On the convergence of iteration processes for semigroups of nonlinear mappings inmodular function spaces
© Bin Dehaish et al.; licensee Springer. 2015
Received: 9 August 2014
Accepted: 8 December 2014
Published: 16 January 2015
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 . In 2011, Khamsi and Kozlowski  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  and then extended to the semigroups of asymptotic pointwise nonexpansivemappings by the authors in . 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 .
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  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  and the existence of common fixed points of semigroups of pointwiseLipschitzian mappings in Banach spaces . Recently, the weak and strong convergence of such processes to commonfixed points of semigroups of mappings in Banach spaces was demonstrated byKozlowski and Sims  and by Kozlowski in .
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.
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 ; 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.
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.
defines a norm, frequently called the Luxemburg norm.
The following notions will be used throughout the paper.
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.
Let us recall the definition of an asymptotic pointwise nonexpansive mapping actingin a modular function space.
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.
The above notation will be consistently used throughout this paper.
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 .
We will need the following result, being a modular equivalent of a norm property inuniformly convex Banach spaces; see e.g..
The following property plays in the theory of modular function space a role similarto the reflexivity in Banach spaces; see e.g..
The next lemma is a generalization of the minimizing sequence property for typesdefined by sequences in Lemma 4.3 in  to the one-parameter semigroup case.
Using Lemma 2.2, the authors proved the following common fixed point result forasymptotic pointwise nonexpansive semigroups.
Assume is (UUC1). LetCbe aρ-closed, ρ-bounded, convex,and nonempty subset. Let be an asymptotic pointwise nonexpansive semigroup onC. Then ℱ has a common fixed point and the set of 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.).
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  or in Orlicz sequence spaces  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 .
Remark 3.1 Note that the ρ-a.e. strong Opial property impliesthe ρ-a.e. Opial property .
Remark 3.2 Also, note that, by virtue of Theorem 2.1 in , 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 .
To begin our discussion of the demiclosedness principle, let us quote the followingversion of this theorem applied to the asymptotic pointwise nonexpansive mappings [, Theorem 4.1].
Theorem 3.1 (Demiclosedness principle)
ρhas the strong Opial property,
ρhas the strong Opial property,
As a corollary to this result, we get the following important result.
ρhas the strong Opial property,
The above results lead us to the following version of the demiclosedness principlefor semigroup of mappings.
Theorem 3.3 (Demiclosedness principle)
ρhas the strong Opial property,
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.
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.
ρhas the strong Opial property,
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.
The following result provides an important technique which will be used in thispaper.
as claimed. □
which completes the proof of the lemma. □
The following theorem is an immediate consequence of Lemma 4.5 andTheorem 4.1.
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.
as claimed. □
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.
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