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

Recently, Khan andAbbas initiated the study of approximating fixed points ofmultivalued nonlinearmappings inmodular 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 ρ-quasi-nonexpansive mappings using the PicardKrasnoselskii hybrid iterative process. We apply our results to solving certain initial value problem.


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 wellknown 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 quasinonexpansive 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
A set  ∈ Σ is said to be -null if (1  ) = 0 for every  ∈ .A property () is said to hold -almost everywhere (a.e.) if the set { ∈ Ω : () does not hold} 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  ∈ M(Ω, Σ, P, ) is actually an equivalence class of functions equal -a.e.rather than an individual function.
Where no confusion exists, we shall write M instead of M(Ω, Σ, P, ).
The following definitions were given in [1].
Definition 2. Let  be a regular function pseudomodular.
(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 Ω.
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 Δ 2 -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 It is known that the norm and modular convergence are also the same when we deal with the Δ 2 -type condition (see, e.g., [15]).
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 V  ∈    (  ) and 0 <  < 1.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.
The following Lemma will be needed in this study.
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,  ∈ .
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.

Iterative Approximation of Fixed Points in Modular Function Spaces
We begin this section with the following proposition.
Proposition 17.Let  satisfy (1) and let  be a nonempty -closed, -bounded, and convex subset of   .Let  :  →   () be a multivalued mapping such that    is a -quasinonexpansive mapping.Then the Picard-Krasnoselskii hybrid iterative process (15) is Fejér monotone with respect to   ().
Next, we prove the following proposition.
Proof.Since {  } is Fejér monotone as shown in Proposition 17, we can easily show (i) and (ii).This completes the proof of Proposition 18.
We next prove that lim Using Lemma 4 and (38), we have This means that Using ( 27) and ( 40 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   () ̸ = 0. Then  has a unique fixed point.Moreover, the Picard-Krasnoselskii hybrid iterative process (15) converges to this fixed point.
Using a similar approach as in the proof of Theorem 19, we see that lim →∞ (  − ) = 0.
Next, we prove the uniqueness of .Suppose that  is another fixed point of , and then we have Hence,  = .The proof of Theorem 20 is completed.
Next, we give the following example.
Example 21.Let   = [0, ∞) 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  →  2 is convex.Consider  = [0, 1] as the closed interval in [0, ∞) which is -closed, -bounded, and -complete, since  is continuous.Then the mapping is a -contraction mapping with  = 1/2.Therefore, by Theorem 20, it has a unique fixed point in , which is   () = {0}.

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).
It is easy to show that an iteration process {  } ∞ =1 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.
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   () ̸ = 0. Then Picard-Krasnoselskii hybrid iterative process (15) is -stable.
Remark 25.Theorem 24 generalizes the results of Mbarki and Hadi [26] to multivalued mappings in modular function spaces.

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  ∈ R, and we consider the following initial value problem for an unknown function  : [0, ] → , where  ∈   .
where  ∈  and  > 0 are fixed and  :  →  is such that    is -quasi-nonexpansive mapping.The following notations will be used in this section.For  > 0 we define for any  = { 0 , . . .,   }, a subdivision of the interval [0, ].
The following lemma which is needed to prove our results in this section can be found in [15].Proof.Since    is -quasi-nonexpansive mapping, the proof of Theorem 27 follows the proof of ( [15], Theorem 5.28).
Next, we obtain the following corollaries as a consequence of Theorem 27.  (72)