# On the convergence of iteration processes for semigroups of nonlinear mappings inmodular function spaces

- Buthinah A Bin Dehaish
^{3}Email author, - Mohamed A Khamsi
^{4, 5}Email author and - Wojciech M Kozlowski
^{6}Email author

**2015**:3

https://doi.org/10.1186/1687-1812-2015-3

© Bin Dehaish et al.; licensee Springer. 2015

**Received: **9 August 2014

**Accepted: **8 December 2014

**Published: **16 January 2015

## Abstract

## Keywords

## 1 Introduction

*i.e.*

where
, for all
. Their main result (Theorem 3.5) states thatevery asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, boundedand convex subset *C* of a uniformly convex Banach space *X* has afixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to bea natural generalization of nonexpansive mappings. The conditions on
can be for instance expressed in terms of thederivatives of iterations of *T* for differentiable *T*. In 2009,these results were generalized by Hussain and Khamsi to metric spaces [2]. In 2011, Khamsi and Kozlowski [3] extended their result proving the existence of fixed points of asymptoticpointwise *ρ*-nonexpansive mappings acting in modular function spaces.The existence of common fixed points of semigroups of nonexpansive (in a modularsense) mappings acting in modular function spaces was first established by Kozlowskiin [4] and then extended to the semigroups of asymptotic pointwise nonexpansivemappings by the authors in [5]. The proof of this important theorem is of the existential nature anddoes not describe any algorithm for constructing a common fixed point of anasymptotic pointwise *ρ*-nonexpansive semigroup. The current paper aimsat filling this gap. The results of this paper generalize the convergence ofgeneralized Mann processes to common fixed points of semigroups of nonexpansivesemigroups studied in the recent paper by Bin Dehaish and Kozlowski [6].

Let us recall that modular function spaces are a natural generalization of bothfunction and sequence variants of many spaces like Lebesgue, Orlicz,Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and manyothers, important from an applications perspective; see the book by Kozlowski [7] for an extensive list of examples and special cases. There exists anextensive literature on the topic of the fixed point theory in modular functionspaces; see *e.g.*[3, 7–18] and the references therein. It is also worthwhile mentioning a growinginterest in applications of the methods of the fixed point theory to semigroups ofnonlinear mappings and applications to the area of differential and integralequations (see *e.g.*[10, 19, 20]).

It is well known that the fixed point construction iteration processes forgeneralized nonexpansive mappings have been successfully used to develop efficientand powerful numerical methods for solving various nonlinear equations andvariational problems, often of great importance for applications in various areas ofpure and applied science. There exists an extensive literature on the subject ofiterative fixed point construction processes for asymptotically nonexpansivemappings in Hilbert, Banach, and metric spaces; see *e.g.*[2, 21–37] and the references therein. Kozlowski proved convergence to a fixed pointof some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banachspaces [38] and the existence of common fixed points of semigroups of pointwiseLipschitzian mappings in Banach spaces [39]. Recently, the weak and strong convergence of such processes to commonfixed points of semigroups of mappings in Banach spaces was demonstrated byKozlowski and Sims [40] and by Kozlowski in [41].

We would like to emphasize that all convergence theorems proved in this paper defineconstructive algorithms that can be actually implemented. When dealing with specificapplications of these theorems, one should take into consideration how additionalproperties of the mappings, sets, and modulars involved can influence the actualimplementation of the algorithms defined in this paper.

## 2 Preliminaries

Let us introduce basic notions related to modular function spaces and relatednotation which will be used in this paper. For further details we refer the readerto preliminary sections of the recent articles [3, 6, 16] or to the survey article [17]; see also [7, 42, 43] for the standard framework of modular function spaces.

Let Ω be a nonempty set and Σ be a nontrivial *σ*-algebra ofsubsets of Ω. Let
be a *δ*-ring of subsets of Ω, such that
for any
and
. Let us assume that there exists an increasingsequence of sets
such that
. By ℰ we denote the linear space of all simplefunctions with supports from
. By
we will denote the space of all extended measurablefunctions, *i.e.* all functions
such that there exists a sequence
,
and
for all
. By
we denote the characteristic function of the set*A*.

**Definition 2.1**[7]

*ρ*is a regular convex function pseudomodular if

- (i)
- (ii)
- (iii)
- (iv)
- (v)

Similarly to the case of measure spaces, we say that a set
is *ρ*-null if
for every
. We say that a property holds *ρ*-almosteverywhere if the exceptional set is *ρ*-null. As usual we identify anypair of measurable sets whose symmetric difference is *ρ*-null as wellas any pair of measurable functions differing only on a *ρ*-null set.With this in mind we define
, where each element is actually an equivalence classof functions equal *ρ*-a.e. rather than an individual function.

**Definition 2.2**[7]

We say that a regular function pseudomodular *ρ* is a regular convexfunction modular if
implies
*ρ*-a.e. The class of all nonzero regular convex function modularsdefined on Ω will be denoted by ℜ.

*ρ*be a convex function modular. A modular function space is thevector space . In the vector space , the following formula:

defines a norm, frequently called the Luxemburg norm.

The following notions will be used throughout the paper.

**Definition 2.4**[8]

Since *ρ* fails in general the triangle identity, many of the knownproperties of limit may not extend to *ρ*-convergence. For example,*ρ*-convergence does not necessarily imply the*ρ*-Cauchy condition. However, it is important to remember that the*ρ*-limit is unique when it exists. The following proposition bringstogether a few facts, which will be often used in the proofs of our results.

**Proposition 2.1**[7]

Let us recall the definition of an asymptotic pointwise nonexpansive mapping actingin a modular function space.

**Definition 2.5**[3]

A point
is called a fixed point of *T* whenever
. The set of fixed points of *T* will bedenoted by
.

The notion of the asymptotic pointwise nonexpansiveness will be now extended to aone-parameter family of mappings. Throughout this paper *J* will be thesemigroup of all nonnegative numbers, that is,
with normal addition.

**Definition 2.7**A one-parameter family of mappings from

*C*into itself is said to bean asymptotic pointwise nonexpansive semigroup on

*C*if ℱ satisfiesthe following conditions:

- (i)
- (ii)
- (iii)

The common fixed points are frequently interpreted as the stationary points of thesemigroup ℱ. Note that without loss of generality we may assume for any and and .

The above notation will be consistently used throughout this paper.

**Definition 2.8**By we will denote the class of all asymptotic pointwisenonexpansive semigroups on

*C*such that

Note that we do not assume that all functions are bounded by a common constant. Therefore, we donot assume that ℱ is uniformly Lipschitzian.

The concept *ρ*-type is a powerful technical tool which is used in theproofs of many fixed point results. The definition of a *ρ*-type isbased on a given sequence. In this work, we generalize this definition to be adaptedto one-parameter family of mappings.

Note that *τ* is convex provided *ρ* is convex.

Let us recall the modular equivalents of uniform convexity introduced in [3].

**Definition 2.11**Let . We define the following uniform convexity (UC) typeproperties of the function modular

*ρ*:

- (i)

*ρ*satisfies (UC) if forevery , , . Note that for every , , for small enough.

- (ii)

We will need the following result, being a modular equivalent of a norm property inuniformly convex Banach spaces; see *e.g.*[26].

**Lemma 2.1**[6]

The following property plays in the theory of modular function space a role similarto the reflexivity in Banach spaces; see *e.g.*[9].

**Definition 2.12** We say that
has property
if and only if every nonincreasing sequence
of nonempty, *ρ*-bounded,*ρ*-closed, and convex subsets of
has nonempty intersection.

Similarly to the Banach space case, the modular uniform convexity implies property .

**Theorem 2.1**[3]

*Let*
*be* (*UUC*1) *then*
*has property*
.

The next lemma is a generalization of the minimizing sequence property for typesdefined by sequences in Lemma 4.3 in [16] to the one-parameter semigroup case.

**Lemma 2.2**[5]

*Assume*

*is*(

*UUC*1).

*Let*

*C*

*be a nonempty*,

*ρ*-

*bounded*,

*ρ*-

*closed*,

*and convex subset of*.

*Let*

*τ*

*be a type defined by a one*-

*parameter family*

*in*

*C*.

Using Lemma 2.2, the authors proved the following common fixed point result forasymptotic pointwise nonexpansive semigroups.

**Theorem 2.2**[5]

*Assume*
*is* (*UUC*1). *Let**C**be a**ρ*-*closed*, *ρ*-*bounded*, *convex*,*and nonempty subset*. *Let*
*be an asymptotic pointwise nonexpansive semigroup on**C*. *Then* ℱ *has a common fixed point and the set*
*of common fixed points is**ρ*-*closed and convex*.

## 3 The demiclosedness principle

In this section we will use the notion the uniform continuity of the function modular*ρ* in the sense of the following definition (see *e.g.*[16]).

Let us mention that the uniform continuity holds for a large class of functionmodulars. For instance, it can be proved that in Orlicz spaces over a finiteatomless measure [44] or in Orlicz sequence spaces [45] the uniform continuity of the Orlicz modular is equivalent to the
-type condition. Recall that *ρ* satisfiesthe
-type condition if and only if there exists
such that
, for any
.

Let us recall the definition of the Opial property and the strong Opial property inmodular function spaces [16, 46].

**Definition 3.2**We say that satisfies the

*ρ*-a.e. Opial property iffor every which is

*ρ*-a.e. convergent to 0 suchthat there exists a for which

**Definition 3.3**We say that satisfies the

*ρ*-a.e. strong Opialproperty if for every which is

*ρ*-a.e. convergent to 0 suchthat there exists a for which

**Remark 3.1** Note that the *ρ*-a.e. strong Opial property impliesthe *ρ*-a.e. Opial property [46].

**Remark 3.2** Also, note that, by virtue of Theorem 2.1 in [46], every convex, orthogonally additive function modular *ρ* hasthe *ρ*-a.e. strong Opial property. Let us recall that *ρ*is called orthogonally additive if
whenever
. Therefore, all Orlicz and Musielak-Orlicz spacesmust have the strong Opial property.

Note that the Opial property in the norm sense does not necessarily hold for severalclassical Banach function spaces. For instance the norm Opial property does not holdfor spaces for , while the modular strong Opial property holds in for all .

**Lemma 3.1**[4]

*Let*
. *Assume that*
*has the**ρ*-*a*.*e*. *strong Opial property*. *Let*
*be a nonempty*, *strongly**ρ*-*bounded*, *and**ρ*-*a*.*e*. *compact convex set*. *Thenany**ρ*-*type defined in C attains its minimum in**C*.

To begin our discussion of the demiclosedness principle, let us quote the followingversion of this theorem applied to the asymptotic pointwise nonexpansive mappings [[6], Theorem 4.1].

**Theorem 3.1** (Demiclosedness principle)

*Let*
*be nonempty*, *convex*, *strongly**ρ*-*bounded*, *and**ρ*-*closed*, *and let*
. *Let*
, *and*
. *If*
*ρ*-*a*.*e*. *and*
, *then*
.

We will need a version of the above theorem without assuming that . This will require a different proof, which issketched below.

*Let*

*be nonempty*,

*convex*,

*strongly*

*ρ*-

*bounded*,

*and*

*ρ*-

*closed*.

*Let*

*be an asymptotic pointwise nonexpansive mapping such that*

*and*
*for any*
. *We will assume the functions*
*are bounded on**C*, *i*.*e*. *T**is uniformly**ρ*-*Lipschitzian mapping*. *Let*
, *and*
. *If*
*ρ*-*a*.*e*. *and*
, *then*
. *In particular*, *we have*
.

Since *T* is *ρ*-continuous, we get
, *i.e.*
. □

As a corollary to this result, we get the following important result.

*Let*

*be nonempty*,

*convex*,

*strongly*

*ρ*-

*bounded*,

*and*

*ρ*-

*closed*.

*Let*

*be asymptotic pointwise nonexpansive mappings such that*

*for any*,

*with*,

*and*,

*for any*.

*We will assume the functions*

*and*

*are bounded on*

*C*.

*Let*,

*and*.

*If*

*ρ*-

*a*.

*e*.

*and*

*then*
. *In particular*, *we have*
.

The above results lead us to the following version of the demiclosedness principlefor semigroup of mappings.

**Theorem 3.3** (Demiclosedness principle)

*Let*

*be nonempty*,

*convex*,

*strongly*

*ρ*-

*bounded*,

*and*

*ρ*-

*closed*,

*and let*

*be continuous*.

*Let*,

*and*.

*Assume*

*ρ*-

*a*.

*e*.

*If there exist*

*such that*

*is irrational and*

*ρ*is uniformly continuous,we have

## 4 Convergence of generalized Krasnosel’skii-Mann iteration processes

Let us start with the precise definition of the generalized Krasnosel’skii-Manniteration process for semigroups of nonlinear mappings.

**Definition 4.1**Let , and . The generalized Krasnosel’skii-Mann iterationprocess generated by the semigroup ℱ, the sequences and , is defined by the following iterative formula:

Arguing exactly like in the proof of Lemma 5.2 in [6] (see also Lemma 22.20 in [40]), we get the following result.

**Lemma 4.1***Let*
*be* (*UUC*1). *Let*
*be a**ρ*-*closed*, *ρ*-*bounded*, *and convexset*. *Let*
,
, *and let*
*be a sequence generated by a generalized Krasnosel’skii*-*Mannprocess*
. *Then there exists an*
*such that*
.

We will prove now a generic version of the convergence theorem for the sequences which are generated by the Krasnosel’skii-Manniteration process and are at the same time approximate fixed point sequences.

*Let*

*be nonempty*,

*ρ*-

*a*.

*e*.

*compact*,

*convex*,

*strongly*

*ρ*-

*bounded*,

*and*

*ρ*-

*closed*,

*and let*.

*Assume that*

*is a well defined Krasnosel’skii*-

*Mann iteration process*.

*If for the sequence*

*generated by*

*we have*

*where*
*are such that*
*is irrational*, *then*
*converges**ρ*-*a*.*e*. *to a common fixed point*
.

*Proof*Observe that by Theorem 2.2 the set of fixed points is nonempty, convex and

*ρ*-closed.Consider , two

*ρ*-a.e. cluster points of . There exits then , subsequences of such that

*ρ*-a.e., and

*ρ*-a.e. It follows from Theorem 3.3 that and . By Lemma 4.1, there exist such that

The contradiction implies that
. Therefore,
has at most one *ρ*-a.e. cluster point.Since *C* is *ρ*-a.e. compact it follows that the sequence
has exactly one *ρ*-a.e. cluster point
, which means that
*ρ*-a.e. Applying the demiclosedness principle again, we get
. By the same argument, we get
(observe that the construction of *w* did notdepend on the selection of
). From the density of
in
, we conclude that
for any
, as claimed. □

Let us apply the above result to some more specific situations. First we need toprove a series of axillary results. Let us start with the following elementarylemma.

**Lemma 4.2**[31]

*Suppose*

*is a bounded sequence of real numbers and*

*is a double index sequence of real numbers which satisfy*

*for each*
. *Then*
*converges to an*
.

The following result provides an important technique which will be used in thispaper.

**Lemma 4.3**

*Let*

*be*(

*UUC*1).

*Let*

*be a*

*ρ*-

*closed*,

*ρ*-

*bounded*,

*and convexset*.

*Let*.

*Assume that*

*w*

*is a common fixed point of*ℱ.

*Let us denote by*

*a sequence generated by the generalized Krasnosel’skii*-

*Mannprocess*.

*Then there exists*

*such that*

Denote for every and . Observe that by the assumptions on the sequence , . By Lemma 4.2, there exists an such that , as claimed. □

**Lemma 4.4**

*Let*

*be*(

*UUC*1).

*Let*

*be a*

*ρ*-

*closed*,

*ρ*-

*bounded*,

*and convexset*,

*and*.

*Let*

*be a generalized Krasnosel’skii*-

*Mann iteration process*.

*Then*

*Proof*By Theorem 2.2, ℱ has at least one common fixed point . In view of Lemma 4.3, there exists such that

as claimed. □

**Lemma 4.5**

*Let*

*be*(

*UUC*1)

*and have the*

*property*.

*Let*

*be a*

*ρ*-

*closed*,

*ρ*-

*bounded*,

*and convexset and let*.

*Denote by*

*the sequence generated by a well defined generalizedKrasnosel’skii*-

*Mann process*.

*Let*

*be such that for every*

*there exists a strictly increasing sequence of natural numbers*

*satisfying the following conditions*:

- (a)
- (b)

*ρ*has the property,

which completes the proof of the lemma. □

The following theorem is an immediate consequence of Lemma 4.5 andTheorem 4.1.

**Theorem 4.2**

*Let*

*be uniformly continuous function modular satisfying*(

*UUC*1).

*Assume in addition that*

*ρ*

*satisfies*

*and has the strong Opial property*.

*Let*

*be a*

*ρ*-

*closed*,

*ρ*-

*bounded*,

*and convexset and let*.

*Denote by*

*the sequence generated by a well defined generalizedKrasnosel’skii*-

*Mann process*.

*Let*

*be such that*

*is irrational and that there exists a strictly increasing sequence of naturalnumbers*

*satisfying the following conditions*:

- (a)
- (b)
- (c)

*Then the sequence*
*converges**ρ*-*a*.*e*. *to a common fixed point*
.

**Remark 4.1** Observe that a sequence
satisfying assumptions of Theorem 4.2 can bealways constructed. The main difficulty is in ensuring that the correspondingprocess
is well defined.

The next result answers the question when the sequence generated by the generalizedKrasnosel’skii-Mann process will converge strongly to a common fixed point.Not surprisingly we need to add a compactness assumption.

**Theorem 4.3**

*Under the assumptions of Theorem*4.2,

*if in addition*

*C*

*is assumed to be*

*ρ*-

*compact*,

*then the sequence*

*generated by*

*converges strongly to a common fixed point*,

*that is*,

*Proof*It follows from Theorem 4.2 that there exists a common fixedpoint such that converges

*ρ*-a.e. By

*ρ*-compactness of

*C*there exist and a subsequence of such that

as claimed. □

**Remark 4.2** Observe that in view of the
assumption, the *ρ*-compactness of theset *C* assumed in Theorem 4.3 is equivalent to the compactness in thesense of the norm defined by *ρ*.

## Declarations

### Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King AbdulazizUniversity, Jeddah, under Grant No. (247-001-D1434). The authors,therefore, acknowledge with thanks technical and financial support of DSR.

## Authors’ Affiliations

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