1 Introduction

In 2008, Kirk and Xu [1] studied existence of fixed points of asymptotic pointwise nonexpansivemappings , 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 setA.

Definition 2.1[7]

Let be a nontrivial, convex and even function. We saythat ρ is a regular convex function pseudomodular if

  1. (i)

    ;

  2. (ii)

    ρ is monotone, i.e. for all implies , where ;

  3. (iii)

    ρ is orthogonally subadditive, i.e. for any such that , ;

  4. (iv)

    ρ has the Fatou property, i.e. for all implies , where ;

  5. (v)

    ρ is order continuous in ℰ, i.e. and implies .

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.

Definition 2.3[7, 42, 43]

Let 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]

Let .

  1. (a)

    We say that is ρ-convergent to f and write if and only if .

  2. (b)

    A sequence , where , is called ρ-Cauchy if as .

  3. (c)

    We say that is ρ-complete if and only if any ρ-Cauchy sequence in is ρ-convergent.

  4. (d)

    A set is called ρ-closed if for any sequence of , the convergence implies that f belongs to B.

  5. (e)

    A set is called ρ-bounded if .

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 theρ-Cauchy condition. However, it is important to remember that theρ-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.

  1. (i)

    isρ-complete.

  2. (ii)

    ρ-Ballsareρ-closed.

  3. (iii)

    Iffor anthen there exists a subsequenceofsuch thatρ-a.e.

  4. (iv)

    wheneverρ-a.e. (note: this property is equivalent to the Fatou property).

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

Definition 2.5[3]

Let and let be nonempty and ρ-closed. A mapping is called

  1. (i)

    a pointwise Lipschitzian mapping if there exists such that

  2. (ii)

    an asymptotic pointwise nonexpansive if there exists a sequence of mappings such that

and for any .

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

Define , where , for any and . Clearly then

(2.1)

Definition 2.6 Define as a class of all asymptotic pointwise nonexpansivemappings T such that

(2.2)
(2.3)

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:

  1. (i)

    for ;

  2. (ii)

    for and ;

  3. (iii)

    for each , is an asymptotic pointwise nonexpansive mapping, i.e. there exists a function such that

    (2.4)

such that for every ;

  1. (iv)

    for each , the mapping is ρ-continuous.

For each let denote the set of its fixed points. Define then theset of all common fixed points set for mappings from ℱ as the followingintersection:

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 .

Denoting and for , we note that without loss of generality we canassume that ℱ is asymptotically nonexpansive if

(2.5)
(2.6)

Define . Note that

(2.7)

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

(2.8)
(2.9)

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

Definition 2.9 We will say that a semigroup is ρ continuous if

for any and .

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.

Definition 2.10 Let be convex and ρ-bounded.

  1. (1)

    A function is called a ρ-type (or shortly a type) if there exists a one-parameter family of elements of K such that for any we have

  2. (2)

    Let τ be a ρ-type. A sequence is called a minimizing sequence of τ if

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:

  1. (i)

    Let , . Define

Let

and if . We say that ρ satisfies (UC) if forevery , , . Note that for every , , for small enough.

  1. (ii)

    We say that ρ satisfies (UUC1) if there exists , for every , and such that

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]

Letbe (UUC1) and letbe bounded away from 0 and 1. If there existssuch that

(2.10)
(2.11)

then

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]

Letbe (UUC1) thenhas 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]

Assumeis (UUC1). LetCbe a nonempty, ρ-bounded,ρ-closed, and convex subset of. Letτbe a type defined by a one-parameter familyinC.

  1. (i)

    If, then.

  2. (ii)

    Any minimizing sequenceofτisρ-convergent. Moreover, theρ-limit ofis independent of the minimizing sequence.

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

Theorem 2.2[5]

Assumeis (UUC1). LetCbe aρ-closed, ρ-bounded, convex,and nonempty subset. Letbe an asymptotic pointwise nonexpansive semigroup onC. Then ℱ has a common fixed point and the setof common fixed points isρ-closed and convex.

3 The demiclosedness principle

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

Definition 3.1 We say that is uniformly continuous if for every and , there exists such that

(3.1)

provided and .

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

(3.2)

the following inequality holds for any not equal to 0:

(3.3)

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

(3.4)

the following equality holds for any :

(3.5)

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 ρis 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 thathas theρ-a.e. strong Opial property. Letbe a nonempty, stronglyρ-bounded, andρ-a.e. compact convex set. Thenanyρ-type defined in C attains its minimum inC.

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. Assume that

  1. (1)

    ρis (UUC1),

  2. (2)

    ρhas the strong Opial property,

  3. (3)

    ρhas theproperty and is uniformly continuous.

Letbe nonempty, convex, stronglyρ-bounded, andρ-closed, and let. Let, and. Ifρ-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.

Theorem 3.2Let. Assume that

  1. (1)

    ρis (UUC1),

  2. (2)

    ρhas the strong Opial property,

  3. (3)

    ρhas theproperty and is uniformly continuous.

Letbe nonempty, convex, stronglyρ-bounded, andρ-closed. Letbe an asymptotic pointwise nonexpansive mapping such that

andfor any. We will assume the functionsare bounded onC, i.e. Tis uniformlyρ-Lipschitzian mapping. Let, and. Ifρ-a.e. and, then. In particular, we have.

Proof Define the ρ-type function

Note that

Indeed, we have

which implies

for any , where . Hence

Since ρ has the property, we get

Since ρ is uniformly continuous, we get

for any and . In particular, we have

In other words, we have , for any . Since ρ has the Opial property, it iseasy to prove that

Since T is an asymptotic pointwise nonexpansive mapping, we get

Since ρ is (UUC1), then arguing similarly to the proof ofTheorem 4.1 in [6], we have

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

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

Corollary 3.1Let. Assume that

  1. (1)

    ρis (UUC1),

  2. (2)

    ρhas the strong Opial property,

  3. (3)

    ρhas theproperty and is uniformly continuous.

Letbe nonempty, convex, stronglyρ-bounded, andρ-closed. Letbe asymptotic pointwise nonexpansive mappings such that

for any, with, and, for any. We will assume the functionsandare bounded onC. Let, and. Ifρ-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. Assume that

  1. (1)

    ρis (UUC1),

  2. (2)

    ρhas the strong Opial property,

  3. (3)

    ρhas theproperty and is uniformly continuous.

Letbe nonempty, convex, stronglyρ-bounded, andρ-closed, and letbe continuous. Let, and. Assumeρ-a.e. If there existsuch thatis irrational and

then.

Proof In view of Corollary 3.1, we know that

Let us denote . Note that for any we have

Combining the above we get

Since is irrational, then the set is dense in [47]. Since ℱ is continuous and ρ is uniformly continuous,we have

Hence as desired. □

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:

(4.1)

and

  1. (1)

    is bounded away from 0 and 1,

  2. (2)

    ,

  3. (3)

    for every .

Definition 4.2 We say that a generalized Krasnosel’skii-Mann iterationprocess is well defined if

(4.2)

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 4.1Letbe (UUC1). Letbe aρ-closed, ρ-bounded, and convexset. Let, , and letbe a sequence generated by a generalized Krasnosel’skii-Mannprocess. Then there exists ansuch 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.

Theorem 4.1Let. Assume that

  1. (1)

    ρis (UUC1),

  2. (2)

    ρhas the strong Opial property,

  3. (3)

    ρhas theproperty and is uniformly continuous.

Letbe nonempty, ρ-a.e. compact,convex, stronglyρ-bounded, andρ-closed, and let. Assume thatis a well defined Krasnosel’skii-Mann iteration process.If for the sequencegenerated bywe have

(4.3)

whereare such thatis irrational, thenconvergesρ-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

(4.4)

We claim that . Assume to the contrary that . Then by the Opial property we have

(4.5)

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. Thenconverges to an.

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

Lemma 4.3Letbe (UUC1). Letbe aρ-closed, ρ-bounded, and convexset. Let. Assume thatwis a common fixed point of ℱ. Let us denote bya sequence generated by the generalized Krasnosel’skii-Mannprocess. Then there existssuch that

(4.6)

Proof Let . Since

it follows that for every ,

(4.7)

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

Lemma 4.4Letbe (UUC1). Letbe aρ-closed, ρ-bounded, and convexset, and. Letbe a generalized Krasnosel’skii-Mann iteration process.Then

(4.8)

and

(4.9)

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

(4.10)

Note that

(4.11)

and that

(4.12)

Set , , and note that by (4.10), and by (4.11). Observe also that

(4.13)

Hence, it follows from Lemma 2.1 that

(4.14)

which by the construction of the sequence is equivalent to

(4.15)

as claimed. □

Lemma 4.5Letbe (UUC1) and have theproperty. Letbe aρ-closed, ρ-bounded, and convexset and let. Denote bythe sequence generated by a well defined generalizedKrasnosel’skii-Mann process. Letbe such that for everythere exists a strictly increasing sequence of natural numberssatisfying the following conditions:

  1. (a)

    as,

  2. (b)

    , where.

Thenis an approximate fixed point sequence for all mappingswhere, that is,

(4.16)

for every.

Proof Let us fix . Note that

(4.17)

By , it suffices to prove that

(4.18)

To this end observe that

(4.19)

in view of the assumption (a) and by (4.9) in Lemma 4.4. Observe that

(4.20)

Indeed,

which tends to zero as because of (4.17), Lemma 4.4, the fact that theprocess is well defined, assumptions (b) and (2.9), and the boundedness of eachfunction . This convergence gives us, via, the required (4.20). On the other hand,

which tends to zero as because of assumption (a), Lemma 4.4, (4.20),the fact that the process is well defined, and the fact that the semigroup isasymptotic pointwise nonexpansive. Since 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 4.2Letbe uniformly continuous function modular satisfying (UUC1).Assume in addition thatρsatisfiesand has the strong Opial property. Letbe aρ-closed, ρ-bounded, and convexset and let. Denote bythe sequence generated by a well defined generalizedKrasnosel’skii-Mann process. Letbe such thatis irrational and that there exists a strictly increasing sequence of naturalnumberssatisfying the following conditions:

  1. (a)

    as,

  2. (b)

    , where,

  3. (c)

    , where.

Then the sequenceconvergesρ-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 4.3Under the assumptions of Theorem 4.2, if in additionCis assumed to beρ-compact, then the sequencegenerated byconverges strongly to a common fixed point, that is,

(4.21)

Proof It follows from Theorem 4.2 that there exists a common fixedpoint such that converges ρ-a.e. Byρ-compactness of C there exist and a subsequence of such that

(4.22)

By Proposition 2.1 there exists a subsequence of such that

(4.23)

By the uniqueness of the ρ-a.e. limit we get . Hence

(4.24)

On the other hand, the limit exists by Lemma 4.3, which implies 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.