Abstract
Let be a uniformly convex modular function space with a strong Opial property. Let be an asymptotic pointwise nonexpansive mapping, where C is a ρa.e. compact convex subset of . In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.
MSC: 47H09, 47H10.
Keywords:
fixed point; nonexpansive mapping; fixed point iteration process; Mann process; Ishikawa process; modular function space; Orlicz space; Opial property; uniform convexity1 Introduction
In 2008, Kirk and Xu [21] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings , i.e.,
where , for all . Their main result (Theorem 3.5) states that every asymptotic pointwise nonexpansive selfmapping of a nonempty, closed, bounded and convex subset C of a uniformly convex Banach space X has a fixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization of nonexpansive mappings. The conditions on can be for instance expressed in terms of the derivatives of iterations of T for differentiable T. In 2009 these results were generalized by Hussain and Khamsi to metric spaces, [9].
In 2011, Khamsi and Kozlowski [18] extended their result proving the existence of fixed points of asymptotic pointwise ρnonexpansive mappings acting in modular function spaces. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise ρnonexpansive mapping. This paper aims at filling this gap.
Let us recall that modular function spaces are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, MusielakOrlicz, Lorentz, OrliczLorentz, CalderonLozanovskii spaces and many others, see the book by Kozlowski [24] for an extensive list of examples and special cases. There exists an extensive literature on the topic of the fixed point theory in modular function spaces, see, e.g., [35,8,13,14,1720,24] and the papers referenced there.
It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. There exists an extensive literature on the subject of iterative fixed point construction processes for asymptotically nonexpansive mappings in Hilbert, Banach and metric spaces, see, e.g., [1,2,6,7,9,12,16,3036,3842] and the works referred there. Kozlowski proved convergence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [25] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [26]. Recently, weak and strong convergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [28].
We would like to emphasize that all convergence theorems proved in this paper define constructive algorithms that can be actually implemented. When dealing with specific applications of these theorems, one should take into consideration how additional properties of the mappings, sets and modulars involved can influence the actual implementation of the algorithms defined in this paper.
The paper is organized as follows:
(a) Section 2 provides necessary preliminary material on modular function spaces.
(b) Section 3 introduces the asymptotic pointwise nonexpansive mappings and related notions.
(c) Section 4 deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.
(d) Section 5 utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.
(e) Section 6 establishes the almost everywhere convergence theorem for generalized Ishikawa process.
(f) Section 7 provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact set C.
2 Preliminaries
Let Ω be a nonempty set and Σ be a nontrivial σalgebra of subsets of Ω. Let be a δring of subsets of Ω such that for any and . Let us assume that there exists an increasing sequence of sets such that . By ℰ we denote the linear space of all simple functions with supports from . By we will denote the space of all extended measurable functions, 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 Let be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:
(ii) ρ is monotone, i.e., for all implies , where ;
(iii) ρ is orthogonally subadditive, i.e., for any such that , ;
(iv) ρ has the Fatou property, i.e., for all implies , where ;
(v) ρ is order continuous in ℰ, i.e., and implies .
Similarly, as in the case of measure spaces, we say that a set is ρnull if for every . We say that a property holds ρalmost everywhere if the exceptional set is ρnull. As usual, we identify any pair of measurable sets whose symmetric difference is ρnull as well as any pair of measurable functions differing only on a ρnull set. With this in mind we define
where each is actually an equivalence class of functions equal ρa.e. rather than an individual function. Where no confusion exists we will write ℳ instead of .
Definition 2.2 Let ρ be a regular function pseudomodular.
(1) We say that ρ is a regular convex function semimodular if for every implies ρa.e.;
(2) We say that ρ is a regular convex function modular if implies ρa.e.;
The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.
Let us denote for , . It is easy to prove that is a function pseudomodular in the sense of Def.2.1.1 in [24] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [2224].
Remark 2.1 We limit ourselves to convex function modulars in this paper. However, omitting convexity in Definition 2.1 or replacing it by sconvexity would lead to the definition of nonconvex or sconvex regular function pseudomodulars, semimodulars and modulars as in [24].
Let ρ be a convex function modular.
(a) A modular function space is the vector space , or briefly , defined by
(b) The following formula defines a norm in (frequently called Luxemurg norm):
In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.
(1) , is complete and the normis monotone w.r.t. the natural order in ℳ.
(3) Iffor anthen there exists a subsequenceofsuch thatρa.e.
(4) Ifconverges uniformly tofon a setthenfor every.
(5) Letρa.e. There exists a nondecreasing sequence of setssuch thatandconverges uniformly tofon every (Egoroff theorem).
(6) wheneverρa.e. (Note: this property is equivalent to the Fatou property.)
(b) has the Lebesgue property, i.e., for, and.
(c) is the closure of ℰ (in the sense of).
The following definition plays an important role in the theory of modular function spaces.
Definition 2.4 Let . We say that ρ has the property if
Theorem 2.2Let. The following conditions are equivalent:
(e) iffor an, then, i.e., the modular convergence is equivalent to the norm convergence.
We will also use another type of convergence which is situated between norm and modular convergence. It is defined, among other important terms, in the following definition.
(a) We say that is ρconvergent to f and write if and only if .
(b) A sequence where is called ρCauchy if as .
(c) A set is called ρclosed if for any sequence of , the convergence implies that f belongs to B.
(d) A set is called ρbounded if .
(e) A set is called strongly ρbounded if there exists such that .
(f) A set is called ρcompact if for any in C there exists a subsequence and an such that .
(g) A set is called ρa.e. closed if for any in C which ρa.e. converges to some f, then we must have .
(h) A set is called ρa.e. compact if for any in C, there exists a subsequence which ρa.e. converges to some .
(i) Let and . The ρdistance between f and C is defined as
Let us note that ρconvergence does not necessarily imply ρCauchy condition. Also, does not imply in general , . Using Theorem 2.1, it is not difficult to prove the following:
(ii) ρballsareρclosed andρa.e. closed.
Let us compare different types of compactness introduced in Definition 2.5.
Proposition 2.2Let. The following relationships hold for sets:
(i) IfCisρcompact, thenCisρa.e. compact.
(ii) IfCiscompact, thenCisρcompact.
(iii) Ifρsatisfies, thencompactness andρcompactness are equivalent in.
Proof
(i) follows from Theorem 2.1 part (3).
(ii) follows from Theorem 2.1 part (2).
(iii) follows from (2.2) and from Theorem 2.2 part (e).
□
3 Asymptotic pointwise nonexpansive mappings
Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [18].
Definition 3.1 Let and let be nonempty and ρclosed. A mapping is called an asymptotic pointwise mapping if there exists a sequence of mappings such that
(i) If for every and every , then T is called ρnonexpansive or shortly nonexpansive.
(ii) If converges pointwise to , then T is called asymptotic pointwise contraction.
(iii) If for any , then T is called asymptotic pointwise nonexpansive.
(iv) If for any with , then T is called strongly asymptotic pointwise contraction.
Denoting , we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if
Define . In view of (3.2), we have
The above notation will be consistently used throughout this paper.
By we will denote the class of all asymptotic pointwise nonexpansive mappings .
In this paper, we will impose some restrictions on the behavior of and . This type of assumptions is typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, e.g., [25].
Definition 3.2 Define as a class of all such that
We recall the following concepts related to the modular uniform convexity introduced in [18]:
Definition 3.3 Let . We define the following uniform convexity type properties of the function modular ρ: Let , , . Define
Let
and if . We will use the following notational convention: .
Definition 3.4 We say that ρ satisfies if for every , , . Note that for every , , for small enough. We say that ρ satisfies if for every , there exists depending only on s and ε such that
We will need the following result whose proof is elementary. Note that for , this result follows directly from Definition 3.4.
Lemma 3.1Letbeand let. Then for every, there existsdepending only onsandεsuch that
The notion of bounded away sequences of real numbers will be used extensively throughout this paper.
Definition 3.5 A sequence is called bounded away from 0 if there exists such that for every . Similarly, is called bounded away from 1 if there exists such that for every .
We will need the following generalization of Lemma 4.1 from [18] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see, e.g., [36].
Lemma 3.2Letbeand letbe bounded away from 0 and 1. If there existssuch that
then
Proof Assume to the contrary that this is not the case and fix an arbitrary . Passing to a subsequence if necessary, we may assume that there exists an such that
while
Since is bounded away from 0 and 1 there exist such that for all natural n. Passing to a subsequence if necessary, we can assume that . For every and , let us define . Observe that the function is a convex function. Hence that the function
is also convex on , and consequently, it is a continuous function on . Noting that
we conclude that is a continuous function of . Hence
By (3.8) and (3.9)
By (3.12) the lefthand side of (3.13) tends to as while the righthand side tends to in view of (3.7). Hence
By and by Lemma 3.1, there exists satisfying
Combining (3.14) with (3.15) we get
Letting we get a contradiction which completes the proof. □
Let us introduce a notion of a ρtype, a powerful technical tool which will be used in the proofs of our fixed point results.
Definition 3.6 Let be convex and ρbounded. A function is called a ρtype (or shortly a type) if there exists a sequence of elements of K such that for any there holds
Note that τ is convex provided ρ is convex. A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists. This will be the subject of Lemma 3.3 below. First, let us recall the definition of the Opial property and the strong Opial property in modular function spaces, [15,17].
Definition 3.7 We say that satisfies the ρa.e. Opial property if for every which is ρa.e. convergent to 0 such that there exists a for which
the following inequality holds for any not equal to 0
Definition 3.8 We say that satisfies the ρa.e. strong Opial property if for every which is ρa.e. convergent to 0 such that there exists a for which
the following equality holds for any
Remark 3.1 Note that the ρa.e. Strong Opial property implies ρa.e. Opial property [15].
Remark 3.2 Also, note that, by virtue of Theorem 2.1 in [15], every convex, orthogonally additive function modular ρ has the ρa.e. strong Opial property. Let us recall that ρ is called orthogonally additive if whenever . Therefore, all Orlicz and MusielakOrlicz spaces must have the strong Opial property.
Note that the Opial property in the norm sense does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for spaces for while the modular strong Opial property holds in for all .
Lemma 3.3[27]
Let. Assume thathas theρa.e. strong Opial property. Letbe a nonempty, stronglyρbounded andρa.e. compact convex set. Then anyρtype defined in C attains its minimum inC.
Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.
Theorem 3.1[18]
Assumeis. LetCbe aρclosedρbounded convex nonempty subset. Then anyasymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed pointsisρclosed.
4 Demiclosedness Principle
The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem 5.1. Our proof the Demiclosedness Principle uses the parallelogram inequality valid in the modular spaces with the property (see Lemma 4.2 in [18]). We start with a technical result which will be used in the proof of Theorem 4.1.
Lemma 4.1Let. Letbe a convex set, and let. Ifis aρapproximate fixed point sequence for T, that is, as, then for every fixedthere holds
Proof It follows from 3.5 that there exists a finite constant such that
Using the convexity of ρ and the ρnonexpansiveness of T, we get
Corollary 4.1If, under the hypothesis of Lemma 4.1, ρsatisfies additionally thecondition, thenas.
The version of the Demiclosedness Principle used in this paper (Theorem 4.1) requires the uniform continuity of the function modular ρ in the sense of the following definition (see, e.g., [17]).
Definition 4.1 We say that is uniformly continuous if to every and , there exists such that
Let us mention that the uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [37] or in sequence Orlicz spaces [11] the uniform continuity of the Orlicz modular is equivalent to the type condition.
Theorem 4.1Demiclosedness Principle. Let. Assume that:
(2) ρhas strong Opial property,
(3) ρhasproperty and is uniformly continuous.
Letbe a nonempty, convex, stronglyρbounded andρclosed, and let. Let, and. Ifρa.e. and, then.
Proof Let us recall that by definition of uniform continuity of ρ to every and , there exists such that
provided and . Fix any . Noting that due to the strong ρboundedness of C and that by Corollary (4.1), it follows from (4.5) with and that
Define the ρtype φ by
By (4.7) we get
Using (4.10) with and by passing with m to infinity, we conclude that
Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since a.e., it follows via the Opial property that for any
which implies that
Combining (4.11) with (4.13), we have
that is,
We claim that
Assume to the contrary that (4.16) does not hold, that is,
By , it follows from (4.17) that does not tend to zero. By passing to a subsequence if necessary, we can assume that there exists such that
for every . Applying the modular parallelogram inequality valid in modular function spaces, see Lemma 4.2 in [18],
where , and for , with , , , , we get
Note that by (4.13)
Combining (4.22) with (4.23), we obtain
which implies
Letting and applying (4.15), we get
Using the properties of Ψ, we conclude that tends to zero itself, which contradicts our assumption (4.17). Hence, as . Clearly, then as , that is, while by ρcontinuity of T. By the uniqueness of the ρlimit, we obtain , that is, . □
5 Convergence of generalized Mann iteration process
The following elementary, easy to prove, lemma will be used in this paper.
Lemma 5.1[2]
Supposeis a bounded sequence of real numbers andis a doublyindex sequence of real numbers which satisfy
for each. Thenconverges to an.
Following Mann [29], let us start with the definition of the generalized Mann iteration process.
Definition 5.1 Let and let be an increasing sequence of natural numbers. Let be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T, the sequence , and the sequence denoted by is defined by the following iterative formula:
Definition 5.2 We say that a generalized Mann iteration process is well defined if
Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness, for every . Hence we can always select a subsequence such that (5.2) holds. In other words, by a suitable choice of , we can always make well defined.
The following result provides an important technique which will be used in this paper.
Lemma 5.2Letbe. Letbe aρclosed, ρbounded and convex set. Letand let. Assume that a sequenceis bounded away from 0 and 1. Letwbe a fixed point ofTandbe a generalized Mann process. Then there existssuch that
Denote for every and . Observe that since , it follows that . By Lemma 5.1, there exists an such that as claimed. □
The next result will be essential for proving the convergence theorems for iterative process.
Lemma 5.3Letbe. Letbe aρclosed, ρbounded and convex set, and. Assume that a sequenceis bounded away from 0 and 1. Letandbe a generalized Mann iteration process. Then
and
Proof By Theorem 3.1, T has at least one fixed point . In view of Lemma 5.2, there exists such that
Note that
and that
Set , , and note that by (5.7), and by (5.8). Observe also that
Hence, it follows from Lemma 3.2 that
which by the construction of the sequence is equivalent to
as claimed. □
In the next lemma, we prove that under suitable assumption the sequence becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.
Definition 5.3 A strictly increasing sequence is called quasiperiodic if the sequence is bounded, or equivalently, if there exists a number such that any block of p consecutive natural numbers must contain a term of the sequence . The smallest of such numbers p will be called a quasiperiod of .
Lemma 5.4Letbesatisfying. Letbe aρclosed, ρbounded and convex set, and. Letbe bounded away from 0 and 1. Letbe such that the generalized Mann processis well defined. If, in addition, the set of indicesis quasiperiodic, thenis an approximate fixed point sequence, i.e.,
Proof Let be a quasiperiod of . Observe that it is enough to prove that as through . Indeed, let us fix . From as through it follows that
for sufficiently large k. By the quasiperiodicity of , to every positive integer k, there exists such that . Assume that (the proof for the other case is identical). Since T is ρLipschitzian with the constant , there exist a such that
Note that by (5.6) and by , for k sufficiently large. This implies that
and therefore,
which demonstrates that
To prove that as through , observe that, since for such k, there holds
which tends to zero in view of (5.5), (5.6) and (5.2). □
The next theorem is the main result of this section.
(2) ρhas Strong Opial Property,
(3) ρhasproperty and is uniformly continuous.
Letbe nonempty, ρa.e. compact, convex, stronglyρbounded andρclosed, and let. Assume that a sequenceis bounded away from 0 and 1. Letandbe a welldefined generalized Mann iteration process. Assume, in addition, that the set of indicesis quasiperiodic. Then there existssuch thatρa.e.
Proof Observe that by Theorem 4.1 in [18], the set of fixed points is nonempty, convex and ρclosed. Note also that by Lemma 3.1 in [27], it follows from the strong Opial property of ρ that any ρtype attains its minimum in C. By Lemma 5.4, the sequence is an approximate fixed point sequence, that is,
as . Consider , two ρa.e. cluster points of . There exits then , subsequences of such that ρa.e. and ρa.e. By Theorem 4.1, and . By Lemma 5.2, there exist such that
We claim that . Assume to the contrary that . Then, by the strong Opial property, we have
The contradiction implies that . Therefore, has at most one ρa.e. cluster point. Since, C is ρa.e. compact it follows that the sequence has exactly one ρa.e. cluster point, which means that ρa.e. Using Theorem 4.1 again, we get as claimed. □
Remark 5.2 It is easy to see that we can always construct a sequence with the quasiperiodic properties specified in the assumptions of Theorem 5.1. When constructing concrete implementations of this algorithm, the difficulty will be to ensure that the constructed sequence is not “too sparse” in the sense that the generalized Mann process remains well defined. The similar quasiperiodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2,25,28].
6 Convergence of generalized Ishikawa iteration process
The twostep Ishikawa iteration process is a generalization of the onestep Mann process. The Ishikawa iteration process, [10], provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.
Definition 6.1 Let and let be an increasing sequence of natural numbers. Let be bounded away from 0 and 1, and be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T, the sequences , , and the sequence denoted by is defined by the following iterative formula:
Definition 6.2 We say that a generalized Ishikawa iteration process is well defined if
Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness, for every . Hence we can always select a subsequence such that (6.2) holds. In other words, by a suitable choice of , we can always make well defined.
Lemma 6.1Letbe. Letbe aρclosed, ρbounded and convex set. Letand let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letandbe a generalized Ishikawa process. There exists then ansuch that.
It is easy to see that and that . Moreover, a straight calculation shows that each satisfies
where
and
Note that , which follows directly from the fact that and from (6.5). Using (6.5) and the fact that , we have
Fix any . Since , it follows that there exists a such that for , . Therefore, using the same argument as in the proof of Lemma 5.2, we deduce that for and
Arguing like in the proof of Lemma 5.2, we conclude that there exists an such that . □
Lemma 6.2Letbe. Letbe aρclosed, ρbounded and convex set. Letand let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe a generalized Ishikawa process. Define
Then
or equivalently
Proof By Theorem 3.1, . Let us fix . By Lemma 6.1, exists. Let us denote it by r. Since , , and by Lemma 6.1, we have the following:
Note that
Applying Lemma 3.2 with and , we obtain the desired equality , while (6.11) follows from (6.10) via the construction formulas for and . □
Lemma 6.3Letbesatisfying. Letbe aρclosed, ρbounded and convex set. Letand let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe a welldefined generalized Ishikawa process. Then
Since is bounded away from 1, there exists such that for every . Hence,
The righthand side of this inequality tends to zero because by Lemma 6.2 and ρ satisfies . □
Lemma 6.4Letbesatisfying. Letbe aρclosed, ρbounded and convex set, and. Letbe bounded away from 0 and 1 andbe bounded away from 1. Letbe such that the generalized Ishikawa processis well defined. If, in addition, the setis quasiperiodic, thenis an approximate fixed point sequence, i.e.,
Proof The proof is analogous to that of Lemma 5.4 with (6.11) used instead of (5.6) and (6.14) replacing (5.5). □
(2) ρhas Strong Opial Property,
(3) ρhasproperty and is uniformly continuous.
Letbe nonempty, ρa.e. compact, convex, stronglyρbounded andρclosed, and let. Let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe such that the generalized Ishikawa processis well defined. If, in addition, the setis quasiperiodic, thengenerated byconvergesρa.e. to a fixed point.
Proof The proof is analogous to that of Theorem 5.1 with Lemma 5.4 replaced by Lemma 6.4, and Lemma 5.2 replaced by Lemma 6.1. □
7 Strong convergence
It is interesting that, provided C is ρcompact, both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property.
Theorem 7.1Letsatisfy conditionsand. Letbe aρcompact, ρbounded and convex set, and let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe such that the generalized Mann process (resp. Ishikawa process) is well defined. Then there exists a fixed pointsuch that thengenerated by (resp. ) converges strongly to a fixed point ofT, that is
Proof By the ρcompactness of C, we can select a subsequence of such that there exists with
Note that
which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By it follows from (7.3) that
Observe that by the convexity of ρ and by ρnonexpansiveness of T, we have
which tends to zero by (7.4) and by Lemma 5.3 (resp. Lemma 6.4). Hence which implies that . Applying Lemma 5.2 (resp. Lemma 6.1), we conclude that exists. By (7.4) this limit must be equal to zero which implies that
□
Remark 7.1 Observe that in view of the assumption, the ρcompactness of the set C assumed in Theorem 7.1 is equivalent to the compactness in the sense of the norm defined by ρ.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors equally participated in all stages of preparations of this article. Both authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.
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