Iterative Approximation of Fixed Point of Multivalued...
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Research Article Iterative Approximation of Fixed Point of Multivalued -Quasi-Nonexpansive Mappings in Modular Function Spaces with Applications Godwin Amechi Okeke , 1 Sheila Amina Bishop, 2 and Safeer Hussain Khan 3 1 Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, PMB 1526, Owerri, Imo State, Nigeria 2 Department of Mathematics, Covenant University, PMB 1023, Ota, Ogun State, Nigeria 3 Department of Mathematics, Statistics and Physics, Qatar University, P.O. Box 2713, Doha, Qatar Correspondence should be addressed to Godwin Amechi Okeke; [email protected] Received 4 July 2017; Accepted 12 December 2017; Published 24 January 2018 Academic Editor: Adrian Petrusel Copyright © 2018 Godwin Amechi Okeke 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. Recently, Khan and Abbas initiated the study of approximating fixed points of multivalued nonlinear mappings in modular function spaces. It is our purpose in this study to continue this recent trend in the study of fixed point theory of multivalued nonlinear mappings in modular function spaces. We prove some interesting theorems for -quasi-nonexpansive mappings using the Picard- Krasnoselskii hybrid iterative process. We apply our results to solving certain initial value problem. 1. Introduction Recently, Khan and Abbas [1] initiated the study of approx- imating fixed points of multivalued nonlinear mappings in modular function spaces. e purpose of this paper is to continue this recent trend in the study of fixed point theory of multivalued nonlinear mappings in mod- ular function spaces. We prove some interesting theo- rems for -quasi-nonexpansive mappings using the Picard- Krasnoselskii hybrid iterative process, recently introduced by Okeke and Abbas [2] as a modification of the Picard- Mann hybrid iterative process, introduced by Khan [3]. We also prove some stability results using this iterative process. Moreover, we apply our results in solving certain initial value problem. For over a century now, the study of fixed point theory of multivalued nonlinear mappings has attracted many well- known mathematicians and mathematical scientists (see, e.g., Brouwer [4], Downing and Kirk [5], Geanakoplos [6], Kakutani [7], Nash [8], Nash [9], Nadler [10], Abbas and Rhoades [11], and Khan et al. [12]). e motivation for such studies stems mainly from the usefulness of fixed point theory results in real-world applications, as in Game eory and Market Economy and in other areas of mathematical sciences such as in Nonsmooth Differential Equations. e theory of modular spaces was initiated in 1950 by Nakano [13] in connection with the theory of ordered spaces which was further generalized by Musielak and Orlicz [14]. Modular function spaces are natural generalizations of both function and sequence variants of several important, from application perspective, spaces like Musielak-Orlicz, Orlicz, Lorentz, Orlicz-Lorentz, Kothe, Lebesgue, and Calderon- Lozanovskii spaces and several others. Interest in quasi- nonexpansive mappings in modular function spaces stems mainly in the richness of structure of modular function spaces that, besides being Banach spaces (or -spaces in a more general settings), are equipped with modular equiv- alents of norm or metric notions and also equipped with almost everywhere convergence and convergence in submea- sure. It is known that modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts, particularly in applications to integral operators, approximation, and fixed point results. Moreover, there are certain fixed point Hindawi Journal of Function Spaces Volume 2018, Article ID 1785702, 9 pages https://doi.org/10.1155/2018/1785702
Transcript of Iterative Approximation of Fixed Point of Multivalued...
Research ArticleIterative Approximation of Fixed Point ofMultivalued 120588-Quasi-Nonexpansive Mappings in ModularFunction Spaces with Applications
Godwin Amechi Okeke 1 Sheila Amina Bishop2 and Safeer Hussain Khan 3
1Department of Mathematics School of Physical Sciences Federal University of Technology Owerri PMB 1526Owerri Imo State Nigeria2Department of Mathematics Covenant University PMB 1023 Ota Ogun State Nigeria3Department of Mathematics Statistics and Physics Qatar University PO Box 2713 Doha Qatar
Correspondence should be addressed to Godwin Amechi Okeke gaokeke1yahoocouk
Received 4 July 2017 Accepted 12 December 2017 Published 24 January 2018
Academic Editor Adrian Petrusel
Copyright copy 2018 Godwin Amechi Okeke et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Recently Khan andAbbas initiated the study of approximating fixed points ofmultivalued nonlinearmappings inmodular functionspaces It is our purpose in this study to continue this recent trend in the study of fixed point theory of multivalued nonlinearmappings in modular function spaces We prove some interesting theorems for 120588-quasi-nonexpansive mappings using the Picard-Krasnoselskii hybrid iterative process We apply our results to solving certain initial value problem
1 Introduction
Recently Khan and Abbas [1] initiated the study of approx-imating fixed points of multivalued nonlinear mappingsin modular function spaces The purpose of this paperis to continue this recent trend in the study of fixedpoint theory of multivalued nonlinear mappings in mod-ular function spaces We prove some interesting theo-rems for 120588-quasi-nonexpansive mappings using the Picard-Krasnoselskii hybrid iterative process recently introducedby Okeke and Abbas [2] as a modification of the Picard-Mann hybrid iterative process introduced by Khan [3] Wealso prove some stability results using this iterative processMoreover we apply our results in solving certain initial valueproblem
For over a century now the study of fixed point theoryof multivalued nonlinear mappings has attracted many well-known mathematicians and mathematical scientists (seeeg Brouwer [4] Downing and Kirk [5] Geanakoplos [6]Kakutani [7] Nash [8] Nash [9] Nadler [10] Abbas andRhoades [11] and Khan et al [12]) The motivation for suchstudies stemsmainly from the usefulness of fixed point theory
results in real-world applications as in Game Theory andMarket Economy and in other areas of mathematical sciencessuch as in Nonsmooth Differential Equations
The theory of modular spaces was initiated in 1950 byNakano [13] in connection with the theory of ordered spaceswhich was further generalized by Musielak and Orlicz [14]Modular function spaces are natural generalizations of bothfunction and sequence variants of several important fromapplication perspective spaces like Musielak-Orlicz OrliczLorentz Orlicz-Lorentz Kothe Lebesgue and Calderon-Lozanovskii spaces and several others Interest in quasi-nonexpansive mappings in modular function spaces stemsmainly in the richness of structure of modular functionspaces that besides being Banach spaces (or 119865-spaces in amore general settings) are equipped with modular equiv-alents of norm or metric notions and also equipped withalmost everywhere convergence and convergence in submea-sure It is known that modular type conditions are muchmore natural asmodular type assumptions can bemore easilyverified than their metric or norm counterparts particularlyin applications to integral operators approximation andfixed point results Moreover there are certain fixed point
HindawiJournal of Function SpacesVolume 2018 Article ID 1785702 9 pageshttpsdoiorg10115520181785702
2 Journal of Function Spaces
results that can be proved only using the apparatus ofmodular function spaces Hence fixed point theory results inmodular function spaces in this perspective which shouldbe considered as complementary to the fixed point theory innormed and metric spaces (see eg [15 16])
Several authors have proved very interesting fixed pointsresults in the framework of modular function spaces (seeeg [15 17ndash19]) Abbas et al [20] proved the existenceand uniqueness of common fixed point of certain nonlinearmappings satisfying some contractive conditions in partiallyordered spaces Ozturk et al [21] established some interestingfixed point results of nonlinear mappings satisfying integraltype contractive conditions in the framework of modularspaces endowed with a graph Recently Khan and Abbas ini-tiated the study of approximating fixed points of multivaluednonlinear mappings in the framework of modular functionspaces [1] A very recent work was given by Khan et al [12]They approximated the fixed points of 120588-quasi-nonexpansivemultivalued mappings in modular function spaces using athree-step iterative process where120588 satisfies the so-calledΔ 2-condition Their results improve and generalize the results ofKhan and Abbas [1]
Motivated by the above results we prove some conver-gence and stability results for 120588-quasi-nonexpansive map-pings using the Picard-Krasnoselskii hybrid iterative processOur results improve extend and generalize several knownresults including the recent results of Khan et al [12] in thesense that the restriction that 120588 satisfies the so-called Δ 2-condition in [12] is removed in the present paper Moreoverit is known (see [2]) that the Picard-Krasnoselskii hybriditerative process converges faster than all of Picard MannKrasnoselskii and Ishikawa iterative processes Furthermorewe apply our results in solving certain initial value problem
2 Preliminaries
In this study we let Ω denote a nonempty set and let Σ bea nontrivial 120590-algebra of subsets of Ω Let P be a 120575-ring ofsubsets of Ω such that 119864 cap 119860 isin P for any 119864 isin P and119860 isin Σ Let us assume that there exists an increasing sequenceof sets 119870119899 isin P such that Ω = ⋃ 119870119899 (eg P can be theclass of sets of finite measure in 120590-finite measure space) By1119860 we denote the characteristic function of the set 119860 in ΩBy 120576 we denote the linear space of all simple functions withsupport fromP ByMinfin we denote the space of all extendedmeasurable functions that is all functions119891 Ω rarr [minusinfin infin]such that there exists a sequence 119892119899 sub 120576 |119892119899| le |119891| and119892119899(120596) rarr 119891(120596) for each 120596 isin ΩDefinition 1 Let 120588 Minfin rarr [0 infin] be a nontrivialconvex and even function One says that 120588 is a regular convexfunction pseudomodular if
(1) 120588(0) = 0(2) 120588 is monotone that is |119891(120596)| le |119892(120596)| for any 120596 isin Ω
implies 120588(119891) le 120588(119892) where 119891 119892 isin Minfin(3) 120588 is orthogonally subadditive that is 120588(1198911119860cup119861) le120588(1198911119860)+120588(1198911119861) for any 119860 119861 isin Σ such that 119860cap119861 = 0119891 isin Minfin
(4) 120588 has Fatou property that is |119891119899(120596)| uarr |119891(120596)| for all120596 isin Ω implies 120588(119891119899) uarr 120588(119891) where 119891 isin Minfin(5) 120588 is order continuous in 120576 that is 119892119899 isin 120576 and |119892119899(120596)| darr0 implies 120588(119892119899) darr 0A set119860 isin Σ is said to be 120588-null if 120588(1198921119860) = 0 for every 119892 isin120576 A property 119901(120596) is said to hold 120588-almost everywhere (120588-
ae) if the set 120596 isin Ω 119901(120596) does not hold is 120588-null As usualwe identify any pair of measurable sets whose symmetricdifference is 120588-null as well as any pair ofmeasurable functionsdiffering only on a 120588-null set With this in mind we define
M (Ω ΣP 120588) = 119891 isin Minfin 1003816100381610038161003816119891 (120596)1003816100381610038161003816 lt infin 120588-ae (1)
where 119891 isin M(Ω ΣP 120588) is actually an equivalence classof functions equal 120588-ae rather than an individual functionWhere no confusion exists we shall write M instead ofM(Ω ΣP 120588)
The following definitions were given in [1]
Definition 2 Let 120588 be a regular function pseudomodular(a) One says that 120588 is a regular convex function modular
if 120588(119891) = 0 implies 119891 = 0 120588-ae(b) One says that120588 is a regular convex function semimod-
ular if 120588(120572119891) = 0 for every 120572 gt 0 implies 119891 = 0 120588-aeIt is known (see eg [15]) that 120588 satisfies the followingproperties
(1) 120588(0) = 0 iff 119891 = 0 120588-ae(2) 120588(120572119891) = 120588(119891) for every scalar 120572 with |120572| = 1 and 119891 isin
M(3) 120588(120572119891 + 120573119892) le 120588(119891) + 120588(119892) if 120572 + 120573 = 1 120572 120573 ge 0 and119891 119892 isin M
120588 is called a convex modular if in addition thefollowing property is satisfied
(31015840) 120588(120572119891 + 120573119892) le 120572120588(119891) + 120573120588(119892) if 120572 + 120573 = 1 120572 120573 ge 0and 119891 119892 isin MThe class of all nonzero regular convex functionmodulars on Ω is denoted byR
Definition 3 The convex function modular 120588 defines themodular function space 119871120588 as
119871120588 = 119891 isin M 120588 (120582119891) 997888rarr 0 as 120582 997888rarr 0 (2)
Generally the modular 120588 is not subadditive and thereforedoes not behave as a norm or a distance However themodular space 119871120588 can be equipped with an 119865-norm definedby
10038171003817100381710038171198911003817100381710038171003817120588 = inf 120572 gt 0 120588 (119891120572 ) le 120572 (3)
In the case that 120588 is convex modular
10038171003817100381710038171198911003817100381710038171003817120588 = inf 120572 gt 0 120588 (119891120572 ) le 1 (4)
defines a norm on the modular space 119871120588 and it is called theLuxemburg norm
Journal of Function Spaces 3
Lemma 4 (see [15]) Let 120588 isin R Defining 1198710120588 = 119891 isin119871120588 120588(119891 sdot) 119894119904 119900119903119889119890119903 119888119900119899119905119894119899119906119900119906119904 and 119864120588 = 119891 isin 119871120588 120582119891 isin1198710120588 for every 120582 gt 0 one has the following(i) 119871120588 sup 1198710120588 sup 119864120588(ii) 119864120588 has the Lebesgue property that is 120588(120572119891 119863119896) rarr 0
for 120572 gt 0 119891 isin 119864120588 and 119863119896 darr 0(iii) 119864120588 is the closure of 120576 (in the sense of sdot 120588)The following uniform convexity type properties of 120588 can
be found in [17]
Definition 5 Let 120588 be a nonzero regular convex functionmodular defined on Ω
(i) Let 119903 gt 0 120598 gt 0 Define1198631 (119903 120598) = (119891 119892) 119891 119892 isin 119871120588 120588 (119891) le 119903 120588 (119892)
le 119903 120588 (119891 minus 119892) ge 120598119903 (5)
Let
1205751 (119903 120598) = inf 1 minus 1119903 120588 (119891 + 1198922 ) (119891 119892) isin 1198631 (119903 120598)if 1198631 (119903 120598) = 0
(6)
and 1205751(119903 120598) = 1 if 1198631(119903 120598) = 0 One says that 120588satisfies (UC1) if for every 119903 gt 0 120598 gt 0 1205751(119903 120598) gt 0Observe that for every 119903 gt 0 1198631(119903 120598) = 0 for 120598 gt 0small enough
(ii) One says that 120588 satisfies (UUC1) if for every 119904 ge 0120598 gt 0 there exists 1205781(119904 120598) gt 0 depending only on 119904and 120598 such that 1205751(119903 120598) gt 1205781(119904 120598) gt 0 for any 119903 gt 119904
(iii) Let 119903 gt 0 120598 gt 0 Define1198632 (119903 120598) = (119891 119892) 119891 119892 isin 119871120588 120588 (119891) le 119903 120588 (119892)
le 119903 120588 (119891 minus 1198922 ) ge 120598119903 (7)
Let
1205752 (119903 120598) = inf 1 minus 1119903 120588 (119891 + 1198922 ) (119891 119892) isin 1198632 (119903 120598) if 1198632 (119903 120598) = 0
(8)
and 1205752(119903 120598) = 1 if1198632(119903 120598) = 0 one says that 120588 satisfies(UC2) if for every 119903 gt 0 120598 gt 0 1205752(119903 120598) gt 0 Observethat for every 119903 gt 0 1198632(119903 120598) = 0 for 120598 gt 0 smallenough
(iv) One says that 120588 satisfies (UUC2) if for every 119904 ge 0120598 gt 0 there exists 1205782(119904 120598) gt 0 depending only on 119904and 120598 such that 1205752(119903 120598) gt 1205782(119904 120598) gt 0 for any 119903 gt 119904
(v) One says that 120588 is strictly convex (SC) if for every119891 119892 isin 119871120588 such that 120588(119891) = 120588(119892) and 120588((119891 + 119892)2) =(120588(119891) + 120588(119892))2 there holds 119891 = 119892Proposition 6 (see [15]) The following conditions character-ize relationship between the above defined notions
(i) (119880119880119862119894) rArr (119880119862119894) for 119894 = 1 2(ii) 1205751(119903 120598) le 1205752(119903 120598)(iii) (1198801198621) rArr (1198801198622)(iv) (1198801198801198621) rArr (1198801198801198622)(v) If 120588 is homogeneous (eg it is a norm) then all the
conditions (1198801198621) (1198801198622) (1198801198801198621) and (1198801198801198622) areequivalent and 1205751(119903 2120598) = 1205751(1 2120598) = 1205752(1 120598) =1205752(119903 120598)
Definition 7 A nonzero regular convex function modular 120588is said to satisfy the Δ 2-condition if sup119899ge1120588(2119891119899 119863119896) rarr 0 as119896 rarr infinwhenever 119863119896 decreases to 0 and sup119899ge1120588(119891119899 119863119896) rarr0 as 119896 rarr infinDefinition 8 A function modular is said to satisfy the Δ 2-type condition if there exists119870 gt 0 such that for any119891 isin 119871120588one has 120588(2119891) le 119870120588(119891)
In general Δ 2-condition and Δ 2-type condition arenot equivalent even though it is easy to see that Δ 2-typecondition impliesΔ 2-condition on themodular space 119871120588 see[22]
Definition 9 Let 119871120588 be a modular spaceThe sequence 119891119899 sub119871120588 is called(1) 120588-convergent to 119891 isin 119871120588 if 120588(119891119899 minus 119891) rarr 0 as 119899 rarr infin(2) 120588-Cauchy if 120588(119891119899 minus 119891119898) rarr 0 as 119899 and 119898 rarr infinObserve that 120588-convergence does not imply 120588-Cauchy
since 120588 does not satisfy the triangle inequality In fact onecan easily show that this will happen if and only if 120588 satisfiesthe Δ 2-condition
Kilmer et al [23] defined 120588-distance from an 119891 isin 119871120588 to aset 119863 sub 119871120588 as follows
dist120588 (119891 119863) = inf 120588 (119891 minus ℎ) ℎ isin 119863 (9)
Definition 10 A subset 119863 sub 119871120588 is called(1) 120588-closed if the 120588-limit of a 120588-convergent sequence of119863 always belongs to 119863(2) 120588-ae closed if the 120588-ae limit of a 120588-ae convergent
sequence of 119863 always belongs to 119863(3) 120588-compact if every sequence in 119863 has a 120588-convergent
subsequence in 119863(4) 120588-ae compact if every sequence in 119863 has a 120588-ae
convergent subsequence in 119863(5) 120588-bounded if
diam120588 (119863) = sup 120588 (119891 minus 119892) 119891 119892 isin 119863 lt infin (10)
4 Journal of Function Spaces
It is known that the norm and modular convergence arealso the same when we deal with the Δ 2-type condition (seeeg [15])
A set 119863 sub 119871120588 is called 120588-proximinal if for each 119891 isin 119871120588there exists an element119892 isin 119863 such that 120588(119891minus119892) = dist120588(119891 119863)We shall denote the family of nonempty 120588-bounded 120588-proximinal subsets of 119863 by 119875120588(119863) the family of nonempty120588-closed 120588-bounded subsets of 119863 by 119862120588(119863) and the familyof 120588-compact subsets of 119863 by 119870120588(119863) Let 119867120588(sdot sdot) be the 120588-Hausdorff distance on 119862120588(119871120588) that is
119867120588 (119860 119861) = maxsup119891isin119860
dist120588 (119891 119861) sup119892isin119861
dist120588 (119892 119860) 119860 119861 isin 119862120588 (119871120588)
(11)
A multivalued map 119879 119863 rarr 119862120588(119871120588) is said to be
(a) 120588-contraction mapping if there exists a constant 119896 isin[0 1) such that
119867120588 (119879119891 119879119892) le 119896120588 (119891 minus 119892) forall119891 119892 isin 119863 (12)
(b) 120588-nonexpansive (see eg Khan and Abbas [1]) if
119867120588 (119879119891 119879119892) le 120588 (119891 minus 119892) forall119891 119892 isin 119863 (13)
(c) 120588-quasi-nonexpansive mapping if
119867120588 (119879119891 119901) le 120588 (119891 minus 119901) forall119891 isin 119863 119901 isin 119865120588 (119879) (14)
A sequence 119905119899 sub (0 1) is called bounded away from 0 ifthere exists 119886 gt 0 such that 119905119899 ge 119886 for every 119899 isin N Similarly119905119899 sub (0 1) is called bounded away from 1 if there exists 119887 lt 1such that 119905119899 le 119887 for every 119899 isin N
Okeke andAbbas [2] introduced the Picard-Krasnoselskiihybrid iterative process The authors proved that this newhybrid iterative process converges faster than all of PicardMann Krasnoselskii and Ishikawa iterative processes whenapplied to contraction mappings We now give the analogueof the Picard-Krasnoselskii hybrid iterative process in mod-ular function spaces as follows let 119879 119863 rarr 119875120588(119863) bea multivalued mapping and 119891119899 sub 119863 be defined by thefollowing iteration process
119891119899+1 isin 119875119879120588 (119892119899)119892119899 = (1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) 119899 isin N (15)
where V119899 isin 119875119879120588 (119891119899) and 0 lt 120582 lt 1 It is our purpose in thepresent paper to prove some new fixed point theorems usingthis iteration process in the framework of modular functionspaces
Definition 11 A sequence 119891119899 sub 119863 is said to be Fejermonotone with respect to subset 119875120588(119863) of 119863 if 120588(119891119899+1 minus 119901) le120588(119891119899 minus 119901) for all 119901 isin 119875119879120588 (119863) of 119863 119899 isin N
The following Lemma will be needed in this study
Lemma 12 (see [22]) Let 120588 be a function modular and 119891119899 and119892119899 be two sequences in 119883120588 Then
lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim sup119899rarrinfin
120588 (119891119899 + 119892119899)= lim sup119899rarrinfin
120588 (119891119899) lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim inf119899rarrinfin
120588 (119891119899 + 119892119899)= lim inf119899rarrinfin
120588 (119891119899)
(16)
Lemma 13 (see [17]) Let120588 satisfy (1198801198801198621) and let 119905119896 sub (0 1)be bounded away from 0 and 1 If there exists 119877 gt 0 such that
lim sup119899rarrinfin
120588 (119891119899) le 119877lim sup119899rarrinfin
120588 (119892119899) le 119877lim119899rarrinfin
120588 (119905119899119891119899 + (1 minus 119905119899) 119892119899) = 119877(17)
and then lim119899rarrinfin120588(119891119899 minus 119892119899) = 0The above lemma is an analogue of a famous lemma due
to Schu [24] in Banach spacesA function 119891 isin 119871120588 is called a fixed point of 119879 119871120588 rarr119875120588(119863) if119891 isin 119879119891The set of all fixed points of119879will be denoted
by 119865120588(119879)Lemma 14 (see [1]) Let 119879 119863 rarr 119875120588(119863) be a multivaluedmapping and
119875119879120588 (119891) = 119892 isin 119879119891 120588 (119891 minus 119892) = dist120588 (119891 119879119891) (18)
Then the following are equivalent
(1) 119891 isin 119865120588(119879) that is 119891 isin 119879119891(2) 119875119879120588 (119891) = 119891 that is 119891 = 119892 for each 119892 isin 119875119879120588 (119891)(3) 119891 isin 119865(119875119879120588 (119891)) that is 119891 isin 119875119879120588 (119891) Further 119865120588(119879) =
119865(119875119879120588 (119891)) where 119865(119875119879120588 (119891)) denotes the set of fixedpoints of 119875119879120588 (119891)
The following examples were presented by Razani et al[25]
Example 15 Let (119883 sdot ) be a norm space then sdot is amodular But the converse is not true
Example 16 Let (119883 sdot ) be a norm space For any 119896 ge 1 sdot 119896is a modular on 1198833 Iterative Approximation of Fixed Points inModular Function Spaces
We begin this section with the following proposition
Proposition 17 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr
Journal of Function Spaces 5
119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Then the Picard-Krasnoselskii hybriditerative process (15) is Fejer monotone with respect to 119865120588(119879)Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (19)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (20)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(21)
Using (21) in (19) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (22)
Hence the Picard-Krasnoselskii hybrid iterative process (15)is Fejer monotone with respect to 119865120588(119879) This completes theproof of Proposition 17
Next we prove the following proposition
Proposition 18 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Let 119891119899 be the Picard-Krasnoselskiihybrid iterative process (15) then
(i) the sequence 119891119899 is bounded(ii) for each 119891 isin 119863 120588(119891119899 minus 119891) converges
Proof Since 119891119899 is Fejer monotone as shown in Proposi-tion 17 we can easily show (i) and (ii) This completes theproof of Proposition 18
Theorem 19 Let 120588 satisfy (1198801198801198621) and let 119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Suppose that 119865120588(119879) = 0 Let119891119899 sub 119863 be the Picard-Krasnoselskii hybrid iterative process(15) Then lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879) andlim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (23)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (24)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(25)
Using (25) in (23) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (26)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Suppose that
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (27)
where 119871 ge 0We next prove that lim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0 Since
dist120588(119891119899 119875119879120588 (119891119899)) le 120588(119891119899 minus V119899) it suffices to show that
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (28)
Now
120588 (V119899 minus 119901) le 119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901)) le 120588 (119891119899 minus 119901) (29)
and this implies that
lim sup119899rarrinfin
120588 (V119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (30)
and by (27) we have
lim sup119899rarrinfin
120588 (V119899 minus 119901) le 119871 (31)
Using (25) we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (32)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (33)
Next we have
119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) le 120588 (119891119899 minus 119901) (34)
and this implies that
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (35)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (36)
6 Journal of Function Spaces
Using (23) and (24) we have
lim119899rarrinfin
120588 (119891119899+1 minus 119901) = lim119899rarrinfin
120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)]= lim119899rarrinfin
120588 (119891119899 minus 119901) le 119871(37)
Moreover
120588 (119891119899+1 minus 119901) le 120588 [(1 minus 120582) 119891119899 + 120582V119899 minus 119901]= 120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)] (38)
Using Lemma 4 and (38) we have
119871 = lim inf119899rarrinfin
120588 (119891119899+1 minus 119901)= lim inf119899rarrinfin
120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)]= lim inf119899rarrinfin
120588 (119891119899 minus 119901) (39)
This means that
119871 = lim inf119899rarrinfin
120588 (119891119899 minus 119901) (40)
Using (27) and (40) we have
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (41)
Using (27) (31) (37) and Lemma 12 we have
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (42)
Hence
lim119899rarrinfin
dist120588 (119891119899 119875119879120588 (119891119899)) = 0 (43)
The proof of Theorem 19 is completed
Next we prove the following theorem
Theorem 20 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then 119879 has a unique fixed pointMoreover the Picard-Krasnoselskii hybrid iterative process (15)converges to this fixed point
Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 119896120588 (119892119899 minus 119901)le 120588 (119892119899 minus 119901) (44)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) minus 119901] (45)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582119896120588 (119891119899 minus 119901)le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(46)
Using (46) in (44) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (47)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Using a similar approach as in the proof of Theorem 19 wesee that lim119899rarrinfin120588(119891119899 minus 119901) = 0
Next we show that 119891119899 is a 120588-Cauchy sequence Sincelim119899rarrinfin(119891119899 minus 119901) = 0 we proceed by contradiction Hencethere exists 120598 gt 0 and two sequences of natural numbers119898(119894) 119899(119894) such that
119899 (119894) gt 119898 (119894) ge 119894120588 (119891119899(119894) minus 119891119898(119894)) gt 120598 (48)
For all integer 119894 let 119899(119894) be the least integer exceeding 119898(119894)which satisfy (48) then
120588 (119891119899(119894) minus 119891119898(119894)) gt 120598120588 (119891119899(119894)minus1 minus 119891119898(119894)) le 120598 (49)
So we have
120598 lt 120588 (119891119899(119894) minus 119891119898(119894)) le 120588 (119891119899(119894) minus 1199012 ) + 120588 (119901 minus 119891119898(119894)2 )
le 12120588 (119891119899(119894) minus 119901) + 12120588 (119901 minus 119891119898(119894))le 120588 (119891119899(119894) minus 119901) + 120588 (119901 minus 119891119898(119894)) 997888rarr 0 as 119899 997888rarr infin
(50)
This is a contradiction Hence 119891119899 is a 120588-Cauchy sequenceTherefore there exists 119901 isin 119863 such that 119891119899 rarr 0 as 119899 rarr infin
Next we have 119879119901 = 119901 Clearly120588 (119901 minus 1198791199012 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1198791199012 )
le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119879119901)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119879119901)= 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119901) 997888rarr 0
as 119899 997888rarr infin
(51)
Hence 120588((119901 minus 119879119901)2) = 0 Therefore 119901 = 119879119901
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
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2 Journal of Function Spaces
results that can be proved only using the apparatus ofmodular function spaces Hence fixed point theory results inmodular function spaces in this perspective which shouldbe considered as complementary to the fixed point theory innormed and metric spaces (see eg [15 16])
Several authors have proved very interesting fixed pointsresults in the framework of modular function spaces (seeeg [15 17ndash19]) Abbas et al [20] proved the existenceand uniqueness of common fixed point of certain nonlinearmappings satisfying some contractive conditions in partiallyordered spaces Ozturk et al [21] established some interestingfixed point results of nonlinear mappings satisfying integraltype contractive conditions in the framework of modularspaces endowed with a graph Recently Khan and Abbas ini-tiated the study of approximating fixed points of multivaluednonlinear mappings in the framework of modular functionspaces [1] A very recent work was given by Khan et al [12]They approximated the fixed points of 120588-quasi-nonexpansivemultivalued mappings in modular function spaces using athree-step iterative process where120588 satisfies the so-calledΔ 2-condition Their results improve and generalize the results ofKhan and Abbas [1]
Motivated by the above results we prove some conver-gence and stability results for 120588-quasi-nonexpansive map-pings using the Picard-Krasnoselskii hybrid iterative processOur results improve extend and generalize several knownresults including the recent results of Khan et al [12] in thesense that the restriction that 120588 satisfies the so-called Δ 2-condition in [12] is removed in the present paper Moreoverit is known (see [2]) that the Picard-Krasnoselskii hybriditerative process converges faster than all of Picard MannKrasnoselskii and Ishikawa iterative processes Furthermorewe apply our results in solving certain initial value problem
2 Preliminaries
In this study we let Ω denote a nonempty set and let Σ bea nontrivial 120590-algebra of subsets of Ω Let P be a 120575-ring ofsubsets of Ω such that 119864 cap 119860 isin P for any 119864 isin P and119860 isin Σ Let us assume that there exists an increasing sequenceof sets 119870119899 isin P such that Ω = ⋃ 119870119899 (eg P can be theclass of sets of finite measure in 120590-finite measure space) By1119860 we denote the characteristic function of the set 119860 in ΩBy 120576 we denote the linear space of all simple functions withsupport fromP ByMinfin we denote the space of all extendedmeasurable functions that is all functions119891 Ω rarr [minusinfin infin]such that there exists a sequence 119892119899 sub 120576 |119892119899| le |119891| and119892119899(120596) rarr 119891(120596) for each 120596 isin ΩDefinition 1 Let 120588 Minfin rarr [0 infin] be a nontrivialconvex and even function One says that 120588 is a regular convexfunction pseudomodular if
(1) 120588(0) = 0(2) 120588 is monotone that is |119891(120596)| le |119892(120596)| for any 120596 isin Ω
implies 120588(119891) le 120588(119892) where 119891 119892 isin Minfin(3) 120588 is orthogonally subadditive that is 120588(1198911119860cup119861) le120588(1198911119860)+120588(1198911119861) for any 119860 119861 isin Σ such that 119860cap119861 = 0119891 isin Minfin
(4) 120588 has Fatou property that is |119891119899(120596)| uarr |119891(120596)| for all120596 isin Ω implies 120588(119891119899) uarr 120588(119891) where 119891 isin Minfin(5) 120588 is order continuous in 120576 that is 119892119899 isin 120576 and |119892119899(120596)| darr0 implies 120588(119892119899) darr 0A set119860 isin Σ is said to be 120588-null if 120588(1198921119860) = 0 for every 119892 isin120576 A property 119901(120596) is said to hold 120588-almost everywhere (120588-
ae) if the set 120596 isin Ω 119901(120596) does not hold is 120588-null As usualwe identify any pair of measurable sets whose symmetricdifference is 120588-null as well as any pair ofmeasurable functionsdiffering only on a 120588-null set With this in mind we define
M (Ω ΣP 120588) = 119891 isin Minfin 1003816100381610038161003816119891 (120596)1003816100381610038161003816 lt infin 120588-ae (1)
where 119891 isin M(Ω ΣP 120588) is actually an equivalence classof functions equal 120588-ae rather than an individual functionWhere no confusion exists we shall write M instead ofM(Ω ΣP 120588)
The following definitions were given in [1]
Definition 2 Let 120588 be a regular function pseudomodular(a) One says that 120588 is a regular convex function modular
if 120588(119891) = 0 implies 119891 = 0 120588-ae(b) One says that120588 is a regular convex function semimod-
ular if 120588(120572119891) = 0 for every 120572 gt 0 implies 119891 = 0 120588-aeIt is known (see eg [15]) that 120588 satisfies the followingproperties
(1) 120588(0) = 0 iff 119891 = 0 120588-ae(2) 120588(120572119891) = 120588(119891) for every scalar 120572 with |120572| = 1 and 119891 isin
M(3) 120588(120572119891 + 120573119892) le 120588(119891) + 120588(119892) if 120572 + 120573 = 1 120572 120573 ge 0 and119891 119892 isin M
120588 is called a convex modular if in addition thefollowing property is satisfied
(31015840) 120588(120572119891 + 120573119892) le 120572120588(119891) + 120573120588(119892) if 120572 + 120573 = 1 120572 120573 ge 0and 119891 119892 isin MThe class of all nonzero regular convex functionmodulars on Ω is denoted byR
Definition 3 The convex function modular 120588 defines themodular function space 119871120588 as
119871120588 = 119891 isin M 120588 (120582119891) 997888rarr 0 as 120582 997888rarr 0 (2)
Generally the modular 120588 is not subadditive and thereforedoes not behave as a norm or a distance However themodular space 119871120588 can be equipped with an 119865-norm definedby
10038171003817100381710038171198911003817100381710038171003817120588 = inf 120572 gt 0 120588 (119891120572 ) le 120572 (3)
In the case that 120588 is convex modular
10038171003817100381710038171198911003817100381710038171003817120588 = inf 120572 gt 0 120588 (119891120572 ) le 1 (4)
defines a norm on the modular space 119871120588 and it is called theLuxemburg norm
Journal of Function Spaces 3
Lemma 4 (see [15]) Let 120588 isin R Defining 1198710120588 = 119891 isin119871120588 120588(119891 sdot) 119894119904 119900119903119889119890119903 119888119900119899119905119894119899119906119900119906119904 and 119864120588 = 119891 isin 119871120588 120582119891 isin1198710120588 for every 120582 gt 0 one has the following(i) 119871120588 sup 1198710120588 sup 119864120588(ii) 119864120588 has the Lebesgue property that is 120588(120572119891 119863119896) rarr 0
for 120572 gt 0 119891 isin 119864120588 and 119863119896 darr 0(iii) 119864120588 is the closure of 120576 (in the sense of sdot 120588)The following uniform convexity type properties of 120588 can
be found in [17]
Definition 5 Let 120588 be a nonzero regular convex functionmodular defined on Ω
(i) Let 119903 gt 0 120598 gt 0 Define1198631 (119903 120598) = (119891 119892) 119891 119892 isin 119871120588 120588 (119891) le 119903 120588 (119892)
le 119903 120588 (119891 minus 119892) ge 120598119903 (5)
Let
1205751 (119903 120598) = inf 1 minus 1119903 120588 (119891 + 1198922 ) (119891 119892) isin 1198631 (119903 120598)if 1198631 (119903 120598) = 0
(6)
and 1205751(119903 120598) = 1 if 1198631(119903 120598) = 0 One says that 120588satisfies (UC1) if for every 119903 gt 0 120598 gt 0 1205751(119903 120598) gt 0Observe that for every 119903 gt 0 1198631(119903 120598) = 0 for 120598 gt 0small enough
(ii) One says that 120588 satisfies (UUC1) if for every 119904 ge 0120598 gt 0 there exists 1205781(119904 120598) gt 0 depending only on 119904and 120598 such that 1205751(119903 120598) gt 1205781(119904 120598) gt 0 for any 119903 gt 119904
(iii) Let 119903 gt 0 120598 gt 0 Define1198632 (119903 120598) = (119891 119892) 119891 119892 isin 119871120588 120588 (119891) le 119903 120588 (119892)
le 119903 120588 (119891 minus 1198922 ) ge 120598119903 (7)
Let
1205752 (119903 120598) = inf 1 minus 1119903 120588 (119891 + 1198922 ) (119891 119892) isin 1198632 (119903 120598) if 1198632 (119903 120598) = 0
(8)
and 1205752(119903 120598) = 1 if1198632(119903 120598) = 0 one says that 120588 satisfies(UC2) if for every 119903 gt 0 120598 gt 0 1205752(119903 120598) gt 0 Observethat for every 119903 gt 0 1198632(119903 120598) = 0 for 120598 gt 0 smallenough
(iv) One says that 120588 satisfies (UUC2) if for every 119904 ge 0120598 gt 0 there exists 1205782(119904 120598) gt 0 depending only on 119904and 120598 such that 1205752(119903 120598) gt 1205782(119904 120598) gt 0 for any 119903 gt 119904
(v) One says that 120588 is strictly convex (SC) if for every119891 119892 isin 119871120588 such that 120588(119891) = 120588(119892) and 120588((119891 + 119892)2) =(120588(119891) + 120588(119892))2 there holds 119891 = 119892Proposition 6 (see [15]) The following conditions character-ize relationship between the above defined notions
(i) (119880119880119862119894) rArr (119880119862119894) for 119894 = 1 2(ii) 1205751(119903 120598) le 1205752(119903 120598)(iii) (1198801198621) rArr (1198801198622)(iv) (1198801198801198621) rArr (1198801198801198622)(v) If 120588 is homogeneous (eg it is a norm) then all the
conditions (1198801198621) (1198801198622) (1198801198801198621) and (1198801198801198622) areequivalent and 1205751(119903 2120598) = 1205751(1 2120598) = 1205752(1 120598) =1205752(119903 120598)
Definition 7 A nonzero regular convex function modular 120588is said to satisfy the Δ 2-condition if sup119899ge1120588(2119891119899 119863119896) rarr 0 as119896 rarr infinwhenever 119863119896 decreases to 0 and sup119899ge1120588(119891119899 119863119896) rarr0 as 119896 rarr infinDefinition 8 A function modular is said to satisfy the Δ 2-type condition if there exists119870 gt 0 such that for any119891 isin 119871120588one has 120588(2119891) le 119870120588(119891)
In general Δ 2-condition and Δ 2-type condition arenot equivalent even though it is easy to see that Δ 2-typecondition impliesΔ 2-condition on themodular space 119871120588 see[22]
Definition 9 Let 119871120588 be a modular spaceThe sequence 119891119899 sub119871120588 is called(1) 120588-convergent to 119891 isin 119871120588 if 120588(119891119899 minus 119891) rarr 0 as 119899 rarr infin(2) 120588-Cauchy if 120588(119891119899 minus 119891119898) rarr 0 as 119899 and 119898 rarr infinObserve that 120588-convergence does not imply 120588-Cauchy
since 120588 does not satisfy the triangle inequality In fact onecan easily show that this will happen if and only if 120588 satisfiesthe Δ 2-condition
Kilmer et al [23] defined 120588-distance from an 119891 isin 119871120588 to aset 119863 sub 119871120588 as follows
dist120588 (119891 119863) = inf 120588 (119891 minus ℎ) ℎ isin 119863 (9)
Definition 10 A subset 119863 sub 119871120588 is called(1) 120588-closed if the 120588-limit of a 120588-convergent sequence of119863 always belongs to 119863(2) 120588-ae closed if the 120588-ae limit of a 120588-ae convergent
sequence of 119863 always belongs to 119863(3) 120588-compact if every sequence in 119863 has a 120588-convergent
subsequence in 119863(4) 120588-ae compact if every sequence in 119863 has a 120588-ae
convergent subsequence in 119863(5) 120588-bounded if
diam120588 (119863) = sup 120588 (119891 minus 119892) 119891 119892 isin 119863 lt infin (10)
4 Journal of Function Spaces
It is known that the norm and modular convergence arealso the same when we deal with the Δ 2-type condition (seeeg [15])
A set 119863 sub 119871120588 is called 120588-proximinal if for each 119891 isin 119871120588there exists an element119892 isin 119863 such that 120588(119891minus119892) = dist120588(119891 119863)We shall denote the family of nonempty 120588-bounded 120588-proximinal subsets of 119863 by 119875120588(119863) the family of nonempty120588-closed 120588-bounded subsets of 119863 by 119862120588(119863) and the familyof 120588-compact subsets of 119863 by 119870120588(119863) Let 119867120588(sdot sdot) be the 120588-Hausdorff distance on 119862120588(119871120588) that is
119867120588 (119860 119861) = maxsup119891isin119860
dist120588 (119891 119861) sup119892isin119861
dist120588 (119892 119860) 119860 119861 isin 119862120588 (119871120588)
(11)
A multivalued map 119879 119863 rarr 119862120588(119871120588) is said to be
(a) 120588-contraction mapping if there exists a constant 119896 isin[0 1) such that
119867120588 (119879119891 119879119892) le 119896120588 (119891 minus 119892) forall119891 119892 isin 119863 (12)
(b) 120588-nonexpansive (see eg Khan and Abbas [1]) if
119867120588 (119879119891 119879119892) le 120588 (119891 minus 119892) forall119891 119892 isin 119863 (13)
(c) 120588-quasi-nonexpansive mapping if
119867120588 (119879119891 119901) le 120588 (119891 minus 119901) forall119891 isin 119863 119901 isin 119865120588 (119879) (14)
A sequence 119905119899 sub (0 1) is called bounded away from 0 ifthere exists 119886 gt 0 such that 119905119899 ge 119886 for every 119899 isin N Similarly119905119899 sub (0 1) is called bounded away from 1 if there exists 119887 lt 1such that 119905119899 le 119887 for every 119899 isin N
Okeke andAbbas [2] introduced the Picard-Krasnoselskiihybrid iterative process The authors proved that this newhybrid iterative process converges faster than all of PicardMann Krasnoselskii and Ishikawa iterative processes whenapplied to contraction mappings We now give the analogueof the Picard-Krasnoselskii hybrid iterative process in mod-ular function spaces as follows let 119879 119863 rarr 119875120588(119863) bea multivalued mapping and 119891119899 sub 119863 be defined by thefollowing iteration process
119891119899+1 isin 119875119879120588 (119892119899)119892119899 = (1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) 119899 isin N (15)
where V119899 isin 119875119879120588 (119891119899) and 0 lt 120582 lt 1 It is our purpose in thepresent paper to prove some new fixed point theorems usingthis iteration process in the framework of modular functionspaces
Definition 11 A sequence 119891119899 sub 119863 is said to be Fejermonotone with respect to subset 119875120588(119863) of 119863 if 120588(119891119899+1 minus 119901) le120588(119891119899 minus 119901) for all 119901 isin 119875119879120588 (119863) of 119863 119899 isin N
The following Lemma will be needed in this study
Lemma 12 (see [22]) Let 120588 be a function modular and 119891119899 and119892119899 be two sequences in 119883120588 Then
lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim sup119899rarrinfin
120588 (119891119899 + 119892119899)= lim sup119899rarrinfin
120588 (119891119899) lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim inf119899rarrinfin
120588 (119891119899 + 119892119899)= lim inf119899rarrinfin
120588 (119891119899)
(16)
Lemma 13 (see [17]) Let120588 satisfy (1198801198801198621) and let 119905119896 sub (0 1)be bounded away from 0 and 1 If there exists 119877 gt 0 such that
lim sup119899rarrinfin
120588 (119891119899) le 119877lim sup119899rarrinfin
120588 (119892119899) le 119877lim119899rarrinfin
120588 (119905119899119891119899 + (1 minus 119905119899) 119892119899) = 119877(17)
and then lim119899rarrinfin120588(119891119899 minus 119892119899) = 0The above lemma is an analogue of a famous lemma due
to Schu [24] in Banach spacesA function 119891 isin 119871120588 is called a fixed point of 119879 119871120588 rarr119875120588(119863) if119891 isin 119879119891The set of all fixed points of119879will be denoted
by 119865120588(119879)Lemma 14 (see [1]) Let 119879 119863 rarr 119875120588(119863) be a multivaluedmapping and
119875119879120588 (119891) = 119892 isin 119879119891 120588 (119891 minus 119892) = dist120588 (119891 119879119891) (18)
Then the following are equivalent
(1) 119891 isin 119865120588(119879) that is 119891 isin 119879119891(2) 119875119879120588 (119891) = 119891 that is 119891 = 119892 for each 119892 isin 119875119879120588 (119891)(3) 119891 isin 119865(119875119879120588 (119891)) that is 119891 isin 119875119879120588 (119891) Further 119865120588(119879) =
119865(119875119879120588 (119891)) where 119865(119875119879120588 (119891)) denotes the set of fixedpoints of 119875119879120588 (119891)
The following examples were presented by Razani et al[25]
Example 15 Let (119883 sdot ) be a norm space then sdot is amodular But the converse is not true
Example 16 Let (119883 sdot ) be a norm space For any 119896 ge 1 sdot 119896is a modular on 1198833 Iterative Approximation of Fixed Points inModular Function Spaces
We begin this section with the following proposition
Proposition 17 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr
Journal of Function Spaces 5
119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Then the Picard-Krasnoselskii hybriditerative process (15) is Fejer monotone with respect to 119865120588(119879)Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (19)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (20)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(21)
Using (21) in (19) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (22)
Hence the Picard-Krasnoselskii hybrid iterative process (15)is Fejer monotone with respect to 119865120588(119879) This completes theproof of Proposition 17
Next we prove the following proposition
Proposition 18 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Let 119891119899 be the Picard-Krasnoselskiihybrid iterative process (15) then
(i) the sequence 119891119899 is bounded(ii) for each 119891 isin 119863 120588(119891119899 minus 119891) converges
Proof Since 119891119899 is Fejer monotone as shown in Proposi-tion 17 we can easily show (i) and (ii) This completes theproof of Proposition 18
Theorem 19 Let 120588 satisfy (1198801198801198621) and let 119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Suppose that 119865120588(119879) = 0 Let119891119899 sub 119863 be the Picard-Krasnoselskii hybrid iterative process(15) Then lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879) andlim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (23)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (24)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(25)
Using (25) in (23) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (26)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Suppose that
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (27)
where 119871 ge 0We next prove that lim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0 Since
dist120588(119891119899 119875119879120588 (119891119899)) le 120588(119891119899 minus V119899) it suffices to show that
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (28)
Now
120588 (V119899 minus 119901) le 119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901)) le 120588 (119891119899 minus 119901) (29)
and this implies that
lim sup119899rarrinfin
120588 (V119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (30)
and by (27) we have
lim sup119899rarrinfin
120588 (V119899 minus 119901) le 119871 (31)
Using (25) we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (32)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (33)
Next we have
119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) le 120588 (119891119899 minus 119901) (34)
and this implies that
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (35)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (36)
6 Journal of Function Spaces
Using (23) and (24) we have
lim119899rarrinfin
120588 (119891119899+1 minus 119901) = lim119899rarrinfin
120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)]= lim119899rarrinfin
120588 (119891119899 minus 119901) le 119871(37)
Moreover
120588 (119891119899+1 minus 119901) le 120588 [(1 minus 120582) 119891119899 + 120582V119899 minus 119901]= 120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)] (38)
Using Lemma 4 and (38) we have
119871 = lim inf119899rarrinfin
120588 (119891119899+1 minus 119901)= lim inf119899rarrinfin
120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)]= lim inf119899rarrinfin
120588 (119891119899 minus 119901) (39)
This means that
119871 = lim inf119899rarrinfin
120588 (119891119899 minus 119901) (40)
Using (27) and (40) we have
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (41)
Using (27) (31) (37) and Lemma 12 we have
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (42)
Hence
lim119899rarrinfin
dist120588 (119891119899 119875119879120588 (119891119899)) = 0 (43)
The proof of Theorem 19 is completed
Next we prove the following theorem
Theorem 20 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then 119879 has a unique fixed pointMoreover the Picard-Krasnoselskii hybrid iterative process (15)converges to this fixed point
Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 119896120588 (119892119899 minus 119901)le 120588 (119892119899 minus 119901) (44)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) minus 119901] (45)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582119896120588 (119891119899 minus 119901)le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(46)
Using (46) in (44) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (47)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Using a similar approach as in the proof of Theorem 19 wesee that lim119899rarrinfin120588(119891119899 minus 119901) = 0
Next we show that 119891119899 is a 120588-Cauchy sequence Sincelim119899rarrinfin(119891119899 minus 119901) = 0 we proceed by contradiction Hencethere exists 120598 gt 0 and two sequences of natural numbers119898(119894) 119899(119894) such that
119899 (119894) gt 119898 (119894) ge 119894120588 (119891119899(119894) minus 119891119898(119894)) gt 120598 (48)
For all integer 119894 let 119899(119894) be the least integer exceeding 119898(119894)which satisfy (48) then
120588 (119891119899(119894) minus 119891119898(119894)) gt 120598120588 (119891119899(119894)minus1 minus 119891119898(119894)) le 120598 (49)
So we have
120598 lt 120588 (119891119899(119894) minus 119891119898(119894)) le 120588 (119891119899(119894) minus 1199012 ) + 120588 (119901 minus 119891119898(119894)2 )
le 12120588 (119891119899(119894) minus 119901) + 12120588 (119901 minus 119891119898(119894))le 120588 (119891119899(119894) minus 119901) + 120588 (119901 minus 119891119898(119894)) 997888rarr 0 as 119899 997888rarr infin
(50)
This is a contradiction Hence 119891119899 is a 120588-Cauchy sequenceTherefore there exists 119901 isin 119863 such that 119891119899 rarr 0 as 119899 rarr infin
Next we have 119879119901 = 119901 Clearly120588 (119901 minus 1198791199012 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1198791199012 )
le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119879119901)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119879119901)= 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119901) 997888rarr 0
as 119899 997888rarr infin
(51)
Hence 120588((119901 minus 119879119901)2) = 0 Therefore 119901 = 119879119901
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
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Journal of Function Spaces 3
Lemma 4 (see [15]) Let 120588 isin R Defining 1198710120588 = 119891 isin119871120588 120588(119891 sdot) 119894119904 119900119903119889119890119903 119888119900119899119905119894119899119906119900119906119904 and 119864120588 = 119891 isin 119871120588 120582119891 isin1198710120588 for every 120582 gt 0 one has the following(i) 119871120588 sup 1198710120588 sup 119864120588(ii) 119864120588 has the Lebesgue property that is 120588(120572119891 119863119896) rarr 0
for 120572 gt 0 119891 isin 119864120588 and 119863119896 darr 0(iii) 119864120588 is the closure of 120576 (in the sense of sdot 120588)The following uniform convexity type properties of 120588 can
be found in [17]
Definition 5 Let 120588 be a nonzero regular convex functionmodular defined on Ω
(i) Let 119903 gt 0 120598 gt 0 Define1198631 (119903 120598) = (119891 119892) 119891 119892 isin 119871120588 120588 (119891) le 119903 120588 (119892)
le 119903 120588 (119891 minus 119892) ge 120598119903 (5)
Let
1205751 (119903 120598) = inf 1 minus 1119903 120588 (119891 + 1198922 ) (119891 119892) isin 1198631 (119903 120598)if 1198631 (119903 120598) = 0
(6)
and 1205751(119903 120598) = 1 if 1198631(119903 120598) = 0 One says that 120588satisfies (UC1) if for every 119903 gt 0 120598 gt 0 1205751(119903 120598) gt 0Observe that for every 119903 gt 0 1198631(119903 120598) = 0 for 120598 gt 0small enough
(ii) One says that 120588 satisfies (UUC1) if for every 119904 ge 0120598 gt 0 there exists 1205781(119904 120598) gt 0 depending only on 119904and 120598 such that 1205751(119903 120598) gt 1205781(119904 120598) gt 0 for any 119903 gt 119904
(iii) Let 119903 gt 0 120598 gt 0 Define1198632 (119903 120598) = (119891 119892) 119891 119892 isin 119871120588 120588 (119891) le 119903 120588 (119892)
le 119903 120588 (119891 minus 1198922 ) ge 120598119903 (7)
Let
1205752 (119903 120598) = inf 1 minus 1119903 120588 (119891 + 1198922 ) (119891 119892) isin 1198632 (119903 120598) if 1198632 (119903 120598) = 0
(8)
and 1205752(119903 120598) = 1 if1198632(119903 120598) = 0 one says that 120588 satisfies(UC2) if for every 119903 gt 0 120598 gt 0 1205752(119903 120598) gt 0 Observethat for every 119903 gt 0 1198632(119903 120598) = 0 for 120598 gt 0 smallenough
(iv) One says that 120588 satisfies (UUC2) if for every 119904 ge 0120598 gt 0 there exists 1205782(119904 120598) gt 0 depending only on 119904and 120598 such that 1205752(119903 120598) gt 1205782(119904 120598) gt 0 for any 119903 gt 119904
(v) One says that 120588 is strictly convex (SC) if for every119891 119892 isin 119871120588 such that 120588(119891) = 120588(119892) and 120588((119891 + 119892)2) =(120588(119891) + 120588(119892))2 there holds 119891 = 119892Proposition 6 (see [15]) The following conditions character-ize relationship between the above defined notions
(i) (119880119880119862119894) rArr (119880119862119894) for 119894 = 1 2(ii) 1205751(119903 120598) le 1205752(119903 120598)(iii) (1198801198621) rArr (1198801198622)(iv) (1198801198801198621) rArr (1198801198801198622)(v) If 120588 is homogeneous (eg it is a norm) then all the
conditions (1198801198621) (1198801198622) (1198801198801198621) and (1198801198801198622) areequivalent and 1205751(119903 2120598) = 1205751(1 2120598) = 1205752(1 120598) =1205752(119903 120598)
Definition 7 A nonzero regular convex function modular 120588is said to satisfy the Δ 2-condition if sup119899ge1120588(2119891119899 119863119896) rarr 0 as119896 rarr infinwhenever 119863119896 decreases to 0 and sup119899ge1120588(119891119899 119863119896) rarr0 as 119896 rarr infinDefinition 8 A function modular is said to satisfy the Δ 2-type condition if there exists119870 gt 0 such that for any119891 isin 119871120588one has 120588(2119891) le 119870120588(119891)
In general Δ 2-condition and Δ 2-type condition arenot equivalent even though it is easy to see that Δ 2-typecondition impliesΔ 2-condition on themodular space 119871120588 see[22]
Definition 9 Let 119871120588 be a modular spaceThe sequence 119891119899 sub119871120588 is called(1) 120588-convergent to 119891 isin 119871120588 if 120588(119891119899 minus 119891) rarr 0 as 119899 rarr infin(2) 120588-Cauchy if 120588(119891119899 minus 119891119898) rarr 0 as 119899 and 119898 rarr infinObserve that 120588-convergence does not imply 120588-Cauchy
since 120588 does not satisfy the triangle inequality In fact onecan easily show that this will happen if and only if 120588 satisfiesthe Δ 2-condition
Kilmer et al [23] defined 120588-distance from an 119891 isin 119871120588 to aset 119863 sub 119871120588 as follows
dist120588 (119891 119863) = inf 120588 (119891 minus ℎ) ℎ isin 119863 (9)
Definition 10 A subset 119863 sub 119871120588 is called(1) 120588-closed if the 120588-limit of a 120588-convergent sequence of119863 always belongs to 119863(2) 120588-ae closed if the 120588-ae limit of a 120588-ae convergent
sequence of 119863 always belongs to 119863(3) 120588-compact if every sequence in 119863 has a 120588-convergent
subsequence in 119863(4) 120588-ae compact if every sequence in 119863 has a 120588-ae
convergent subsequence in 119863(5) 120588-bounded if
diam120588 (119863) = sup 120588 (119891 minus 119892) 119891 119892 isin 119863 lt infin (10)
4 Journal of Function Spaces
It is known that the norm and modular convergence arealso the same when we deal with the Δ 2-type condition (seeeg [15])
A set 119863 sub 119871120588 is called 120588-proximinal if for each 119891 isin 119871120588there exists an element119892 isin 119863 such that 120588(119891minus119892) = dist120588(119891 119863)We shall denote the family of nonempty 120588-bounded 120588-proximinal subsets of 119863 by 119875120588(119863) the family of nonempty120588-closed 120588-bounded subsets of 119863 by 119862120588(119863) and the familyof 120588-compact subsets of 119863 by 119870120588(119863) Let 119867120588(sdot sdot) be the 120588-Hausdorff distance on 119862120588(119871120588) that is
119867120588 (119860 119861) = maxsup119891isin119860
dist120588 (119891 119861) sup119892isin119861
dist120588 (119892 119860) 119860 119861 isin 119862120588 (119871120588)
(11)
A multivalued map 119879 119863 rarr 119862120588(119871120588) is said to be
(a) 120588-contraction mapping if there exists a constant 119896 isin[0 1) such that
119867120588 (119879119891 119879119892) le 119896120588 (119891 minus 119892) forall119891 119892 isin 119863 (12)
(b) 120588-nonexpansive (see eg Khan and Abbas [1]) if
119867120588 (119879119891 119879119892) le 120588 (119891 minus 119892) forall119891 119892 isin 119863 (13)
(c) 120588-quasi-nonexpansive mapping if
119867120588 (119879119891 119901) le 120588 (119891 minus 119901) forall119891 isin 119863 119901 isin 119865120588 (119879) (14)
A sequence 119905119899 sub (0 1) is called bounded away from 0 ifthere exists 119886 gt 0 such that 119905119899 ge 119886 for every 119899 isin N Similarly119905119899 sub (0 1) is called bounded away from 1 if there exists 119887 lt 1such that 119905119899 le 119887 for every 119899 isin N
Okeke andAbbas [2] introduced the Picard-Krasnoselskiihybrid iterative process The authors proved that this newhybrid iterative process converges faster than all of PicardMann Krasnoselskii and Ishikawa iterative processes whenapplied to contraction mappings We now give the analogueof the Picard-Krasnoselskii hybrid iterative process in mod-ular function spaces as follows let 119879 119863 rarr 119875120588(119863) bea multivalued mapping and 119891119899 sub 119863 be defined by thefollowing iteration process
119891119899+1 isin 119875119879120588 (119892119899)119892119899 = (1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) 119899 isin N (15)
where V119899 isin 119875119879120588 (119891119899) and 0 lt 120582 lt 1 It is our purpose in thepresent paper to prove some new fixed point theorems usingthis iteration process in the framework of modular functionspaces
Definition 11 A sequence 119891119899 sub 119863 is said to be Fejermonotone with respect to subset 119875120588(119863) of 119863 if 120588(119891119899+1 minus 119901) le120588(119891119899 minus 119901) for all 119901 isin 119875119879120588 (119863) of 119863 119899 isin N
The following Lemma will be needed in this study
Lemma 12 (see [22]) Let 120588 be a function modular and 119891119899 and119892119899 be two sequences in 119883120588 Then
lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim sup119899rarrinfin
120588 (119891119899 + 119892119899)= lim sup119899rarrinfin
120588 (119891119899) lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim inf119899rarrinfin
120588 (119891119899 + 119892119899)= lim inf119899rarrinfin
120588 (119891119899)
(16)
Lemma 13 (see [17]) Let120588 satisfy (1198801198801198621) and let 119905119896 sub (0 1)be bounded away from 0 and 1 If there exists 119877 gt 0 such that
lim sup119899rarrinfin
120588 (119891119899) le 119877lim sup119899rarrinfin
120588 (119892119899) le 119877lim119899rarrinfin
120588 (119905119899119891119899 + (1 minus 119905119899) 119892119899) = 119877(17)
and then lim119899rarrinfin120588(119891119899 minus 119892119899) = 0The above lemma is an analogue of a famous lemma due
to Schu [24] in Banach spacesA function 119891 isin 119871120588 is called a fixed point of 119879 119871120588 rarr119875120588(119863) if119891 isin 119879119891The set of all fixed points of119879will be denoted
by 119865120588(119879)Lemma 14 (see [1]) Let 119879 119863 rarr 119875120588(119863) be a multivaluedmapping and
119875119879120588 (119891) = 119892 isin 119879119891 120588 (119891 minus 119892) = dist120588 (119891 119879119891) (18)
Then the following are equivalent
(1) 119891 isin 119865120588(119879) that is 119891 isin 119879119891(2) 119875119879120588 (119891) = 119891 that is 119891 = 119892 for each 119892 isin 119875119879120588 (119891)(3) 119891 isin 119865(119875119879120588 (119891)) that is 119891 isin 119875119879120588 (119891) Further 119865120588(119879) =
119865(119875119879120588 (119891)) where 119865(119875119879120588 (119891)) denotes the set of fixedpoints of 119875119879120588 (119891)
The following examples were presented by Razani et al[25]
Example 15 Let (119883 sdot ) be a norm space then sdot is amodular But the converse is not true
Example 16 Let (119883 sdot ) be a norm space For any 119896 ge 1 sdot 119896is a modular on 1198833 Iterative Approximation of Fixed Points inModular Function Spaces
We begin this section with the following proposition
Proposition 17 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr
Journal of Function Spaces 5
119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Then the Picard-Krasnoselskii hybriditerative process (15) is Fejer monotone with respect to 119865120588(119879)Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (19)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (20)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(21)
Using (21) in (19) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (22)
Hence the Picard-Krasnoselskii hybrid iterative process (15)is Fejer monotone with respect to 119865120588(119879) This completes theproof of Proposition 17
Next we prove the following proposition
Proposition 18 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Let 119891119899 be the Picard-Krasnoselskiihybrid iterative process (15) then
(i) the sequence 119891119899 is bounded(ii) for each 119891 isin 119863 120588(119891119899 minus 119891) converges
Proof Since 119891119899 is Fejer monotone as shown in Proposi-tion 17 we can easily show (i) and (ii) This completes theproof of Proposition 18
Theorem 19 Let 120588 satisfy (1198801198801198621) and let 119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Suppose that 119865120588(119879) = 0 Let119891119899 sub 119863 be the Picard-Krasnoselskii hybrid iterative process(15) Then lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879) andlim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (23)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (24)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(25)
Using (25) in (23) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (26)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Suppose that
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (27)
where 119871 ge 0We next prove that lim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0 Since
dist120588(119891119899 119875119879120588 (119891119899)) le 120588(119891119899 minus V119899) it suffices to show that
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (28)
Now
120588 (V119899 minus 119901) le 119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901)) le 120588 (119891119899 minus 119901) (29)
and this implies that
lim sup119899rarrinfin
120588 (V119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (30)
and by (27) we have
lim sup119899rarrinfin
120588 (V119899 minus 119901) le 119871 (31)
Using (25) we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (32)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (33)
Next we have
119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) le 120588 (119891119899 minus 119901) (34)
and this implies that
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (35)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (36)
6 Journal of Function Spaces
Using (23) and (24) we have
lim119899rarrinfin
120588 (119891119899+1 minus 119901) = lim119899rarrinfin
120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)]= lim119899rarrinfin
120588 (119891119899 minus 119901) le 119871(37)
Moreover
120588 (119891119899+1 minus 119901) le 120588 [(1 minus 120582) 119891119899 + 120582V119899 minus 119901]= 120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)] (38)
Using Lemma 4 and (38) we have
119871 = lim inf119899rarrinfin
120588 (119891119899+1 minus 119901)= lim inf119899rarrinfin
120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)]= lim inf119899rarrinfin
120588 (119891119899 minus 119901) (39)
This means that
119871 = lim inf119899rarrinfin
120588 (119891119899 minus 119901) (40)
Using (27) and (40) we have
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (41)
Using (27) (31) (37) and Lemma 12 we have
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (42)
Hence
lim119899rarrinfin
dist120588 (119891119899 119875119879120588 (119891119899)) = 0 (43)
The proof of Theorem 19 is completed
Next we prove the following theorem
Theorem 20 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then 119879 has a unique fixed pointMoreover the Picard-Krasnoselskii hybrid iterative process (15)converges to this fixed point
Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 119896120588 (119892119899 minus 119901)le 120588 (119892119899 minus 119901) (44)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) minus 119901] (45)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582119896120588 (119891119899 minus 119901)le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(46)
Using (46) in (44) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (47)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Using a similar approach as in the proof of Theorem 19 wesee that lim119899rarrinfin120588(119891119899 minus 119901) = 0
Next we show that 119891119899 is a 120588-Cauchy sequence Sincelim119899rarrinfin(119891119899 minus 119901) = 0 we proceed by contradiction Hencethere exists 120598 gt 0 and two sequences of natural numbers119898(119894) 119899(119894) such that
119899 (119894) gt 119898 (119894) ge 119894120588 (119891119899(119894) minus 119891119898(119894)) gt 120598 (48)
For all integer 119894 let 119899(119894) be the least integer exceeding 119898(119894)which satisfy (48) then
120588 (119891119899(119894) minus 119891119898(119894)) gt 120598120588 (119891119899(119894)minus1 minus 119891119898(119894)) le 120598 (49)
So we have
120598 lt 120588 (119891119899(119894) minus 119891119898(119894)) le 120588 (119891119899(119894) minus 1199012 ) + 120588 (119901 minus 119891119898(119894)2 )
le 12120588 (119891119899(119894) minus 119901) + 12120588 (119901 minus 119891119898(119894))le 120588 (119891119899(119894) minus 119901) + 120588 (119901 minus 119891119898(119894)) 997888rarr 0 as 119899 997888rarr infin
(50)
This is a contradiction Hence 119891119899 is a 120588-Cauchy sequenceTherefore there exists 119901 isin 119863 such that 119891119899 rarr 0 as 119899 rarr infin
Next we have 119879119901 = 119901 Clearly120588 (119901 minus 1198791199012 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1198791199012 )
le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119879119901)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119879119901)= 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119901) 997888rarr 0
as 119899 997888rarr infin
(51)
Hence 120588((119901 minus 119879119901)2) = 0 Therefore 119901 = 119879119901
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
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4 Journal of Function Spaces
It is known that the norm and modular convergence arealso the same when we deal with the Δ 2-type condition (seeeg [15])
A set 119863 sub 119871120588 is called 120588-proximinal if for each 119891 isin 119871120588there exists an element119892 isin 119863 such that 120588(119891minus119892) = dist120588(119891 119863)We shall denote the family of nonempty 120588-bounded 120588-proximinal subsets of 119863 by 119875120588(119863) the family of nonempty120588-closed 120588-bounded subsets of 119863 by 119862120588(119863) and the familyof 120588-compact subsets of 119863 by 119870120588(119863) Let 119867120588(sdot sdot) be the 120588-Hausdorff distance on 119862120588(119871120588) that is
119867120588 (119860 119861) = maxsup119891isin119860
dist120588 (119891 119861) sup119892isin119861
dist120588 (119892 119860) 119860 119861 isin 119862120588 (119871120588)
(11)
A multivalued map 119879 119863 rarr 119862120588(119871120588) is said to be
(a) 120588-contraction mapping if there exists a constant 119896 isin[0 1) such that
119867120588 (119879119891 119879119892) le 119896120588 (119891 minus 119892) forall119891 119892 isin 119863 (12)
(b) 120588-nonexpansive (see eg Khan and Abbas [1]) if
119867120588 (119879119891 119879119892) le 120588 (119891 minus 119892) forall119891 119892 isin 119863 (13)
(c) 120588-quasi-nonexpansive mapping if
119867120588 (119879119891 119901) le 120588 (119891 minus 119901) forall119891 isin 119863 119901 isin 119865120588 (119879) (14)
A sequence 119905119899 sub (0 1) is called bounded away from 0 ifthere exists 119886 gt 0 such that 119905119899 ge 119886 for every 119899 isin N Similarly119905119899 sub (0 1) is called bounded away from 1 if there exists 119887 lt 1such that 119905119899 le 119887 for every 119899 isin N
Okeke andAbbas [2] introduced the Picard-Krasnoselskiihybrid iterative process The authors proved that this newhybrid iterative process converges faster than all of PicardMann Krasnoselskii and Ishikawa iterative processes whenapplied to contraction mappings We now give the analogueof the Picard-Krasnoselskii hybrid iterative process in mod-ular function spaces as follows let 119879 119863 rarr 119875120588(119863) bea multivalued mapping and 119891119899 sub 119863 be defined by thefollowing iteration process
119891119899+1 isin 119875119879120588 (119892119899)119892119899 = (1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) 119899 isin N (15)
where V119899 isin 119875119879120588 (119891119899) and 0 lt 120582 lt 1 It is our purpose in thepresent paper to prove some new fixed point theorems usingthis iteration process in the framework of modular functionspaces
Definition 11 A sequence 119891119899 sub 119863 is said to be Fejermonotone with respect to subset 119875120588(119863) of 119863 if 120588(119891119899+1 minus 119901) le120588(119891119899 minus 119901) for all 119901 isin 119875119879120588 (119863) of 119863 119899 isin N
The following Lemma will be needed in this study
Lemma 12 (see [22]) Let 120588 be a function modular and 119891119899 and119892119899 be two sequences in 119883120588 Then
lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim sup119899rarrinfin
120588 (119891119899 + 119892119899)= lim sup119899rarrinfin
120588 (119891119899) lim119899rarrinfin
120588 (119892119899) = 0 997904rArr lim inf119899rarrinfin
120588 (119891119899 + 119892119899)= lim inf119899rarrinfin
120588 (119891119899)
(16)
Lemma 13 (see [17]) Let120588 satisfy (1198801198801198621) and let 119905119896 sub (0 1)be bounded away from 0 and 1 If there exists 119877 gt 0 such that
lim sup119899rarrinfin
120588 (119891119899) le 119877lim sup119899rarrinfin
120588 (119892119899) le 119877lim119899rarrinfin
120588 (119905119899119891119899 + (1 minus 119905119899) 119892119899) = 119877(17)
and then lim119899rarrinfin120588(119891119899 minus 119892119899) = 0The above lemma is an analogue of a famous lemma due
to Schu [24] in Banach spacesA function 119891 isin 119871120588 is called a fixed point of 119879 119871120588 rarr119875120588(119863) if119891 isin 119879119891The set of all fixed points of119879will be denoted
by 119865120588(119879)Lemma 14 (see [1]) Let 119879 119863 rarr 119875120588(119863) be a multivaluedmapping and
119875119879120588 (119891) = 119892 isin 119879119891 120588 (119891 minus 119892) = dist120588 (119891 119879119891) (18)
Then the following are equivalent
(1) 119891 isin 119865120588(119879) that is 119891 isin 119879119891(2) 119875119879120588 (119891) = 119891 that is 119891 = 119892 for each 119892 isin 119875119879120588 (119891)(3) 119891 isin 119865(119875119879120588 (119891)) that is 119891 isin 119875119879120588 (119891) Further 119865120588(119879) =
119865(119875119879120588 (119891)) where 119865(119875119879120588 (119891)) denotes the set of fixedpoints of 119875119879120588 (119891)
The following examples were presented by Razani et al[25]
Example 15 Let (119883 sdot ) be a norm space then sdot is amodular But the converse is not true
Example 16 Let (119883 sdot ) be a norm space For any 119896 ge 1 sdot 119896is a modular on 1198833 Iterative Approximation of Fixed Points inModular Function Spaces
We begin this section with the following proposition
Proposition 17 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr
Journal of Function Spaces 5
119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Then the Picard-Krasnoselskii hybriditerative process (15) is Fejer monotone with respect to 119865120588(119879)Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (19)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (20)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(21)
Using (21) in (19) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (22)
Hence the Picard-Krasnoselskii hybrid iterative process (15)is Fejer monotone with respect to 119865120588(119879) This completes theproof of Proposition 17
Next we prove the following proposition
Proposition 18 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Let 119891119899 be the Picard-Krasnoselskiihybrid iterative process (15) then
(i) the sequence 119891119899 is bounded(ii) for each 119891 isin 119863 120588(119891119899 minus 119891) converges
Proof Since 119891119899 is Fejer monotone as shown in Proposi-tion 17 we can easily show (i) and (ii) This completes theproof of Proposition 18
Theorem 19 Let 120588 satisfy (1198801198801198621) and let 119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Suppose that 119865120588(119879) = 0 Let119891119899 sub 119863 be the Picard-Krasnoselskii hybrid iterative process(15) Then lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879) andlim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (23)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (24)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(25)
Using (25) in (23) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (26)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Suppose that
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (27)
where 119871 ge 0We next prove that lim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0 Since
dist120588(119891119899 119875119879120588 (119891119899)) le 120588(119891119899 minus V119899) it suffices to show that
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (28)
Now
120588 (V119899 minus 119901) le 119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901)) le 120588 (119891119899 minus 119901) (29)
and this implies that
lim sup119899rarrinfin
120588 (V119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (30)
and by (27) we have
lim sup119899rarrinfin
120588 (V119899 minus 119901) le 119871 (31)
Using (25) we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (32)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (33)
Next we have
119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) le 120588 (119891119899 minus 119901) (34)
and this implies that
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (35)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (36)
6 Journal of Function Spaces
Using (23) and (24) we have
lim119899rarrinfin
120588 (119891119899+1 minus 119901) = lim119899rarrinfin
120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)]= lim119899rarrinfin
120588 (119891119899 minus 119901) le 119871(37)
Moreover
120588 (119891119899+1 minus 119901) le 120588 [(1 minus 120582) 119891119899 + 120582V119899 minus 119901]= 120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)] (38)
Using Lemma 4 and (38) we have
119871 = lim inf119899rarrinfin
120588 (119891119899+1 minus 119901)= lim inf119899rarrinfin
120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)]= lim inf119899rarrinfin
120588 (119891119899 minus 119901) (39)
This means that
119871 = lim inf119899rarrinfin
120588 (119891119899 minus 119901) (40)
Using (27) and (40) we have
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (41)
Using (27) (31) (37) and Lemma 12 we have
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (42)
Hence
lim119899rarrinfin
dist120588 (119891119899 119875119879120588 (119891119899)) = 0 (43)
The proof of Theorem 19 is completed
Next we prove the following theorem
Theorem 20 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then 119879 has a unique fixed pointMoreover the Picard-Krasnoselskii hybrid iterative process (15)converges to this fixed point
Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 119896120588 (119892119899 minus 119901)le 120588 (119892119899 minus 119901) (44)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) minus 119901] (45)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582119896120588 (119891119899 minus 119901)le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(46)
Using (46) in (44) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (47)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Using a similar approach as in the proof of Theorem 19 wesee that lim119899rarrinfin120588(119891119899 minus 119901) = 0
Next we show that 119891119899 is a 120588-Cauchy sequence Sincelim119899rarrinfin(119891119899 minus 119901) = 0 we proceed by contradiction Hencethere exists 120598 gt 0 and two sequences of natural numbers119898(119894) 119899(119894) such that
119899 (119894) gt 119898 (119894) ge 119894120588 (119891119899(119894) minus 119891119898(119894)) gt 120598 (48)
For all integer 119894 let 119899(119894) be the least integer exceeding 119898(119894)which satisfy (48) then
120588 (119891119899(119894) minus 119891119898(119894)) gt 120598120588 (119891119899(119894)minus1 minus 119891119898(119894)) le 120598 (49)
So we have
120598 lt 120588 (119891119899(119894) minus 119891119898(119894)) le 120588 (119891119899(119894) minus 1199012 ) + 120588 (119901 minus 119891119898(119894)2 )
le 12120588 (119891119899(119894) minus 119901) + 12120588 (119901 minus 119891119898(119894))le 120588 (119891119899(119894) minus 119901) + 120588 (119901 minus 119891119898(119894)) 997888rarr 0 as 119899 997888rarr infin
(50)
This is a contradiction Hence 119891119899 is a 120588-Cauchy sequenceTherefore there exists 119901 isin 119863 such that 119891119899 rarr 0 as 119899 rarr infin
Next we have 119879119901 = 119901 Clearly120588 (119901 minus 1198791199012 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1198791199012 )
le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119879119901)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119879119901)= 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119901) 997888rarr 0
as 119899 997888rarr infin
(51)
Hence 120588((119901 minus 119879119901)2) = 0 Therefore 119901 = 119879119901
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
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Journal of Function Spaces 5
119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Then the Picard-Krasnoselskii hybriditerative process (15) is Fejer monotone with respect to 119865120588(119879)Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (19)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (20)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(21)
Using (21) in (19) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (22)
Hence the Picard-Krasnoselskii hybrid iterative process (15)is Fejer monotone with respect to 119865120588(119879) This completes theproof of Proposition 17
Next we prove the following proposition
Proposition 18 Let120588 satisfy (1198801198801198621) and let119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Let 119891119899 be the Picard-Krasnoselskiihybrid iterative process (15) then
(i) the sequence 119891119899 is bounded(ii) for each 119891 isin 119863 120588(119891119899 minus 119891) converges
Proof Since 119891119899 is Fejer monotone as shown in Proposi-tion 17 we can easily show (i) and (ii) This completes theproof of Proposition 18
Theorem 19 Let 120588 satisfy (1198801198801198621) and let 119863 be a nonempty120588-closed 120588-bounded and convex subset of 119871120588 Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping such that 119875119879120588 is a 120588-quasi-nonexpansive mapping Suppose that 119865120588(119879) = 0 Let119891119899 sub 119863 be the Picard-Krasnoselskii hybrid iterative process(15) Then lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879) andlim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) (23)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901] (24)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(25)
Using (25) in (23) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (26)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Suppose that
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (27)
where 119871 ge 0We next prove that lim119899rarrinfindist120588(119891119899 119875119879120588 (119891119899)) = 0 Since
dist120588(119891119899 119875119879120588 (119891119899)) le 120588(119891119899 minus V119899) it suffices to show that
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (28)
Now
120588 (V119899 minus 119901) le 119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901)) le 120588 (119891119899 minus 119901) (29)
and this implies that
lim sup119899rarrinfin
120588 (V119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (30)
and by (27) we have
lim sup119899rarrinfin
120588 (V119899 minus 119901) le 119871 (31)
Using (25) we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (32)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (33)
Next we have
119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 120588 (119892119899 minus 119901) le 120588 (119891119899 minus 119901) (34)
and this implies that
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le lim sup119899rarrinfin
120588 (119891119899 minus 119901) (35)
and hence we have
lim sup119899rarrinfin
120588 (119892119899 minus 119901) le 119871 (36)
6 Journal of Function Spaces
Using (23) and (24) we have
lim119899rarrinfin
120588 (119891119899+1 minus 119901) = lim119899rarrinfin
120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)]= lim119899rarrinfin
120588 (119891119899 minus 119901) le 119871(37)
Moreover
120588 (119891119899+1 minus 119901) le 120588 [(1 minus 120582) 119891119899 + 120582V119899 minus 119901]= 120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)] (38)
Using Lemma 4 and (38) we have
119871 = lim inf119899rarrinfin
120588 (119891119899+1 minus 119901)= lim inf119899rarrinfin
120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)]= lim inf119899rarrinfin
120588 (119891119899 minus 119901) (39)
This means that
119871 = lim inf119899rarrinfin
120588 (119891119899 minus 119901) (40)
Using (27) and (40) we have
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (41)
Using (27) (31) (37) and Lemma 12 we have
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (42)
Hence
lim119899rarrinfin
dist120588 (119891119899 119875119879120588 (119891119899)) = 0 (43)
The proof of Theorem 19 is completed
Next we prove the following theorem
Theorem 20 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then 119879 has a unique fixed pointMoreover the Picard-Krasnoselskii hybrid iterative process (15)converges to this fixed point
Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 119896120588 (119892119899 minus 119901)le 120588 (119892119899 minus 119901) (44)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) minus 119901] (45)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582119896120588 (119891119899 minus 119901)le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(46)
Using (46) in (44) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (47)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Using a similar approach as in the proof of Theorem 19 wesee that lim119899rarrinfin120588(119891119899 minus 119901) = 0
Next we show that 119891119899 is a 120588-Cauchy sequence Sincelim119899rarrinfin(119891119899 minus 119901) = 0 we proceed by contradiction Hencethere exists 120598 gt 0 and two sequences of natural numbers119898(119894) 119899(119894) such that
119899 (119894) gt 119898 (119894) ge 119894120588 (119891119899(119894) minus 119891119898(119894)) gt 120598 (48)
For all integer 119894 let 119899(119894) be the least integer exceeding 119898(119894)which satisfy (48) then
120588 (119891119899(119894) minus 119891119898(119894)) gt 120598120588 (119891119899(119894)minus1 minus 119891119898(119894)) le 120598 (49)
So we have
120598 lt 120588 (119891119899(119894) minus 119891119898(119894)) le 120588 (119891119899(119894) minus 1199012 ) + 120588 (119901 minus 119891119898(119894)2 )
le 12120588 (119891119899(119894) minus 119901) + 12120588 (119901 minus 119891119898(119894))le 120588 (119891119899(119894) minus 119901) + 120588 (119901 minus 119891119898(119894)) 997888rarr 0 as 119899 997888rarr infin
(50)
This is a contradiction Hence 119891119899 is a 120588-Cauchy sequenceTherefore there exists 119901 isin 119863 such that 119891119899 rarr 0 as 119899 rarr infin
Next we have 119879119901 = 119901 Clearly120588 (119901 minus 1198791199012 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1198791199012 )
le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119879119901)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119879119901)= 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119901) 997888rarr 0
as 119899 997888rarr infin
(51)
Hence 120588((119901 minus 119879119901)2) = 0 Therefore 119901 = 119879119901
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
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[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
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6 Journal of Function Spaces
Using (23) and (24) we have
lim119899rarrinfin
120588 (119891119899+1 minus 119901) = lim119899rarrinfin
120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 V119899 minus 119901]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))]le lim119899rarrinfin
[(1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)]= lim119899rarrinfin
120588 (119891119899 minus 119901) le 119871(37)
Moreover
120588 (119891119899+1 minus 119901) le 120588 [(1 minus 120582) 119891119899 + 120582V119899 minus 119901]= 120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)] (38)
Using Lemma 4 and (38) we have
119871 = lim inf119899rarrinfin
120588 (119891119899+1 minus 119901)= lim inf119899rarrinfin
120588 [(119891119899 minus 119901) + 120582 (V119899 minus 119891119899)]= lim inf119899rarrinfin
120588 (119891119899 minus 119901) (39)
This means that
119871 = lim inf119899rarrinfin
120588 (119891119899 minus 119901) (40)
Using (27) and (40) we have
lim119899rarrinfin
120588 (119891119899 minus 119901) = 119871 (41)
Using (27) (31) (37) and Lemma 12 we have
lim119899rarrinfin
120588 (119891119899 minus V119899) = 0 (42)
Hence
lim119899rarrinfin
dist120588 (119891119899 119875119879120588 (119891119899)) = 0 (43)
The proof of Theorem 19 is completed
Next we prove the following theorem
Theorem 20 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then 119879 has a unique fixed pointMoreover the Picard-Krasnoselskii hybrid iterative process (15)converges to this fixed point
Proof Suppose 119901 isin 119865120588(119879) By Lemma 13 119875119879120588 (119901) = 119901 and119865120588(119879) = 119865(119875119879120588 ) Using (15) we have the following estimate
120588 (119891119899+1 minus 119901) le 119867120588 (119875119879120588 (119892119899) 119875119879120588 (119901)) le 119896120588 (119892119899 minus 119901)le 120588 (119892119899 minus 119901) (44)
Next we have
120588 (119892119899 minus 119901) = 120588 [(1 minus 120582) 119891119899 + 120582119875119879120588 (V119899) minus 119901] (45)
By convexity of 120588 we have120588 (119892119899 minus 119901) le (1 minus 120582) 120588 (119891119899 minus 119901)
+ 120582119867120588 (119875119879120588 (119891119899) 119875119879120588 (119901))le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582119896120588 (119891119899 minus 119901)le (1 minus 120582) 120588 (119891119899 minus 119901) + 120582120588 (119891119899 minus 119901)= 120588 (119891119899 minus 119901)
(46)
Using (46) in (44) we have
120588 (119891119899+1 minus 119901) le 120588 (119891119899 minus 119901) (47)
This shows that lim119899rarrinfin120588(119891119899 minus 119901) exists for all 119901 isin 119865120588(119879)Using a similar approach as in the proof of Theorem 19 wesee that lim119899rarrinfin120588(119891119899 minus 119901) = 0
Next we show that 119891119899 is a 120588-Cauchy sequence Sincelim119899rarrinfin(119891119899 minus 119901) = 0 we proceed by contradiction Hencethere exists 120598 gt 0 and two sequences of natural numbers119898(119894) 119899(119894) such that
119899 (119894) gt 119898 (119894) ge 119894120588 (119891119899(119894) minus 119891119898(119894)) gt 120598 (48)
For all integer 119894 let 119899(119894) be the least integer exceeding 119898(119894)which satisfy (48) then
120588 (119891119899(119894) minus 119891119898(119894)) gt 120598120588 (119891119899(119894)minus1 minus 119891119898(119894)) le 120598 (49)
So we have
120598 lt 120588 (119891119899(119894) minus 119891119898(119894)) le 120588 (119891119899(119894) minus 1199012 ) + 120588 (119901 minus 119891119898(119894)2 )
le 12120588 (119891119899(119894) minus 119901) + 12120588 (119901 minus 119891119898(119894))le 120588 (119891119899(119894) minus 119901) + 120588 (119901 minus 119891119898(119894)) 997888rarr 0 as 119899 997888rarr infin
(50)
This is a contradiction Hence 119891119899 is a 120588-Cauchy sequenceTherefore there exists 119901 isin 119863 such that 119891119899 rarr 0 as 119899 rarr infin
Next we have 119879119901 = 119901 Clearly120588 (119901 minus 1198791199012 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1198791199012 )
le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119879119901)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119879119901)= 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119901) 997888rarr 0
as 119899 997888rarr infin
(51)
Hence 120588((119901 minus 119879119901)2) = 0 Therefore 119901 = 119879119901
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Journal of Function Spaces 7
Next we prove the uniqueness of 119901 Suppose that 119902 isanother fixed point of 119879 and then we have
120588 (119901 minus 1199022 ) le 120588 (119901 minus 1198911198992 ) + 120588 (119891119899 minus 1199022 )le 12120588 (119901 minus 119891119899) + 12120588 (119891119899 minus 119902)le 120588 (119901 minus 119891119899) + 120588 (119891119899 minus 119902) 997888rarr 0
as 119899 997888rarr infin
(52)
Hence 119901 = 119902 The proof of Theorem 20 is completed
Next we give the following example
Example 21 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr 1199052 (53)
We see that 120588 is not a norm However it is a modular sincethe function 119905 rarr 1199052 is convex Consider 119863 = [0 1] as theclosed interval in [0 infin) which is 120588-closed 120588-bounded and120588-complete since 120588 is continuous Then the mapping
119879 119863 997888rarr 119875120588 (119863)119905 997888rarr 1199052
(54)
is a 120588-contraction mapping with 119896 = 12 Therefore byTheorem 20 it has a unique fixed point in119863 which is119865120588(119879) =04 Stability Results
We begin this section by defining the concept of 119879-stable andalmost 119879-stable of an iterative process in modular functionspaces Moreover we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15)
Definition 22 Let 119863 be a nonempty convex subset of amodular function space 119871120588 and 119879 119863 rarr 119863 be an operatorAssume that 1199091 isin 119863 and 119909119899+1 = 119891(119879 119909119899) defines an iterationscheme which produces a sequence 119909119899infin119899=1 sub 119863 Supposefurthermore that 119909119899infin119899=1 converges strongly to 119909lowast isin 119865120588(119879) =0 Let 119910119899infin119899=1 be any bounded sequence in 119863 and put 120576119899 =120588(119910119899+1 minus 119891(119879 119910119899))
(1) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be 119879-stable on 119863 if lim119899rarrinfin120576119899 = 0implies that lim119899rarrinfin119910119899 = 119909lowast
(2) The iteration scheme 119909119899infin119899=1 defined by 119909119899+1 =119891(119879 119909119899) is said to be almost119879-stable on119863 ifsuminfin119899=1 120576119899 ltinfin implies that lim119899rarrinfin119910119899 = 119909lowastIt is easy to show that an iteration process 119909119899infin119899=1 which
is 119879-stable on 119862 is almost 119879-stable on 119863
Next we provide the following numerical example toshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stableExample 23 Let 119871120588 = [0 infin) be a vector space and 120588 be anapplication defined as follows
120588 119871120588 997888rarr 119871120588119905 997888rarr |119905| (55)
Let 119863 = [0 1] be the closed interval in [0 infin) which is 120588-closed 120588-bounded and 120588-complete Let 119879 [0 1] rarr [0 1]be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping satisfying contractive condition 119879119909 = 1199092 We nowshow that Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable and hence almost 119879-stable with 119896 = 12 and119865120588(119879) = 0 Suppose that 119910119899 = 1119899 is an arbitrary sequencein 119871120588 Take 120582 = 12 Then lim119899rarrinfin119910119899 = 0 Put
120576119899 = 120588 (119910119899+1 minus 119891 (119879 119910119899))= dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))
(56)
and we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)= 1003816100381610038161003816119910119899+1 minus (1 minus 120582) 119910119899 minus 1205821199101198991003816100381610038161003816 = 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 12119899 minus 12119899
1003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816 1119899 + 1 minus 1119899
1003816100381610038161003816100381610038161003816
(57)
Hence
lim119899rarrinfin
120576119899 = 0 (58)
Therefore Picard-Krasnoselskii hybrid iterative process (15)is 119879-stable Clearly (15) is almost 119879-stable
Next we prove the following stability results
Theorem 24 Let 119863 be a 120588-closed 120588-bounded and convexsubset of a 120588-complete modular space 119871120588 and 119879 119863 rarr 119875120588(119863)be a multivalued mapping such that 119875119879120588 is a 120588-contractionmapping and 119865120588(119879) = 0 Then Picard-Krasnoselskii hybriditerative process (15) is 119879-stableProof Suppose 119910119899 sub 119871120588 and define 120576119899 = 120588(119910119899+1 minus 119891(119879 119910119899))Let 119901 be the unique fixed point of 119879 We want to show thatlim119899rarrinfin119910119899 = 119901 if and only if lim119899rarrinfin120576119899 = 0 Suppose that 119910119899converges to 119901 Using (15) and the convexity of 120588 we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Journal of Function Spaces
le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(59)
Hence
lim119899rarrinfin
120576119899 = 0 (60)
Conversely suppose that lim119899rarrinfin120576119899 = 0 Then we have
120576119899 = dist120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899))le 119867120588 (119875119879120588 (119910119899+1) 119875119879120588 (119892119899)) le 120588 (119910119899+1 minus 119892119899)le 120588 (119910119899+1 minus (1 minus 120582) 119910119899 minus 120582119910119899) = 120588 (119910119899+1 minus 119910119899)le 120588 (119910119899+1 minus 119901) + 120588 (119901 minus 119910119899)
(61)
Since lim119899rarrinfin120576119899 = 0 it follows from relation (61) thatlim119899rarrinfin119910119899 = 119901 The proof of Theorem 24 is completed
Remark 25 Theorem 24 generalizes the results of Mbarkiand Hadi [26] to multivalued mappings in modular functionspaces
5 Applications to Differential Equations
In this section we apply our results to differential equationsThe results of this section follow similar applications in [15]Let 120588 isin R and we consider the following initial valueproblem for an unknown function 119906 [0 119860] rarr 119862 where119862 isin 119864120588
119906 (0) = 1198911199061015840 (119905) + (119868 minus 119879) 119906 (119905) = 0 (62)
where 119891 isin 119862 and 119860 gt 0 are fixed and 119879 119862 rarr 119862 issuch that119875119879120588 is120588-quasi-nonexpansivemappingThe followingnotations will be used in this section For 119905 gt 0 we define
119870 (119905) = 1 minus 119890minus119905 = int1199050
119890119904minus119905119889119904 (63)
For any function ] [0 119860] rarr 119871120588 where 119860 gt 0 and any119905 isin [0 119860] we define119878 (]) (119905) = int119905
0119890119904minus119905] (119904) 119889119904 (64)
We also denote
119878120591 (]) (119905) = 119899minus1sum119894=0
(119905119894+1 minus 119905119894) 119890119905119894minus119905] (119905119894) (65)
for any 120591 = 1199050 119905119899 a subdivision of the interval [0 119860]The following lemmawhich is needed to prove our results
in this section can be found in [15]
Lemma 26 Let 120588 isin R be separable Let 119909 119910 [0 119860] rarr 119871120588 betwo Bochner-integrable sdot 120588-bounded functions where 119860 gt 0Then for every 119905 isin [0 119860] one has
120588 (119890minus119905119910 (119905) + int1199050
119890119904minus119905119909 (119904) 119889119904)le 119890minus119905120588 (119910 (119905)) + 119870 (119905) sup
119904isin[0119905]
120588 (119909 (119904)) (66)
We now state our results for this section
Theorem 27 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-quasi-nonexpansive mapping Let one fix 119891 isin 119862and119860 gt 0 and define the sequence of functions119906119899 [0 119860] rarr 119862by the following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (67)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (68)
and the function 119906 [0 119860] rarr 119862 defined by (68) is a solutionof initial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (69)
Proof Since 119875119879120588 is 120588-quasi-nonexpansive mapping the proofof Theorem 27 follows the proof of ([15] Theorem 528)
Next we obtain the following corollaries as a consequenceof Theorem 27
Corollary 28 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-nonexpansive mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (70)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (71)
and the function 119906 [0 119860] rarr 119862 defined by (71) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (72)
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Journal of Function Spaces 9
Corollary 29 Let 120588 isin R be separable Let 119863 sub 119864120588 bea nonempty convex 120588-bounded 120588-closed set with the Vitaliproperty Let 119879 119863 rarr 119875120588(119863) be a multivalued mapping suchthat 119875119879120588 is a 120588-contraction mapping Let one fix 119891 isin 119862 and119860 gt 0 and define the sequence of functions 119906119899 [0 119860] rarr 119862 bythe following inductive formula
1199060 (119905) = 119891119906119899+1 (119905) = 119890minus119905119891 + int119905
0119890119904minus119905119879 (119906119899 (119904)) 119889119904 (73)
Then for every 119905 isin [0 119860] there exists 119906(119905) isin 119862 such that
120588 (119906119899 (119905) minus 119906 (119905)) 997888rarr 0 (74)
and the function 119906 [0 119860] rarr 119862 defined by (74) is a solution ofinitial value problem (62) Moreover
120588 (119891 minus 119906119899 (119905)) le 119870119899+1 (119860) 120575120588 (119862) (75)
Remark 30 Corollary 28 generalizes the results of Khamsiand Kozlowski ([15] Theorem 528) to a multivalued map-ping
Conflicts of Interest
The authors declare that they do not have any conflicts ofinterest
Authorsrsquo Contributions
All authors contributed equally to writing this research paperEach author read and approved the final manuscript
References
[1] S H Khan and M Abbas ldquoApproximating fixed points ofmultivalued 120588-nonexpansive mappings in modular functionspacesrdquo Fixed Point Theory and Applications vol 2014 articleno 34 2014
[2] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017
[3] S H Khan ldquoA Picard-Mann hybrid iterative processrdquo FixedPoint Theory and Applications vol 2013 article no 69 2013
[4] L E J Brouwer ldquoUber Abbildung von MannigfaltigkeitenrdquoMathematische Annalen vol 71 no 4 598 pages 1912
[5] D Downing and W A Kirk ldquoFixed point theorems for set-valued mappings in metric and banach spacesrdquo MathematicaJaponica vol 22 no 1 pp 99ndash112 1977
[6] J Geanakoplos ldquoNash and Walras equilibrium via BrouwerrdquoEconomic Theory vol 21 no 2-3 pp 585ndash603 2003
[7] S Kakutani ldquoA generalization of Brouwerrsquos fixed point theo-remrdquo Duke Mathematical Journal vol 8 pp 457ndash459 1941
[8] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 pp 286ndash295 1951
[9] J Nash ldquoEquilibrium points inN-person gamesrdquo Proceedings ofthe National Acadamy of Sciences of the United States of Americavol 36 pp 48-49 1950
[10] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969
[11] M Abbas and B E Rhoades ldquoFixed point theorems for two newclasses of multivalued mappingsrdquo Applied Mathematics Lettersvol 22 no 9 pp 1364ndash1368 2009
[12] S H Khan M Abbas and S Ali ldquoFixed point approximationof multivalued 120588-quasi-nonexpansive mappings in modularfunction spacesrdquo Journal of Nonlinear Sciences and Applicationsvol 10 no 6 pp 3168ndash3179 2017
[13] H Nakano Modular Semi-Ordered Spaces Maruzen Tokyo1950
[14] J Musielak and W Orlicz ldquoOn modular spacesrdquo Studia Mathe-matica vol 18 pp 591ndash597 1959
[15] M A Khamsi and W M Kozlowski ldquoFixed point theoryin modular function spacesrdquo Fixed Point Theory in ModularFunction Spaces pp 1ndash245 2015
[16] W M Kozlowski ldquoAdvancements in fixed point theory inmodular function spacesrdquo Arabian Journal of Mathematics vol1 no 4 pp 477ndash494 2012
[17] B A Bin Dehaish and W M Kozlowski ldquoFixed point iterationprocesses for asymptotic pointwise nonexpansive mapping inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2012 article no 118 2012
[18] F Golkarmanesh and S Saeidi ldquoAsymptotic pointwise contrac-tive type in modular function spacesrdquo Fixed Point Theory andApplications vol 2013 article no 101 2013
[19] M A Kutbi and A Latif ldquoFixed points of multivalued maps inmodular function spacesrdquo Fixed Point Theory and Applicationsvol 2009 Article ID 786357 2009
[20] M Abbas S Ali and P Kumam ldquoCommon fixed points in par-tially ordered modular function spacesrdquo Journal of Inequalitiesand Applications vol 2014 no 1 article no 78 2014
[21] M Ozturk M Abbas and E Girgin ldquoFixed points of mappingssatisfying contractive condition of integral type in modularspaces endowed with a graphrdquo Fixed Point Theory and Appli-cations vol 2014 no 1 article no 220 2014
[22] T Dominguez-Benavides M A Khamsi and S SamadildquoAsymptotically nonexpansive mappings in modular functionspacesrdquo Journal of Mathematical Analysis and Applications vol265 no 2 pp 249ndash263 2002
[23] S J Kilmer W M Kozlowski and G Lewicki ldquoSigma ordercontinuity and best approximation in 119871120588-spacesrdquo CommentMath Univ Carol vol 32 no 2 pp 2241ndash2250 1991
[24] J Schu ldquoWeak and strong convergence to fixed points of asymp-totically nonexpansive mappingsrdquo Bulletin of the AustralianMathematical Society vol 43 no 1 pp 153ndash159 1991
[25] A Razani V Rakocevic and Z Goodarzi ldquoNon-self mappingsin modular spaces and common fixed point theoremsrdquo CentralEuropean Journal of Mathematics vol 8 no 2 pp 357ndash3662010
[26] A Mbarki and I Hadi ldquoSome theorems on 120601-contractive map-pings in modular spacerdquo Rendicontidel Seminario MatematicoUniversita e Politecnico di Torino vol 72 no 3-4 pp 245ndash2542014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
