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

Authors: Godwin Amechi Okeke; Sheila Amina Bishop; Safeer Hussain Khan
Journal: Journal of Function Spaces
 (2018)

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## 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 forρ-quasi-nonexpansive mappings using the Picard-Krasnoselskii hybrid iterative process. We apply our results to solving certain initial value problem.


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## Body

## 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 ρ-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 inGame Theory and Market Economy and in other areas of mathematical sciences such as inNonsmooth 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 F-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 ρ-quasi-nonexpansive multivalued mappings in modular function spaces using a three-step iterative process, where ρ satisfies the so-called Δ2-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 forρ-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 ρ satisfies the so-called Δ2- 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 letΩ denote a nonempty set and let Σ be a nontrivial σ-algebra of subsets of Ω. Let P be a δ-ring of subsets of Ω, such that E∩A∈P for any E∈P and A∈Σ. Let us assume that there exists an increasing sequence of sets Kn∈P such that Ω=⋃Kn (e.g., P can be the class of sets of finite measure in σ-finite measure space). By 1A, we denote the characteristic function of the set A in Ω. By ε we denote the linear space of all simple functions with support from P. By M∞ we denote the space of all extended measurable functions, that is, all functions f:Ω→[-∞,∞] such that there exists a sequence {gn}⊂ε, |gn|≤|f|, and gn(ω)→f(ω) for each ω∈Ω.Definition 1.
Letρ:M∞→[0,∞] be a nontrivial, convex, and even function. One says that ρ is a regular convex function pseudomodular if(1)
ρ(0)=0;(2)
ρ is monotone, that is, |f(ω)|≤|g(ω)| for any ω∈Ω implies ρ(f)≤ρ(g), where f,g∈M∞;(3)
ρ is orthogonally subadditive, that is, ρ(f1A∪B)≤ρ(f1A)+ρ(f1B) for any A,B∈Σ such that A∩B≠∅, f∈M∞;(4)
ρ has Fatou property, that is, |fn(ω)|↑|f(ω)| for all ω∈Ω implies ρ(fn)↑ρ(f), where f∈M∞;(5)
ρ is order continuous in ε, that is, gn∈ε and |gn(ω)|↓0 implies ρ(gn)↓0.A setA∈Σ is said to be ρ-null if ρ(g1A)=0 for every g∈ε. A property p(ω) is said to hold ρ-almost everywhere (ρ-a.e.) if the set ω∈Ω:p(ω) does not hold is ρ-null. As usual, we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind we define (1)MΩ,Σ,P,ρ=f∈M∞:fω<∞ρ-a.e.,where f∈M(Ω,Σ,P,ρ) is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists, we shall write M instead of M(Ω,Σ,P,ρ).The following definitions were given in [1].Definition 2.
Letρ be a regular function pseudomodular.(a)
One says thatρ is a regular convex function modular if ρ(f)=0 implies f=0  ρ-a.e.(b)
One says thatρ is a regular convex function semimodular if ρ(αf)=0 for every α>0 implies f=0  ρ-a.e.
It is known (see, e.g., [15]) that ρ satisfies the following properties:(1)
ρ(0)=0 iff f=0  ρ-a.e.(2)
ρ(αf)=ρ(f) for every scalar α with |α|=1 and f∈M.(3)
ρ(αf+βg)≤ρ(f)+ρ(g) if α+β=1, α,β≥0, and f,g∈M.ρ is called a convex modular if, in addition, the following property is satisfied:(3′)
ρ(αf+βg)≤αρ(f)+βρ(g) if α+β=1, α,β≥0, and f,g∈M.The class of all nonzero regular convex function modulars onΩ is denoted by R.Definition 3.
The convex function modularρ defines the modular function space Lρ as (2)Lρ=f∈M;ρλf⟶0asλ⟶0.Generally, the modularρ is not subadditive and therefore does not behave as a norm or a distance. However, the modular space Lρ can be equipped with an F-norm defined by (3)fρ=inf⁡α>0:ρfα≤α.In the case that ρ is convex modular, (4)fρ=inf⁡α>0:ρfα≤1defines a norm on the modular space Lρ, and it is called the Luxemburg norm.Lemma 4 (see [15]).
Letρ∈R. Defining Lρ0={f∈Lρ;ρ(f,·)isordercontinuous} and Eρ={f∈Lρ;λf∈Lρ0for  everyλ>0}, one has the following:(i)
Lρ⊃Lρ0⊃Eρ.(ii)
Eρ has the Lebesgue property; that is, ρ(αf,Dk)→0, for α>0, f∈Eρ, and Dk↓∅.(iii)
Eρ is the closure of ε (in the sense of ·ρ).The following uniform convexity type properties ofρ can be found in [17].Definition 5.
Letρ be a nonzero regular convex function modular defined on Ω.(i)
Letr>0, ϵ>0. Define (5)D1r,ϵ=f,g:f,g∈Lρ,ρf≤r,ρg≤r,ρf-g≥ϵr.Let(6)δ1r,ϵ=inf⁡1-1rρf+g2:f,g∈D1r,ϵifD1r,ϵ≠∅,andδ1(r,ϵ)=1 if D1(r,ϵ)=∅. One says that ρ satisfies (UC1) if for every r>0, ϵ>0, δ1(r,ϵ)>0. Observe that for every r>0, D1(r,ϵ)≠∅, for ϵ>0 small enough.(ii)
One says thatρ satisfies (UUC1) if for every s≥0, ϵ>0, there exists η1(s,ϵ)>0 depending only on s and ϵ such that δ1(r,ϵ)>η1(s,ϵ)>0 for any r>s.(iii)
Letr>0, ϵ>0. Define (7)D2r,ϵ=f,g:f,g∈Lρ,ρf≤r,ρg≤r,ρf-g2≥ϵr.Let(8)δ2r,ϵ=inf⁡1-1rρf+g2:f,g∈D2r,ϵ,ifD2r,ϵ≠∅,andδ2(r,ϵ)=1 if D2(r,ϵ)=∅. one says that ρ satisfies (UC2) if for every r>0, ϵ>0, δ2(r,ϵ)>0. Observe that for every r>0, D2(r,ϵ)≠∅, for ϵ>0 small enough.(iv)
One says thatρ satisfies (UUC2) if for every s≥0, ϵ>0, there exists η2(s,ϵ)>0 depending only on s and ϵ such that δ2(r,ϵ)>η2(s,ϵ)>0 for any r>s.(v)
One says thatρ is strictly convex (SC), if for every f,g∈Lρ such that ρ(f)=ρ(g) and ρf+g/2=ρ(f)+ρ(g)/2, there holds f=g.Proposition 6 (see [15]).
The following conditions characterize relationship between the above defined notions:(i)
(UUCi)⇒(UCi) for i=1,2.(ii)
δ1(r,ϵ)≤δ2(r,ϵ).(iii)
(UC1)⇒(UC2).(iv)
UUC1⇒UUC2.(v)
Ifρ is homogeneous (e.g., it is a norm), then all the conditions (UC1), (UC2), (UUC1), and (UUC2) are equivalent and δ1(r,2ϵ)=δ1(1,2ϵ)=δ2(1,ϵ)=δ2(r,ϵ).Definition 7.
A nonzero regular convex function modularρ is said to satisfy the Δ2-condition, if supn≥1ρ(2fn,Dk)→0 as k→∞ whenever {Dk} decreases to ∅ and supn≥1ρ(fn,Dk)→0 as k→∞.Definition 8.
A function modular is said to satisfy theΔ2-type condition, if there exists K>0 such that, for any f∈Lρ, one has ρ(2f)≤Kρ(f).In general,Δ2-condition and Δ2-type condition are not equivalent, even though it is easy to see that Δ2-type condition implies Δ2-condition on the modular space Lρ; see [22].Definition 9.
LetLρ be a modular space. The sequence {fn}⊂Lρ is called(1)
ρ-convergent to f∈Lρ if ρ(fn-f)→0 as n→∞;(2)
ρ-Cauchy, if ρ(fn-fm)→0 as n and m→∞.Observe thatρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangle inequality. In fact, one can easily show that this will happen if and only if ρ satisfies the Δ2-condition.Kilmer et al. [23] defined ρ-distance from an f∈Lρ to a set D⊂Lρ as follows: (9)distρf,D=inf⁡ρf-h:h∈D.Definition 10.
A subsetD⊂Lρ is called(1)
ρ-closed if the ρ-limit of a ρ-convergent sequence of D always belongs to D;(2)
ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent sequence of D always belongs to D;(3)
ρ-compact if every sequence in D has a ρ-convergent subsequence in D;(4)
ρ-a.e. compact if every sequence in D has a ρ-a.e. convergent subsequence in D;(5)
ρ-bounded if (10)diamρD=sup⁡ρf-g:f,g∈D<∞.It is known that the norm and modular convergence are also the same when we deal with theΔ2-type condition (see, e.g., [15]).A setD⊂Lρ is called ρ-proximinal if for each f∈Lρ there exists an element g∈D such that ρ(f-g)=distρ(f,D). We shall denote the family of nonempty ρ-bounded ρ-proximinal subsets of D by Pρ(D), the family of nonempty ρ-closed ρ-bounded subsets of D by Cρ(D), and the family of ρ-compact subsets of D by Kρ(D). Let Hρ(·,·) be the ρ-Hausdorff distance on CρLρ; that is, (11)HρA,B=max⁡supf∈Adistρf,B,supg∈Bdistρg,A,A,B∈CρLρ. A multivalued map T:D→Cρ(Lρ) is said to be(a)
ρ-contraction mapping if there exists a constant k∈[0,1) such that (12)HρTf,Tg≤kρf-g,∀f,g∈D,(b)
ρ-nonexpansive (see, e.g., Khan and Abbas [1]) if (13)HρTf,Tg≤ρf-g,∀f,g∈D,(c)
ρ-quasi-nonexpansive mapping if (14)HρTf,p≤ρf-p∀f∈D,p∈FρT.A sequence {tn}⊂(0,1) is called bounded away from 0 if there exists a>0 such that tn≥a for every n∈N. Similarly, {tn}⊂(0,1) is called bounded away from 1 if there exists b<1 such that tn≤b for every n∈N.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 T:D→Pρ(D) be a multivalued mapping and {fn}⊂D be defined by the following iteration process:(15)fn+1∈PρTgngn=1-λfn+λPρTvn,n∈N,where vn∈PρT(fn) and 0<λ<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.Definition 11.
A sequence{fn}⊂D is said to be Fejér monotone with respect to subset Pρ(D) of D if ρ(fn+1-p)≤ρ(fn-p), for all p∈PρT(D) of D, n∈N.The following Lemma will be needed in this study.Lemma 12 (see [22]).
Letρ be a function modular and fn and gn be two sequences in Xρ. Then (16)limn→∞⁡ρgn=0⟹limsupn→∞⁡ρfn+gn=limsupn→∞⁡ρfn,limn→∞ρgn=0⟹liminfn→∞ρfn+gn=liminfn→∞⁡ρfn.Lemma 13 (see [17]).
Letρ satisfy (UUC1) and let {tk}⊂(0,1) be bounded away from 0 and 1. If there exists R>0 such that(17)limsupn→∞⁡ρfn≤R,limsupn→∞⁡ρgn≤R,limn→∞⁡ρtnfn+1-tngn=R,and then limn→∞⁡ρ(fn-gn)=0.The above lemma is an analogue of a famous lemma due to Schu [24] in Banach spaces.A functionf∈Lρ is called a fixed point of T:Lρ→Pρ(D) if f∈Tf. The set of all fixed points of T will be denoted by Fρ(T).Lemma 14 (see [1]).
LetT:D→Pρ(D) be a multivalued mapping and (18)PρTf=g∈Tf:ρf-g=distρf,Tf. Then the following are equivalent:(1)
f∈Fρ(T), that is, f∈Tf.(2)
PρT(f)={f}, that is, f=g for each g∈PρT(f).(3)
f∈F(PρT(f)), that is, f∈PρT(f). Further Fρ(T)=F(PρT(f)), where F(PρT(f)) denotes the set of fixed points of PρT(f).The following examples were presented by Razani et al. [25].Example 15.
Let(X,·) be a norm space; then · is a modular. But the converse is not true.Example 16.
Let(X,·) be a norm space. For any k≥1, ·k is a modular on X.


## 3. Iterative Approximation of Fixed Points in Modular Function Spaces

We begin this section with the following proposition.Proposition 17.
Letρ satisfy (UUC1) and let D be a nonempty ρ-closed, ρ-bounded, and convex subset of Lρ. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-quasi-nonexpansive mapping. Then the Picard-Krasnoselskii hybrid iterative process (15) is Fejér monotone with respect to Fρ(T).Proof.
Supposep∈Fρ(T). By Lemma 13, PρT(p)={p} and Fρ(T)=F(PρT). Using (15), we have the following estimate: (19)ρfn+1-p≤HρPρTgn,PρTp≤ρgn-p.Next, we have (20)ρgn-p=ρ1-λfn+λPρTvn-p.By convexity of ρ, we have (21)ρgn-p≤1-λρfn-p+λHρPρTfn,PρTp≤1-λρfn-p+λρfn-p=ρfn-p.Using (21) in (19), we have(22)ρfn+1-p≤ρfn-p.Hence, the Picard-Krasnoselskii hybrid iterative process (15) is Fejér monotone with respect to Fρ(T). This completes the proof of Proposition 17.Next, we prove the following proposition.Proposition 18.
Letρ satisfy (UUC1) and let D be a nonempty ρ-closed, ρ-bounded, and convex subset of Lρ. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-quasi-nonexpansive mapping. Let {fn} be the Picard-Krasnoselskii hybrid iterative process (15); then(i)
the sequence{fn} is bounded;(ii)
for eachf∈D, {ρ(fn-f)} converges.Proof.
Since{fn} is Fejér monotone as shown in Proposition 17, we can easily show (i) and (ii). This completes the proof of Proposition 18.Theorem 19.
Letρ satisfy (UUC1) and let D be a nonempty ρ-closed, ρ-bounded, and convex subset of Lρ. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-quasi-nonexpansive mapping. Suppose that Fρ(T)≠∅. Let {fn}⊂D be the Picard-Krasnoselskii hybrid iterative process (15). Then limn→∞ρ(fn-p) exists for all p∈Fρ(T) and limn→∞distρ(fn,PρT(fn))=0.Proof.
Supposep∈Fρ(T). By Lemma 13, PρT(p)={p} and Fρ(T)=F(PρT). Using (15), we have the following estimate: (23)ρfn+1-p≤HρPρTgn,PρTp≤ρgn-p.Next, we have (24)ρgn-p=ρ1-λfn+λPρTvn-p.By convexity of ρ, we have (25)ρgn-p≤1-λρfn-p+λHρPρTfn,PρTp≤1-λρfn-p+λρfn-p=ρfn-p.Using (25) in (23), we have (26)ρfn+1-p≤ρfn-p.This shows that limn→∞ρ(fn-p) exists for all p∈Fρ(T).
Suppose that(27)limn→∞⁡ρfn-p=L,where L≥0.
We next prove thatlimn→∞distρ(fn,PρT(fn))=0. Since distρ(fn,PρT(fn))≤ρ(fn-vn), it suffices to show that (28)limn→∞⁡ρfn-vn=0.Now, (29)ρvn-p≤HρPρTfn,PρTp≤ρfn-p,and this implies that (30)limsupn→∞⁡ρvn-p≤limsupn→∞⁡ρfn-p,and, by (27), we have (31)limsupn→∞⁡ρvn-p≤L.Using (25), we have (32)limsupn→∞⁡ρgn-p≤limsupn→∞⁡ρfn-p,and, hence, we have (33)limsupn→∞⁡ρgn-p≤L.Next, we have (34)HρPρTgn,PρTp≤ρgn-p≤ρfn-p,and this implies that (35)limsupn→∞⁡ρgn-p≤limsupn→∞⁡ρfn-p,and, hence, we have (36)limsupn→∞⁡ρgn-p≤L.Using (23) and (24), we have (37)limn→∞ρfn+1-p=limn→∞ρ1-λfn+λPρTvn-p≤limn→∞1-λρfn-p+λHρPρTfn,PρTp≤limn→∞1-λρfn-p+λρfn-p=limn→∞ρfn-p≤L.Moreover, (38)ρfn+1-p≤ρ1-λfn+λvn-p=ρfn-p+λvn-fn.Using Lemma 4 and (38), we have (39)L=liminfn→∞⁡ρfn+1-p=liminfn→∞⁡ρfn-p+λvn-fn=liminfn→∞⁡ρfn-p.This means that (40)L=liminfn→∞⁡ρfn-p.Using (27) and (40), we have (41)limn→∞⁡ρfn-p=L.Using (27), (31), (37), and Lemma 12, we have (42)limn→∞⁡ρfn-vn=0.Hence, (43)limn→∞distρfn,PρTfn=0.The proof of Theorem 19 is completed.Next, we prove the following theorem.Theorem 20.
LetD be a ρ-closed, ρ-bounded, and convex subset of a ρ-complete modular space Lρ and T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-contraction mapping and Fρ(T)≠∅. Then T has a unique fixed point. Moreover, the Picard-Krasnoselskii hybrid iterative process (15) converges to this fixed point.Proof.
Supposep∈Fρ(T). By Lemma 13, PρT(p)={p} and Fρ(T)=F(PρT). Using (15), we have the following estimate: (44)ρfn+1-p≤HρPρTgn,PρTp≤kρgn-p≤ρgn-p.Next, we have (45)ρgn-p=ρ1-λfn+λPρTvn-p.By convexity of ρ, we have (46)ρgn-p≤1-λρfn-p+λHρPρTfn,PρTp≤1-λρfn-p+λkρfn-p≤1-λρfn-p+λρfn-p=ρfn-p.Using (46) in (44), we have (47)ρfn+1-p≤ρfn-p.This shows that limn→∞ρ(fn-p) exists for all p∈Fρ(T). Using a similar approach as in the proof of Theorem 19, we see that limn→∞ρ(fn-p)=0.
Next, we show that{fn} is a ρ-Cauchy sequence. Since limn→∞(fn-p)=0, we proceed by contradiction. Hence, there exists ϵ>0 and two sequences of natural numbers {m(i)},{n(i)} such that (48)ni>mi≥i,ρfni-fmi>ϵ.For all integer i, let n(i) be the least integer exceeding m(i) which satisfy (48); then (49)ρfni-fmi>ϵ,ρfni-1-fmi≤ϵ.So, we have (50)ϵ<ρfni-fmi≤ρfni-p2+ρp-fmi2≤12ρfni-p+12ρp-fmi≤ρfni-p+ρp-fmi⟶0asn⟶∞.This is a contradiction. Hence, {fn} is a ρ-Cauchy sequence. Therefore, there exists p∈D such that fn→0 as n→∞.
Next, we haveTp=p. Clearly, (51)ρp-Tp2≤ρp-fn2+ρfn-Tp2≤12ρp-fn+12ρfn-Tp≤ρp-fn+ρfn-Tp=ρp-fn+ρfn-p⟶0asn⟶∞.Hence, ρp-Tp/2=0. Therefore, p=Tp.
Next, we prove the uniqueness ofp. Suppose that q is another fixed point of T, and then we have(52)ρp-q2≤ρp-fn2+ρfn-q2≤12ρp-fn+12ρfn-q≤ρp-fn+ρfn-q⟶0asn⟶∞.Hence, p=q. The proof of Theorem 20 is completed.Next, we give the following example.Example 21.
LetLρ=[0,∞) be a vector space and ρ be an application defined as follows: (53)ρ:Lρ⟶Lρt⟶t2. We see that ρ is not a norm. However, it is a modular since the function t→t2 is convex. Consider D=[0,1] as the closed interval in [0,∞) which is ρ-closed, ρ-bounded, and ρ-complete, since ρ is continuous. Then the mapping (54)T:D⟶PρDt⟶t2is a ρ-contraction mapping with k=1/2. Therefore, by Theorem 20, it has a unique fixed point in D, which is Fρ(T)={0}.


## 4. Stability Results

We begin this section by defining the concept ofT-stable and almost T-stable of an iterative process in modular function spaces. Moreover, we prove some stability results for Picard-Krasnoselskii hybrid iterative process (15).Definition 22.
LetD be a nonempty convex subset of a modular function space Lρ and T:D→D be an operator. Assume that x1∈D and xn+1=f(T,xn) defines an iteration scheme which produces a sequence xnn=1∞⊂D. Suppose, furthermore, that {xn}n=1∞ converges strongly to x∗∈Fρ(T)≠∅. Let ynn=1∞ be any bounded sequence in D and put εn=ρ(yn+1-f(T,yn)).(1)
The iteration scheme{xn}n=1∞ defined by xn+1=f(T,xn) is said to be T-stable on D if limn→∞εn=0 implies that limn→∞yn=x∗.(2)
The iteration schemexnn=1∞ defined by xn+1=f(T,xn) is said to be almost T-stable on D if ∑n=1∞εn<∞ implies that limn→∞yn=x∗.It is easy to show that an iteration processxnn=1∞ which is T-stable on C is almost T-stable on D.Next, we provide the following numerical example to show that Picard-Krasnoselskii hybrid iterative process (15) is T-stable.Example 23.
LetLρ=[0,∞) be a vector space and ρ be an application defined as follows(55)ρ:Lρ⟶Lρt⟶t. Let D=[0,1] be the closed interval in [0,∞) which is ρ-closed, ρ-bounded, and ρ-complete. Let T:[0,1]→[0,1] be a multivalued mapping such that PρT is a ρ-contraction mapping satisfying contractive condition Tx=x/2. We now show that Picard-Krasnoselskii hybrid iterative process (15) is T-stable and hence almost T-stable with k=1/2 and Fρ(T)={0}. Suppose that {yn}=1/n is an arbitrary sequence in Lρ. Take λ=1/2. Then limn→∞yn=0. Put (56)εn=ρyn+1-fT,yn=distρPρTyn+1,PρTgn≤HρPρTyn+1,PρTgn,
and we have(57)εn=distρPρTyn+1,PρTgn≤HρPρTyn+1,PρTgn≤ρyn+1-gn=yn+1-1-λyn-λyn=1n+1-12n-12n=1n+1-1n. Hence, (58)limn→∞εn=0.Therefore, Picard-Krasnoselskii hybrid iterative process (15) is T-stable. Clearly, (15) is almost T-stable.Next, we prove the following stability results.Theorem 24.
LetD be a ρ-closed, ρ-bounded, and convex subset of a ρ-complete modular space Lρ and T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-contraction mapping and Fρ(T)≠∅. Then Picard-Krasnoselskii hybrid iterative process (15) is T-stable.Proof.
Suppose{yn}⊂Lρ, and define εn=ρ(yn+1-f(T,yn)). Let p be the unique fixed point of T. We want to show that limn→∞yn=p if and only if limn→∞εn=0. Suppose that {yn} converges to p. Using (15) and the convexity of ρ, we have (59)εn=distρPρTyn+1,PρTgn≤HρPρTyn+1,PρTgn≤ρyn+1-gn≤ρyn+1-1-λyn-λyn=ρyn+1-yn≤ρyn+1-p+ρp-yn.Hence, (60)limn→∞εn=0.
Conversely, suppose thatlimn→∞εn=0. Then we have (61)εn=distρPρTyn+1,PρTgn≤HρPρTyn+1,PρTgn≤ρyn+1-gn≤ρyn+1-1-λyn-λyn=ρyn+1-yn≤ρyn+1-p+ρp-yn. Since limn→∞εn=0, it follows from relation (61) that limn→∞yn=p. The proof of Theorem 24 is completed.Remark 25.
Theorem24 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 ρ∈R, and we consider the following initial value problem for an unknown function u:[0,A]→C, where C∈Eρ.(62)u0=fu′t+I-Tut=0, where f∈C and A>0 are fixed and T:C→C is such that PρT is ρ-quasi-nonexpansive mapping. The following notations will be used in this section. For t>0 we define (63)Kt=1-e-t=∫0tes-tds.For any function ν:[0,A]→Lρ, where A>0, and any t∈[0,A], we define (64)Sνt=∫0tes-tνsds.We also denote (65)Sτνt=∑i=0n-1ti+1-tieti-tνti,for any τ={t0,…,tn}, a subdivision of the interval [0,A].The following lemma which is needed to prove our results in this section can be found in [15].Lemma 26.
Letρ∈R be separable. Let x,y:[0,A]→Lρ be two Bochner-integrable ·ρ-bounded functions, where A>0. Then for every t∈[0,A] one has (66)ρe-tyt+∫0tes-txsds≤e-tρyt+Ktsups∈0,tρxs.We now state our results for this section.Theorem 27.
Letρ∈R be separable. Let D⊂Eρ be a nonempty, convex, ρ-bounded, ρ-closed set with the Vitali property. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-quasi-nonexpansive mapping. Let one fix f∈C and A>0 and define the sequence of functions un:[0,A]→C by the following inductive formula: (67)u0t=fun+1t=e-tf+∫0tes-tTunsds. Then for every t∈[0,A] there exists u(t)∈C such that (68)ρunt-ut⟶0and the function u:[0,A]→C defined by (68) is a solution of initial value problem (62). Moreover, (69)ρf-unt≤Kn+1AδρC.Proof.
SincePρT is ρ-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 Theorem27.Corollary 28.
Letρ∈R be separable. Let D⊂Eρ be a nonempty, convex, ρ-bounded, ρ-closed set with the Vitali property. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-nonexpansive mapping. Let one fix f∈C and A>0 and define the sequence of functions un:[0,A]→C by the following inductive formula: (70)u0t=fun+1t=e-tf+∫0tes-tTunsds. Then for every t∈[0,A] there exists u(t)∈C such that (71)ρunt-ut⟶0and the function u:[0,A]→C defined by (71) is a solution of initial value problem (62). Moreover, (72)ρf-unt≤Kn+1AδρC.Corollary 29.
Letρ∈R be separable. Let D⊂Eρ be a nonempty, convex, ρ-bounded, ρ-closed set with the Vitali property. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-contraction mapping. Let one fix f∈C and A>0 and define the sequence of functions un:[0,A]→C by the following inductive formula: (73)u0t=fun+1t=e-tf+∫0tes-tTunsds. Then for every t∈[0,A] there exists u(t)∈C such that (74)ρunt-ut⟶0and the function u:[0,A]→C defined by (74) is a solution of initial value problem (62). Moreover, (75)ρf-unt≤Kn+1AδρC.Remark 30.
Corollary28 generalizes the results of Khamsi and Kozlowski ([15], Theorem 5.28) to a multivalued mapping.



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