Iterative Approximation of Fixed Point of Multivalued <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 9.41541 12.3952" width="9.41541pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-239" d="M516 277C516 378 460 448 376 448C335 448 288 434 240 401C158 345 127 277 107 159C53 -161 38 -221 23 -247L28 -261C43 -260 95 -250 114 -226C122 -216 129 -199 146 -75C152 -29 157 0 164 31C182 9 206 -12 237 -12C267 -12 325 8 376 46C451 102 516 179 516 277ZM432 259C432 134 366 40 259 40C236 40 204 48 171 76L195 203C219 327 253 400 326 400C398 400 432 336 432 259Z"/></g></svg>-Quasi-Nonexpansive Mappings in Modular Function Spaces with Applications

Okeke, Godwin Amechi; Bishop, Sheila Amina; Khan, Safeer Hussain

doi:https://doi.org/10.1155/2018/1785702

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Research Article | Open Access

Volume 2018 | Article ID 1785702 | https://doi.org/10.1155/2018/1785702

Iterative Approximation of Fixed Point of Multivalued -Quasi-Nonexpansive Mappings in Modular Function Spaces with Applications

Godwin Amechi Okeke,¹Sheila Amina Bishop,²and Safeer Hussain Khan³

Academic Editor: Adrian Petrusel

Received04 Jul 2017

Accepted12 Dec 2017

Published24 Jan 2018

Abstract

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 approximating fixed points of multivalued nonlinear mappings in modular function spaces. The purpose of this paper is 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, 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]). The motivation for such studies stems mainly from the usefulness of fixed point theory results in real-world applications, as in Game Theory and Market Economy and in other areas of mathematical sciences such as in Nonsmooth Differential Equations.

The 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 equivalents of norm or metric notions and also equipped with almost everywhere convergence and convergence in submeasure. 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 results that can be proved only using the apparatus of modular function spaces. Hence, fixed point theory results in modular function spaces, in this perspective, which should be considered as complementary to the fixed point theory in normed and metric spaces (see, e.g., [15, 16]).

Several authors have proved very interesting fixed points results in the framework of modular function spaces (see, e.g., [15, 17–19]). Abbas et al. [20] proved the existence and uniqueness of common fixed point of certain nonlinear mappings satisfying some contractive conditions in partially ordered spaces. Öztürk et al. [21] established some interesting fixed point results of nonlinear mappings satisfying integral type contractive conditions in the framework of modular spaces endowed with a graph. Recently, Khan and Abbas initiated the study of approximating fixed points of multivalued nonlinear mappings in the framework of modular function spaces [1]. A very recent work was given by Khan et al. [12]. They approximated the fixed points of -quasi-nonexpansive multivalued mappings in modular function spaces using a three-step iterative process, where satisfies the so-called -condition. Their results improve and generalize the results of Khan and Abbas [1].

Motivated by the above results, we prove some convergence and stability results for -quasi-nonexpansive mappings using the Picard-Krasnoselskii hybrid iterative process. Our results improve, extend, and generalize several known results, including the recent results of Khan et al. [12], in the sense that the restriction that satisfies the so-called - condition in [12] is removed in the present paper. Moreover, it is known (see, [2]) that the Picard-Krasnoselskii hybrid iterative process converges faster than all of Picard, Mann, Krasnoselskii, and Ishikawa iterative processes. Furthermore, we apply our results in solving certain initial value problem.

2. Preliminaries

In this study, we let denote a nonempty set and let be a nontrivial -algebra of subsets of Let be a -ring of subsets of , such that for any and Let us assume that there exists an increasing sequence of sets such that (e.g., can be the class of sets of finite measure in -finite measure space). By , we denote the characteristic function of the set in By we denote the linear space of all simple functions with support from By we denote the space of all extended measurable functions, that is, all functions such that there exists a sequence , , and for each

Definition 1. Let be a nontrivial, convex, and even function. One says that is a regular convex function pseudomodular if(1);(2) is monotone, that is, for any implies , where ;(3) is orthogonally subadditive, that is, for any such that , ;(4) has Fatou property, that is, for all implies , where ;(5) is order continuous in , that is, and implies

A set is said to be -null if for every A property is said to hold -almost everywhere -a.e.) if the set does not is -null. As usual, we identify any pair of measurable sets whose symmetric difference is -null as well as any pair of measurable functions differing only on a -null set. With this in mind we define where is actually an equivalence class of functions equal -a.e. rather than an individual function. Where no confusion exists, we shall write instead of

The following definitions were given in [1].

Definition 2. Let be a regular function pseudomodular.(a)One says that is a regular convex function modular if implies -a.e.(b)One says that is a regular convex function semimodular if for every implies -a.e. It is known (see, e.g., [15]) that satisfies the following properties:(1) iff -a.e.(2) for every scalar with and (3) if , , and is called a convex modular if, in addition, the following property is satisfied:(3′) if , , and The class of all nonzero regular convex function modulars on is denoted by

Definition 3. The convex function modular defines the modular function space as

Generally, the modular is not subadditive and therefore does not behave as a norm or a distance. However, the modular space can be equipped with an -norm defined by In the case that is convex modular, defines a norm on the modular space , and it is called the Luxemburg norm.

Lemma 4 (see [15]). Let Defining and , one has the following:(i)(ii) has the Lebesgue property; that is, , for , , and (iii) is the closure of (in the sense of ).

The following uniform convexity type properties of can be found in [17].

Definition 5. Let be a nonzero regular convex function modular defined on (i)Let , Define Let and if One says that satisfies if for every , , Observe that for every , , for small enough.(ii)One says that satisfies if for every , , there exists depending only on and such that for any (iii)Let , Define Let and if one says that satisfies if for every , , Observe that for every , , for small enough.(iv)One says that satisfies if for every , , there exists depending only on and such that for any (v)One says that is strictly convex , if for every such that and , there holds

Proposition 6 (see [15]). The following conditions characterize relationship between the above defined notions:(i) for (ii)(iii)(iv)(v)If is homogeneous (e.g., it is a norm), then all the conditions , , , and are equivalent and

Definition 7. A nonzero regular convex function modular is said to satisfy the -condition, if as whenever decreases to and as

Definition 8. A function modular is said to satisfy the -type condition, if there exists such that, for any , one has

In general, -condition and -type condition are not equivalent, even though it is easy to see that -type condition implies -condition on the modular space ; see [22].

Definition 9. Let be a modular space. The sequence is called(1)-convergent to if as ;(2)-Cauchy, if as and

Observe that -convergence does not imply -Cauchy since does not satisfy the triangle inequality. In fact, one can easily show that this will happen if and only if satisfies the -condition.

Kilmer et al. [23] defined -distance from an to a set as follows:

Definition 10. A subset is called(1)-closed if the -limit of a -convergent sequence of always belongs to ;(2)-a.e. closed if the -a.e. limit of a -a.e. convergent sequence of always belongs to ;(3)-compact if every sequence in has a -convergent subsequence in ;(4)-a.e. compact if every sequence in has a -a.e. convergent subsequence in ;(5)-bounded if

It is known that the norm and modular convergence are also the same when we deal with the -type condition (see, e.g., [15]).

A set is called -proximinal if for each there exists an element such that We shall denote the family of nonempty -bounded -proximinal subsets of by , the family of nonempty -closed -bounded subsets of by , and the family of -compact subsets of by Let be the -Hausdorff distance on ; that is, A multivalued map is said to be(a)-contraction mapping if there exists a constant such that (b)-nonexpansive (see, e.g., Khan and Abbas [1]) if (c)-quasi-nonexpansive mapping if A sequence is called bounded away from 0 if there exists such that for every Similarly, is called bounded away from 1 if there exists such that for every

Okeke and Abbas [2] introduced the Picard-Krasnoselskii hybrid iterative process. The authors proved that this new hybrid iterative process converges faster than all of Picard, Mann, Krasnoselskii, and Ishikawa iterative processes when applied to contraction mappings. We now give the analogue of the Picard-Krasnoselskii hybrid iterative process in modular function spaces as follows: let be a multivalued mapping and be defined by the following iteration process:where and It is our purpose in the present paper to prove some new fixed point theorems using this iteration process in the framework of modular function spaces.

Definition 11. A sequence is said to be Fejér monotone with respect to subset of if , for all of ,

The following Lemma will be needed in this study.

Lemma 12 (see [22]). Let be a function modular and and be two sequences in Then

Lemma 13 (see [17]). Let satisfy and let be bounded away from 0 and 1. If there exists such thatand then

The above lemma is an analogue of a famous lemma due to Schu [24] in Banach spaces.

A function is called a fixed point of if The set of all fixed points of will be denoted by

Lemma 14 (see [1]). Let be a multivalued mapping and Then the following are equivalent:(1), that is, (2), that is, for each (3), that is, Further , where denotes the set of fixed points of

The following examples were presented by Razani et al. [25].

Example 15. Let be a norm space; then is a modular. But the converse is not true.

Example 16. Let be a norm space. For any , is a modular on

3. Iterative Approximation of Fixed Points in Modular Function Spaces

We begin this section with the following proposition.

Proposition 17. Let satisfy and let be a nonempty -closed, -bounded, and convex subset of Let be a multivalued mapping such that is a -quasi-nonexpansive mapping. Then the Picard-Krasnoselskii hybrid iterative process (15) is Fejér monotone with respect to

Proof. Suppose By Lemma 13, and Using (15), we have the following estimate: Next, we have By convexity of , we have Using (21) in (19), we haveHence, the Picard-Krasnoselskii hybrid iterative process (15) is Fejér monotone with respect to This completes the proof of Proposition 17.

Next, we prove the following proposition.

Proposition 18. Let satisfy and let be a nonempty -closed, -bounded, and convex subset of Let be a multivalued mapping such that is a -quasi-nonexpansive mapping. Let be the Picard-Krasnoselskii hybrid iterative process (15); then(i)the sequence is bounded;(ii)for each , converges.

Proof. Since is Fejér monotone as shown in Proposition 17, we can easily show (i) and (ii). This completes the proof of Proposition 18.

Theorem 19. Let satisfy and let be a nonempty -closed, -bounded, and convex subset of Let be a multivalued mapping such that is a -quasi-nonexpansive mapping. Suppose that Let be the Picard-Krasnoselskii hybrid iterative process (15). Then exists for all and

Proof. Suppose By Lemma 13, and Using (15), we have the following estimate: Next, we have By convexity of , we have Using (25) in (23), we have This shows that exists for all
Suppose that where
We next prove that Since , it suffices to show that Now, and this implies that and, by (27), we have Using (25), we have and, hence, we have Next, we have and this implies that and, hence, we have Using (23) and (24), we have Moreover, Using Lemma 4 and (38), we have This means that Using (27) and (40), we have Using (27), (31), (37), and Lemma 12, we have Hence, The proof of Theorem 19 is completed.

Next, we prove the following theorem.

Theorem 20. Let be a -closed, -bounded, and convex subset of a -complete modular space and be a multivalued mapping such that is a -contraction mapping and Then has a unique fixed point. Moreover, the Picard-Krasnoselskii hybrid iterative process (15) converges to this fixed point.

Proof. Suppose By Lemma 13, and Using (15), we have the following estimate: Next, we have By convexity of , we have Using (46) in (44), we have This shows that exists for all Using a similar approach as in the proof of Theorem 19, we see that
Next, we show that is a -Cauchy sequence. Since , we proceed by contradiction. Hence, there exists and two sequences of natural numbers such that For all integer , let be the least integer exceeding which satisfy (48); then So, we have This is a contradiction. Hence, is a -Cauchy sequence. Therefore, there exists such that as
Next, we have Clearly, Hence, Therefore,
Next, we prove the uniqueness of Suppose that is another fixed point of , and then we haveHence, The proof of Theorem 20 is completed.

Next, we give the following example.

Example 21. Let be a vector space and be an application defined as follows: We see that is not a norm. However, it is a modular since the function is convex. Consider as the closed interval in which is -closed, -bounded, and -complete, since is continuous. Then the mapping is a -contraction mapping with Therefore, by Theorem 20, it has a unique fixed point in , which is

4. Stability Results

We begin this section by defining the concept of -stable and almost -stable of an iterative process in modular function spaces. Moreover, we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15).

Definition 22. Let be a nonempty convex subset of a modular function space and be an operator. Assume that and defines an iteration scheme which produces a sequence Suppose, furthermore, that converges strongly to Let be any bounded sequence in and put (1)The iteration scheme defined by is said to be -stable on if implies that (2)The iteration scheme defined by is said to be almost -stable on if implies that

It is easy to show that an iteration process which is -stable on is almost -stable on

Next, we provide the following numerical example to show that Picard-Krasnoselskii hybrid iterative process (15) is -stable.

Example 23. Let be a vector space and be an application defined as follows Let be the closed interval in which is -closed, -bounded, and -complete. Let be a multivalued mapping such that is a -contraction mapping satisfying contractive condition We now show that Picard-Krasnoselskii hybrid iterative process (15) is -stable and hence almost -stable with and Suppose that is an arbitrary sequence in Take Then Put and we have Hence, Therefore, Picard-Krasnoselskii hybrid iterative process (15) is -stable. Clearly, (15) is almost -stable.

Next, we prove the following stability results.

Theorem 24. Let be a -closed, -bounded, and convex subset of a -complete modular space and be a multivalued mapping such that is a -contraction mapping and Then Picard-Krasnoselskii hybrid iterative process (15) is -stable.

Proof. Suppose , and define Let be the unique fixed point of We want to show that if and only if Suppose that converges to Using (15) and the convexity of , we have Hence, Conversely, suppose that Then we have Since , it follows from relation (61) that The proof of Theorem 24 is completed.

Remark 25. Theorem 24 generalizes the results of Mbarki and Hadi [26] to multivalued mappings in modular function spaces.

5. Applications to Differential Equations

In this section, we apply our results to differential equations. The results of this section follow similar applications in [15]. Let , and we consider the following initial value problem for an unknown function , where where and are fixed and is such that is -quasi-nonexpansive mapping. The following notations will be used in this section. For we define For any function , where , and any , we define We also denote for any , a subdivision of the interval

The following lemma which is needed to prove our results in this section can be found in [15].

Lemma 26. Let be separable. Let be two Bochner-integrable -bounded functions, where Then for every one has

We now state our results for this section.

Theorem 27. Let be separable. Let be a nonempty, convex, -bounded, -closed set with the Vitali property. Let be a multivalued mapping such that is a -quasi-nonexpansive mapping. Let one fix and and define the sequence of functions by the following inductive formula: Then for every there exists such that and the function defined by (68) is a solution of initial value problem (62). Moreover,

Proof. Since is -quasi-nonexpansive mapping, the proof of Theorem 27 follows the proof of ([15], Theorem ).

Next, we obtain the following corollaries as a consequence of Theorem 27.

Corollary 28. Let be separable. Let be a nonempty, convex, -bounded, -closed set with the Vitali property. Let be a multivalued mapping such that is a -nonexpansive mapping. Let one fix and and define the sequence of functions by the following inductive formula: Then for every there exists such that and the function defined by (71) is a solution of initial value problem (62). Moreover,

Corollary 29. Let be separable. Let be a nonempty, convex, -bounded, -closed set with the Vitali property. Let be a multivalued mapping such that is a -contraction mapping. Let one fix and and define the sequence of functions by the following inductive formula: Then for every there exists such that and the function defined by (74) is a solution of initial value problem (62). Moreover,

Remark 30. Corollary 28 generalizes the results of Khamsi and Kozlowski ([15], Theorem ) to a multivalued mapping.

Conflicts of Interest

The authors declare that they do not have any conflicts of interest.

Authors’ Contributions

All authors contributed equally to writing this research paper. Each author read and approved the final manuscript.

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Copyright © 2018 Godwin Amechi Okeke et al. This 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.

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