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

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.

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

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;

• (a)

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

• (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

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

In the case $ρ$ is convex modular,

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 $Eρ={f∈Lρ;λf∈Lρ0 for everyλ>0}$, 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

Let

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

Let

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:

• (i)

  $(UUCi)⇒(UCi)$ for $i=1,2$⁠.

• (ii)

  $δ1(r,ϵ)≤δ2(r,ϵ)$⁠.

• (iii)

  $(UC1)⇒(UC2)$⁠.

• (iv)

  $(UUC1)⇒(UUC2)$⁠.

• (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:

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

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

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]:

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,

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

  $Hρ(Tf,Tg)≤kρ(f−g),forallf,g∈D.$

  (2.1)

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

  $Hρ(Tf,Tg)≤ρ(f−g),for allf,g∈D.$

  (2.2)

• (c) $ρ$-quasi-nonexpansive mapping if

  $Hρ(Tf,p)≤ρ(f−p)for allf∈Dandp∈Fρ(T).$

  (2.3)

• (d) $ρ$-quasi-contractive mapping if

  $Hρ(Tf,Tg)≤δρ(f−g)+Lρ(Tf−f),for allf,g∈D,0≤δ<1andL≥0.$

  (2.4)

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:

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.

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

and

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

and

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

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

• (i) If $limn→∞bn=0$, then $limn→∞an=0$.

• (ii) If $∑n=0∞bn<∞$, then $∑n=0∞an<∞$.

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 to$Fρ(T)$.

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:

The convexity of $ρ$ implies

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):

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

Next, we have

By convexity of $ρ$⁠, we have

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

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

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 $Lρ$. 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

• (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:

The convexity of $ρ$ implies (3.11)

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):

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

Next, we have

By convexity of $ρ$⁠, we have

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

Using (3.17) in (3.14), we have

Using (3.18), we inductively obtain

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

which implies that (3.19) becomes:

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:

The convexity of $ρ$ implies

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):