreasing and continuous satisfying 120601(119905) gt 0 for119905 isin (0infin) and 120601(0) = 0
Definition 8 (see [17]) A function 120601 [0infin) rarr [0infin) iscalled a comparison function if it satisfies the following
(i) 120601 is increasing that is 1199051
le 1199052
implies 120601(1199051
) le 120601(1199052
)for 1199051
1199052
isin [0infin)(ii) 120601119899(119905)
119899isinN converges to 0 as 119899 rarr infin for all 119905 isin[0infin)
Definition 9 (see [22]) A function 120601 [0infin) rarr [0infin) iscalled a (119888)-comparison function if
(i) 120601 is increasing(ii) there exist 119896
0
isin N 119886 isin (0 1) and a convergent seriesof nonnegative terms suminfin
119896=1
V119896
such that
120601119896+1
(119905) le 119886120601119896
(119905) + V119896
(2)
for 119896 ge 1198960
and any 119905 isin [0infin)
Let C be the collection of all (119888)-comparison functions120601 [0infin) rarr [0infin) defined in Definition 9
Lemma 10 (see [22]) If 120601 [0infin) rarr [0infin) is a (119888)-comparison function then the following hold
(i) 120601 is comparison function(ii) 120601(119905) lt 119905 for any 119905 isin [0infin)(iii) 120601 is continuous at 0
(iv) the series suminfin119896=0
120601119896(119905) converges for any 119905 isin [0infin)
In 1922 Banach proved that every contraction in a com-plete metric space has a unique fixed point This celebratedresult has been generalized and improved bymany authors inthe context of different abstract spaces for various operators(see [1ndash28] and the references therein) Recently fixed pointtheorems for operators 119879 defined on a complete metric space119883 with a cyclic representation of 119883 with respect to 119879 haveappeared in the literature (see eg [13ndash17]) Now we presenta modification of themain result of [16] For this we need thefollowing definitions
Definition 11 (see [13]) Let119883 be a nonempty set119898 a positiveinteger and 119879 119883 rarr 119883 a mapping 119883 = ⋃
119898
119894=1
119860119894
is said tobe a cyclic representation of119883 with respect to 119879 if
Abstract and Applied Analysis 3
(i) 119860119894
119894 = 1 2 119898 are nonempty sets(ii) 119879(119860
1
) sub 1198602
119879(119860119898minus1
) sub 119860119898
119879(119860119898
) sub 1198601
Definition 12 Let (119883 119889) be a metric space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119883 =⋃119898
119894=1
119860119894
An operator 119879 119883 rarr 119883 is a cyclic (120601)-contractionif
(i) 119883 = ⋃119898
119894=1
119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119889(119879119909 119879119910) le 120601(119889(119909 119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin F
The main result of [14] is the following
Theorem 13 (Theorem 6 of [14]) Let (119883 119889) be a completemetric space m a positive integer 119860
1
1198602
119860119898
nonemptysubsets of 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic(120601 minus120595)-contraction with 120601 120595 isin F Then 119879 has a unique fixedpoint 119911 isin ⋂119898
119894=1
119860119894
Themain aimof this paper is to prove results similar to theabovementioned theorems in uniform spaces and to presentmodifications of Theorem 21 [16] Theorems 31-32 in [18]and other related results
2 Main Result
First we present the following definition
Definition 14 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119883 =⋃119898
119894=1
119860119894
An operator 119879 119883 rarr 119883 is a cyclic (120601)-contractionif
(i) 119883 = ⋃119898
119894=1
119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898
119901 (119879119909 119879119910) le 120601 (119901 (119909 119910)) (3)
where 119860119898+1
= 1198601
and 120601 isin C
Our main result is the following
Theorem 15 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integerand119860
1
1198602
119860119898
nonempty closed subsets of119883 with respectto the topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contractionThen119879 has a unique fixedpoint 119909 isin ⋂119898
119894=1
119860119894
Proof We first show that the fixed point of 119879 is unique (if itexists) Suppose on the contrary that 119910 119911 isin 119883 with 119910 = 119911 arefixed points of 119879 The cyclic character of 119879 and the fact that119910 119911 isin 119883 are fixed points of119879 imply that 119910 119911 isin ⋂119898
119894=1
119860119894
Usingthe contractive condition we obtain
119901 (119910 119911) = 119901 (119879119910 119879119911) le 120601 (119901 (119910 119911)) lt 119901 (119910 119911) (4)
and from the last inequality
119901 (119910 119911) = 0 (5)
Similarly we can show that 119901(119910 119910) = 0 and consequently119910 = 119911
Now we prove the existence of a fixed point Note that119901 is not symmetric To show that the sequence 119909
119899
isCauchy we will show that both lim
119899rarrinfin
119901(119909119899
119909119899+119902
) = 0 andlim119899rarrinfin
119901(119909119899+119902
119909119899
) = 0 for any 119902 gt 1For this aim take 119909
0
isin 119883 and consider the sequence givenby
119909119899+1
= 119879119909119899
119899 = 0 1 2 (6)
If there exists 1198990
isin N such that 1199091198990+1
= 1199091198990 then the proof is
completed In this case 1199091198990is the required fixed point of 119879
Throughout the proof we assume that
119909119899+1
= 119909119899
for any 119899 = 0 1 2 (7)
Notice that for any 119899 gt 0 there exists 119894119899
isin 1 2 119898 suchthat 119909
119899minus1
isin 119860119894119899and 119909
119899
isin 119860119894119899+1
since119883 = ⋃119898
119894=1
119860119894
Due to thefact that 119879 is a cyclic (120601)-contraction we have
119901 (119909119899
119909119899+1
) = 119901 (119879119909119899minus1
119879119909119899
) le 120601 (119901 (119909119899minus1
119909119899
)) (8)
by taking 119909 = 119909119899
and 119910 = 119909119899+1
in (3) From (8) and taking themonotonicity of 120601 into account we derive by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(9)
As 119901 is an 119864-distance we obtain that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (10)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(11)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenoting
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (12)
implies that
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(13)
As 120601 is a (119888)-comparison function supposing 119901(1199090
1199091
) gt 0by Lemma 10 (iv) it follows that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (14)
so there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (15)
4 Abstract and Applied Analysis
Then by (13) we obtain that
lim119899rarrinfin
119901 (119909119899
119909119899+119902
) = 0 (16)
By repeating the same arguments in the proof of (16) weconclude that
lim119899rarrinfin
119901 (119909119899+119902
119909119899
) = 0 (17)
Consequently we get that the sequence 119909119899
119899ge0
is a 119901-Cauchy in the 119878-complete space 119883 = ⋃
119898
119894=1
119860119894
Thus thereexists 119909 isin 119883 such that lim
119899rarrinfin
119909119899
= 119909 In what follows weprove that119909 is a fixed point of119879 In fact since lim
119899rarrinfin
119909119899
= 119909as 119883 = ⋃
119898
119894=1
119860119894
is a cyclic representation of 119883 with respectto 119879 the sequence 119909
119899
has infinite terms in each 119860119894
for119894 isin 1 2 119898
Since 119860119894
is closed for every 119894 it follows that 119909 isin ⋂119898
119894=1
119860119894
thus we take a subsequence 119909
119899119896of 119909119899
with 119909119899119896isin 119860119894minus1
Usingthe contractive condition we can obtain
119901 (119909 119879119909) le 119901 (119909 119909119899119896+1
) + 119901 (119909119899119896+1
119879119909)
= 119901 (119909 119909119899119896+1
) + 119901 (119879119909119899119896 119879119909)
le 119901 (119909 119909119899119896+1
) + 120601 (119901 (119909119899119896 119909))
(18)
and since 119909119899119896
rarr 119909 and 120601 belong toC letting 119896 rarr infin in thelast inequality we have 119901(119909 119879119909) = 0 Analogously we canderive that 119901(119909 119909) = 0 and therefore 119909 is a fixed point of 119879This finishes the proof
Corollary 16 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integer1198601
1198602
119860119898
nonempty closed subsets of 119883 with respect tothe topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let operator119879 119883 rarr 119883 satisfy
(i) 119901(119879119909 119879119910) le 119896119901(119909 119910) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 =1 2 119898 where 119860
119898+1
= 1198601
and 0 lt 119896 lt 1
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120601(119905) = 119896119905
Corollary 17 (cf [16]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contraction Then 119879 has a unique fixed point 119911 isin ⋂119898
119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 18 (cf [13]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof119883 and119883 = ⋃
119898
119894=1
119860119894
a cyclic representation of119883with respectto 119879 Let 119879 119883 rarr 119883 satisfy
119901 (119879119909 119879119910) le 119896119901 (119909 119910) (19)
for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119896 isin (0 1) and119860119898+1
= 1198601
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Definition 19 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119879 119892 119883 rarr 119883 self-mappings An operator 119879 is a cyclic (120601)-119892-contraction if
(i) 119892119883 = ⋃119898
119894=1
119892119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119901(119879119909 119879119910) le 120601(119901(119892119909 119892119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin C
Inspired by [28] we now prove a common fixed pointtheorem as an application of our Theorem 15
Theorem 20 Let (119883 120599) be a uniform space 119879 119892 119883 rarr 119883self-maps such that 119879 is cyclic (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform space together with 119901 being an 119864-distance on119883 Suppose that 119892119860
1
1198921198602
119892119860119898
are nonemptyclosed subsets of 119892119883 with respect to the uniform topology and119879119883 sub 119892119883 = ⋃
119898
119894=1
119892119860119894
Then 119879 and 119892 have a uniquecoincidence point Moreover if 119879 and 119892 are weakly compatiblethen they have a unique common fixed point 119911 isin ⋂119898
119894=1
119892119860119894
Proof As 119892 119883 rarr 119883 so there exists 119864 sub 119883 such that 119892119864 =119892119883 and 119892 119864 rarr 119883 is one-to-one Now since 119879119883 sub 119892119883we define mappings ℎ 119892119864 rarr 119892119864 by ℎ(119892119909) = 119879119909 Since 119892is one-to-one on 119864 so ℎ is well defined As 119879 is cyclic (120601)-119892-contraction so
119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910)) (20)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 Thus
119901 (ℎ (119892119909) ℎ (119892119910)) = 119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910))
(21)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 whichimplies that ℎ is cyclic (120601)-contraction on 119892119883 Hence all theconditions ofTheorem 15 are satisfied by ℎ so ℎ has a uniquefixed point 119911 = 119892119909 in 119892119883 That is 119892119909 = 119911 = ℎ(119911) = ℎ(119892119909) =119879119909 so 119879 and 119892 have a unique coincidence point as requiredMoreover if 119879 and 119892 are weakly compatible then they have aunique common fixed point
Corollary 21 (cf Theorem 32 [18]) Let (119883 120599) be a uniformspace 119879 119892 119883 rarr 119883 self-maps such that 119879 is (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform spacetogether with 119901 being an 119864-distance on 119883 Suppose that 119879119883 sub119892119883 and 119879 and 119892 are commuting Then 119879 and 119892 have a uniquecommon fixed point 119911 isin 119883
Proof Take 119860119894
= 119883 for all 119894 = 1 119898 in Theorem 20
Example 22 Let (119883 119889) be a metric space where119883 = 1119899 cup0 and 119889 = | | Set 119860
1
= 13119899 cup 0 1 1198602
= 1(3119899 + 1) cup0 1 and 119860
3
= 1(3119899 + 2) cup 0 1 Define 120599 = 119880120598
| 120598 gt 0It is easy to see that (119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin) by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119879 119883 rarr 119883
Abstract and Applied Analysis 5
by 119879(0) = 119879(1) = 0 and 119879(1119899) = 1(4119899 + 1) then for every119909 119910 = 0 1 we have
119889 (119879119909 119879119910) = 119889 (1
4119899 + 1
1
4119898 + 1)
=|4 (119898 minus 119899)|
|4119899 + 1| |4119898 + 1|le|119898 minus 119899|
4119899 sdot 119898
le1
4
1003816100381610038161003816100381610038161003816
1
119899minus1
119898
1003816100381610038161003816100381610038161003816=1
4119889 (119909 119910)
(22)
Also for 119909 119910 = 0 1 the above inequality obviously holdsThis shows that the contractive condition of Corollary 16 issatisfied and 0 is fixed point 119879
Definition 23 Let (119883 120599) be a uniform space let 119891 119892 119883 rarr119883 be two mappings and let 119860 and 119861 be nonempty closedsubsets of 119883 The 119883 = 119860 cup 119861 is said to be a cyclicrepresentation of119883with respect to the pair (119891 119892) if 119891(119860) sub 119861and 119892(119861) sub 119860
Definition 24 Let (119883 120599) be a uniform space 119860 119861 nonemptysubsets of 119883 and 119883 = 119860 cup 119861 Two self-maps 119891 119892 119883 rarr 119883are called cyclic (120601)-contraction pair if
(i) 119883 = 119860cup119861 is a cyclic representation of119883 with respectto the pair (119891 119892)
(ii) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le120601(max119901(119909 119891119909) 119901(119910 119892119910)) for any 119909 isin 119860 119910 isin 119861where 120601 isin C
Theorem 25 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that 119891 119892 119883 rarr 119883are cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119909 isin 119860 cap 119861
Proof Take 1199090
isin 119883 and consider the sequence given by
1198911199092119899
= 1199092119899+1
1198921199092119899+1
= 1199092119899+2
119899 = 0 1 2
(23)
Since 119883 = 119860 cup 119861 for any 119899 gt 0 1199092119899
isin 119860 and 1199092119899+1
isin 119861 and(119891 119892) are cyclic (120601)-contraction pair we have
119901 (1199092119899+1
1199092119899+2
) = 119901 (1198911199092119899
1198921199092119899+1
)
le max 119901 (1198911199092119899
1198921199092119899+1
) 119901 (1198921199092119899+1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1198911199092119899
) 119901 (1199092119899+1
1198921199092119899+1
))
= 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899+1
1199092119899+2
))
(24)
Hence
119901 (1199092119899+1
1199092119899+2
) le 120601 (119901 (1199092119899
1199092119899+1
)) (25)
Similarly we have
119901 (1199092119899
1199092119899+1
) = 119901 (1198921199092119899minus1
1198911199092119899
)
le max 119901 (1198911199092119899
1198921199092119899minus1
) 119901 (1198921199092119899minus1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899minus1
1199092119899
))
(26)
Hence
119901 (1199092119899
1199092119899+1
) le 120601 (119901 (1199092119899minus1
1199092119899
)) (27)
From inequalities (25) and (27) and taking into accountthe monotonicity of 120601 we get by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(28)
Since 119901 is an 119864-distance we find that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (29)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(30)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenote
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (31)
By relation (31) we have
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(32)
Regarding 120601 isin C together with Lemma 10(iv) we get that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (33)
since 119901(1199090
1199091
) gt 0 Thus there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (34)
Then by (32) we obtain that
119901 (119909119899
119909119899+119902
) 997888rarr 0 as 119899 997888rarr infin (35)
In an analogous way we derive that
119901 (119909119899+119902
119909119899
) 997888rarr 0 as 119899 997888rarr infin (36)
Hence we get that 119909119899
119899ge0
is a 119901-Cauchy sequence in the119878-complete space 119883 = 119860 cup 119861 So there exists 119909 isin 119883such that lim
119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 In whatfollows we prove that 119909 is a fixed point of 119891 119892 In fact sincelim119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 and as 119883 = 119860 cup 119861 is a
6 Abstract and Applied Analysis
cyclic representation of 119883 with respect to 119891 119892 the sequence119909119899
has infinite terms in each 119860 119861Since 119860 119861 are closed it follows that 119909 isin 119860 cap 119861 thus we
take subsequences 1199092119899
1199092119899+1
of 119909119899
with 1199092119899
isin 119860 and 1199092119899+1
isin119861 Using the contractive condition we can obtain
119901 (119909 119891119909) le 119901 (119909 1199092119899+2
) + 119901 (1199092119899+2
119891119909)
= 119901 (119909 1199092119899+2
) + 119901 (1198921199092119899+1
119891119909)
le 119901 (119909 1199092119899+2
) +max 119901 (1198921199092119899+1
119891119909)
119901 (119891119909 1198921199092119899+1
)
le 119901 (119909 1199092119899+2
) + 120601 (max 119901 (119909 119891119909)
119901 (1199092119899+1
1199092119899+2
))
(37)
and since 119909119899
rarr 119909 and 120601 belong toC letting 119899 rarr infin in thelast inequality we have 119901(119909 119891119909) le 120601(119901(119909 119891119909)) lt 119901(119909 119891119909)hence119901(119909 119891119909) = 0 Similarly we can show that 119901(119909 119909) = 0and therefore 119909 is a fixed point of 119891 Similarly we can showthat 119909 is a fixed point of 119892 Finally in order to prove theuniqueness of the fixed point we have 119910 119911 isin 119883 with 119910 and119911 fixed points of 119891 119892 The cyclic character of 119891 119892 and the factthat 119910 119911 isin 119883 are fixed points of 119891 119892 imply that 119910 119911 isin 119860 cap 119861Using the contractive condition we obtain
119901 (119910 119911) = 119901 (119891119910 119892119911) le max 119901 (119891119910 119892119911) 119901 (119892119911 119891119910)
le 120601 (max 119901 (119910 119891119910) 119901 (119911 119892119911)) = 0
(38)
and from the last inequality we get
119901 (119910 119911) = 0 (39)
Using the same arguments above we can show that 119901(119910 119910) =0 and consequently 119910 = 119911 This finishes the proof
Corollary 26 Let (119883 119889) be a complete metric space and 119860 119861nonempty closed subsets of 119883 and 119883 = 119860 cup 119861 Let f 119892 119883 rarr119883 be cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119911 isin 119860 cap 119861
Proof By Theorem 25 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 27 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that the maps 119891 119892 119883 rarr 119883 satisfy the following inequality
(i) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le119896max119901(119909 119891119909) 119901(119910 119892119910) for any 119909 isin 119860 119910 isin 119861where 0 lt 119896 lt 1
Then 119891 119892 have a unique common fixed point 119911 isin 119860 cap 119861
Proof ByTheorem 25 it is enough to set 120601(119905) = 119896119905
Example 28 Let (119883 119901) be a partial metric space where 119883 =1119899 cup 0 1 and 119901(119909 119910) = max119909 119910 Set119860 = 12119899 cup 0 1and 119861 = 1(2119899 + 1) cup 0 1 Define 120599 = 119880
120598
| 120598 gt 0 where119880120598
= (119909 119910) isin 1198832 119901(119909 119910) lt 119901(119909 119909)+120598 It is easy to see that(119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin)by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119891 119883 rarr 119883 by 119891(12119899) =1(4119899 + 1) 119891(0) = 119891(1) = 0 and 119892(1(2119899 + 1)) = 1(4119899 + 2)119892(0) = 119892(1) = 0 Then for every 119909 119910 = 0 1 we have
max 119901 (119891119909 119892119910) 119901 (119892119910 119891119909)
= max 119901( 1
4119899 + 1
1
4119898 + 2)
119901 (1
4119898 + 2
1
4119899 + 1)
= max 1
4119899 + 1
1
4119898 + 2
le1
2max 1
2119899
1
2119898 + 1
=1
2max 119901 (119909 119891119909) 119901 (119910 119892119910)
(40)
for any 119909 isin 119860 119910 isin 119861 Also for 119909 119910 = 0 1 the above inequalityobviously holds This shows that the contractive condition ofCorollary 27 is satisfied and 0 is a common fixed point of 119891and 119892
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanksDSR KAU for the financialsupport
References
[1] N Bourbaki Elements de mathematique Fasc II Livre IIITopologie generale Chapitre 1 Structures Topologiques Chapitre2 Structures Uniformes vol 1142 of Actualites Scientifiques etIndustrielles Hermann Paris France 1965
[2] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 1 Springer New York NY USA 1986
[3] V Berinde Iterative Approximation of Fixed Points SpringerBerlin Germany 2007
[4] J Jachymski ldquoFixed point theorems for expansive mappingsrdquoMathematica Japonica vol 42 no 1 pp 131ndash136 1995
[5] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996
[6] M Cherichi and B Samet ldquoFixed point theorems on orderedgauge spaces with applications to nonlinear integral equationsrdquoFixed Point Theory and Applications vol 2012 article 13 2012
Abstract and Applied Analysis 7
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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
2 Abstract and Applied Analysis
in 119883 is called a Cauchy sequence for 120599 if for any 119881 isin 120599 thereexists 119873 ge 1 such that 119909
119899
and 119909119898
are 119881-close for 119899119898 ge 119873For (119883 120599) there is a unique topology 120591(120599) on119883 generated by119881(119909) = 119910 isin 119883 | (119909 119910) isin 119881 where 119881 isin 120599
A sequence 119909119899
in119883 is convergent to 119909 for 120599 denoted bylim119899rarrinfin
119909119899
= 119909 if for any 119881 isin 120599 there exists 1198990
isin N such that119909119899
isin 119881(119909) for every 119899 ge 1198990
A uniform space (119883 120599) is calledHausdorff if the intersection of all the 119881 isin 120599 is equal to Δ of119883 that is if (119909 119910) isin 119881 for all119881 isin 120599 implies 119909 = 119910 If119881 = 119881minus1then we say that a subset 119881 isin 120599 is symmetrical Throughoutthe paper we assume that each 119881 isin 120599 is symmetrical Formore details see for example [1 18ndash21]
Now we recall the notions of 119860-distance and 119864-distance
Definition 1 (see eg [18 19]) Let (119883 120599) be a uniform spaceA function 119901 119883 times 119883 rarr [0infin) is said to be an 119860-distanceif for any 119881 isin 120599 there exists 120575 gt 0 such that if 119901(119911 119909) le 120575 and119901(119911 119910) le 120575 for some 119911 isin 119883 then (119909 119910) isin 119881
Definition 2 (see eg [18 19]) Let (119883 120599) be a uniform spaceA function 119901 119883times119883 rarr [0infin) is said to be an 119864-distance if
(1199011
) 119901 is an 119860-distance(1199012
) 119901(119909 119910) le 119901(119909 119911) + 119901(119911 119910) forall119909 119910 119911 isin 119883
Example 3 (see eg [18 19]) Let (119883 120599) be a uniform spaceand let 119889 be a metric on 119883 It is evident that (119883 120599
119889
) is auniform space where 120599
119889
is the set of all subsets of 119883 times 119883containing a ldquobandrdquo 119861
120598
= (119909 119910) isin 1198832 | 119889(119909 119910) lt 120598 forsome 120598 gt 0 Moreover if 120599 sube 120599
119889
then 119889 is an 119864-distance on(119883 120599)
Lemma 4 (see eg [18 19]) Let (119883 120599) be a Hausdorffuniform space and let 119901 be an 119860-distance on X Let 119909
119899
and119910119899
be sequences in 119883 and 120572119899
and let 120573119899
be sequences in[0infin) converging to 0 Then for 119909 119910 119911 isin 119883 the followinghold
(a) If 119901(119909119899
119910) le 120572119899
and 119901(119909119899
119911) le 120573119899
for all 119899 isin N then119910 = 119911 In particular if 119901(119909 119910) = 0 and 119901(119909 119911) = 0then 119910 = 119911
(b) If 119901(119909119899
119910119899
) le 120572119899
and 119901(119909119899
119911) le 120573119899
for all 119899 isin N then119910119899
converges to 119911(c) If 119901(119909
119899
119909119898
) le 120572119899
for all 119899119898 isin N with 119898 gt 119899 then119909119899
is a Cauchy sequence in (119883 120599)
Let 119901 be an 119860-distance A sequence in a uniform space(119883 120599)with an119860-distance is said to be a 119901-Cauchy if for every120598 gt 0 there exists 119899
0
isin N such that 119901(119909119899
119909119898
) lt 120598 for all119899119898 ge 119899
0
Definition 5 (see eg [18 19]) Let (119883 120599) be a uniform spaceand let 119901 be an 119860-distance on119883
(1) 119883 is 119878-complete if for every 119901-Cauchy sequence 119909119899
there exists 119909 in119883 with lim
119899rarrinfin
119901(119909119899
119909) = 0(2) 119883 is 119901-Cauchy complete if for every 119901-Cauchy
sequence 119909119899
there exists 119909 in119883with lim119899rarrinfin
119909119899
= 119909with respect to 120591(120599)
Remark 6 Let (119883 120599) be a Hausdorff uniform space which is119878-complete If a sequence 119909
119899
is a 119901-Cauchy sequence thenwe have lim
119899rarrinfin
119901(119909119899
119909) = 0 Regarding Lemma 4(b) wederive that lim
119899rarrinfin
119909119899
= 119909 with respect to the topology 120591(120599)and hence 119878-completeness implies 119901-Cauchy completeness
Definition 7 Let (119883 120599) be a Hausdorff uniform space andlet 119901 be an 119860-distance on 119883 Two self-mappings 119891 and 119892 of119883 are said to be weak compatible if they commute at theircoincidence points that is 119891119909 = 119892119909 implies that 119891119892119909 = 119892119891119909
We denote by F the class of functions 120601 [0infin) rarr[0infin) nondecreasing and continuous satisfying 120601(119905) gt 0 for119905 isin (0infin) and 120601(0) = 0
Definition 8 (see [17]) A function 120601 [0infin) rarr [0infin) iscalled a comparison function if it satisfies the following
(i) 120601 is increasing that is 1199051
le 1199052
implies 120601(1199051
) le 120601(1199052
)for 1199051
1199052
isin [0infin)(ii) 120601119899(119905)
119899isinN converges to 0 as 119899 rarr infin for all 119905 isin[0infin)
Definition 9 (see [22]) A function 120601 [0infin) rarr [0infin) iscalled a (119888)-comparison function if
(i) 120601 is increasing(ii) there exist 119896
0
isin N 119886 isin (0 1) and a convergent seriesof nonnegative terms suminfin
119896=1
V119896
such that
120601119896+1
(119905) le 119886120601119896
(119905) + V119896
(2)
for 119896 ge 1198960
and any 119905 isin [0infin)
Let C be the collection of all (119888)-comparison functions120601 [0infin) rarr [0infin) defined in Definition 9
Lemma 10 (see [22]) If 120601 [0infin) rarr [0infin) is a (119888)-comparison function then the following hold
(i) 120601 is comparison function(ii) 120601(119905) lt 119905 for any 119905 isin [0infin)(iii) 120601 is continuous at 0
(iv) the series suminfin119896=0
120601119896(119905) converges for any 119905 isin [0infin)
In 1922 Banach proved that every contraction in a com-plete metric space has a unique fixed point This celebratedresult has been generalized and improved bymany authors inthe context of different abstract spaces for various operators(see [1ndash28] and the references therein) Recently fixed pointtheorems for operators 119879 defined on a complete metric space119883 with a cyclic representation of 119883 with respect to 119879 haveappeared in the literature (see eg [13ndash17]) Now we presenta modification of themain result of [16] For this we need thefollowing definitions
Definition 11 (see [13]) Let119883 be a nonempty set119898 a positiveinteger and 119879 119883 rarr 119883 a mapping 119883 = ⋃
119898
119894=1
119860119894
is said tobe a cyclic representation of119883 with respect to 119879 if
Abstract and Applied Analysis 3
(i) 119860119894
119894 = 1 2 119898 are nonempty sets(ii) 119879(119860
1
) sub 1198602
119879(119860119898minus1
) sub 119860119898
119879(119860119898
) sub 1198601
Definition 12 Let (119883 119889) be a metric space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119883 =⋃119898
119894=1
119860119894
An operator 119879 119883 rarr 119883 is a cyclic (120601)-contractionif
(i) 119883 = ⋃119898
119894=1
119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119889(119879119909 119879119910) le 120601(119889(119909 119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin F
The main result of [14] is the following
Theorem 13 (Theorem 6 of [14]) Let (119883 119889) be a completemetric space m a positive integer 119860
1
1198602
119860119898
nonemptysubsets of 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic(120601 minus120595)-contraction with 120601 120595 isin F Then 119879 has a unique fixedpoint 119911 isin ⋂119898
119894=1
119860119894
Themain aimof this paper is to prove results similar to theabovementioned theorems in uniform spaces and to presentmodifications of Theorem 21 [16] Theorems 31-32 in [18]and other related results
2 Main Result
First we present the following definition
Definition 14 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119883 =⋃119898
119894=1
119860119894
An operator 119879 119883 rarr 119883 is a cyclic (120601)-contractionif
(i) 119883 = ⋃119898
119894=1
119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898
119901 (119879119909 119879119910) le 120601 (119901 (119909 119910)) (3)
where 119860119898+1
= 1198601
and 120601 isin C
Our main result is the following
Theorem 15 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integerand119860
1
1198602
119860119898
nonempty closed subsets of119883 with respectto the topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contractionThen119879 has a unique fixedpoint 119909 isin ⋂119898
119894=1
119860119894
Proof We first show that the fixed point of 119879 is unique (if itexists) Suppose on the contrary that 119910 119911 isin 119883 with 119910 = 119911 arefixed points of 119879 The cyclic character of 119879 and the fact that119910 119911 isin 119883 are fixed points of119879 imply that 119910 119911 isin ⋂119898
119894=1
119860119894
Usingthe contractive condition we obtain
119901 (119910 119911) = 119901 (119879119910 119879119911) le 120601 (119901 (119910 119911)) lt 119901 (119910 119911) (4)
and from the last inequality
119901 (119910 119911) = 0 (5)
Similarly we can show that 119901(119910 119910) = 0 and consequently119910 = 119911
Now we prove the existence of a fixed point Note that119901 is not symmetric To show that the sequence 119909
119899
isCauchy we will show that both lim
119899rarrinfin
119901(119909119899
119909119899+119902
) = 0 andlim119899rarrinfin
119901(119909119899+119902
119909119899
) = 0 for any 119902 gt 1For this aim take 119909
0
isin 119883 and consider the sequence givenby
119909119899+1
= 119879119909119899
119899 = 0 1 2 (6)
If there exists 1198990
isin N such that 1199091198990+1
= 1199091198990 then the proof is
completed In this case 1199091198990is the required fixed point of 119879
Throughout the proof we assume that
119909119899+1
= 119909119899
for any 119899 = 0 1 2 (7)
Notice that for any 119899 gt 0 there exists 119894119899
isin 1 2 119898 suchthat 119909
119899minus1
isin 119860119894119899and 119909
119899
isin 119860119894119899+1
since119883 = ⋃119898
119894=1
119860119894
Due to thefact that 119879 is a cyclic (120601)-contraction we have
119901 (119909119899
119909119899+1
) = 119901 (119879119909119899minus1
119879119909119899
) le 120601 (119901 (119909119899minus1
119909119899
)) (8)
by taking 119909 = 119909119899
and 119910 = 119909119899+1
in (3) From (8) and taking themonotonicity of 120601 into account we derive by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(9)
As 119901 is an 119864-distance we obtain that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (10)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(11)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenoting
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (12)
implies that
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(13)
As 120601 is a (119888)-comparison function supposing 119901(1199090
1199091
) gt 0by Lemma 10 (iv) it follows that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (14)
so there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (15)
4 Abstract and Applied Analysis
Then by (13) we obtain that
lim119899rarrinfin
119901 (119909119899
119909119899+119902
) = 0 (16)
By repeating the same arguments in the proof of (16) weconclude that
lim119899rarrinfin
119901 (119909119899+119902
119909119899
) = 0 (17)
Consequently we get that the sequence 119909119899
119899ge0
is a 119901-Cauchy in the 119878-complete space 119883 = ⋃
119898
119894=1
119860119894
Thus thereexists 119909 isin 119883 such that lim
119899rarrinfin
119909119899
= 119909 In what follows weprove that119909 is a fixed point of119879 In fact since lim
119899rarrinfin
119909119899
= 119909as 119883 = ⋃
119898
119894=1
119860119894
is a cyclic representation of 119883 with respectto 119879 the sequence 119909
119899
has infinite terms in each 119860119894
for119894 isin 1 2 119898
Since 119860119894
is closed for every 119894 it follows that 119909 isin ⋂119898
119894=1
119860119894
thus we take a subsequence 119909
119899119896of 119909119899
with 119909119899119896isin 119860119894minus1
Usingthe contractive condition we can obtain
119901 (119909 119879119909) le 119901 (119909 119909119899119896+1
) + 119901 (119909119899119896+1
119879119909)
= 119901 (119909 119909119899119896+1
) + 119901 (119879119909119899119896 119879119909)
le 119901 (119909 119909119899119896+1
) + 120601 (119901 (119909119899119896 119909))
(18)
and since 119909119899119896
rarr 119909 and 120601 belong toC letting 119896 rarr infin in thelast inequality we have 119901(119909 119879119909) = 0 Analogously we canderive that 119901(119909 119909) = 0 and therefore 119909 is a fixed point of 119879This finishes the proof
Corollary 16 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integer1198601
1198602
119860119898
nonempty closed subsets of 119883 with respect tothe topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let operator119879 119883 rarr 119883 satisfy
(i) 119901(119879119909 119879119910) le 119896119901(119909 119910) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 =1 2 119898 where 119860
119898+1
= 1198601
and 0 lt 119896 lt 1
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120601(119905) = 119896119905
Corollary 17 (cf [16]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contraction Then 119879 has a unique fixed point 119911 isin ⋂119898
119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 18 (cf [13]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof119883 and119883 = ⋃
119898
119894=1
119860119894
a cyclic representation of119883with respectto 119879 Let 119879 119883 rarr 119883 satisfy
119901 (119879119909 119879119910) le 119896119901 (119909 119910) (19)
for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119896 isin (0 1) and119860119898+1
= 1198601
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Definition 19 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119879 119892 119883 rarr 119883 self-mappings An operator 119879 is a cyclic (120601)-119892-contraction if
(i) 119892119883 = ⋃119898
119894=1
119892119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119901(119879119909 119879119910) le 120601(119901(119892119909 119892119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin C
Inspired by [28] we now prove a common fixed pointtheorem as an application of our Theorem 15
Theorem 20 Let (119883 120599) be a uniform space 119879 119892 119883 rarr 119883self-maps such that 119879 is cyclic (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform space together with 119901 being an 119864-distance on119883 Suppose that 119892119860
1
1198921198602
119892119860119898
are nonemptyclosed subsets of 119892119883 with respect to the uniform topology and119879119883 sub 119892119883 = ⋃
119898
119894=1
119892119860119894
Then 119879 and 119892 have a uniquecoincidence point Moreover if 119879 and 119892 are weakly compatiblethen they have a unique common fixed point 119911 isin ⋂119898
119894=1
119892119860119894
Proof As 119892 119883 rarr 119883 so there exists 119864 sub 119883 such that 119892119864 =119892119883 and 119892 119864 rarr 119883 is one-to-one Now since 119879119883 sub 119892119883we define mappings ℎ 119892119864 rarr 119892119864 by ℎ(119892119909) = 119879119909 Since 119892is one-to-one on 119864 so ℎ is well defined As 119879 is cyclic (120601)-119892-contraction so
119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910)) (20)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 Thus
119901 (ℎ (119892119909) ℎ (119892119910)) = 119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910))
(21)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 whichimplies that ℎ is cyclic (120601)-contraction on 119892119883 Hence all theconditions ofTheorem 15 are satisfied by ℎ so ℎ has a uniquefixed point 119911 = 119892119909 in 119892119883 That is 119892119909 = 119911 = ℎ(119911) = ℎ(119892119909) =119879119909 so 119879 and 119892 have a unique coincidence point as requiredMoreover if 119879 and 119892 are weakly compatible then they have aunique common fixed point
Corollary 21 (cf Theorem 32 [18]) Let (119883 120599) be a uniformspace 119879 119892 119883 rarr 119883 self-maps such that 119879 is (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform spacetogether with 119901 being an 119864-distance on 119883 Suppose that 119879119883 sub119892119883 and 119879 and 119892 are commuting Then 119879 and 119892 have a uniquecommon fixed point 119911 isin 119883
Proof Take 119860119894
= 119883 for all 119894 = 1 119898 in Theorem 20
Example 22 Let (119883 119889) be a metric space where119883 = 1119899 cup0 and 119889 = | | Set 119860
1
= 13119899 cup 0 1 1198602
= 1(3119899 + 1) cup0 1 and 119860
3
= 1(3119899 + 2) cup 0 1 Define 120599 = 119880120598
| 120598 gt 0It is easy to see that (119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin) by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119879 119883 rarr 119883
Abstract and Applied Analysis 5
by 119879(0) = 119879(1) = 0 and 119879(1119899) = 1(4119899 + 1) then for every119909 119910 = 0 1 we have
119889 (119879119909 119879119910) = 119889 (1
4119899 + 1
1
4119898 + 1)
=|4 (119898 minus 119899)|
|4119899 + 1| |4119898 + 1|le|119898 minus 119899|
4119899 sdot 119898
le1
4
1003816100381610038161003816100381610038161003816
1
119899minus1
119898
1003816100381610038161003816100381610038161003816=1
4119889 (119909 119910)
(22)
Also for 119909 119910 = 0 1 the above inequality obviously holdsThis shows that the contractive condition of Corollary 16 issatisfied and 0 is fixed point 119879
Definition 23 Let (119883 120599) be a uniform space let 119891 119892 119883 rarr119883 be two mappings and let 119860 and 119861 be nonempty closedsubsets of 119883 The 119883 = 119860 cup 119861 is said to be a cyclicrepresentation of119883with respect to the pair (119891 119892) if 119891(119860) sub 119861and 119892(119861) sub 119860
Definition 24 Let (119883 120599) be a uniform space 119860 119861 nonemptysubsets of 119883 and 119883 = 119860 cup 119861 Two self-maps 119891 119892 119883 rarr 119883are called cyclic (120601)-contraction pair if
(i) 119883 = 119860cup119861 is a cyclic representation of119883 with respectto the pair (119891 119892)
(ii) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le120601(max119901(119909 119891119909) 119901(119910 119892119910)) for any 119909 isin 119860 119910 isin 119861where 120601 isin C
Theorem 25 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that 119891 119892 119883 rarr 119883are cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119909 isin 119860 cap 119861
Proof Take 1199090
isin 119883 and consider the sequence given by
1198911199092119899
= 1199092119899+1
1198921199092119899+1
= 1199092119899+2
119899 = 0 1 2
(23)
Since 119883 = 119860 cup 119861 for any 119899 gt 0 1199092119899
isin 119860 and 1199092119899+1
isin 119861 and(119891 119892) are cyclic (120601)-contraction pair we have
119901 (1199092119899+1
1199092119899+2
) = 119901 (1198911199092119899
1198921199092119899+1
)
le max 119901 (1198911199092119899
1198921199092119899+1
) 119901 (1198921199092119899+1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1198911199092119899
) 119901 (1199092119899+1
1198921199092119899+1
))
= 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899+1
1199092119899+2
))
(24)
Hence
119901 (1199092119899+1
1199092119899+2
) le 120601 (119901 (1199092119899
1199092119899+1
)) (25)
Similarly we have
119901 (1199092119899
1199092119899+1
) = 119901 (1198921199092119899minus1
1198911199092119899
)
le max 119901 (1198911199092119899
1198921199092119899minus1
) 119901 (1198921199092119899minus1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899minus1
1199092119899
))
(26)
Hence
119901 (1199092119899
1199092119899+1
) le 120601 (119901 (1199092119899minus1
1199092119899
)) (27)
From inequalities (25) and (27) and taking into accountthe monotonicity of 120601 we get by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(28)
Since 119901 is an 119864-distance we find that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (29)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(30)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenote
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (31)
By relation (31) we have
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(32)
Regarding 120601 isin C together with Lemma 10(iv) we get that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (33)
since 119901(1199090
1199091
) gt 0 Thus there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (34)
Then by (32) we obtain that
119901 (119909119899
119909119899+119902
) 997888rarr 0 as 119899 997888rarr infin (35)
In an analogous way we derive that
119901 (119909119899+119902
119909119899
) 997888rarr 0 as 119899 997888rarr infin (36)
Hence we get that 119909119899
119899ge0
is a 119901-Cauchy sequence in the119878-complete space 119883 = 119860 cup 119861 So there exists 119909 isin 119883such that lim
119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 In whatfollows we prove that 119909 is a fixed point of 119891 119892 In fact sincelim119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 and as 119883 = 119860 cup 119861 is a
6 Abstract and Applied Analysis
cyclic representation of 119883 with respect to 119891 119892 the sequence119909119899
has infinite terms in each 119860 119861Since 119860 119861 are closed it follows that 119909 isin 119860 cap 119861 thus we
take subsequences 1199092119899
1199092119899+1
of 119909119899
with 1199092119899
isin 119860 and 1199092119899+1
isin119861 Using the contractive condition we can obtain
119901 (119909 119891119909) le 119901 (119909 1199092119899+2
) + 119901 (1199092119899+2
119891119909)
= 119901 (119909 1199092119899+2
) + 119901 (1198921199092119899+1
119891119909)
le 119901 (119909 1199092119899+2
) +max 119901 (1198921199092119899+1
119891119909)
119901 (119891119909 1198921199092119899+1
)
le 119901 (119909 1199092119899+2
) + 120601 (max 119901 (119909 119891119909)
119901 (1199092119899+1
1199092119899+2
))
(37)
and since 119909119899
rarr 119909 and 120601 belong toC letting 119899 rarr infin in thelast inequality we have 119901(119909 119891119909) le 120601(119901(119909 119891119909)) lt 119901(119909 119891119909)hence119901(119909 119891119909) = 0 Similarly we can show that 119901(119909 119909) = 0and therefore 119909 is a fixed point of 119891 Similarly we can showthat 119909 is a fixed point of 119892 Finally in order to prove theuniqueness of the fixed point we have 119910 119911 isin 119883 with 119910 and119911 fixed points of 119891 119892 The cyclic character of 119891 119892 and the factthat 119910 119911 isin 119883 are fixed points of 119891 119892 imply that 119910 119911 isin 119860 cap 119861Using the contractive condition we obtain
119901 (119910 119911) = 119901 (119891119910 119892119911) le max 119901 (119891119910 119892119911) 119901 (119892119911 119891119910)
le 120601 (max 119901 (119910 119891119910) 119901 (119911 119892119911)) = 0
(38)
and from the last inequality we get
119901 (119910 119911) = 0 (39)
Using the same arguments above we can show that 119901(119910 119910) =0 and consequently 119910 = 119911 This finishes the proof
Corollary 26 Let (119883 119889) be a complete metric space and 119860 119861nonempty closed subsets of 119883 and 119883 = 119860 cup 119861 Let f 119892 119883 rarr119883 be cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119911 isin 119860 cap 119861
Proof By Theorem 25 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 27 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that the maps 119891 119892 119883 rarr 119883 satisfy the following inequality
(i) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le119896max119901(119909 119891119909) 119901(119910 119892119910) for any 119909 isin 119860 119910 isin 119861where 0 lt 119896 lt 1
Then 119891 119892 have a unique common fixed point 119911 isin 119860 cap 119861
Proof ByTheorem 25 it is enough to set 120601(119905) = 119896119905
Example 28 Let (119883 119901) be a partial metric space where 119883 =1119899 cup 0 1 and 119901(119909 119910) = max119909 119910 Set119860 = 12119899 cup 0 1and 119861 = 1(2119899 + 1) cup 0 1 Define 120599 = 119880
120598
| 120598 gt 0 where119880120598
= (119909 119910) isin 1198832 119901(119909 119910) lt 119901(119909 119909)+120598 It is easy to see that(119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin)by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119891 119883 rarr 119883 by 119891(12119899) =1(4119899 + 1) 119891(0) = 119891(1) = 0 and 119892(1(2119899 + 1)) = 1(4119899 + 2)119892(0) = 119892(1) = 0 Then for every 119909 119910 = 0 1 we have
max 119901 (119891119909 119892119910) 119901 (119892119910 119891119909)
= max 119901( 1
4119899 + 1
1
4119898 + 2)
119901 (1
4119898 + 2
1
4119899 + 1)
= max 1
4119899 + 1
1
4119898 + 2
le1
2max 1
2119899
1
2119898 + 1
=1
2max 119901 (119909 119891119909) 119901 (119910 119892119910)
(40)
for any 119909 isin 119860 119910 isin 119861 Also for 119909 119910 = 0 1 the above inequalityobviously holds This shows that the contractive condition ofCorollary 27 is satisfied and 0 is a common fixed point of 119891and 119892
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanksDSR KAU for the financialsupport
References
[1] N Bourbaki Elements de mathematique Fasc II Livre IIITopologie generale Chapitre 1 Structures Topologiques Chapitre2 Structures Uniformes vol 1142 of Actualites Scientifiques etIndustrielles Hermann Paris France 1965
[2] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 1 Springer New York NY USA 1986
[3] V Berinde Iterative Approximation of Fixed Points SpringerBerlin Germany 2007
[4] J Jachymski ldquoFixed point theorems for expansive mappingsrdquoMathematica Japonica vol 42 no 1 pp 131ndash136 1995
[5] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996
[6] M Cherichi and B Samet ldquoFixed point theorems on orderedgauge spaces with applications to nonlinear integral equationsrdquoFixed Point Theory and Applications vol 2012 article 13 2012
Abstract and Applied Analysis 7
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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 3
(i) 119860119894
119894 = 1 2 119898 are nonempty sets(ii) 119879(119860
1
) sub 1198602
119879(119860119898minus1
) sub 119860119898
119879(119860119898
) sub 1198601
Definition 12 Let (119883 119889) be a metric space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119883 =⋃119898
119894=1
119860119894
An operator 119879 119883 rarr 119883 is a cyclic (120601)-contractionif
(i) 119883 = ⋃119898
119894=1
119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119889(119879119909 119879119910) le 120601(119889(119909 119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin F
The main result of [14] is the following
Theorem 13 (Theorem 6 of [14]) Let (119883 119889) be a completemetric space m a positive integer 119860
1
1198602
119860119898
nonemptysubsets of 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic(120601 minus120595)-contraction with 120601 120595 isin F Then 119879 has a unique fixedpoint 119911 isin ⋂119898
119894=1
119860119894
Themain aimof this paper is to prove results similar to theabovementioned theorems in uniform spaces and to presentmodifications of Theorem 21 [16] Theorems 31-32 in [18]and other related results
2 Main Result
First we present the following definition
Definition 14 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119883 =⋃119898
119894=1
119860119894
An operator 119879 119883 rarr 119883 is a cyclic (120601)-contractionif
(i) 119883 = ⋃119898
119894=1
119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898
119901 (119879119909 119879119910) le 120601 (119901 (119909 119910)) (3)
where 119860119898+1
= 1198601
and 120601 isin C
Our main result is the following
Theorem 15 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integerand119860
1
1198602
119860119898
nonempty closed subsets of119883 with respectto the topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contractionThen119879 has a unique fixedpoint 119909 isin ⋂119898
119894=1
119860119894
Proof We first show that the fixed point of 119879 is unique (if itexists) Suppose on the contrary that 119910 119911 isin 119883 with 119910 = 119911 arefixed points of 119879 The cyclic character of 119879 and the fact that119910 119911 isin 119883 are fixed points of119879 imply that 119910 119911 isin ⋂119898
119894=1
119860119894
Usingthe contractive condition we obtain
119901 (119910 119911) = 119901 (119879119910 119879119911) le 120601 (119901 (119910 119911)) lt 119901 (119910 119911) (4)
and from the last inequality
119901 (119910 119911) = 0 (5)
Similarly we can show that 119901(119910 119910) = 0 and consequently119910 = 119911
Now we prove the existence of a fixed point Note that119901 is not symmetric To show that the sequence 119909
119899
isCauchy we will show that both lim
119899rarrinfin
119901(119909119899
119909119899+119902
) = 0 andlim119899rarrinfin
119901(119909119899+119902
119909119899
) = 0 for any 119902 gt 1For this aim take 119909
0
isin 119883 and consider the sequence givenby
119909119899+1
= 119879119909119899
119899 = 0 1 2 (6)
If there exists 1198990
isin N such that 1199091198990+1
= 1199091198990 then the proof is
completed In this case 1199091198990is the required fixed point of 119879
Throughout the proof we assume that
119909119899+1
= 119909119899
for any 119899 = 0 1 2 (7)
Notice that for any 119899 gt 0 there exists 119894119899
isin 1 2 119898 suchthat 119909
119899minus1
isin 119860119894119899and 119909
119899
isin 119860119894119899+1
since119883 = ⋃119898
119894=1
119860119894
Due to thefact that 119879 is a cyclic (120601)-contraction we have
119901 (119909119899
119909119899+1
) = 119901 (119879119909119899minus1
119879119909119899
) le 120601 (119901 (119909119899minus1
119909119899
)) (8)
by taking 119909 = 119909119899
and 119910 = 119909119899+1
in (3) From (8) and taking themonotonicity of 120601 into account we derive by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(9)
As 119901 is an 119864-distance we obtain that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (10)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(11)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenoting
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (12)
implies that
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(13)
As 120601 is a (119888)-comparison function supposing 119901(1199090
1199091
) gt 0by Lemma 10 (iv) it follows that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (14)
so there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (15)
4 Abstract and Applied Analysis
Then by (13) we obtain that
lim119899rarrinfin
119901 (119909119899
119909119899+119902
) = 0 (16)
By repeating the same arguments in the proof of (16) weconclude that
lim119899rarrinfin
119901 (119909119899+119902
119909119899
) = 0 (17)
Consequently we get that the sequence 119909119899
119899ge0
is a 119901-Cauchy in the 119878-complete space 119883 = ⋃
119898
119894=1
119860119894
Thus thereexists 119909 isin 119883 such that lim
119899rarrinfin
119909119899
= 119909 In what follows weprove that119909 is a fixed point of119879 In fact since lim
119899rarrinfin
119909119899
= 119909as 119883 = ⋃
119898
119894=1
119860119894
is a cyclic representation of 119883 with respectto 119879 the sequence 119909
119899
has infinite terms in each 119860119894
for119894 isin 1 2 119898
Since 119860119894
is closed for every 119894 it follows that 119909 isin ⋂119898
119894=1
119860119894
thus we take a subsequence 119909
119899119896of 119909119899
with 119909119899119896isin 119860119894minus1
Usingthe contractive condition we can obtain
119901 (119909 119879119909) le 119901 (119909 119909119899119896+1
) + 119901 (119909119899119896+1
119879119909)
= 119901 (119909 119909119899119896+1
) + 119901 (119879119909119899119896 119879119909)
le 119901 (119909 119909119899119896+1
) + 120601 (119901 (119909119899119896 119909))
(18)
and since 119909119899119896
rarr 119909 and 120601 belong toC letting 119896 rarr infin in thelast inequality we have 119901(119909 119879119909) = 0 Analogously we canderive that 119901(119909 119909) = 0 and therefore 119909 is a fixed point of 119879This finishes the proof
Corollary 16 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integer1198601
1198602
119860119898
nonempty closed subsets of 119883 with respect tothe topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let operator119879 119883 rarr 119883 satisfy
(i) 119901(119879119909 119879119910) le 119896119901(119909 119910) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 =1 2 119898 where 119860
119898+1
= 1198601
and 0 lt 119896 lt 1
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120601(119905) = 119896119905
Corollary 17 (cf [16]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contraction Then 119879 has a unique fixed point 119911 isin ⋂119898
119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 18 (cf [13]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof119883 and119883 = ⋃
119898
119894=1
119860119894
a cyclic representation of119883with respectto 119879 Let 119879 119883 rarr 119883 satisfy
119901 (119879119909 119879119910) le 119896119901 (119909 119910) (19)
for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119896 isin (0 1) and119860119898+1
= 1198601
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Definition 19 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119879 119892 119883 rarr 119883 self-mappings An operator 119879 is a cyclic (120601)-119892-contraction if
(i) 119892119883 = ⋃119898
119894=1
119892119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119901(119879119909 119879119910) le 120601(119901(119892119909 119892119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin C
Inspired by [28] we now prove a common fixed pointtheorem as an application of our Theorem 15
Theorem 20 Let (119883 120599) be a uniform space 119879 119892 119883 rarr 119883self-maps such that 119879 is cyclic (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform space together with 119901 being an 119864-distance on119883 Suppose that 119892119860
1
1198921198602
119892119860119898
are nonemptyclosed subsets of 119892119883 with respect to the uniform topology and119879119883 sub 119892119883 = ⋃
119898
119894=1
119892119860119894
Then 119879 and 119892 have a uniquecoincidence point Moreover if 119879 and 119892 are weakly compatiblethen they have a unique common fixed point 119911 isin ⋂119898
119894=1
119892119860119894
Proof As 119892 119883 rarr 119883 so there exists 119864 sub 119883 such that 119892119864 =119892119883 and 119892 119864 rarr 119883 is one-to-one Now since 119879119883 sub 119892119883we define mappings ℎ 119892119864 rarr 119892119864 by ℎ(119892119909) = 119879119909 Since 119892is one-to-one on 119864 so ℎ is well defined As 119879 is cyclic (120601)-119892-contraction so
119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910)) (20)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 Thus
119901 (ℎ (119892119909) ℎ (119892119910)) = 119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910))
(21)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 whichimplies that ℎ is cyclic (120601)-contraction on 119892119883 Hence all theconditions ofTheorem 15 are satisfied by ℎ so ℎ has a uniquefixed point 119911 = 119892119909 in 119892119883 That is 119892119909 = 119911 = ℎ(119911) = ℎ(119892119909) =119879119909 so 119879 and 119892 have a unique coincidence point as requiredMoreover if 119879 and 119892 are weakly compatible then they have aunique common fixed point
Corollary 21 (cf Theorem 32 [18]) Let (119883 120599) be a uniformspace 119879 119892 119883 rarr 119883 self-maps such that 119879 is (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform spacetogether with 119901 being an 119864-distance on 119883 Suppose that 119879119883 sub119892119883 and 119879 and 119892 are commuting Then 119879 and 119892 have a uniquecommon fixed point 119911 isin 119883
Proof Take 119860119894
= 119883 for all 119894 = 1 119898 in Theorem 20
Example 22 Let (119883 119889) be a metric space where119883 = 1119899 cup0 and 119889 = | | Set 119860
1
= 13119899 cup 0 1 1198602
= 1(3119899 + 1) cup0 1 and 119860
3
= 1(3119899 + 2) cup 0 1 Define 120599 = 119880120598
| 120598 gt 0It is easy to see that (119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin) by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119879 119883 rarr 119883
Abstract and Applied Analysis 5
by 119879(0) = 119879(1) = 0 and 119879(1119899) = 1(4119899 + 1) then for every119909 119910 = 0 1 we have
119889 (119879119909 119879119910) = 119889 (1
4119899 + 1
1
4119898 + 1)
=|4 (119898 minus 119899)|
|4119899 + 1| |4119898 + 1|le|119898 minus 119899|
4119899 sdot 119898
le1
4
1003816100381610038161003816100381610038161003816
1
119899minus1
119898
1003816100381610038161003816100381610038161003816=1
4119889 (119909 119910)
(22)
Also for 119909 119910 = 0 1 the above inequality obviously holdsThis shows that the contractive condition of Corollary 16 issatisfied and 0 is fixed point 119879
Definition 23 Let (119883 120599) be a uniform space let 119891 119892 119883 rarr119883 be two mappings and let 119860 and 119861 be nonempty closedsubsets of 119883 The 119883 = 119860 cup 119861 is said to be a cyclicrepresentation of119883with respect to the pair (119891 119892) if 119891(119860) sub 119861and 119892(119861) sub 119860
Definition 24 Let (119883 120599) be a uniform space 119860 119861 nonemptysubsets of 119883 and 119883 = 119860 cup 119861 Two self-maps 119891 119892 119883 rarr 119883are called cyclic (120601)-contraction pair if
(i) 119883 = 119860cup119861 is a cyclic representation of119883 with respectto the pair (119891 119892)
(ii) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le120601(max119901(119909 119891119909) 119901(119910 119892119910)) for any 119909 isin 119860 119910 isin 119861where 120601 isin C
Theorem 25 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that 119891 119892 119883 rarr 119883are cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119909 isin 119860 cap 119861
Proof Take 1199090
isin 119883 and consider the sequence given by
1198911199092119899
= 1199092119899+1
1198921199092119899+1
= 1199092119899+2
119899 = 0 1 2
(23)
Since 119883 = 119860 cup 119861 for any 119899 gt 0 1199092119899
isin 119860 and 1199092119899+1
isin 119861 and(119891 119892) are cyclic (120601)-contraction pair we have
119901 (1199092119899+1
1199092119899+2
) = 119901 (1198911199092119899
1198921199092119899+1
)
le max 119901 (1198911199092119899
1198921199092119899+1
) 119901 (1198921199092119899+1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1198911199092119899
) 119901 (1199092119899+1
1198921199092119899+1
))
= 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899+1
1199092119899+2
))
(24)
Hence
119901 (1199092119899+1
1199092119899+2
) le 120601 (119901 (1199092119899
1199092119899+1
)) (25)
Similarly we have
119901 (1199092119899
1199092119899+1
) = 119901 (1198921199092119899minus1
1198911199092119899
)
le max 119901 (1198911199092119899
1198921199092119899minus1
) 119901 (1198921199092119899minus1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899minus1
1199092119899
))
(26)
Hence
119901 (1199092119899
1199092119899+1
) le 120601 (119901 (1199092119899minus1
1199092119899
)) (27)
From inequalities (25) and (27) and taking into accountthe monotonicity of 120601 we get by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(28)
Since 119901 is an 119864-distance we find that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (29)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(30)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenote
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (31)
By relation (31) we have
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(32)
Regarding 120601 isin C together with Lemma 10(iv) we get that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (33)
since 119901(1199090
1199091
) gt 0 Thus there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (34)
Then by (32) we obtain that
119901 (119909119899
119909119899+119902
) 997888rarr 0 as 119899 997888rarr infin (35)
In an analogous way we derive that
119901 (119909119899+119902
119909119899
) 997888rarr 0 as 119899 997888rarr infin (36)
Hence we get that 119909119899
119899ge0
is a 119901-Cauchy sequence in the119878-complete space 119883 = 119860 cup 119861 So there exists 119909 isin 119883such that lim
119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 In whatfollows we prove that 119909 is a fixed point of 119891 119892 In fact sincelim119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 and as 119883 = 119860 cup 119861 is a
6 Abstract and Applied Analysis
cyclic representation of 119883 with respect to 119891 119892 the sequence119909119899
has infinite terms in each 119860 119861Since 119860 119861 are closed it follows that 119909 isin 119860 cap 119861 thus we
take subsequences 1199092119899
1199092119899+1
of 119909119899
with 1199092119899
isin 119860 and 1199092119899+1
isin119861 Using the contractive condition we can obtain
119901 (119909 119891119909) le 119901 (119909 1199092119899+2
) + 119901 (1199092119899+2
119891119909)
= 119901 (119909 1199092119899+2
) + 119901 (1198921199092119899+1
119891119909)
le 119901 (119909 1199092119899+2
) +max 119901 (1198921199092119899+1
119891119909)
119901 (119891119909 1198921199092119899+1
)
le 119901 (119909 1199092119899+2
) + 120601 (max 119901 (119909 119891119909)
119901 (1199092119899+1
1199092119899+2
))
(37)
and since 119909119899
rarr 119909 and 120601 belong toC letting 119899 rarr infin in thelast inequality we have 119901(119909 119891119909) le 120601(119901(119909 119891119909)) lt 119901(119909 119891119909)hence119901(119909 119891119909) = 0 Similarly we can show that 119901(119909 119909) = 0and therefore 119909 is a fixed point of 119891 Similarly we can showthat 119909 is a fixed point of 119892 Finally in order to prove theuniqueness of the fixed point we have 119910 119911 isin 119883 with 119910 and119911 fixed points of 119891 119892 The cyclic character of 119891 119892 and the factthat 119910 119911 isin 119883 are fixed points of 119891 119892 imply that 119910 119911 isin 119860 cap 119861Using the contractive condition we obtain
119901 (119910 119911) = 119901 (119891119910 119892119911) le max 119901 (119891119910 119892119911) 119901 (119892119911 119891119910)
le 120601 (max 119901 (119910 119891119910) 119901 (119911 119892119911)) = 0
(38)
and from the last inequality we get
119901 (119910 119911) = 0 (39)
Using the same arguments above we can show that 119901(119910 119910) =0 and consequently 119910 = 119911 This finishes the proof
Corollary 26 Let (119883 119889) be a complete metric space and 119860 119861nonempty closed subsets of 119883 and 119883 = 119860 cup 119861 Let f 119892 119883 rarr119883 be cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119911 isin 119860 cap 119861
Proof By Theorem 25 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 27 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that the maps 119891 119892 119883 rarr 119883 satisfy the following inequality
(i) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le119896max119901(119909 119891119909) 119901(119910 119892119910) for any 119909 isin 119860 119910 isin 119861where 0 lt 119896 lt 1
Then 119891 119892 have a unique common fixed point 119911 isin 119860 cap 119861
Proof ByTheorem 25 it is enough to set 120601(119905) = 119896119905
Example 28 Let (119883 119901) be a partial metric space where 119883 =1119899 cup 0 1 and 119901(119909 119910) = max119909 119910 Set119860 = 12119899 cup 0 1and 119861 = 1(2119899 + 1) cup 0 1 Define 120599 = 119880
120598
| 120598 gt 0 where119880120598
= (119909 119910) isin 1198832 119901(119909 119910) lt 119901(119909 119909)+120598 It is easy to see that(119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin)by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119891 119883 rarr 119883 by 119891(12119899) =1(4119899 + 1) 119891(0) = 119891(1) = 0 and 119892(1(2119899 + 1)) = 1(4119899 + 2)119892(0) = 119892(1) = 0 Then for every 119909 119910 = 0 1 we have
max 119901 (119891119909 119892119910) 119901 (119892119910 119891119909)
= max 119901( 1
4119899 + 1
1
4119898 + 2)
119901 (1
4119898 + 2
1
4119899 + 1)
= max 1
4119899 + 1
1
4119898 + 2
le1
2max 1
2119899
1
2119898 + 1
=1
2max 119901 (119909 119891119909) 119901 (119910 119892119910)
(40)
for any 119909 isin 119860 119910 isin 119861 Also for 119909 119910 = 0 1 the above inequalityobviously holds This shows that the contractive condition ofCorollary 27 is satisfied and 0 is a common fixed point of 119891and 119892
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanksDSR KAU for the financialsupport
References
[1] N Bourbaki Elements de mathematique Fasc II Livre IIITopologie generale Chapitre 1 Structures Topologiques Chapitre2 Structures Uniformes vol 1142 of Actualites Scientifiques etIndustrielles Hermann Paris France 1965
[2] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 1 Springer New York NY USA 1986
[3] V Berinde Iterative Approximation of Fixed Points SpringerBerlin Germany 2007
[4] J Jachymski ldquoFixed point theorems for expansive mappingsrdquoMathematica Japonica vol 42 no 1 pp 131ndash136 1995
[5] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996
[6] M Cherichi and B Samet ldquoFixed point theorems on orderedgauge spaces with applications to nonlinear integral equationsrdquoFixed Point Theory and Applications vol 2012 article 13 2012
Abstract and Applied Analysis 7
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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
4 Abstract and Applied Analysis
Then by (13) we obtain that
lim119899rarrinfin
119901 (119909119899
119909119899+119902
) = 0 (16)
By repeating the same arguments in the proof of (16) weconclude that
lim119899rarrinfin
119901 (119909119899+119902
119909119899
) = 0 (17)
Consequently we get that the sequence 119909119899
119899ge0
is a 119901-Cauchy in the 119878-complete space 119883 = ⋃
119898
119894=1
119860119894
Thus thereexists 119909 isin 119883 such that lim
119899rarrinfin
119909119899
= 119909 In what follows weprove that119909 is a fixed point of119879 In fact since lim
119899rarrinfin
119909119899
= 119909as 119883 = ⋃
119898
119894=1
119860119894
is a cyclic representation of 119883 with respectto 119879 the sequence 119909
119899
has infinite terms in each 119860119894
for119894 isin 1 2 119898
Since 119860119894
is closed for every 119894 it follows that 119909 isin ⋂119898
119894=1
119860119894
thus we take a subsequence 119909
119899119896of 119909119899
with 119909119899119896isin 119860119894minus1
Usingthe contractive condition we can obtain
119901 (119909 119879119909) le 119901 (119909 119909119899119896+1
) + 119901 (119909119899119896+1
119879119909)
= 119901 (119909 119909119899119896+1
) + 119901 (119879119909119899119896 119879119909)
le 119901 (119909 119909119899119896+1
) + 120601 (119901 (119909119899119896 119909))
(18)
and since 119909119899119896
rarr 119909 and 120601 belong toC letting 119896 rarr infin in thelast inequality we have 119901(119909 119879119909) = 0 Analogously we canderive that 119901(119909 119909) = 0 and therefore 119909 is a fixed point of 119879This finishes the proof
Corollary 16 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 119898 a positive integer1198601
1198602
119860119898
nonempty closed subsets of 119883 with respect tothe topological space (119883 120591(120599)) and 119883 = ⋃
119898
119894=1
119860119894
Let operator119879 119883 rarr 119883 satisfy
(i) 119901(119879119909 119879119910) le 119896119901(119909 119910) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 =1 2 119898 where 119860
119898+1
= 1198601
and 0 lt 119896 lt 1
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120601(119905) = 119896119905
Corollary 17 (cf [16]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof 119883 and 119883 = ⋃
119898
119894=1
119860119894
Let 119879 119883 rarr 119883 be a cyclic (120601)-contraction Then 119879 has a unique fixed point 119911 isin ⋂119898
119894=1
119860119894
Proof ByTheorem 15 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 18 (cf [13]) Let (119883 119889) be a complete metric space119898 a positive integer 119860
1
1198602
119860119898
nonempty closed subsetsof119883 and119883 = ⋃
119898
119894=1
119860119894
a cyclic representation of119883with respectto 119879 Let 119879 119883 rarr 119883 satisfy
119901 (119879119909 119879119910) le 119896119901 (119909 119910) (19)
for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119896 isin (0 1) and119860119898+1
= 1198601
Then 119879 has a unique fixed point 119911 isin ⋂119898119894=1
119860119894
Definition 19 Let (119883 120599) be a uniform space 119898 a positiveinteger 119860
1
1198602
119860119898
nonempty subsets of 119883 and 119879 119892 119883 rarr 119883 self-mappings An operator 119879 is a cyclic (120601)-119892-contraction if
(i) 119892119883 = ⋃119898
119894=1
119892119860119894
is a cyclic representation of 119883 withrespect to 119879
(ii) 119901(119879119909 119879119910) le 120601(119901(119892119909 119892119910)) for any 119909 isin 119860119894
119910 isin 119860119894+1
119894 = 1 2 119898 where 119860
119898+1
= 1198601
and 120601 isin C
Inspired by [28] we now prove a common fixed pointtheorem as an application of our Theorem 15
Theorem 20 Let (119883 120599) be a uniform space 119879 119892 119883 rarr 119883self-maps such that 119879 is cyclic (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform space together with 119901 being an 119864-distance on119883 Suppose that 119892119860
1
1198921198602
119892119860119898
are nonemptyclosed subsets of 119892119883 with respect to the uniform topology and119879119883 sub 119892119883 = ⋃
119898
119894=1
119892119860119894
Then 119879 and 119892 have a uniquecoincidence point Moreover if 119879 and 119892 are weakly compatiblethen they have a unique common fixed point 119911 isin ⋂119898
119894=1
119892119860119894
Proof As 119892 119883 rarr 119883 so there exists 119864 sub 119883 such that 119892119864 =119892119883 and 119892 119864 rarr 119883 is one-to-one Now since 119879119883 sub 119892119883we define mappings ℎ 119892119864 rarr 119892119864 by ℎ(119892119909) = 119879119909 Since 119892is one-to-one on 119864 so ℎ is well defined As 119879 is cyclic (120601)-119892-contraction so
119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910)) (20)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 Thus
119901 (ℎ (119892119909) ℎ (119892119910)) = 119901 (119879119909 119879119910) le 120601 (119901 (119892119909 119892119910))
(21)
for any 119892119909 isin 119892119860119894
119892119910 isin 119892119860119894+1
119894 = 1 2 119898 whichimplies that ℎ is cyclic (120601)-contraction on 119892119883 Hence all theconditions ofTheorem 15 are satisfied by ℎ so ℎ has a uniquefixed point 119911 = 119892119909 in 119892119883 That is 119892119909 = 119911 = ℎ(119911) = ℎ(119892119909) =119879119909 so 119879 and 119892 have a unique coincidence point as requiredMoreover if 119879 and 119892 are weakly compatible then they have aunique common fixed point
Corollary 21 (cf Theorem 32 [18]) Let (119883 120599) be a uniformspace 119879 119892 119883 rarr 119883 self-maps such that 119879 is (120601)-119892-contraction and 119892119883119878-complete Hausdorff uniform spacetogether with 119901 being an 119864-distance on 119883 Suppose that 119879119883 sub119892119883 and 119879 and 119892 are commuting Then 119879 and 119892 have a uniquecommon fixed point 119911 isin 119883
Proof Take 119860119894
= 119883 for all 119894 = 1 119898 in Theorem 20
Example 22 Let (119883 119889) be a metric space where119883 = 1119899 cup0 and 119889 = | | Set 119860
1
= 13119899 cup 0 1 1198602
= 1(3119899 + 1) cup0 1 and 119860
3
= 1(3119899 + 2) cup 0 1 Define 120599 = 119880120598
| 120598 gt 0It is easy to see that (119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin) by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119879 119883 rarr 119883
Abstract and Applied Analysis 5
by 119879(0) = 119879(1) = 0 and 119879(1119899) = 1(4119899 + 1) then for every119909 119910 = 0 1 we have
119889 (119879119909 119879119910) = 119889 (1
4119899 + 1
1
4119898 + 1)
=|4 (119898 minus 119899)|
|4119899 + 1| |4119898 + 1|le|119898 minus 119899|
4119899 sdot 119898
le1
4
1003816100381610038161003816100381610038161003816
1
119899minus1
119898
1003816100381610038161003816100381610038161003816=1
4119889 (119909 119910)
(22)
Also for 119909 119910 = 0 1 the above inequality obviously holdsThis shows that the contractive condition of Corollary 16 issatisfied and 0 is fixed point 119879
Definition 23 Let (119883 120599) be a uniform space let 119891 119892 119883 rarr119883 be two mappings and let 119860 and 119861 be nonempty closedsubsets of 119883 The 119883 = 119860 cup 119861 is said to be a cyclicrepresentation of119883with respect to the pair (119891 119892) if 119891(119860) sub 119861and 119892(119861) sub 119860
Definition 24 Let (119883 120599) be a uniform space 119860 119861 nonemptysubsets of 119883 and 119883 = 119860 cup 119861 Two self-maps 119891 119892 119883 rarr 119883are called cyclic (120601)-contraction pair if
(i) 119883 = 119860cup119861 is a cyclic representation of119883 with respectto the pair (119891 119892)
(ii) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le120601(max119901(119909 119891119909) 119901(119910 119892119910)) for any 119909 isin 119860 119910 isin 119861where 120601 isin C
Theorem 25 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that 119891 119892 119883 rarr 119883are cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119909 isin 119860 cap 119861
Proof Take 1199090
isin 119883 and consider the sequence given by
1198911199092119899
= 1199092119899+1
1198921199092119899+1
= 1199092119899+2
119899 = 0 1 2
(23)
Since 119883 = 119860 cup 119861 for any 119899 gt 0 1199092119899
isin 119860 and 1199092119899+1
isin 119861 and(119891 119892) are cyclic (120601)-contraction pair we have
119901 (1199092119899+1
1199092119899+2
) = 119901 (1198911199092119899
1198921199092119899+1
)
le max 119901 (1198911199092119899
1198921199092119899+1
) 119901 (1198921199092119899+1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1198911199092119899
) 119901 (1199092119899+1
1198921199092119899+1
))
= 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899+1
1199092119899+2
))
(24)
Hence
119901 (1199092119899+1
1199092119899+2
) le 120601 (119901 (1199092119899
1199092119899+1
)) (25)
Similarly we have
119901 (1199092119899
1199092119899+1
) = 119901 (1198921199092119899minus1
1198911199092119899
)
le max 119901 (1198911199092119899
1198921199092119899minus1
) 119901 (1198921199092119899minus1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899minus1
1199092119899
))
(26)
Hence
119901 (1199092119899
1199092119899+1
) le 120601 (119901 (1199092119899minus1
1199092119899
)) (27)
From inequalities (25) and (27) and taking into accountthe monotonicity of 120601 we get by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(28)
Since 119901 is an 119864-distance we find that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (29)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(30)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenote
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (31)
By relation (31) we have
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(32)
Regarding 120601 isin C together with Lemma 10(iv) we get that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (33)
since 119901(1199090
1199091
) gt 0 Thus there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (34)
Then by (32) we obtain that
119901 (119909119899
119909119899+119902
) 997888rarr 0 as 119899 997888rarr infin (35)
In an analogous way we derive that
119901 (119909119899+119902
119909119899
) 997888rarr 0 as 119899 997888rarr infin (36)
Hence we get that 119909119899
119899ge0
is a 119901-Cauchy sequence in the119878-complete space 119883 = 119860 cup 119861 So there exists 119909 isin 119883such that lim
119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 In whatfollows we prove that 119909 is a fixed point of 119891 119892 In fact sincelim119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 and as 119883 = 119860 cup 119861 is a
6 Abstract and Applied Analysis
cyclic representation of 119883 with respect to 119891 119892 the sequence119909119899
has infinite terms in each 119860 119861Since 119860 119861 are closed it follows that 119909 isin 119860 cap 119861 thus we
take subsequences 1199092119899
1199092119899+1
of 119909119899
with 1199092119899
isin 119860 and 1199092119899+1
isin119861 Using the contractive condition we can obtain
119901 (119909 119891119909) le 119901 (119909 1199092119899+2
) + 119901 (1199092119899+2
119891119909)
= 119901 (119909 1199092119899+2
) + 119901 (1198921199092119899+1
119891119909)
le 119901 (119909 1199092119899+2
) +max 119901 (1198921199092119899+1
119891119909)
119901 (119891119909 1198921199092119899+1
)
le 119901 (119909 1199092119899+2
) + 120601 (max 119901 (119909 119891119909)
119901 (1199092119899+1
1199092119899+2
))
(37)
and since 119909119899
rarr 119909 and 120601 belong toC letting 119899 rarr infin in thelast inequality we have 119901(119909 119891119909) le 120601(119901(119909 119891119909)) lt 119901(119909 119891119909)hence119901(119909 119891119909) = 0 Similarly we can show that 119901(119909 119909) = 0and therefore 119909 is a fixed point of 119891 Similarly we can showthat 119909 is a fixed point of 119892 Finally in order to prove theuniqueness of the fixed point we have 119910 119911 isin 119883 with 119910 and119911 fixed points of 119891 119892 The cyclic character of 119891 119892 and the factthat 119910 119911 isin 119883 are fixed points of 119891 119892 imply that 119910 119911 isin 119860 cap 119861Using the contractive condition we obtain
119901 (119910 119911) = 119901 (119891119910 119892119911) le max 119901 (119891119910 119892119911) 119901 (119892119911 119891119910)
le 120601 (max 119901 (119910 119891119910) 119901 (119911 119892119911)) = 0
(38)
and from the last inequality we get
119901 (119910 119911) = 0 (39)
Using the same arguments above we can show that 119901(119910 119910) =0 and consequently 119910 = 119911 This finishes the proof
Corollary 26 Let (119883 119889) be a complete metric space and 119860 119861nonempty closed subsets of 119883 and 119883 = 119860 cup 119861 Let f 119892 119883 rarr119883 be cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119911 isin 119860 cap 119861
Proof By Theorem 25 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 27 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that the maps 119891 119892 119883 rarr 119883 satisfy the following inequality
(i) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le119896max119901(119909 119891119909) 119901(119910 119892119910) for any 119909 isin 119860 119910 isin 119861where 0 lt 119896 lt 1
Then 119891 119892 have a unique common fixed point 119911 isin 119860 cap 119861
Proof ByTheorem 25 it is enough to set 120601(119905) = 119896119905
Example 28 Let (119883 119901) be a partial metric space where 119883 =1119899 cup 0 1 and 119901(119909 119910) = max119909 119910 Set119860 = 12119899 cup 0 1and 119861 = 1(2119899 + 1) cup 0 1 Define 120599 = 119880
120598
| 120598 gt 0 where119880120598
= (119909 119910) isin 1198832 119901(119909 119910) lt 119901(119909 119909)+120598 It is easy to see that(119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin)by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119891 119883 rarr 119883 by 119891(12119899) =1(4119899 + 1) 119891(0) = 119891(1) = 0 and 119892(1(2119899 + 1)) = 1(4119899 + 2)119892(0) = 119892(1) = 0 Then for every 119909 119910 = 0 1 we have
max 119901 (119891119909 119892119910) 119901 (119892119910 119891119909)
= max 119901( 1
4119899 + 1
1
4119898 + 2)
119901 (1
4119898 + 2
1
4119899 + 1)
= max 1
4119899 + 1
1
4119898 + 2
le1
2max 1
2119899
1
2119898 + 1
=1
2max 119901 (119909 119891119909) 119901 (119910 119892119910)
(40)
for any 119909 isin 119860 119910 isin 119861 Also for 119909 119910 = 0 1 the above inequalityobviously holds This shows that the contractive condition ofCorollary 27 is satisfied and 0 is a common fixed point of 119891and 119892
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanksDSR KAU for the financialsupport
References
[1] N Bourbaki Elements de mathematique Fasc II Livre IIITopologie generale Chapitre 1 Structures Topologiques Chapitre2 Structures Uniformes vol 1142 of Actualites Scientifiques etIndustrielles Hermann Paris France 1965
[2] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 1 Springer New York NY USA 1986
[3] V Berinde Iterative Approximation of Fixed Points SpringerBerlin Germany 2007
[4] J Jachymski ldquoFixed point theorems for expansive mappingsrdquoMathematica Japonica vol 42 no 1 pp 131ndash136 1995
[5] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996
[6] M Cherichi and B Samet ldquoFixed point theorems on orderedgauge spaces with applications to nonlinear integral equationsrdquoFixed Point Theory and Applications vol 2012 article 13 2012
Abstract and Applied Analysis 7
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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 5
by 119879(0) = 119879(1) = 0 and 119879(1119899) = 1(4119899 + 1) then for every119909 119910 = 0 1 we have
119889 (119879119909 119879119910) = 119889 (1
4119899 + 1
1
4119898 + 1)
=|4 (119898 minus 119899)|
|4119899 + 1| |4119898 + 1|le|119898 minus 119899|
4119899 sdot 119898
le1
4
1003816100381610038161003816100381610038161003816
1
119899minus1
119898
1003816100381610038161003816100381610038161003816=1
4119889 (119909 119910)
(22)
Also for 119909 119910 = 0 1 the above inequality obviously holdsThis shows that the contractive condition of Corollary 16 issatisfied and 0 is fixed point 119879
Definition 23 Let (119883 120599) be a uniform space let 119891 119892 119883 rarr119883 be two mappings and let 119860 and 119861 be nonempty closedsubsets of 119883 The 119883 = 119860 cup 119861 is said to be a cyclicrepresentation of119883with respect to the pair (119891 119892) if 119891(119860) sub 119861and 119892(119861) sub 119860
Definition 24 Let (119883 120599) be a uniform space 119860 119861 nonemptysubsets of 119883 and 119883 = 119860 cup 119861 Two self-maps 119891 119892 119883 rarr 119883are called cyclic (120601)-contraction pair if
(i) 119883 = 119860cup119861 is a cyclic representation of119883 with respectto the pair (119891 119892)
(ii) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le120601(max119901(119909 119891119909) 119901(119910 119892119910)) for any 119909 isin 119860 119910 isin 119861where 120601 isin C
Theorem 25 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that 119891 119892 119883 rarr 119883are cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119909 isin 119860 cap 119861
Proof Take 1199090
isin 119883 and consider the sequence given by
1198911199092119899
= 1199092119899+1
1198921199092119899+1
= 1199092119899+2
119899 = 0 1 2
(23)
Since 119883 = 119860 cup 119861 for any 119899 gt 0 1199092119899
isin 119860 and 1199092119899+1
isin 119861 and(119891 119892) are cyclic (120601)-contraction pair we have
119901 (1199092119899+1
1199092119899+2
) = 119901 (1198911199092119899
1198921199092119899+1
)
le max 119901 (1198911199092119899
1198921199092119899+1
) 119901 (1198921199092119899+1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1198911199092119899
) 119901 (1199092119899+1
1198921199092119899+1
))
= 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899+1
1199092119899+2
))
(24)
Hence
119901 (1199092119899+1
1199092119899+2
) le 120601 (119901 (1199092119899
1199092119899+1
)) (25)
Similarly we have
119901 (1199092119899
1199092119899+1
) = 119901 (1198921199092119899minus1
1198911199092119899
)
le max 119901 (1198911199092119899
1198921199092119899minus1
) 119901 (1198921199092119899minus1
1198911199092119899
)
le 120601 (max 119901 (1199092119899
1199092119899+1
) 119901 (1199092119899minus1
1199092119899
))
(26)
Hence
119901 (1199092119899
1199092119899+1
) le 120601 (119901 (1199092119899minus1
1199092119899
)) (27)
From inequalities (25) and (27) and taking into accountthe monotonicity of 120601 we get by induction that
119901 (119909119899
119909119899+1
) le 120601119899
(119901 (1199090
1199091
)) for any 119899 = 1 2
(28)
Since 119901 is an 119864-distance we find that
119901 (119909119899
119909119898
) le 119901 (119909119899
119909119899+1
) + sdot sdot sdot + 119901 (119909119898minus1
119909119898
) (29)
so for 119902 ge 1 we have that
119901 (119909119899
119909119899+119902
) le 120601119899
(119901 (1199090
1199091
)) + sdot sdot sdot + 120601119899+119902minus1
(119901 (1199090
1199091
))
(30)
In the sequel we will prove that 119909119899
is a 119901-Cauchy sequenceDenote
119878119899
=119899
sum119896=0
120601119896
(119901 (1199090
1199091
)) 119899 ge 0 (31)
By relation (31) we have
119901 (119909119899
119909119899+119902
) le 119878119899+119902minus1
minus 119878119899minus1
(32)
Regarding 120601 isin C together with Lemma 10(iv) we get that
infin
sum119896=0
120601119896
(119901 (1199090
1199091
)) lt infin (33)
since 119901(1199090
1199091
) gt 0 Thus there is 119878 isin [0infin) such that
lim119899rarrinfin
119878119899
= 119878 (34)
Then by (32) we obtain that
119901 (119909119899
119909119899+119902
) 997888rarr 0 as 119899 997888rarr infin (35)
In an analogous way we derive that
119901 (119909119899+119902
119909119899
) 997888rarr 0 as 119899 997888rarr infin (36)
Hence we get that 119909119899
119899ge0
is a 119901-Cauchy sequence in the119878-complete space 119883 = 119860 cup 119861 So there exists 119909 isin 119883such that lim
119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 In whatfollows we prove that 119909 is a fixed point of 119891 119892 In fact sincelim119899rarrinfin
1198911199092119899
= lim119899rarrinfin
1198921199092119899+1
= 119909 and as 119883 = 119860 cup 119861 is a
6 Abstract and Applied Analysis
cyclic representation of 119883 with respect to 119891 119892 the sequence119909119899
has infinite terms in each 119860 119861Since 119860 119861 are closed it follows that 119909 isin 119860 cap 119861 thus we
take subsequences 1199092119899
1199092119899+1
of 119909119899
with 1199092119899
isin 119860 and 1199092119899+1
isin119861 Using the contractive condition we can obtain
119901 (119909 119891119909) le 119901 (119909 1199092119899+2
) + 119901 (1199092119899+2
119891119909)
= 119901 (119909 1199092119899+2
) + 119901 (1198921199092119899+1
119891119909)
le 119901 (119909 1199092119899+2
) +max 119901 (1198921199092119899+1
119891119909)
119901 (119891119909 1198921199092119899+1
)
le 119901 (119909 1199092119899+2
) + 120601 (max 119901 (119909 119891119909)
119901 (1199092119899+1
1199092119899+2
))
(37)
and since 119909119899
rarr 119909 and 120601 belong toC letting 119899 rarr infin in thelast inequality we have 119901(119909 119891119909) le 120601(119901(119909 119891119909)) lt 119901(119909 119891119909)hence119901(119909 119891119909) = 0 Similarly we can show that 119901(119909 119909) = 0and therefore 119909 is a fixed point of 119891 Similarly we can showthat 119909 is a fixed point of 119892 Finally in order to prove theuniqueness of the fixed point we have 119910 119911 isin 119883 with 119910 and119911 fixed points of 119891 119892 The cyclic character of 119891 119892 and the factthat 119910 119911 isin 119883 are fixed points of 119891 119892 imply that 119910 119911 isin 119860 cap 119861Using the contractive condition we obtain
119901 (119910 119911) = 119901 (119891119910 119892119911) le max 119901 (119891119910 119892119911) 119901 (119892119911 119891119910)
le 120601 (max 119901 (119910 119891119910) 119901 (119911 119892119911)) = 0
(38)
and from the last inequality we get
119901 (119910 119911) = 0 (39)
Using the same arguments above we can show that 119901(119910 119910) =0 and consequently 119910 = 119911 This finishes the proof
Corollary 26 Let (119883 119889) be a complete metric space and 119860 119861nonempty closed subsets of 119883 and 119883 = 119860 cup 119861 Let f 119892 119883 rarr119883 be cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119911 isin 119860 cap 119861
Proof By Theorem 25 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 27 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that the maps 119891 119892 119883 rarr 119883 satisfy the following inequality
(i) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le119896max119901(119909 119891119909) 119901(119910 119892119910) for any 119909 isin 119860 119910 isin 119861where 0 lt 119896 lt 1
Then 119891 119892 have a unique common fixed point 119911 isin 119860 cap 119861
Proof ByTheorem 25 it is enough to set 120601(119905) = 119896119905
Example 28 Let (119883 119901) be a partial metric space where 119883 =1119899 cup 0 1 and 119901(119909 119910) = max119909 119910 Set119860 = 12119899 cup 0 1and 119861 = 1(2119899 + 1) cup 0 1 Define 120599 = 119880
120598
| 120598 gt 0 where119880120598
= (119909 119910) isin 1198832 119901(119909 119910) lt 119901(119909 119909)+120598 It is easy to see that(119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin)by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119891 119883 rarr 119883 by 119891(12119899) =1(4119899 + 1) 119891(0) = 119891(1) = 0 and 119892(1(2119899 + 1)) = 1(4119899 + 2)119892(0) = 119892(1) = 0 Then for every 119909 119910 = 0 1 we have
max 119901 (119891119909 119892119910) 119901 (119892119910 119891119909)
= max 119901( 1
4119899 + 1
1
4119898 + 2)
119901 (1
4119898 + 2
1
4119899 + 1)
= max 1
4119899 + 1
1
4119898 + 2
le1
2max 1
2119899
1
2119898 + 1
=1
2max 119901 (119909 119891119909) 119901 (119910 119892119910)
(40)
for any 119909 isin 119860 119910 isin 119861 Also for 119909 119910 = 0 1 the above inequalityobviously holds This shows that the contractive condition ofCorollary 27 is satisfied and 0 is a common fixed point of 119891and 119892
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanksDSR KAU for the financialsupport
References
[1] N Bourbaki Elements de mathematique Fasc II Livre IIITopologie generale Chapitre 1 Structures Topologiques Chapitre2 Structures Uniformes vol 1142 of Actualites Scientifiques etIndustrielles Hermann Paris France 1965
[2] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 1 Springer New York NY USA 1986
[3] V Berinde Iterative Approximation of Fixed Points SpringerBerlin Germany 2007
[4] J Jachymski ldquoFixed point theorems for expansive mappingsrdquoMathematica Japonica vol 42 no 1 pp 131ndash136 1995
[5] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996
[6] M Cherichi and B Samet ldquoFixed point theorems on orderedgauge spaces with applications to nonlinear integral equationsrdquoFixed Point Theory and Applications vol 2012 article 13 2012
Abstract and Applied Analysis 7
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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
cyclic representation of 119883 with respect to 119891 119892 the sequence119909119899
has infinite terms in each 119860 119861Since 119860 119861 are closed it follows that 119909 isin 119860 cap 119861 thus we
take subsequences 1199092119899
1199092119899+1
of 119909119899
with 1199092119899
isin 119860 and 1199092119899+1
isin119861 Using the contractive condition we can obtain
119901 (119909 119891119909) le 119901 (119909 1199092119899+2
) + 119901 (1199092119899+2
119891119909)
= 119901 (119909 1199092119899+2
) + 119901 (1198921199092119899+1
119891119909)
le 119901 (119909 1199092119899+2
) +max 119901 (1198921199092119899+1
119891119909)
119901 (119891119909 1198921199092119899+1
)
le 119901 (119909 1199092119899+2
) + 120601 (max 119901 (119909 119891119909)
119901 (1199092119899+1
1199092119899+2
))
(37)
and since 119909119899
rarr 119909 and 120601 belong toC letting 119899 rarr infin in thelast inequality we have 119901(119909 119891119909) le 120601(119901(119909 119891119909)) lt 119901(119909 119891119909)hence119901(119909 119891119909) = 0 Similarly we can show that 119901(119909 119909) = 0and therefore 119909 is a fixed point of 119891 Similarly we can showthat 119909 is a fixed point of 119892 Finally in order to prove theuniqueness of the fixed point we have 119910 119911 isin 119883 with 119910 and119911 fixed points of 119891 119892 The cyclic character of 119891 119892 and the factthat 119910 119911 isin 119883 are fixed points of 119891 119892 imply that 119910 119911 isin 119860 cap 119861Using the contractive condition we obtain
119901 (119910 119911) = 119901 (119891119910 119892119911) le max 119901 (119891119910 119892119911) 119901 (119892119911 119891119910)
le 120601 (max 119901 (119910 119891119910) 119901 (119911 119892119911)) = 0
(38)
and from the last inequality we get
119901 (119910 119911) = 0 (39)
Using the same arguments above we can show that 119901(119910 119910) =0 and consequently 119910 = 119911 This finishes the proof
Corollary 26 Let (119883 119889) be a complete metric space and 119860 119861nonempty closed subsets of 119883 and 119883 = 119860 cup 119861 Let f 119892 119883 rarr119883 be cyclic (120601)-contraction pair Then 119891 and 119892 have a uniquecommon fixed point 119911 isin 119860 cap 119861
Proof By Theorem 25 it is enough to set 120599 = 119880120598
| 120598 gt 0
Corollary 27 Let (119883 120599) be an 119878-complete Hausdorff uniformspace such that 119901 is an 119864-distance on 119883 and let 119860 119861 benonempty closed subsets of 119883 with respect to the topologicalspace (119883 120591(120599)) and 119883 = 119860 cup 119861 Suppose that the maps 119891 119892 119883 rarr 119883 satisfy the following inequality
(i) max119901(119891119909 119892119910) 119901(119892119910 119891119909) le119896max119901(119909 119891119909) 119901(119910 119892119910) for any 119909 isin 119860 119910 isin 119861where 0 lt 119896 lt 1
Then 119891 119892 have a unique common fixed point 119911 isin 119860 cap 119861
Proof ByTheorem 25 it is enough to set 120601(119905) = 119896119905
Example 28 Let (119883 119901) be a partial metric space where 119883 =1119899 cup 0 1 and 119901(119909 119910) = max119909 119910 Set119860 = 12119899 cup 0 1and 119861 = 1(2119899 + 1) cup 0 1 Define 120599 = 119880
120598
| 120598 gt 0 where119880120598
= (119909 119910) isin 1198832 119901(119909 119910) lt 119901(119909 119909)+120598 It is easy to see that(119883 120599) is a uniform space If we define 120601 [0infin) rarr [0infin)by 120601(119905) = 119896119905 for 0 lt 119896 lt 1 and 119891 119883 rarr 119883 by 119891(12119899) =1(4119899 + 1) 119891(0) = 119891(1) = 0 and 119892(1(2119899 + 1)) = 1(4119899 + 2)119892(0) = 119892(1) = 0 Then for every 119909 119910 = 0 1 we have
max 119901 (119891119909 119892119910) 119901 (119892119910 119891119909)
= max 119901( 1
4119899 + 1
1
4119898 + 2)
119901 (1
4119898 + 2
1
4119899 + 1)
= max 1
4119899 + 1
1
4119898 + 2
le1
2max 1
2119899
1
2119898 + 1
=1
2max 119901 (119909 119891119909) 119901 (119910 119892119910)
(40)
for any 119909 isin 119860 119910 isin 119861 Also for 119909 119910 = 0 1 the above inequalityobviously holds This shows that the contractive condition ofCorollary 27 is satisfied and 0 is a common fixed point of 119891and 119892
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanksDSR KAU for the financialsupport
References
[1] N Bourbaki Elements de mathematique Fasc II Livre IIITopologie generale Chapitre 1 Structures Topologiques Chapitre2 Structures Uniformes vol 1142 of Actualites Scientifiques etIndustrielles Hermann Paris France 1965
[2] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 1 Springer New York NY USA 1986
[3] V Berinde Iterative Approximation of Fixed Points SpringerBerlin Germany 2007
[4] J Jachymski ldquoFixed point theorems for expansive mappingsrdquoMathematica Japonica vol 42 no 1 pp 131ndash136 1995
[5] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996
[6] M Cherichi and B Samet ldquoFixed point theorems on orderedgauge spaces with applications to nonlinear integral equationsrdquoFixed Point Theory and Applications vol 2012 article 13 2012
Abstract and Applied Analysis 7
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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
[7] M O Olatinwo ldquoSome common fixed point theorems forselfmappings in uniform spacerdquo Acta Mathematica vol 23 no1 pp 47ndash54 2007
[8] I Altun andM Imdad ldquoSome fixed point theorems on ordereduniform spacesrdquo Filomat vol 23 pp 15ndash22 2009
[9] E Tarafdar ldquoAn approach to fixed-point theorems on uniformspacesrdquo Transactions of the AmericanMathematical Society vol191 pp 209ndash225 1974
[10] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[11] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001
[12] S Z Wang B Y Li Z M Gao and K Iseki ldquoSome fixed pointtheorems on expansion mappingsrdquo Mathematica Japonica vol29 no 4 pp 631ndash636 1984
[13] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[14] E Karapınar and K Sadarangani ldquoFixed point theory for cyclic(120601 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[15] E Karapınar andH K Nashine ldquoFixed point theorem for cyclicChatterjea type contractionsrdquo Journal of Applied Mathematicsvol 2012 Article ID 165698 15 pages 2012
[16] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 3-4 pp 1181ndash1187 2010
[17] I A Rus ldquoCyclic representations and fixed pointsrdquo Annalsof the Tiberiu Popoviciu Seminar of Functional EquationsApproximation and Convexity vol 3 pp 171ndash178 2005
[18] M Aamri and D El Moutawakil ldquoCommon fixed point theo-rems for119864-contractive or119864-expansivemaps in uniform spacesrdquoActa Mathematica vol 20 no 1 pp 83ndash91 2004
[19] M Aamri andD ElMoutawakil ldquoWeak compatibility and com-mon fixed point theorems for A-contractive and E-expansivemaps in uniform spacesrdquo Serdica vol 31 no 1-2 pp 75ndash862005
[20] M Aamri S Bennani and D El Moutawakil ldquoFixed pointsand variational principle in uniform spacesrdquo Siberian ElectronicMathematical Reports vol 3 pp 137ndash142 2006
[21] R P Agarwal D OrsquoRegan and N S Papageorgiou ldquoCommonfixed point theory for multivalued contractive maps of Reichtype in uniform spacesrdquo Applicable Analysis vol 83 no 1 pp37ndash47 2004
[22] V Berinde Contractii Generalizate si Aplicatii vol 22 EdituraCub Press Baia Mare Romania 1997
[23] V Popa ldquoA general fixed point theorem for two pairs of map-pings on two metric spacesrdquo Novi Sad Journal of Mathematicsvol 35 no 2 pp 79ndash83 2005
[24] N Hussain Z Kadelburg S Radenovic and F Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012
[25] N Hussain and H K Pathak ldquoCommon fixed point andapproximation results for 119867-operator pair with applicationsrdquoApplied Mathematics and Computation vol 218 no 22 pp11217ndash11225 2012
[26] N Hussain G Jungck and M A Khamsi ldquoNonexpansiveretracts and weak compatible pairs in metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 100 2012
[27] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011
[28] R H Haghi Sh Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods amp Applications vol 74 no 5 pp 1799ndash18032011
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
