Approximation of fixed point of multivalued ρ-quasi-contractive mappings in modular function spaces

Abstract

The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.

Citation

Okeke, G.A. and Khan, S.H. (2020), "Approximation of fixed point of multivalued ρ-quasi-contractive mappings in modular function spaces", Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 75-93. https://doi.org/10.1016/j.ajmsc.2019.02.001

1. Introduction

It is known that there is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractive type operator (see, e.g. [4,17]). Hence, there is a practical and theoretical interest in approximating fixed points of several contractive type operators. 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. Khan et al. [13]). 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.

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, 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, should be considered as complementary to the fixed point theory in normed and metric spaces (see, e.g. [10]). Several authors have proved very interesting fixed points results in the framework of modular function spaces, (see, e.g. [10,11,15,18]).

It is our purpose in the present paper to extend the recent results of Okeke et al. [17] to the class of multivalued ρ-quasi-contractive mappings, which is known to be wider than the class of Zamfirescu operators (see, e.g. [5]) in modular function spaces. We approximate the fixed point of these classes of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces. Consequently, we define the concepts of ρ-T-stable, ρ-almost T-stable and ρ-summably almost T-stable in modular function spaces. We prove that some fixed point iterative processes are ρ-summably almost T-stable with respect to T, where T is a multivalued ρ-quasi-contractive mapping in modular function spaces.

2. Preliminaries

In this study, we let Ω denote a nonempty set and Σ 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 (for instance, P can be the class of sets of finite measure in a 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 supports from P. By M∞ we denote the space of all extended measurable functions, i.e., all functions f:Ω→[−∞,∞] such that there exists a sequence {gn}⊂ε, |gn|≤|f| and gn(ω)→f(ω) for each ω∈Ω.

Definition 2.1. Let ρ:M∞→[0,∞] be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if

1. ρ(0)=0;

2. ρ is monotone, i.e.,|f(ω)|≤|g(ω)| for any ω∈Ω implies ρ(f)≤ρ(g), where f,g∈M∞;

3. ρ is orthogonally subadditive, i.e., ρ(f1A∪B)≤ρ(f1A)+ρ(f1B) for any A,B∈Σ such that A∩B≠Ø, f∈M∞;

4. ρ has Fatou property, i.e.,|fn(ω)|↑|f(ω)| for all ω∈Ω implies ρ(fn)↑ρ(f), where f∈M∞;

5. ρ is order continuous in ε, i.e., gn∈ε and |gn(ω)|↓0 implies ρ(gn)↓0.

A set A∈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

M(Ω,Σ,P,ρ)={f∈M∞:|f(ω)|<∞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 [12].

Definition 2.2. Let ρ be a regular function pseudomodular;

1. (a)

   we say that ρ is a regular convex function modular if ρ(f)=0 implies f=0 ρ-a.e.

2. (b)

   we say that ρ 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. [10]) 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 on Ω is denoted by ℜ.

Definition 2.3. The convex function modular ρ defines the modular function space Lρ as

Lρ={f∈M;ρ(λf)→0 as λ→0}.

Generally, the modular ρ 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

‖f‖ρ=inf{α>0:ρ(fα)≤α}.

In the case ρ is convex modular,

‖f‖ρ=inf{α>0:ρ(fα)≤1}.

defines a norm on the modular space Lρ, and it is called the Luxemburg norm.

Lemma 2.1 ([10]). Let ρ∈ℜ. Defining Lρ0={f∈Lρ;ρ(f,.)is order continuous} and Eρ={f∈Lρ;λf∈Lρ0 for every λ>0}, we have

• (i)Lρ⊃Lρ0⊃Eρ;

• (ii)Eρ has the Lebesgue property, i.e., ρ(αf,Dk)→0, for α>0, f∈Eρ and Dk↓∅;

• (iii)Eρ is the closure of ε (in the sense of ‖.‖ρ ).

Definition 2.4. A nonzero regular convex function ρ is said to satisfy the Δ2-condition, if supn≥1ρ(2fn,Dk)→0 as k→∞ whenever {Dk} decreases to ø and supn≥1ρ(fn,Dk)→0 as k→∞.

If ρ is convex and satisfies Δ2-condition, then Lρ=Eρ.

The following uniform convexity type properties of ρ can be found in [6].

Definition 2.5. Let ρ be a nonzero regular convex function modular defined on Ω

(i) Let r>0, ϵ>0. Define

D1(r,ϵ)={(f,g):f,g∈Lρ,ρ(f)≤r,ρ(g)≤r,ρ(f−g)≥ϵr}.

Let

δ1(r,ϵ)=inf{1−1rρ(f+g2):(f,g)∈D1(r,ϵ)}  if D1(r,ϵ)≠Ø,

and δ1(r,ϵ)=1 if D1(r,ϵ)=Ø. We say that ρ satisfies (UC1) if for every r>0, ϵ>0, δ1(r,ϵ)>0. Observe that for every r>0, D1(r,ϵ)≠Ø, for ϵ>0 small enough.

(ii) We say that ρ 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) Let r>0, ϵ>0. Define

D2(r,ϵ)={(f,g):f,g∈Lρ,ρ(f)≤r,ρ(g)≤r,ρ(f−g2)≥ϵr}.

Let

δ2(r,ϵ)=inf{1−1rρ(f+g2):(f,g)∈D2(r,ϵ)}, if D2(r,ϵ)≠Ø,

and δ2(r,ϵ)=1 if D2(r,ϵ)=Ø. We say that ρ satisfies (UC2) if for every r>0, ϵ>0, δ2(r,ϵ)>0. Observe that for every r>0, D2(r,ϵ)≠Ø, for ϵ>0 small enough.

(iv) We say that ρ 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) We say that ρ is strictly convex (SC), if for every f,g∈Lρ such that ρ(f)=ρ(g) and ρ(f+g2)=ρ(f)+ρ(g)2, there holds f=g.

Proposition 2.1. ([10]). The following conditions characterize relationship between the above defined notions:

1. (i)

   (UUCi)⇒(UCi) for i=1,2.

2. (ii)

   δ1(r,ϵ)≤δ2(r,ϵ).

3. (iii)

   (UC1)⇒(UC2).

4. (iv)

   (UUC1)⇒(UUC2).

5. (v)

   If ρ is homogeneous (e.g. it is a norm), then all the conditions (UC1),(UC2),(UUC1), (UUC2) are equivalent and δ1(r,2ϵ)=δ1(1,2ϵ)=δ2(1,ϵ)=δ2(r,ϵ).

Definition 2.6. Let Lρ 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 that ρ-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. [14] defined ρ-distance from an f∈Lρ to a set D⊂Lρ as follows:

distρ(f,D)=inf{ρ(f−h):h∈D}.

Definition 2.7. A subset D⊂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

diamρ(D)=sup{ρ(f−g):f,g∈D}<∞.

The following famous result was proved by Zamfirescu [19]

Theorem 2.1. ([19]). Let (X,d) be a complete metric space, and let T:X→X be a mapping for which there exist real numbers a,b and c satisfying 0<a<1, 0<b,c<12 such that for each pair x,y∈X at least one of the following is true:

(z1) d(Tx,Ty)≤ad(x,y),

(z2) d(Tx,Ty)≤b[d(x,Tx)+d(y,Ty)],

(z3) d(Tx,Ty)≤c[d(x,Ty)+d(y,Tx)].

Then T has a unique fixed point p and the Picard iteration process {xn} defined by

xn+1=Txn, n=0,1,2,…

converges to p for any x0∈X.

Remark 2.1. Any operator T which satisfies the contractive conditions (z1)–(z3) of Theorem 2.1 is called a Zamfirescu operator (see e.g. [5]) and is denoted by Z.

The following class of quasi-contractive operators was introduced on a normed space E by Berinde [5]:

‖Tx−Ty‖≤δ‖x−y‖+L‖Tx−x‖,

for any x,y∈E, 0≤δ<1 and L≥0. He proved that this class is wider than the class of Zamfirescu operators.

A set D⊂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 Cρ(Lρ), that is,

Hρ(A,B)=max{supf∈Adistρ(f,B),supg∈Bdistρ(g,A)},A,B∈Cρ(Lρ).

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

  (2.1)Hρ(Tf,Tg)≤kρ(f−g), for all f,g∈D.

• (b) ρ-nonexpansive (see, e.g. Khan and Abbas [12]) if

  (2.2)Hρ(Tf,Tg)≤ρ(f−g), for  all f,g∈D.

• (c) ρ-quasi-nonexpansive mapping if

  (2.3)Hρ(Tf,p)≤ρ(f−p) for  all f∈D and p∈Fρ(T).

• (d) ρ-quasi-contractive mapping if

  (2.4)Hρ(Tf,Tg)≤δρ(f−g)+Lρ(Tf−f), for  all f,g∈D,0≤δ<1 and L≥0.

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.

Recently, Okeke et al. [17] approximated the fixed point of multivalued ρ-quasi-nonexpansive mappings using the Picard–Krasnoselskii hybrid iterative process. It is known that this iteration process converges faster than all of Picard, Mann, Krasnoselskii and Ishikawa iterative processes when applied to contraction mappings (see, Okeke and Abbas [16]). The following is the analogue of the Picard–Krasnoselskii hybrid iterative process in modular function spaces: Let T:D→Pρ(D) be a multivalued mapping and {fn}⊂D be defined by the following iteration process:

(2.5){fn+1∈PρT(gn)gn=(1−λ)fn+λPρT(vn), 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.

The following is the analogue of the S-iteration, introduced by Agarwal et al. [1] in modular function spaces.

(2.6){f0∈Dfn+1=(1−αn)un+αnvngn=(1−βn)fn+βnun,

where un∈PρT(fn), vn∈PρT(gn), the sequences {αn},{βn}⊂(0,1) are bounded away from both 0 and 1. It is known (see, e.g. [9]) that the S-iteration converges faster than the Mann iteration process and the Ishikawa iteration process for Zamfirescu operators.

Definition 2.8. 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.

Definition 2.9. ([12]). A multivalued mapping T:D→Cρ(D) is said to satisfy condition (I) if there exists a nondecreasing function l:[0,∞)→[0,∞) with l(0)=0, l(r)>0 for all r∈(0,∞) such that distρ(f,Tf)≥l(distρ(f,Fρ(T))) for all f∈D.

The following Lemma will be needed in this study.

Lemma 2.2. ([2]). Let ρ∈ℜ satisfy the Δ2-condition. Let {fn} and{gn} be two sequences in Lρ. Then

limn→∞ρ(gn)=0⇒limsupn→∞ρ(fn+gn)=limsupn→∞ρ(fn)

and

limn→∞ρ(gn)=0⇒liminfn→∞ρ(fn+gn)=liminfn→∞ρ(fn).

Lemma 2.3. ([6]). Let ρ satisfy(UUC1) and let{tk}⊂(0,1) be bounded away from 0 and 1. If there exists R>0 such that

limsupn→∞ρ(fn)≤R, limsupn→∞ρ(gn)≤R

and

limn→∞ρ(tnfn+(1−tn)gn)=R,

then limn→∞ρ(fn−gn)=0.

A function f∈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 2.4. ([12]). Let T:D→Pρ(D) be a multivalued mapping and

PρT(f)={g∈Tf:ρ(f−g)=distρ(f,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).

Lemma 2.5. ([3]). Let {an}n=0∞,{bn}n=0∞ be sequences of nonnegative numbers and 0≤q<1, such that

an+1≤qan+bn, for all n≥0.

• (i) If limn→∞bn=0, then limn→∞an=0.

• (ii) If ∑n=0∞bn<∞, then ∑n=0∞an<∞.

3. Approximation of fixed points in modular function spaces

We begin this section with the following proposition

Proposition 3.1. Let ρ satisfy (UUC1) and Δ2-condition. 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-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process (2.6), such that the sequences{αn}⊂(0,1) and{βn}⊂(0,1) are bounded away from both 0 and 1. Then the S-iterative process (2.6) is Fejér monotone with respect toFρ(T).

Proof. Let p∈Fρ(T). By Lemma 2.4, PρT(p)={p} and Fρ(T)=F(PρT). Using relation (2.4) and (2.6), we obtain the following estimate:

(3.1)ρ(fn+1−p)=ρ[(1−αn)un+αnvn−p]=ρ[(1−αn)(un−p)+αn(vn−p)].

The convexity of ρ implies

(3.2)ρ(fn+1−p)≤(1−αn)ρ(un−p)+αnρ(vn−p)≤(1−αn)Hρ(PρT(fn),PρT(p))+αnHρ(PρT(gn),PρT(p)).

From relation (2.4), with f=p, g=fn and also f=p, g=gn, then we obtain the following estimates from relation (3.2):

(3.3)Hρ(PρT(fn),PρT(p))≤δρ(fn−p).

(3.4)Hρ(PρT(gn),PρT(p))≤δρ(gn−p).

Using (3.3), (3.4) and the fact that 0≤δ<1 in (3.2), we have

(3.5)ρ(fn+1−p)≤(1−αn)δρ(fn−p)+αnδρ(gn−p)≤(1−αn)ρ(fn−p)+αnρ(gn−p).

Next, we have

(3.6)ρ(gn−p)=ρ[(1−βn)fn+βnun−p]=ρ[(1−βn)(fn−p)+βn(un−p)].

By convexity of ρ, we have

(3.7)ρ(gn−p)≤(1−βn)ρ(fn−p)+βnHρ(PρT(fn),PρT(p)).

Using (2.4) with f=p and g=fn and the fact that 0≤δ<1, relation (3.7) yields:

(3.8)ρ(gn−p)≤(1−βn)ρ(fn−p)+βnδρ(fn−p)≤(1−βn)ρ(fn−p)+βnρ(fn−p)=ρ(fn−p).

Using (3.8) in (3.5), we obtain: (3.9)

(3.9)ρ(fn+1−p)≤(1−αn)ρ(fn−p)+αnρ(fn−p)=ρ(fn−p).

Hence, the S-iteration (2.6) is Fejér monotone with respect to Fρ(T). The proof of Proposition 3.1 is completed. □

Next, we prove the following proposition.

Proposition 3.2. Let ρ satisfy the (UUC1) and Δ2-condition. Suppose that D is a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. LetT:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process (2.6), such that the sequences{αn}⊂(0,1) and{βn}⊂(0,1) are bounded away from both 0 and 1. Then

• (i) the sequence {fn} is bounded.

• (ii) for each f∈D,{ρ(fn−f)} converges.

Proof. Since {fn} is Fejér monotone as shown in Proposition 3.1. Using the fact that ρ satisfies the Δ2-condition, we can easily show (i) and (ii). This completes the proof of Proposition 3.2. □

Theorem 3.1. Let ρ satisfy (UUC1) and Δ2-condition. Let D 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-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process (2.6) and f0∈D, where the sequences {αn},{βn}⊂(0,1) are bounded away from both 0 and 1, satisfying ∑n=0∞αn=∞. Then {fn} converges strongly to the fixed point of T.

Proof. Let p∈Fρ(T). By Lemma 2.4, PρT(p)={p} and Fρ(T)=F(PρT). Using relation (2.4) and (2.6), we obtain the following estimate:

(3.10)ρ(fn+1−p)=ρ[(1−αn)un+αnvn−p]=ρ[(1−αn)(un−p)+αn(vn−p)].

The convexity of ρ implies (3.11)

(3.11)ρ(fn+1−p)≤(1−αn)ρ(un−p)+αnρ(vn−p)≤(1−αn)Hρ(PρT(fn),PρT(p))+αnHρ(PρT(gn),PρT(p)).

From relation (2.4), with f=p, g=fn and also f=p, g=gn, then we obtain the following estimates from relation (3.11):

(3.12)Hρ(PρT(fn),PρT(p))≤δρ(fn−p).

(3.13)Hρ(PρT(gn),PρT(p))≤δρ(gn−p).

Using (3.12) and (3.13) in (3.11), we have

(3.14)ρ(fn+1−p)≤(1−αn)δρ(fn−p)+αnδρ(gn−p).

Next, we have

(3.15)ρ(gn−p)=ρ[(1−βn)fn+βnun−p]=ρ[(1−βn)(fn−p)+βn(un−p)].

By convexity of ρ, we have

(3.16)ρ(gn−p)≤(1−βn)ρ(fn−p)+βnHρ(PρT(fn),PρT(p)).

Using (2.4) with f=p and g=fn, then relation (3.16) yields:

(3.17)ρ(gn−p)≤(1−βn)ρ(fn−p)+βnδρ(fn−p).

Using (3.17) in (3.14), we have

(3.18)ρ(fn+1−p)≤(1−αn)δρ(fn−p)+αnδ(1−βn(1−δ))ρ(fn−p)≤[1−αn(1−δ(1−βn(1−δ)))]ρ(fn−p).

Using (3.18), we inductively obtain

(3.19)ρ(fn+1−p)≤∏k=0n[1−αk(1−δ(1−βk(1−δ)))]ρ(f0−p),n=0,1,2,3,…

Using the fact that 0≤δ<1, {αn},{βn}⊂(0,1) are bounded away from both 0 and 1, satisfying ∑n=0∞αn=∞, relation (3.19) yields

(3.20)limn→∞∏k=0n[1−αk(1−δ(1−βk(1−δ)))]=0,

which implies that (3.19) becomes:

(3.21)limn→∞ρ(fn+1−p)=0.

Consequently, fn→p∈Fρ(T). The proof of Theorem 3.1 is completed. □

Theorem 3.2. Let ρ satisfy (UUC1) and Δ2-condition. 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-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process (2.6) and f0∈D, where the sequences {αn},{βn}⊂(0,1) are bounded away from both 0 and 1. Then limn→∞ρ(fn−p) exists for all p∈Fρ(T) and limn→∞distρ(fn,PρT(fn))=0.

Proof. Let p∈Fρ(T). By Lemma 2.4, PρT(p)={p} and Fρ(T)=F(PρT). Using relation (2.4) and (2.6), we obtain the following estimate:

(3.22)ρ(fn+1−p)=ρ[(1−αn)un+αnvn−p]=ρ[(1−αn)(un−p)+αn(vn−p)].

The convexity of ρ implies

(3.23)ρ(fn+1−p)≤(1−αn)ρ(un−p)+αnρ(vn−p)≤(1−αn)Hρ(PρT(fn),PρT(p))+αnHρ(PρT(gn),PρT(p)).

From relation (2.4), with f=p, g=fn and also f=p, g=gn, then we obtain the following estimates from relation (3.23):

(3.24)Hρ(PρT(fn),PρT(p))≤δρ(fn−p).

(3.25)Hρ(PρT(gn),PρT(p))≤δρ(gn−p).

Using (3.24), (3.25) and the fact that 0≤δ<1 in (3.23), we have

(3.26)ρ(fn+1−p)≤(1−αn)δρ(fn−p)+αnδρ(gn−p)≤(1−αn)ρ(fn−p)+αnρ(gn−p).

Next, we have

(3.27)ρ(gn−p)=ρ[(1−βn)fn+βnun−p]=ρ[(1−βn)(fn−p)+βn(un−p)].

By convexity of ρ, we have

(3.28)ρ(gn−p)≤(1−βn)ρ(fn−p)+βnHρ(PρT(fn),PρT(p)).

Using (3.25) with f=p and g=fn and the fact that 0≤δ<1, relation (3.28) yields:

(3.29)ρ(gn−p)≤(1−βn)ρ(fn−p)+βnδρ(fn−p)≤(1−βn)ρ(fn−p)+βnρ(fn−p)=ρ(fn−p).

Using (3.29) in (3.26), we obtain:

(3.30)ρ(fn+1−p)≤(1−αn)ρ(fn−p)+αnρ(fn−p)=ρ(fn−p).

This implies that limn→∞ρ(fn−p) exists for all p∈Fρ(T).

Let

(3.31)limn→∞ρ(fn−p)=K, where K≥0.

Now, we show that

(3.32)limn→∞distρ(fn,PρT(fn))=0.

Since distρ(fn,PρT(fn))≤ρ(fn−un), it suffices to show that

(3.33)limn→∞ρ(fn−un)=0.

Now,

(3.34)ρ(un−p)≤Hρ(PρT(fn),PρT(p))≤ρ(fn−p).

This implies that

(3.35)limsupn→∞ρ(un−p)≤limsupn→∞ρ(fn−p).

By (3.31), we have

(3.36)limn→∞supρ(un−p)≤K.

Also from (3.29), we have

(3.37)limsupn→∞ρ(gn−p)≤limsupn→∞ρ(fn−p),

so that

(3.38)limsupn→∞ρ(gn−p)≤K.

Moreover, the inequality

(3.39)ρ(vn−p)≤Hρ(PρT(gn),PρT(p))≤ρ(gn−p)≤ρ(fn−p),

this implies that

(3.40)limsupn→∞ρ(vn−p)≤limsupn→∞ρ(fn−p),

hence,

(3.41)limsupn→∞ρ(vn−p)≤K.

Now,

(3.42)limn→∞ρ(fn+1+p)=limn→∞ρ[(1−αn)un+αnvn−p]      =limn→∞ρ[(1−αn)(un−p)+αn(vn−p)]      =K.

Using (3.35), (3.41), (3.42) and Lemma 2.3, we have

(3.43)limn→∞ρ(vn−un)=0.

Now,

(3.44)ρ(fn+1−p)=ρ[(1−αn)un+αnvn−p]=ρ[(un−p)+αn(vn−un)].

Using Lemma 2.2 and (3.44), we have

(3.45)K=liminfn→∞ρ(fn+1−p)=liminfn→∞ρ[(un−p)+αn(vn−un)] =liminfn→∞ρ(un−p).

This means that

(3.46)K=liminfn→∞ρ(un−p).

Using (3.35) and (3.46), we have

(3.47)limn→∞ρ(un−p)=K.

Using (3.43), we have

(3.48)liminfn→∞ρ(un−p)=liminfn→∞ρ[(un−vn)+(vn−p)]=liminfn→∞ρ(vn−p).

But

(3.49)ρ(vn−p)≤Hρ(PρT(gn),PρT(p))≤ρ(gn−p).

Hence,

(3.50)liminfn→∞ρ(vn−p)≤liminfn→∞ρ(gn−p).

By (3.41), we have

(3.51)K≤liminfn→∞ρ(gn−p).

From (3.41) and (3.51), we have

(3.52)limn→∞ρ(gn−p)=K.

Since

(3.53)limn→∞ρ(gn−p)=limn→∞ρ[(1−βn)fn+βnun−p]       =limn→∞ρ[(1−βn)(fn−p)+βn(un−p)]=K.

Using (3.31), (3.35) and Lemma 2.3, we have

(3.54)limn→∞ρ(fn−un)=0.

Hence,

(3.55)limn→∞distρ(fn,PρT(fn))=0.

The proof of Theorem 3.2 is completed. □

Theorem 3.3. Let ρ satisfy (UUC1) and Δ2-condition. Let D be a nonempty ρ-compact, ρ-bounded and convex subset of Lρ. Let T:D→Pρ(D) be a multivalued mapping such that PρT is a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process (2.6) and f0∈D, where the sequences {αn},{βn}⊂(0,1) are bounded away from both 0 and 1. Then {fn} ρ -converges to a fixed point of T.

Proof. Using relation (2.4) with f=q, g=fnk and the fact that 0≤δ<1. Since D is ρ-compact, there exists a subsequence {fnk} of {fn} such that limn→∞(fnk–q)=0 for some q∈D. Next, we show that q is a fixed point of T. Suppose t is an arbitrary point in PρT(q) and f∈PρT(fnk). Observe that

(3.56)ρ(q−t3)=ρ(q−fnk3+fnk−f3+f−t3)    ≤13ρ(q−fnk)+13ρ(fnk−f)+13ρ(f−t)    ≤ρ(q−fnk)+distρ(fnk,PρT(fnk))+distρ(PρT(fnk),t)    ≤ρ(q−fnk)+distρ(fnk,PρT(fnk))+Hρ(PρT(fnk),PρT(q))    ≤ρ(q−fnk)+distρ(fnk,PρT(fnk))+δρ(q−fnk)    ≤ρ(q−fnk)+distρ(fnk,PρT(fnk))+ρ(q−fnk).

By Theorem 3.2, we obtain limn→∞distρ(fn,PρT(fn))=0. So that ρ(q−t3)=0. Therefore, q is a fixed point of PρT. By Lemma 2.4, we see that the set of fixed points of PρT is the same as that of T, hence, we have that {fn} ρ-converges to a fixed point of T. The proof of Theorem 3.3 is completed. □

Theorem 3.4. Let ρ satisfy (UUC1) and Δ2-condition. Let D be a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. Let T:D→Pρ(D) be a multivalued mapping satisfying condition (I) such that PρT is a ρ-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process (2.6) and f0∈D, where the sequences {αn},{βn}⊂(0,1) are bounded away from both 0 and 1. Then {fn} ρ-converges to a fixed point of T.

Proof. The proof of Theorem 3.4 is similar to the proof of Theorem 3 of Khan and Abbas [12]. □

4. ρ-Stability of fixed point iterations in modular function spaces

In this section, we define the concepts of ρ-T-stable, ρ-almost T-stable and ρ-summably almost T-stable in modular function spaces. We prove that some fixed point iterative processes are ρ-summably almost T-stable with respect to T, where T is a multivalued ρ-quasi-contractive mapping in modular function spaces.

Let ρ satisfy (UUC1) and D a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. Let T:D→Pρ(D) be a mapping with Fρ(T)≠Ø. Suppose that {fn}n=0∞ is a fixed point iterative process, i.e. a sequence {fn}n=0∞ defined by f0∈D and (4.1)

(4.1)fn+1=F(T,fn), n=0,1,2,3,…,

where F is a given function.

Several fixed point iterations exist in literature. For instance, Mann iteration, with F(T,fn)=(1−αn)fn+αnTfn, where {αn}⊂[0,1] such that {αn} is bounded away from both 0 and 1. The Ishikawa iteration, with F(T,fn)=(1−αn)fn+αnT[(1−βn)fn+βnTfn], such that {αn}n=0∞,{βn}n=0∞⊂[0,1] are both bounded away from both 0 and 1.

Let {fn}n=0∞ converge strongly to some p∈Fρ(T). In practice, we compute {fn}n=0∞ as follows:

• (i) Choose the initial guess (approximation) f0∈D;

• (ii) Compute f1=F(T,f0). However, as a result of various errors that occur during computations (numerical approximations of functions, rounding errors, derivatives, integration, etc.), we do not obtain the exact value of f1, but a different one, say , which is close enough to f1, this means that h1≈f1;

• (iii) Therefore, during the computation of f2=F(T,f1) we have

(4.2)f2=F(T,h1).

This means that instead of the theoretical value of f2, we expect another value h2 will be obtained, and h2 being close enough to f2, i.e. h2≈f2, and so on.

Continuing this process, we see that instead of the theoretical sequence {fn}n=0∞ defined by the fixed point iteration (4.1), we obtain practically an approximate sequence {hn}n=0∞.

The fixed point iteration (4.1) is considered to be numerically stable if and only if for hn close enough to fn at each stage, we have that the approximate {hn}n=0∞ still converges to the fixed point p of Fρ(T).

Next, we give the following definition, which is the analogue of the concept of T-stability introduced by Harder and Hicks (see, [7,8]) in modular function spaces.

Definition 4.1. Let ρ satisfy (UUC1) and D a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. Let T:D→Pρ(D) be a mapping with Fρ(T)≠Ø. Suppose that the fixed point iterative process (4.1) converges to a fixed point p of T. Let {hn}n=0∞ be an arbitrary sequence in D and set

(4.3)εn=ρ(hn+1−F(T,hn)), n=0,1,2,3,…

The fixed point iterative process (4.1) is said to be ρ-T-stable, or ρ-stable or ρ-stable with respect to T if and only if

(4.4)limn→∞εn=0⇒limn→∞hn=p.

Definition 4.2. Let ρ satisfy (UUC1) and D a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. Let T:D→Pρ(D) be a mapping with Fρ(T)≠Ø. Suppose that the fixed point iterative process (4.1) converges to a fixed point p of T. Let {hn}n=0∞ be an arbitrary sequence in D and let {εn}n=0∞ be defined by (4.3). The fixed point iterative process (4.1) is said to be ρ-almost T-stable or ρ-almost stable with respect to T if and only if

(4.5)∑n=0∞εn<∞⇒limx→∞hn=p.

Remark 4.1. It is clear from the definitions that any ρ-stable fixed point iteration {fn} is also ρ-almost stable.

A sharper concept of almost stability was introduced by Berinde [4]. He showed some almost stable fixed point iterations which are also summably almost stable with respect to some classes of contractive operators. We next define the analogue of this concept in modular function spaces.

Definition 4.3. Let ρ satisfy (UUC1) and D a nonempty ρ-closed, ρ-bounded and convex subset of Lρ. Let T:D→Pρ(D) be a mapping with Fρ(T)≠Ø. Suppose that the fixed point iterative process (4.1) converges to a fixed point p of T. Let {hn}n=0∞ be an arbitrary sequence in D and let {εn}n=0∞ be defined by (4.3). The fixed point iterative process (4.1) is said to be ρ-summably almost T-stable or ρ-summably almost stable with respect to T if and only if

(4.6)∑n=0∞εn<∞⇒∑n=0∞ρ(hn−p)<∞.

Remark 4.2. Clearly, any fixed point iteration {fn} that is ρ-almost stable is also ρ-summably almost stable, since

∑n=0∞ρ(hn−p)<∞⇒limn→∞hn=p.

However, we show that the converse is generally not true (see Example 4.1 below).

Example 4.1. Let the real number system ℝ be the space modulared as follows:

ρ(f)=|f|k, k≥1.

Let D={f∈Lρ:0≤f(x)≤1}. Let T:D→Pρ(D) be a multivalued mapping such that PρT is ρ-nonexpansive satisfying Tf=f. Let {fn} be the Picard iteration. Then {fn} is not ρ-summably almost T-stable.

Clearly, D is a nonempty ρ-compact, ρ-bounded and convex subset of Lρ=ℝ which satisfies UC1 condition. Moreover, ρ(f)=|f|k, k≥1 is homogeneous and it is of degree k, hence by Proposition 2.1 (UUC1) hold. Clearly, Fρ(T)=[0,1]. Suppose p=0. Take hn=1n, for each n≥1. Hence, limn→∞hn=0, we see that

εn=ρ(hn+1−F(T,hn))=distρ(1n+1,1n) =|1n+1−1n|k=|1n(n+1)|=1n(n+1).

Hence, ∑n=0∞εn<∞.

However, we have

∑n=0∞ρ(hn−p)=∑n=0∞distρ(1n,0)=∑n=0∞|1n−0|k=∑n=0∞|1n|=∑n=0∞1n=∞.

This means that the Picard iteration {fn} is not ρ-summably almost T-stable.

It is known that the Picard iteration is not T-stable and hence not almost T-stable (see, e.g. [4]).

Next, we prove the following results.

Theorem 4.1. Let ρ satisfy(UUC1) and Δ2-condition. Let D 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-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process as follows

(4.7){f0∈Dfn+1=(1−αn)un+αnvngn=(1−βn)fn+βnun,

where un∈PρT(fn),vn∈PρT(gn), the sequences{αn},{βn}⊂(0,1) are bounded away from both 0 and 1. Then {fn} is ρ -summably almost stable with respect to T.

Proof. Suppose p∈Fρ(T) and {hn} is an arbitrary sequence. Define

(4.8){sn=(1−βn)hn+βnwn,εn=ρ(hn+1−(1−αn)wn−αnzn),

where wn∈PρT(hn), zn∈PρT(sn), the sequences {αn},{βn}⊂(0,1) are bounded away from both 0 and 1.

Using the convexity of ρ, we have the following estimates:

(4.9)ρ(hn+1−p)=ρ(hn+1−(1−αn)wn−αnzn+(1−αn)(wn−p)+αn(zn−p))≤εn+(1−αn)ρ(wn−p)+αnρ(zn−p)≤εn+(1−αn)Hρ(PρT(hn),PρT(p))+αnHρ(PρT(sn),PρT(p)).

Using (4.9), relation (2.4) with f=p, g=hn and also f=p, g=sn, we have

(4.10)ρ(hn+1−p)≤εn+(1−αn)δρ(hn−p)+αnδρ(sn−p).

Next, by convexity of ρ we have

(4.11)ρ(sn−p)=ρ((1−βn)hn+βnwn−p)≤(1−βn)ρ(hn−p)+βnHρ(PρT(hn),PρT(p))≤(1−βn)ρ(hn−p)+βnδρ(hn−p)≤(1−βn)ρ(hn−p)+βnρ(hn−p)=ρ(hn−p).

Using (4.11) in (4.10), we obtain

(4.12)ρ(hn+1−p)≤εn+(1−αn)δρ(hn−p)+αnδρ(hn−p)=εn+δρ(hn−p).

By Lemma 2.5, we have that the two step S-iteration (4.7) is ρ-summably almost stable with respect to T. The proof of Theorem 4.1 is completed. □

Theorem 4.2. Let ρ satisfy(UUC1) and Δ2-condition. Let D 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-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the following iterative process

(4.13){f0∈Dfn+1∈PρT(un)

where un∈PρT(fn). Then {fn} is ρ -summably almost stable with respect to T.

Proof. Let p∈Fρ(T) and {hn} be an arbitrary sequence. Define

(4.14)εn=ρ(hn+1−mn),

where mn∈PρT(hn). Using (4.13), (4.14), relation (2.4) with f=p, g=hn and the convexity of ρ, we have the following estimate:

(4.15)ρ(hn+1−p)=ρ(hn+1−mn+mn−p)≤ρ(hn+1−mn)+ρ(mn−p)≤εn+Hρ(PρT(hn),PρT(p))≤εn+δρ(hn−p).

By Lemma 2.5, it follows that the fixed point iteration (4.13) is ρ-summably almost stable with respect to T. The proof of Theorem 4.2 is completed. □

Theorem 4.3. Let ρ satisfy (UUC1) and Δ2-condition. Let D 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-quasi-contractive mapping, satisfying contractive condition (2.4) and Fρ(T)≠Ø. Let {fn}⊂D be defined by the two step S-iterative process as follows

(4.16){f0∈Dfn+1∈∑i=0kαiuni,n≥0,αi≥0,α1>0,∑i=0kαi=1.

where uni∈PρTi(fn). Then {fn} is ρ-summably almost stable with respect to T.

Proof. Let p∈Fρ(T) and {hn} be any given sequence in D and define

(4.17)εn=ρ(hn+1−∑i=0kαizni),

where zni∈PρTi(hn). Using (4.16), (4.17), relation (2.4) with f=p, g=hn and the convexity of ρ, we have the following estimate:

(4.18)ρ(hn+1−p)=ρ(hn+1−∑i=0kαizni+∑i=0kαizni−p)≤ρ(hn+1−∑i=0kαizni)+ρ(∑i=0kαizni−p)≤εn+ρ(∑i=0kαizni−p)≤εn+Hρ(∑i=0kαiPρTi(hn),PρT(p))≤εn+∑i=0kαiHρ(PρTi(hn),PρT(p))≤εn+(∑i=0kαiδi)ρ(hn−p)=εn+qρ(hn−p),

where q=∑i=0kαiδi<1. Hence, by Lemma 2.5 it follows that the fixed point iteration (4.16) is ρ-summably almost stable with respect to T. The proof of Theorem 4.3 is completed. □