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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 316813, 14 pages
http://dx.doi.org/10.1155/2013/316813
Research Article

Bregman Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces

1Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan
2Department of Mathematics, Yasouj University, Yasouj 75918, Iran
3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

Received 20 September 2013; Accepted 11 November 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Chin-Tzong Pang and Eskandar Naraghirad. 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

We first introduce a new class of mappings called Bregman asymptotic pointwise nonexpansive mappings and investigate the existence and the approximation of fixed points of such mappings defined on a nonempty, bounded, closed, and convex subset C of a real Banach space E. Without using the original Opial property of a Banach space E, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes for Bregman asymptotic pointwise nonexpansive mappings in a reflexive Banach space E. Our results are applicable in the function spaces , where is a real number.

1. Introduction

Throughout this paper, we denote the set of real numbers and the set of positive integers by and , respectively. Let be a Banach space with the norm and the dual space . For any , we denote the value of at by . Let be a sequence in ; we denote the strong convergence of to as by and the weak convergence by . The modulus of convexity of is denoted by for every with . A Banach space is said to be uniformly convex if for every . Let . The norm of is said to be Gâteaux differentiable if for each , the limit exists. In this case, is called smooth. If the limit (2) is attained uniformly for all , then is called uniformly smooth. The Banach space is said to be strictly convex if whenever and . It is well known that is uniformly convex if and only if is uniformly smooth. It is also known that if is reflexive, then is strictly convex if and only if is smooth; for more details, see [1, 2].

Let be a nonempty subset of . Let be a mapping. We denote the set of fixed points of by ; that is, . A mapping is said to be nonexpansive if for all . A mapping is said to be quasi-nonexpansive if and for all and . The nonexpansivity plays an important role in the study of Mann iteration [3] for finding fixed points of a mapping . Recall that the Mann iteration is given by the following formula: Here, is a sequence of real numbers in satisfying some appropriate conditions. A more general iteration is the Ishikawa iteration [4], given by where the sequences and satisfy some appropriate conditions. When all , the Ishikawa iteration reduces to the classical Mann iteration. Construction of fixed points of nonexpansive mappings via Mann’s and Ishikawa’s algorithms [3] has been extensively investigated in the literature (see, e.g., [5] and the references therein). A powerful tool in deriving weak or strong convergence of iterative sequences is due to Opial [6]. A Banach space is said to satisfy the Opial property [6] if for any weakly convergent sequence in with weak limit , we have for all in with . It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces () satisfy the Opial property. However, not every Banach space satisfies the Opial property; see, for example, [7].

Let be a smooth, strictly convex, and reflexive Banach space and let be the normalized duality mapping of . Let be a nonempty, closed, and convex subset of . The generalized projection from onto [8] is defined and denoted by where . Let be a nonempty, closed, and convex subset of a smooth Banach space , and let be a mapping from into itself.

1.1. Some Facts about Gradients

For any convex function we denote the domain of by dom . For any intdom and any , we denote by the right-hand derivative of at in the direction ; that is, The function is said to be Gâteaux differentiable at if exists for any . In this case coincides with , the value of the gradient of at . The function is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function is said to be Fréchet differentiable at if this limit is attained uniformly in . The function is Fréchet differentiable at (see, e.g., [9, page 13] or [10, page 508]) if for all , there exists such that implies that The function is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function is Gâteaux differentiable, then is norm-to-weak* continuous (see, e.g., [9, Proposition ]). Also, it is known that if is Fréchet differentiable, then is norm-to-norm continuous (see, [10, page 508]). The mapping is said to be weakly sequentially continuous if as implies that as (for more details, see [9, Theorem ] or [10, page 508]). The function is said to be strongly coercive if It is also said to be bounded on bounded subsets of if is bounded for each bounded subset of . Finally, is said to be uniformly Fréchet differentiable on a subset of if the limit (7) is attained uniformly for all and .

Let be a reflexive Banach space. For any proper, lower semicontinuous, and convex function , the conjugate function of is defined by for all . It is well known that for all . It is also known that is equivalent to Here, is the subdifferential of [11, 12]. We also know that if is a proper, lower semicontinuous, and convex function, then is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.

1.2. Some Facts about Bregman Distances

Let be a Banach space and let be the dual space of . Let be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] corresponding to is the function defined by It is clear that for all . In that case when is a smooth Banach space, setting for all , we obtain that for all and hence for all .

Let be a Banach space and let be a nonempty and convex subset of . Let be a convex and Gâteaux differentiable function. Then, we know from [15] that for and , if and only if Furthermore, if is a nonempty, closed, and convex subset of a reflexive Banach space and is a strongly coercive Bregman function, then for each , there exists a unique such that The Bregman projection from onto is defined by for all . It is also well known that has the following property: for all and (see [9] for more details).

For any bounded subset of a reflexive Banach space , we denote the Bregman diameter of by

1.3. Some Facts about Uniformly Convex Functions

Let be a Banach space and let for all . Then a function is said to be uniformly convex on bounded subsets of ([16, Pages 203, 221]) if for all , , where is defined by for all . The function is called the gauge of uniform convexity of . The function is also said to be uniformly smooth on bounded subsets of ([16, Pages 207, 221]) if for all , where is defined by for all . The function is said to be uniformly convex if the function , defined by satisfies that .

Remark 1. Let be a Banach space, let be a constant, and let be a convex function which is uniformly convex on bounded subsets. Then for all and , where is the gauge of uniform convexity of .

Definition 2. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of . A mapping is said to be Bregman asymptotic pointwise nonexpansive if there exists a sequence of mappings such that Denoting , we note that without loss of generality we can assume that is Bregman asymptotic pointwise nonexpansive if Define . In view of (23), we obtain Next, we denote by the class of all Bregman asymptotic pointwise nonexpansive mappings .
Imposing some restrictions on the behavior of and , we can define the following subclass of Bregman asymptotic pointwise nonexpansive mappings.

Definition 3. Let and be as in Definition 2. We define as a class of all such that Kirk and Xu [17] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings with respect to the norm of a Banach space . Recently, Kozlowski [18] proved weak and strong convergence theorems for asymptotic pointwise nonexpansive mappings in a Banach space. To see some other related works, we refer the reader to [19, 20].

In this paper, we first investigate the approximation of fixed points of a new class of Bregman asymptotic pointwise nonexpansive mappings defined on a nonempty, bounded, closed, and convex subset of a real Banach space . Without using the Opial property of a Banach space , we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes. Our results improve and generalize many known results in the current literature; see, for example, [18, 21].

2. Preliminaries

In this section, we begin by recalling some preliminaries and lemmas which will be used in the sequel.

Definition 4 (see [10]). Let be a Banach space. The function is said to be a Bregman function if the following conditions are satisfied:(1)is continuous, strictly convex, and Gâteaux differentiable;(2)the set is bounded for all and .

Lemma 5 (see [9, 16]). Let be a reflexive Banach space and a strongly coercive Bregman function. Then (1) is one-to-one, onto, and norm-to-weak* continuous;(2) if and only if ;(3) is bounded for all and ;(4) is Gâteaux differentiable and .

We know the following two results; see [16, Proposition ].

Theorem 6. Let be a reflexive Banach space and a convex function which is bounded on bounded subsets of . Then the following assertions are equivalent:(1) is strongly coercive and uniformly convex on bounded subsets of ;(2) is bounded on bounded subsets and uniformly smooth on bounded subsets of ;(3) is Frchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .

Theorem 7. Let be a reflexive Banach space and a continuous convex function which is strongly coercive. Then the following assertions are equivalent:(1) is bounded on bounded subsets and uniformly smooth on bounded subsets of ;(2) is Frchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of ;(3) is strongly coercive and uniformly convex on bounded subsets of .

Let be a Banach space and let be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] does not satisfy the well known properties of a metric, but it does have the following important property, which is called the three point identity [22]: In particular, it can easily be seen that Indeed, by letting in (26) and taking into account that , we get the desired result.

Lemma 8 (see [23]). Let be a Banach space and a Gâteaux differentiable function which is uniformly convex on bounded subsets of . Let and be bounded sequences in . Then the following assertions are equivalent:(1);(2).

Lemma 9 (see [10, 24]). Let be a reflexive Banach space, a strongly coercive Bregman function, and the function defined by Then the following assertions hold:(1) for all and ;(2) for all and .

Let and be nonempty subsets of a real Banach space with . A mapping is said to be sunny if for each and . A mapping is said to be a retraction if for each .

Lemma 10 (see [25]). Suppose is a bounded sequence of real numbers and is a doubly index sequence of real numbers which satisfy for each . Then converges to an .

Let be a reflexive Banach space and let be an admissible function, that is, a proper, lower-semicontinuous, convex, and Gâteaux differentiable function. Let be a nonempty, closed, and convex subset of and let be a bounded sequence in . For any in , we set The Bregman asymptotic radius of relative to is defined by The Bregman asymptotic center of relative to is the set

The following Bregman Opial-like inequality has been proved in [26]. It is worth mentioning that the Bregman Opial-like inequality is different from the ordinary Opial inequality [6] and can be applied in uniformly convex Banach spaces.

Lemma 11 (see [26]). Let be a Banach space and let be a proper strictly convex function so that it is Gâteaux differentiable on . Suppose is a sequence in such that for some . Then

Theorem 12 (see [16]). Let be a function. Then the following assertions are equivalent:(1) is convex and lower semicontinuous;(2) is convex and weakly lower semicontinuous;(3) is convex and closed;(4) is convex and weakly closed,
where denotes the epigraph of .

3. Fixed Point Theorems and Demiclosedness Principle

Proposition 13. Let be a reflexive Banach space and let be a strongly admissible function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of and let . Then has a fixed point. Moreover, is closed and convex.

Proof. We first show that is nonempty. Let in be fixed. We define a function by In view of Remark 1, it is easy to see that is convex. Since is continuous, by Theorem 12 we conclude that the Bregman distance is weakly lower-semicontinuous in the first argument. Since is a uniformly convex Banach space and is weakly compact, in view of [1] there exists a unique point such that We show that is convergent in norm. To this end, let and be the gauge of uniform convexity of . For any , , put . Then we have . In view of Remark 1, we obtain Applying to both sides of the above inequalities we obtain This, together with , implies that Letting , in (39) we conclude that From the properties of , we deduce that . Thus, is a norm-Cauchy sequence and hence convergent. Let Since is a Bregman asymptotic pointwise nonexpansive mapping, we have, for all , Letting in (42), we conclude that . This shows that
Now, we show that is closed. Let be a sequence in such that as . Then we have that is a bounded sequence in . We claim that . Since is continuous, we conclude that as . This implies that In view of the definition of , we obtain This implies that It follows from Lemma 8 that . Thus we have .
Let us show that is convex. For any , , , and , we set and . We prove that . By the definition of Bregman distance (see (12)), we get This implies that . Thus for each , there exists such that This means that the sequence is bounded. In view of Definition 4, we conclude that the sequence is bounded. Then, by Lemma 8, we obtain . Thus we have ; that is, . On the other hand, in view of three-point identity (see (26)), we deduce that Letting in the above inequalities we deduce that and hence by Lemma 8 we conclude that , which completes the proof.

Lemma 14. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of and let . If is a sequence of such that , then, for any , .

Proof. In view of (25), there exists a finite constant such that It follows from three-point identity (see (26)) that where . This, together with Lemma 8, implies that . This completes the proof.

Theorem 15. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of and let . If converges weakly to and , then . That is, is demiclosed at zero, where is the identity mapping on .

Proof. Let the function be defined by For any with , in view of three-point identity (see (26)), we obtain where . This, together with Lemma 8, implies that In view of (25), there exists a finite constant such that where . This, together with Lemma 8, implies that . Employing Lemma 8, we conclude that This means that Since is a Bregman asymptotic pointwise nonexpansive mapping, it follows that In view of (57), we deduce that By the Bregman Opial-like inequality ((34)) we obtain that for any This shows that . Thus we have Put , , and for all . Then we have . In view of Remark 1, we obtain a continuous strictly increasing convex function with such that Applying to both sides and remembering that we obtain This implies that Letting in (63) we conclude that From the properties of , we deduce that . Since is uniformly norm-to-norm continuous on bounded subsets of , we arrive at . Thus we have ; that is, . On the other hand, in view of three-point identity (see (26)), we deduce that Letting in the above inequalities we deduce that and hence by Lemma 8 we conclude that , which completes the proof.

4. Weak Convergence Theorems of Generalized Mann Iteration Process

In this section, we prove weak convergence theorems of generalized Mann iteration process in a reflexive Banach space.

Definition 16. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of and let . Let be an increasing sequence in and let be such that . The generalized Mann iteration process generated by the mapping , the sequence , and the sequence , denoted by , is defined by the following iterative formula:

Definition 17. We say that a generalized Mann iteration process is well defined if

Remark 18. Observe that by the definition of Bregman asymptotic pointwise nonexpansive, for every . Hence we can always select a subsequence of such that (67) holds. In other words, by a suitable choice of we can always make well defined.

We will prove a series of lemmas necessary for the proof of the generalized Mann process convergence theorem.

Lemma 19. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of . Let and let . Let such that . Let and let be a generalized Mann process. Then there exists such that .

Proof. Let be arbitrary chosen. In view of (66), we obtain This implies that for every , Put for every and . Since , we obtain that . In view of Lemma 11, there exists such that . This completes the proof.

Lemma 20. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of . Let and let . Let such that and let be a generalized Mann process. Then

Proof. In view of Proposition 13, we conclude that Let be fixed. It follows from Lemma 19 that there exists such that . Let and let be the gauge of uniform convexity of . By the definition of , we obtain This implies that Letting in (72) we conclude that From the properties of , we deduce that . Employing Lemma 8, we conclude that This completes the proof.

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 21. A strictly increasing sequence is called quasiperiodic if the sequence is bounded or equivalently if there exists a number such that any block of consecutive natural numbers must contain a term of the sequence . The smallest of such numbers will be called a quasi period of .

Lemma 22. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, bounded, closed, and convex subset of . Let and let . Let such that . Let be such that the generalized Mann process is well defined. If, in addition, the set of indices is quasi-periodic, then is an approximate fixed point sequence; that is,

Proof. In view of (66), we have This, together with Lemmas 8 and 20, implies that In view of Lemma 8, we conclude that Let be a quasi-period of . We first prove that as through . Since for such , we obtain where . This implies that