# Asymptotic pointwise contractive type in modular function spaces

- Farhan Golkarmanesh
^{1}Email author and - Shahram Saeidi
^{2}

**2013**:101

https://doi.org/10.1186/1687-1812-2013-101

© Golkarmanesh and Saeidi; licensee Springer 2013

**Received: **9 September 2012

**Accepted: **11 March 2013

**Published: **17 April 2013

## Abstract

In this paper, we introduce asymptotic pointwise contractive type conditions in modular function spaces and present fixed point results for mappings under such conditions.

**MSC:**47H09, 47H10, 54H25.

## Keywords

*ρ*-contraction typemodular function space

## 1 Introduction

where ${\alpha}_{n}\to \alpha $ pointwise on *M*. Moreover, Kirk and Xu [2] proved that if *C* is a weakly compact convex subset of a Banach space *E* and $T:C\to C$ an asymptotic pointwise contraction, then *T* has a unique fixed point $v\in C$, and for each $x\in C$, the sequence of Picard iterates $\{{T}^{n}x\}$ converges in norm to *v*.

*x*in

*M*,

where ${\alpha}_{n}\to \alpha $ pointwise on *M*.

It is easy to see that an asymptotic pointwise contraction is of asymptotic pointwise contraction type, but the converse is not true [3]. The following result was proved in [3].

**Theorem 1.1** [3]*Let* *C* *be a nonempty weakly compact subset of a Banach space* *E*, *and let* $T:C\to C$ *be a mapping of weak asymptotic pointwise contraction type*. *Then* *T* *has a unique fixed point* $v\in C$ *and*, *for each* $x\in C$, *the sequence of Picard iterates* $\{{T}^{n}x\}$ *converges in norm to* *v*.

On the other hand, Khamsi and Kozlowski [4] studied the concept of asymptotic pointwise contractions in modular function spaces.

In this paper, motivated by Khamsi and Kozlowski [4, 5] and Saeidi [3], we study the notion of asymptotic pointwise contraction type in a modular function space. Moreover, we present fixed results which extend the earlier results in [3, 4].

## 2 Preliminaries

Let Ω be a nonempty set, and let Σ be a nontrivial *σ*-algebra of subsets of Ω. Let $\mathcal{P}$ be a *δ*-ring of subsets of Ω such that $E\cap A\in \mathcal{P}$ for any $E\in \mathcal{P}$ and $A\in \mathrm{\Sigma}$. Let us assume that there exists an increasing sequence of sets ${K}_{n}\in \mathcal{P}$ such that $\mathrm{\Omega}=\bigcup {K}_{n}$. By *ξ* we denote the linear space of all simple functions with supports from $\mathcal{P}$. By ${\mathcal{M}}_{\mathrm{\infty}}$ we denote the space of all extended measurable function, *i.e.*, all function $f:\mathrm{\Omega}\to [-\mathrm{\infty},+\mathrm{\infty}]$ such that there exists a sequence $\{{g}_{n}\}\in \xi $, $|{g}_{n}|\le |f|$ and ${g}_{n}(\omega )\to f(\omega )$ for all $\omega \in \mathrm{\Omega}$.

By ${1}_{A}$ we denote the characteristic function of the set *A*.

**Definition 2.1** [6]

*ρ*is a regular convex function pseudomodular if:

- (a)
$\rho (0)=0$;

- (b)
*ρ*is monotone,*i.e.*, $|f(\omega )|\le |g(\omega )|$ for all $\omega \in \mathrm{\Omega}$ implies $\rho (f)\le \rho (g)$, where $f,g\in {\mathcal{M}}_{\mathrm{\infty}}$; - (c)
*ρ*is orthogonally subadditive,*i.e.*, $\rho (f{1}_{A\cup B})\le \rho (f{1}_{A})+\rho (f{1}_{B})$ for any $A,B\in \mathrm{\Sigma}$ such that $A\cap B\ne \mathrm{\varnothing}$, $f\in {\mathcal{M}}_{\mathrm{\infty}}$; - (d)
*ρ*has the Fatou property,*i.e.*, $|{f}_{n}(\omega )|\uparrow |f(\omega )|$ for all $\omega \in \mathrm{\Omega}$ implies $\rho ({f}_{n})\uparrow \rho (f)$, where $f\in {\mathcal{M}}_{\mathrm{\infty}}$; - (e)
*ρ*is order continuous in*ξ*,*i.e.*, ${g}_{n}\in \xi $ and $|{g}_{n}(\omega )|\downarrow 0$ implies $\rho ({g}_{n})\downarrow 0$.

*ρ*-null if $\rho (g{1}_{A})=0$ for every $g\in \xi $. We say that a property holds

*ρ*-almost everywhere if the exceptional set 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 each $f\in \mathcal{M}(\mathrm{\Omega},\mathrm{\Sigma},\mathcal{P},\rho )$ is actually an equivalence class of functions equal *ρ*-a.e. rather than an individual function. Where no confusion exists, we write ℳ instead of $\mathcal{M}(\mathrm{\Omega},\mathrm{\Sigma},\mathcal{P},\rho )$.

*ρ*be a regular function pseudomodular;

- (a)
we say that

*ρ*is a regular convex function semimodular if $\rho (\alpha f)=0$ for every $\alpha >0$ implies $f=0$*ρ*-a.e.; - (b)
we say that

*ρ*is a regular convex function modular if $\rho (f)=0$ implies $f=0$*ρ*-a.e.

The class of all nonzero regular convex function modulars on Ω is denoted by ℜ.

*ρ*be a convex function modular.

- (a)A modular function space is the vector space ${L}_{\rho}(\mathrm{\Omega},\mathrm{\Sigma})$, or briefly ${L}_{\rho}$, defined by${L}_{\rho}=\{f\in \mathcal{M}:\rho (\lambda f)\to 0\text{as}\lambda \to 0\}.$
- (b)The following formula defines a norm in ${L}_{\rho}$ (frequently called Luxemburg norm):${\parallel f\parallel}_{\rho}=inf\{\alpha >0;\rho (f/\alpha )\le 1\}.$

In the following theorem, we recall some of the properties of modular function spaces that will be used later on in this paper.

*Let*$\rho \in \mathfrak{R}$.

*Defining*${L}_{\rho}^{0}=\{f\in {L}_{\rho};\rho (f,)\mathit{\text{is order continuous}}\}$

*and*${E}_{\rho}=\{f\in {L}_{\rho};\lambda f\in {L}_{\rho}^{0}\mathit{\text{for every}}\lambda >0\}$,

*we have*

- (i)
${L}_{\rho}\supset {L}_{\rho}^{0}\supset {E}_{\rho}$;

- (ii)
${E}_{\rho}$

*has the Lebesgue property*,*i*.*e*., $\rho (\alpha f,{D}_{k})\to 0$,*for*$\alpha >0$, $f\in {E}_{\rho}$*and*${D}_{k}\downarrow \mathrm{\varnothing}$; - (iii)
${E}_{\rho}$

*is the closure of**ξ*(*in the sense of*${\parallel \parallel}_{\rho}$).

- (a)
We say that $\{{f}_{n}\}$ is

*ρ*-convergent to*f*and write ${f}_{n}\to f(\rho )$ if and only if $\rho ({f}_{n}-f)\to 0$. - (b)
A sequence $\{{f}_{n}\}$ where ${f}_{n}\in {L}_{\rho}$ is called

*ρ*-Cauchy if $\rho ({f}_{n}-{f}_{m})\to 0$ as $m,n\to \mathrm{\infty}$. - (c)
A set $C\subset {L}_{\rho}$ is called

*ρ*-closed if for any sequence $\{{f}_{n}\}$ in*C*, the convergence ${f}_{n}\to f(\rho )$ implies that*f*belongs to*C*. - (d)
A set $C\subset {L}_{\rho}$ is called

*ρ*-bounded if $sup\{\rho (f-g);f\in C,g\in C\}<\mathrm{\infty}$. - (e)
For a set $C\subset {L}_{\rho}$, the mapping $T:C\to C$ is called

*ρ*-continuous if ${f}_{n}\to f(\rho )$, then $T({f}_{n})\to T(f)(\rho )$. - (f)
A set $C\subset {L}_{\rho}$ is called

*ρ*-a.e. closed if for any sequence $\{{f}_{n}\}$ in*C*which*ρ*-a.e. converges to some*f*, then we must have $f\in C$. - (g)
A set $C\subset {L}_{\rho}$ is called

*ρ*-a.e. compact if for any sequence $\{{f}_{n}\}$ in*C*, there exists a subsequence $\{{f}_{{n}_{k}}\}$ which*ρ*-a.e. converges to some $f\in C$. - (h)Let $f\in {L}_{\rho}$ and $C\subset {L}_{\rho}$. The
*ρ*-distance between*f*and*C*is defined as${d}_{\rho}(f,C)=inf\{\rho (f-g);g\in C\}.$

Let us recall that *ρ*-convergence does not necessarily imply *ρ*-Cauchy condition. Also, ${f}_{n}\to f$ does not imply in general $\lambda {f}_{n}\to \lambda f$, $\lambda >1$.

**Definition 2.6** [4]

We say that ${L}_{\rho}$ has the property $(R)$ if and only if every nonincreasing sequence $\{{C}_{n}\}$ of nonempty, *ρ*-bounded, *ρ*-closed, convex subsets of ${L}_{\rho}$ has nonempty intersection.

**Definition 2.7** [4]

*ρ*is uniformly continuous if for every $\u03f5>0$ and $L>0$, there exists $\delta >0$ such that

**Definition 2.8** [4]

A function $\lambda :C\to [0,\mathrm{\infty}]$, where $C\subset {L}_{\rho}$ is nonempty and *ρ*-closed, is called *ρ*-lower semicontinuous if for any $\alpha >0$, the set ${C}_{\alpha}=\{f\in C;\lambda (f)\le \alpha \}$ is *ρ*-closed.

*ρ*-lower semicontinuity is equivalent to the condition

The following result plays an important role in the proof of the main results.

**Lemma 2.9** [4]

*Assume that*$\rho \in \mathfrak{R}$

*has the property*$(R)$.

*Let*$C\subset {L}_{\rho}$

*be nonempty*,

*convex*,

*ρ*-

*closed and*

*ρ*-

*bounded*.

*If*$\phi :C\to [0,\mathrm{\infty})$

*is a*

*ρ*-

*lower semicontinuous convex function*,

*then there exists*${x}_{0}\in C$

*such that*

Let us recall the notion of *ρ*-type.

**Definition 2.10** [4]

*ρ*-bounded. A function $\tau :C\to [0,\mathrm{\infty})$ is called a $(\rho )$-type (or shortly a type) if there exists a sequence $\{{y}_{m}\}$ of elements of

*C*such that for any $z\in C$, the following holds:

**Lemma 2.11** [4]

*Let* $\rho \in \mathfrak{R}$ *be uniformly continuous*. *Let* $C\subset {L}_{\rho}$ *be nonempty*, *convex*, *ρ*-*closed and* *ρ*-*bounded*. *Then any* *ρ*-*type* $\tau :C\to [0,\mathrm{\infty})$ *is* *ρ*-*lower semicontinuous in C*.

## 3 Asymptotic pointwise contractive type conditions in modular function spaces

**Definition 3.1** [4]

*ρ*-closed. A mapping $T:C\to C$ is called an asymptotic pointwise mapping if there exists a sequence of mappings ${\alpha}_{n}:C\to [0,1]$ such that

- (a)
If $\{{\alpha}_{n}\}$ converges pointwise to $\alpha :C\to [0,1)$, then

*T*is called asymptotic pointwise*ρ*-contraction. - (b)
If ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le 1$ for any $f\in C$, then

*T*is called asymptotic pointwise nonexpansive. - (c)
If ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le k$ for any $f\in C$ with $0<k<1$, then

*T*is called strongly asymptotic pointwise*ρ*-contraction.

Khamsi and Kozlowski proved the following results in modular function spaces.

**Theorem 3.2** [4]

*Let* $C\subset {L}_{\rho}$ *be nonempty*, *ρ*-*closed and* *ρ*-*bounded*. *Let* $T:C\to C$ *be an asymptotic pointwise* *ρ*-*contraction*. *Then* *T* *has at most one fixed point in* *C*. *Moreover*, *if* ${x}_{0}$ *is a fixed point of* *T*, *then the orbit* $\{{T}^{n}x\}$ *is* *ρ*-*convergent to* ${x}_{0}$ *for any* $x\in C$.

**Theorem 3.3** [4]

*Let us assume that* $\rho \in \mathfrak{R}$ *is uniformly continuous and has the property* $(R)$. *Let* $C\subset {L}_{\rho}$ *be nonempty*, *convex*, *ρ*-*closed and* *ρ*-*bounded*. *Let* $T:C\to C$ *be an asymptotic pointwise* *ρ*-*contraction*. *Then* *T* *has a unique fixed point* ${x}_{0}\in C$. *Moreover*, *the orbit* $\{{T}^{n}x\}$ *is* *ρ*-*convergent to* ${x}_{0}$ *for any* $x\in C$.

Below, we introduce the notion of asymptotic pointwise *ρ*-contraction type in modular function spaces.

**Definition 3.4**Let $C\subset {L}_{\rho}$ be nonempty,

*ρ*-bounded and

*ρ*-closed. A mapping $T:C\to C$ is said to be of asymptotic pointwise

*ρ*-contraction type (resp. of weak asymptotic pointwise

*ρ*-contraction type) if ${T}^{N}$ is

*ρ*-continuous for some integer $N\ge 1$ and there exists a function $\alpha :C\to [0,1)$ such that, for each

*x*in

*C*,

where ${\alpha}_{n}\to \alpha $ pointwise on *M*.

We will obtain fixed point results for these mappings in modular function spaces.

*ρ*-limit of any

*ρ*-convergent sequence in ${L}_{\rho}$ is unique. This fact follows from the following reasoning: Assume that $\rho ({u}_{n}-u)\to 0$ and $\rho ({u}_{n}-v)\to 0$. Then

which implies that $u=v$.

The following theorem is our main result.

**Theorem 3.5** *Let* $\rho \in \mathfrak{R}$ *be uniformly continuous and have the property* (*R*). *Let* $C\subset {L}_{\rho}$ *be nonempty*, *convex*, *ρ*-*closed and* *ρ*-*bounded*. *Let* $T:C\to C$ *be a mapping of weak asymptotic pointwise* *ρ*-*contraction type*. *Then* *T* *has a unique fixed point* $v\in C$ *and*, *for each* $x\in C$, *the sequence of Picard iterates* $\{{T}^{n}x\}$ *is* *ρ*-*convergent to* *v*.

*Proof*Fix an $x\in C$ and define a function

*τ*by

*τ*is

*ρ*-lower semicontinuous in

*C*. By Lemma 2.9, then there exists ${x}_{0}\in C$ such that

*T*is of weak asymptotic pointwise

*ρ*-contraction type, by (3.4) we have ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{m}({x}_{0})\le 0$. Thus, for a subsequence $\{{r}_{{m}_{k}}({x}_{0})\}$ of $\{{r}_{m}({x}_{0})\}$, we have

which forces $\tau ({x}_{0})=0$ as $\alpha ({x}_{0})<1$. Hence $\rho ({T}^{n}x-{x}_{0})\to 0$ as $n\to \mathrm{\infty}$. From this and the continuity of ${T}^{N}$, for some $N\ge 1$, it follows that $\rho ({T}^{N+n}x-{T}^{N}{x}_{0})\to 0$ as $n\to \mathrm{\infty}$. Since the *ρ*-limit of any *ρ*-convergent sequence is unique, we must have ${T}^{N}{x}_{0}={x}_{0}$, namely, ${x}_{0}$ is a fixed point of ${T}^{N}$. Now, repeating the above proof for ${x}_{0}$ instead of *x*, we deduce that ${T}^{n}{x}_{0}$ is *ρ*-convergent to a member *v* of *C*; *i.e.*, $\rho ({T}^{n}{x}_{0}-v)\to 0$. But ${T}^{kN}{x}_{0}={x}_{0}$ for all $k\ge 1$. Hence, $v={x}_{0}$ and then ${T}^{n}{x}_{0}\to {x}_{0}(\rho )$.

Since the choice of $\u03f5>0$ is arbitrary and $\rho \in \mathfrak{R}$, we get $T{x}_{0}={x}_{0}$.

*T*can have only one fixed point. Indeed, if $u,v\in C$ are fixed points of

*T*, then by (3.5), we have

Since $\alpha (u)<1$ and $\rho \in \mathfrak{R}$, we immediately get $u=v$. □

Next, using the *ρ*-a.e. strong Opial property of the function modular, we prove a fixed point theorem which does not assume the uniform continuity of *ρ*.

*ρ*-a.e. strong Opial property (or shortly SO-property) if for every $\{{f}_{n}\}\in {L}_{\rho}$ which is

*ρ*-a.e. convergent to zero such that there exists a $\beta >1$ for which

**Lemma 3.7** [4]

*Let*$\rho \in \mathfrak{R}$.

*Assume that*${L}_{\rho}$

*has the*

*ρ*-

*a*.

*e*.

*strong Opial property*.

*Let*$C\subset {E}_{\rho}$

*be a nonempty*,

*ρ*-

*a*.

*e*.

*compact subset such that there exists*$\beta >1$

*such that*${\delta}_{\rho}(\beta C)=sup\{\rho (\beta (x-y));x,y\in C\}<\mathrm{\infty}$.

*Let*$D\subset C$

*be a nonempty*

*ρ*-

*a*.

*e*.

*closed subset*.

*For any*$n\ge 1$,

*let*${\lambda}_{n}:D\to [0,\mathrm{\infty})$

*be such that for any*$y\in D$,

*there exists a sequence*$\{{y}_{n}\}\subset C$

*such that*,

*for every*$n\ge 1$,

*the following holds*:

*and*$\rho (x-{y}_{n})\le {\lambda}_{n}(x)$

*for every*$x\in D$

*and*$n\ge 1$.

*Let*$\lambda (x)={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\lambda}_{n}(x)$

*for any*$x\in D$.

*Then there exists*${x}_{0}\in D$

*at which*

*λ*

*attains infimum*,

*i*.

*e*.,

**Theorem 3.8** *Let* $\rho \in \mathfrak{R}$. *Assume that* ${L}_{\rho}$ *has the* *ρ*-*a*.*e*. *strong Opial property*. *Let* $C\subset {E}_{\rho}$ *be a nonempty* *ρ*-*a*.*e*. *compact convex subset such that* ${\delta}_{\rho}(\beta C)=sup\{\rho (\beta (x-y));x,y\in C\}<\mathrm{\infty}$ *for some* $\beta >1$. *Then any* $T:C\to C$ *of weak asymptotic pointwise* *ρ*-*contraction type has a unique fixed point* ${x}_{0}\in C$. *Moreover*, *the orbit* $\{{T}^{n}x\}$ *is* *ρ*-*convergent to* ${x}_{0}$ *for any* $x\in C$.

*Proof*Fix an $x\in C$ and define a function

*τ*by

The rest of the proof is like the one used for Theorem 3.5. □

## Declarations

### Acknowledgements

The authors are grateful to the referees for their careful reading of the paper and several helpful comments.

## Authors’ Affiliations

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