Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces
© Dehaish and Kozlowski; licensee Springer 2012
Received: 4 April 2012
Accepted: 2 July 2012
Published: 20 July 2012
Let be a uniformly convex modular function space with a strong Opial property. Let be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.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.
Keywordsfixed point nonexpansive mapping fixed point iteration process Mann process Ishikawa process modular function space Orlicz space Opial property uniform convexity
where , for all . Their main result (Theorem 3.5) states that every asymptotic pointwise nonexpansive self-mapping 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, .
In 2011, Khamsi and Kozlowski  extended their result proving the existence of fixed points of asymptotic pointwise ρ-nonexpansive 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 ρ-nonexpansive 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, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others, see the book by Kozlowski  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., [3–5, 8, 13, 14, 17–20, 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, 30–36, 38–42] and the works referred there. Kozlowski proved convergence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces  and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces . 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 .
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.
Section 2 provides necessary preliminary material on modular function spaces.
Section 3 introduces the asymptotic pointwise nonexpansive mappings and related notions.
Section 4 deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.
Section 5 utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.
Section 6 establishes the almost everywhere convergence theorem for generalized Ishikawa process.
Section 7 provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact set C.
Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring 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.
ρ is monotone, i.e., for all implies , where ;
ρ is orthogonally subadditive, i.e., for any such that , ;
ρ has the Fatou property, i.e., for all implies , where ;
ρ is order continuous in ℰ, i.e., and implies .
where each is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists we will write ℳ instead of .
We say that ρ is a regular convex function semimodular if for every implies ρ-a.e.;
We say that ρ is a regular convex function modular if implies ρ-a.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  (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 [22–24].
Remark 2.1 We limit ourselves to convex function modulars in this paper. However, omitting convexity in Definition 2.1 or replacing it by s-convexity would lead to the definition of nonconvex or s-convex regular function pseudomodulars, semimodulars and modulars as in .
- (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.
, is complete and the norm is monotone w.r.t. the natural order in ℳ.
if and only if for every .
If for an then there exists a subsequence of such that ρ-a.e.
If converges uniformly to f on a set then for every .
Let ρ-a.e. There exists a nondecreasing sequence of sets such that and converges uniformly to f on every (Egoroff theorem).
whenever ρ-a.e. (Note: this property is equivalent to the Fatou property.)
Defining and we have:
has the Lebesgue property, i.e., for , and .
is the closure of ℰ (in the sense of ).
The following definition plays an important role in the theory of modular function spaces.
whenever and .
ρ has ,
is a linear subspace of ,
if , then ,
if for 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.
We say that is ρ-convergent to f and write if and only if .
A sequence where is called ρ-Cauchy if as .
A set is called ρ-closed if for any sequence of , the convergence implies that f belongs to B.
A set is called ρ-bounded if .
A set is called strongly ρ-bounded if there exists such that .
A set is called ρ-compact if for any in C there exists a subsequence and an such that .
A set is called ρ-a.e. closed if for any in C which ρ-a.e. converges to some f, then we must have .
A set is called ρ-a.e. compact if for any in C, there exists a subsequence which ρ-a.e. converges to some .
- (i)Let and . The ρ-distance between f and C is defined as
Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, does not imply in general , . Using Theorem 2.1, it is not difficult to prove the following:
ρ-balls are ρ-closed and ρ-a.e. closed.
Let us compare different types of compactness introduced in Definition 2.5.
If C is ρ-compact, then C is ρ-a.e. compact.
If C is -compact, then C is ρ-compact.
If ρ satisfies , then -compactness and ρ-compactness are equivalent in .
follows from Theorem 2.1 part (3).
follows from Theorem 2.1 part (2).
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, .
If for every and every , then T is called ρ-nonexpansive or shortly nonexpansive.
If converges pointwise to , then T is called asymptotic pointwise contraction.
If for any , then T is called asymptotic pointwise nonexpansive.
If for any with , then T is called strongly asymptotic pointwise contraction.
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., .
We recall the following concepts related to the modular uniform convexity introduced in :
and if . We will use the following notational convention: .
We will need the following result whose proof is elementary. Note that for , this result follows directly from Definition 3.4.
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 .
Letting we get a contradiction which completes the proof. □
Let us introduce a notion of a ρ-type, a powerful technical tool which will be used in the proofs of our fixed point results.
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].
Remark 3.1 Note that the ρ-a.e. Strong Opial property implies 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 has the ρ-a.e. strong Opial property. Let us recall that ρ is called orthogonally additive if whenever . Therefore, all Orlicz and Musielak-Orlicz 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 .
Let. Assume thathas the ρ-a.e. strong Opial property. Letbe a nonempty, strongly ρ-bounded and ρ-a.e. compact convex set. Then any ρ-type defined in C attains its minimum in C.
Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.
Assumeis. Let C be a ρ-closed ρ-bounded convex nonempty subset. Then anyasymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed pointsis ρ-closed.
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 ). We start with a technical result which will be used in the proof of Theorem 4.1.
as . □
Corollary 4.1 If, under the hypothesis of Lemma 4.1, ρ satisfies 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., ).
provided and .
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  or in sequence Orlicz spaces  the uniform continuity of the Orlicz modular is equivalent to the -type condition.
ρ is ,
ρ has strong Opial property,
ρ has property and is uniformly continuous.
Letbe a nonempty, convex, strongly ρ-bounded and ρ-closed, and let. Let, and. Ifρ-a.e. and, then.
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 ρ-continuity of T. By the uniqueness of the ρ-limit, we obtain , that is, . □
5 Convergence of generalized Mann iteration process
The following elementary, easy to prove, lemma will be used in this paper.
for each. Thenconverges to an.
Following Mann , let us start with the definition of the generalized Mann iteration process.
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.
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.
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 quasi-periodic 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 quasi-period of .
as . By again, we get .
which tends to zero in view of (5.5), (5.6) and (5.2). □
The next theorem is the main result of this section.
ρ is ,
ρ has Strong Opial Property,
ρ has property and is uniformly continuous.
Letbe nonempty, ρ-a.e. compact, convex, strongly ρ-bounded and ρ-closed, and let. Assume that a sequenceis bounded away from 0 and 1. Letandbe a well-defined generalized Mann iteration process. Assume, in addition, that the set of indicesis quasi-periodic. Then there existssuch thatρ-a.e.
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. 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 quasi-periodic 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 quasi-periodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2, 25, 28].
6 Convergence of generalized Ishikawa iteration process
The two-step Ishikawa iteration process is a generalization of the one-step Mann process. The Ishikawa iteration process, , provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.
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 6.1 Letbe. Letbe a ρ-closed, ρ-bounded 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.
Arguing like in the proof of Lemma 5.2, we conclude that there exists an such 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 . □
The right-hand side of this inequality tends to zero because by Lemma 6.2 and ρ satisfies . □
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). □
ρ is ,
ρ has Strong Opial Property,
ρ has property and is uniformly continuous.
Letbe nonempty, ρ-a.e. compact, convex, strongly ρ-bounded and ρ-closed, 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 quasi-periodic, thengenerated byconverges ρ-a.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 ρ-compact, both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property.
Remark 7.1 Observe that in view of the assumption, the ρ-compactness of the set C assumed in Theorem 7.1 is equivalent to the compactness in the sense of the norm defined by ρ.
The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.
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