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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 183174, 14 pages
http://dx.doi.org/10.1155/2013/183174
Research Article

On Best Proximity Point Theorems and Fixed Point Theorems for -Cyclic Hybrid Self-Mappings in Banach Spaces

Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia), P.O. Box 644 Bilbao, 48090, Spain

Received 26 December 2012; Accepted 12 February 2013

Academic Editor: Yisheng Song

Copyright © 2013 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper relies on the study of fixed points and best proximity points of a class of so-called generalized point-dependent -hybrid -cyclic self-mappings relative to a Bregman distance , associated with a Gâteaux differentiable proper strictly convex function in a smooth Banach space, where the real functions and quantify the point-to-point hybrid and nonexpansive (or contractive) characteristics of the Bregman distance for points associated with the iterations through the cyclic self-mapping. Weak convergence results to weak cluster points are obtained for certain average sequences constructed with the iterates of the cyclic hybrid self-mappings.

1. Introduction and Preliminaries

The following objects are considered through the paper.(1) The Hilbert space on the field (in particular, or ) is endowed with the inner product which maps to , for all which maps to , where is a Banach space when endowed with a norm induced by the inner product and defined by , for all . It is wellknown that all Hilbert spaces are uniformly convex Banach spaces and that Banach spaces are always reflexive.(2) The (≥2)-cyclic self-mapping with is subject to , where are subsets of , for all , that is, a self-mapping satisfying , (3) The function is a proper convex function which is Gâteaux differentiable in the topological interior of the convex set ; , that is, and convex since is proper with since is convex, and for each , there is (the topological dual of ) such that since   is Gâteaux differentiable in where denotes the Gâteaux derivative of at if . On the other hand, is said to be strictly convex if (4) The Bregman distance (or Bregman divergence) associated with the proper convex function   , where , is defined by provided that it is Gâteaux differentiable everywhere in . If is not Gâteaux differentiable at , then (4) is replaced by where and is finite if and only if , the algebraic interior of defined by The topological interior of is , where is the boundary of . It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances while they are always nonnegative because of the convexity of the function . The Bregman distance between sets is defined as . If for , then . Through the paper, sequences with are simply denoted by for the sake of notation simplicity.

Fixed points and best proximity points of cyclic self-mappings in uniformly convex Banach spaces have been widely studied along the last decades for the cases when the involved sets intersect or not. See, for instance, [13] and references therein. In parallel, interesting results have been obtained for both nonspreading, nonexpansive, and hybrid maps in Hilbert spaces including also to focus the related problems via iterative methods supported by fixed point theory and the use of more general mappings such as nonspreading and pseudocontractive mappings. See, for instance, recent background [47] and references therein. Let be a nonempty subset of a Hilbert space . On the other hand, it has to be pointed out that the characterization of several classes of iterative computations by invoking results of fixed point theory has received much attention in the background literature. See, for instance, [811] and references therein. In [1218], the existence of fixed points of mappings is discussed when is: nonexpansive; that is, , for all , nonspreading; that is, , for all ,-hybrid [17]; that is, , for all . If , then is referred to as hybrid [14, 15], and if and (1.3) is changed to, , for all ,

where is a Gâteaux differentiable convex function, then is referred to as being point-dependent -hybrid relative to the Bregman distance , [16]. A well-known result is that a nonspreading mapping, and then a nonexpansive one, on a nonempty closed convex subset of a Hilbert space has a fixed point if and only it has a bounded sequence on such a subset [18]. The result has been later on extended to -hybrid mappings, [17] and to point-dependent -hybrid ones [16]. As pointed out in [16], what follows directly from the previous definitions, is nonexpansive if and only if it is 0-hybrid while it is nonspreading if and only if it is 2-hybrid; is hybrid if and only if it is 1-hybrid.

This paper is focused on the study of fixed points and best proximity points of a class of generalized point-dependent -hybrid (≥2)-cyclic self-mappings , relative to a Bregman distance in a smooth Banach space, where is a point-dependent real function in (1.4) quantifying the “hybrideness” of the (≥2) cyclic self-mapping and is added as a weighting factor in the first right-hand-side term of (1.4). Such a function is defined through a point-dependent product of the particular point -functions while quantifies either the “nonexpansiveness” or the “contractiveness” of the Bregman distance for points associated with the iterates of the cyclic self-mapping in each of the sets for , where are nonempty closed and convex. Thus, the generalization of the hybrid map studied in this paper has two main characteristics, namely, (a) a weighting point-dependent term is introduced in the contractive condition; (b) the hybrid self-mapping is a cyclic self-mappings. Precise definitions and meaning of those functions are given in Definition 2 of Section 2 which are then used to get the main results obtained in the paper. In most of the results obtained in this paper, the Bregman distance is defined associated with a Gâteaux differentiable proper strictly convex function whose domain includes the union of the subsets of the -hybrid (≥2)-cyclic self-mapping which are not assumed, in general, to intersect. Weak convergence results to weak cluster points of certain average sequences built with the iterates of the cyclic hybrid self-mappings are also obtained. In particular, such weak cluster points are proven to be also fixed points of the composite self-mappings on the sets , even if such sets do not intersect, while they are simultaneously best proximity points of the point-dependent -hybrid (≥2)-cyclic self-mapping relative to .

2. Some Fixed Point Theorems for Cyclic Hybrid Self-Mappings on the Union of Intersecting Subsets

The Bregman distance is not properly a distance, since it does not satisfy symmetry and the triangle inequality, but it is always nonnegative and leads to the following interesting result towards its use in applications of fixed point theory.

Lemma 1. If is a proper strictly convex function being Gâteaux differentiable in , then

Proof. By using (4) for and defining , for all by interchanging and in the definition of in (4), which leads to (9) since , for all , [16, 17], if is proper strictly convex, and the fact that , for all .
Equation (7) follows from (9) for leading to . To prove (8), take and proceed by contradiction using (4) by assuming that for such so that which contradicts . Then, , and, hence, (8) follows And, hence, (10) via (7) and (9).

The following definition is then used.

Definition 2. If , for all , and is a proper convex function which is Gâteaux differentiable in , then the -cyclic self-mapping , where and , for all , is said to be a generalized contractive point-dependent -hybrid -cyclic self-mapping relative to if for some given functions and with , for all , where defined by for any , for all .
If, furthermore, , for all , then is said to be a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to .

If , it is possible to characterize as a trivial -cyclic self-mapping with which does not need to be specifically referred to as 1-cyclic.

Although depends on , the whole does not depend on so that the cyclic self-mapping is referred to as generalized point-dependent -hybrid in the definition.

The following concepts are useful. is said to be totally convex if the modulus of total convexity ; that is, is positive for . is said to be uniformly convex if the modulus of uniform convexity ; that is, is positive for . It holds that , for all [16]. The following result holds.

Theorem 3. Assume that(1) is a lower-semicontinuous proper strictly totally convex function which is Gâteaux differentiable in ; (2), for all , are bounded, closed, and convex subsets of which intersect and is a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to for some given functions and , defined by for any , for all , and some functions , for all , with being bounded;(3)there is a convergent sequence to some for some .
Then, is the unique fixed point of to which all sequences converge for any , for all .

Proof. The recursive use of (14) yields with , with , , for all , where is the identity mapping on . Now, define so that one gets since , for all , since is a generalized contractive point-dependent -hybrid (≥2)-cyclic self-mapping relative to , implies that , for all , since for any , for all (then for any ), where where < 1, since , for all , so that since is bounded, is lower-semicontinuous then with all subgradients in any bounded subsets of being bounded, and and , for all , for all , converge so that they are Cauchy sequences being then bounded, for all , for all , where , since is nonempty and closed, is some fixed point of . As a result, , for all , for all , for all . From a basic property of Bregman distance, , as , for all , for all , for all , if is sequentially consistent. But, since is closed, is sequentially consistent if and only if it is totally convex [19]. Thus, converges also to for any and , for all , so that is a fixed point of . Assume not and proceed by contradiction so as then obtaining ; as from a basic property of Bregman distance. Thus, as since as . As a result, , as from the continuity of , and is a fixed point of . Now, take any so that , then, since is a fixed point of and is a proper strictly totally convex function. As a result, converges to , for all .
It is now proven that is the unique fixed point of . Assume not so that there is . Then, as from (17) for since is a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to with and so that , since if since is proper and totally strictly convex, and since , and . Since is closed and convex, it turns out that is the unique fixed point of .
Note that the result also holds for any for all since maps to for some nonnegative integer through the self-mapping so that as since is the unique fixed point of and converges to for any .

The subsequent result directly extends Theorem 3 to the -composite self-mappings , , defined as ; for all , subject to , for all . The subsets ,   are not required to intersect since the restricted composite mappings as defined earlier are self-mappings on nonempty, closed, and convex sets.

Corollary 4. Assume that (1)   is a proper strictly totally convex function which is lower-semicontinuous and Gâteaux differentiable in , and, furthermore, it is bounded on any bounded subsets of ;  (2)   is bounded and closed, for all , is a -cyclic self-mapping so that   for some is a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to for some given functions   and   for some defined by for any , for all , and for some , for all , where , being bounded and being, furthermore, convex for the given ;  (3)  there is a convergent sequence to some for some and .
Then, is a unique fixed point of to which all sequences converge for any for .
Also, if conditions (1)–(3) are satisfied with all the subsets , for all , being nonempty, closed, and convex for some proper strictly convex function which is Gâteaux differentiable in , then , for all , is a unique fixed point of , for all , to which all sequences converge for any , for all . The unique fixed points of each generalized point-dependent -hybrid -cyclic composite self-mappings , for all , fulfil the relations for , for all , for all .

Outline of Proof. Note that . Equation (14) is now extended to for the given leading to since is a trivial 1-cyclic self-mapping on for . The previous relation leads recursively to. with , , for the given with , where is independent of the particular for . One gets by using very close arguments to those used in the proof of Theorem 3 that . Then, converges to some which is proven to be a unique fixed point in the nonempty, closed, and convex set for . The remaining of the proof is similar to that of Theorem 3. The last part of the result follows by applying its first part to each of the generalized point-dependent -hybrid -cyclic composite self-mappings relative to , for all .

Remark 5. If is totally convex if it is a continuous strictly convex function which is Gâteaux differentiable in , and is closed, [20]. In view of this result, Theorem 3 and Corollary 4 are still valid if the condition of its strict total convexity of is replaced by its continuity and its strict convexity if the Banach space is finite dimensional. Since , for all , it turns out that if is uniformly convex, then it is totally convex. Therefore, Theorem 3 and Corollary 4 still hold if the condition of strict total convexity is replaced with the sufficient one of strict uniform convexity. Note that if a convex function is totally convex then it is sequentially consistent in the sense that as if as for any sequences and in .

Some results on weak cluster points of average sequences built with the iterated sequences generated from hybrid cyclic self-mappings relative to a Bregman distance , for and some , are investigated in the following results related to the fixed points of .

Theorem 6. Assume that(1) is a reflexive space and is a lower-semicontinuous strictly convex function, so that it is Gâteaux differentiable in , and it is bounded on any bounded subsets of ;(2)a -cyclic self-mapping is given defining a composite self-mapping with being bounded, convex, and closed, for all , so that its restricted composite mapping to , for some given , is generalized point-dependent -hybrid relative to for some   and the given .

Define the sequence for , where is the identity mapping on so that , for all , and assume that is bounded for . Then, the following properties hold.(i)Every weak cluster point of for is a fixed point of of for the given . Under the conditions of Theorem 3, there is a unique fixed point of which coincides with the unique cluster point of .(ii)Define sequences for any integer and where are bounded, closed, and convex, for all . Thus, converges weakly to for , where is a fixed point of and a weak cluster point of for and () is both a fixed point of and a weak cluster point of for . Furthermore, if is continuous.

Proof. Using (14) with being a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to for , with , for all , yields Summing up from to and taking yields since is bounded for , its subsequence is then bounded for , and is also bounded on the bounded subset of . Then, converges to zero since is bounded for . Since is lower-semicontinuous, the set of subgradients is bounded in all bounded subsets , for all . As a result, is bounded for and has a subsequence being weakly convergent to some for since is reflexive. One gets by taking that Hence, it follows from Lemma 1 that is a fixed point of   for the given . Now, consider ,  , for any integers so that for . Hence, Property (i) follows with the uniqueness and the coincidence of the weak cluster point of and fixed point of under Theorem 3. The previous reasoning remains valid so that is weakly convergent to some since is bounded for (since is bounded for and is finite), its subsequence for is also bounded so that converges to zero since in is bounded for , . Now, take so that , and then , , is both a fixed point of and a weak cluster point of . Now, take so that . Then, as for and, in addition, if is continuous. Hence, Property (ii) follows.

3. Extensions for Generalized Point-Dependent Cyclic Hybrid Self-Mappings on Nonintersecting Subsets: Weak Convergence to Weak Cluster Points of a Class of Sequences

Some of the results of Section 2 are now generalized to the case when the subsets of the cyclic mapping do not intersect , in general, by taking advantage of the fact that best proximity points of such a self-mapping are fixed points of the restricted composite mapping for . Weak convergence of averaging sequences to weak cluster points and their links with the best proximity points in the various subsets of the -cyclic self-mappings is discussed. Firstly, the following result follows from a close proof to that of Theorem 6 which is omitted.

Theorem 7. Let be a reflexive space, and let be a lower-semicontinuous strictly convex function so that it is Gâteaux differentiable in and it is bounded on any bounded subsets of . Consider the generalized point-dependent -cyclic hybrid self-mapping being relative to for some such that are all bounded, convex, closed, and with nonempty intersection. Define the sequence for , where is the identity mapping on , and assume that is bounded for . Then, the following properties hold. (i) Every weak cluster point of for is a fixed point of .  (ii) Define the sequence for which is bounded, closed, and convex, for all and any integer . Thus, converges weakly to the fixed point of for which is also a weak cluster point of .

Remark 8. The results of Theorems 6 and 7 are extendable without difficulty to the weak cluster points of other related sequences to the considered ones.
Define sequences , , for any given finite non-negative integer under all the hypotheses of Theorem 7. With this notation, the sequence considered in such a corollary is . Direct calculation yields for as since , and then , is bounded. Then, weakly which is the same fixed point of in which is a weak cluster point of for for any finite non-negative integer .
Consider all the hypotheses of Theorem 7 and now define sequences , , for any given finite non-negative integer . With this notation, the sequence considered in the corollary is . Direct calculation yields weakly for as since , and then , is bounded. Then, weakly which is the same fixed point of in which is a weak cluster point of for and for any finite non-negative integer .
Now consider the hypotheses of Theorem 6. It turns out that the sequence for satisfies for any integer , weakly as , where since , is bounded, and for and . Thus, is a fixed point of which is also a weak cluster point of the sequences for . However, it is not guaranteed that without additional hypotheses on such as its continuity, or at least that of the composite mapping allowing to equalize the function of the limit with the limit of the function at such a fixed point.
Now, define for . Note that for , , weakly as since is finite, which is a fixed point in of the composite mapping and a weak cluster point of for finite .

Note that Theorem 6 are supported by boundedness constraints for the sequences of iterates obtained through the cyclic self-mapping which is generalized point-dependent with respect to some convex function. The results of identification of weak cluster points of some average sequences with fixed points of the cyclic self-mapping or its composite mappings do not guarantee uniqueness of fixed points and weak cluster points because the cyclic self-mapping is not restricted to be contractive. By incorporating some background contractive-type conditions for the cyclic self-mapping, the previous results can be extended to include uniqueness of fixed points as follows.

Theorem 9. Assume that.(1)Assumption 1 of Theorem 6 holds with the restriction of to be a uniformly convex Banach space;(2)Assumption 2 of Theorem 6 holds, and, furthermore, all the -cyclic composite mappings with restricted domain ; for all are either contractive or Meir-Keeler contractions.

Then, the following properties hold.(i)Theorem 6 holds. Furthermore, each of the mappings has a unique fixed point which are also best proximity points of in so that ; for all , for all .(ii)If, in addition, , then, there is a unique fixed point of and , for all .

Proof. Note that uniformly convex Banach spaces are also reflexive spaces required by Theorem 6. Each mapping has a unique fixed point , for all , irrespective of being empty or not if is either a cyclic contraction or a Meir-Keeler contraction [13], since are non-empty, closed, and convex, and is a uniformly convex Banach space so that each ; for all is a best proximity point in of . It follows from the hypothesis that there is a unique weak cluster point of for which is the unique fixed point of , for all , and also the unique best proximity point of in for .
It is now proven that if then , for all . Take some for some . Thus, and as since is the unique fixed point of and is the unique fixed point of . Then, , for all ; for all