The Scientific World Journal

Volume 2014 (2014), Article ID 493450, 8 pages

http://dx.doi.org/10.1155/2014/493450

Research Article

Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Bregman Weak Relativity Nonexpansive Mappings in Reflexive Banach Spaces

Habtu Zegeye¹ and Naseer Shahzad²

¹Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana

²Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 28 August 2013; Accepted 22 January 2014; Published 13 March 2014

Academic Editors: L. B. Ćirić, B. Lamichhane, T. Li, A. Moudafi, and C. Zhai

Copyright © 2014 Habtu Zegeye and Naseer Shahzad. 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 introduce an iterative process for finding an element of a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Throughout this paper, is a proper, lower semicontinuous, and convex function, where is a real reflexive Banach space with as its dual. Denote the domain of by ; that is, . Let be a Gâteaux differentiable (see Section 2) function. The function defined by

where is the gradient of , is called the Bregman distance with respect to [1]. The following property of Bregman distance function is known:

for and (see [1, 2]).

A Bregman projection with respect to [1] of onto the nonempty closed and convex set is the unique vector satisfying

Remark 1. If is a smooth Banach space, setting , for all we have , for all , where is the normalized duality mapping from onto , and hence we have the following.

(i) reduces to , for all , which is the Lyapunov function introduced by Alber [3]. If , a Hilbert space, is identity mapping and hence becomes , for .(ii)The Bregman projection reduces to the generalized projection (see, e.g., [3]) which is defined by

Let be a nonempty and convex subset of and let be a mapping. A mapping is said to be nonexpansive if , for all . is said to be quasi-nonexpansive if and , for all and , where stands for the fixed point set of ; that is, . A point is called an asymptotic fixed point of (see [4]) if contains a sequence which converges weakly to such that . We denote by the set of asymptotic fixed points of . A point is called a strong asymptotic fixed point of if there exists a sequence in which converges strongly to and . We denote the set of all strong asymptotic fixed points of by .

A mapping with is called

(i)Bregman quasi-nonexpansive [5] if

(ii)Bregman relatively nonexpansive [5] if

(iii)Bregman firmly nonexpansive [6] if, for all ,

or, equivalently,

(iv)Bregman weak relatively nonexpansive if

Remark 2. We observe from the above definitions that every relatively nonexpansive mapping is Bregman relatively nonexpansive mapping with respect to , for all , where is called relatively nonexpansive mapping if the following conditions are satisfied:

If , a real Hilbert space, then relatively nonexpansive mappings are demiclosed quasi-nonexpansive mappings which include the class of nonexpansive mappings with fixed point nonempty.

Remark 3. It is shown in [6] that if is Bregman firmly nonexpansive then and hence it is Bregman relatively nonexpansive provided that the Legendre function is uniformly Fréchet differentiable and bounded on bounded sets of .

Remark 4. We observe from the above facts that the class of Bregman weak relatively nonexpansive mappings includes the class of Bregman relatively nonexpansive mappings and hence the class of Bregman firmly nonexpansive mappings. In addition, we also have that every continuous Bregman quasi-nonexpansive is Bregman weak relatively nonexpansive mapping.

The following example by Chen et al. [7] shows that the inclusion is proper.

Example 5. Let , for all . Let be defined by , , , , where

Let be defined by

Then, it is shown in [7] that is Bregman weak relatively nonexpansive but not Bregman relatively nonexpansive.

Construction of fixed points of nonexpansive mappings and relatively nonexpansive mappings and their generalizations is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [8–14] and the references therein). Mann [15] and Ishikawa [16] iteration process for approximating fixed point iteration process for nonexpansive mappings and relatively nonexpansive mappings in Hilbert spaces and Banach spaces have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities (see, e.g., [15–21]). However, both iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [15, 16]). Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have been made.

In 2003, Nakajo and Takahashi [22] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space :

where and denotes the metric projection from onto a closed convex subset of . They proved that the sequence defined by (13) converges strongly to a fixed point of under some suitable conditions.

In 2005, Matsushita and Takahashi [23] introduced the following modification of the Mann iteration method for a relatively nonexpansive mapping in a Banach space as follows:

where and is as shown in Remark 1. They showed that generated by (14) converges strongly to a fixed point of under some suitable assumptions.

In [24], Reich and Sabach proposed an algorithm for finding a common fixed point of finitely many Bregman firmly nonexpansive mappings satisfying in a reflexive Banach space as follows:

They proved that, under suitable conditions, the sequence generated by (15) converges strongly to and applied it to the solution of convex feasibility and equilibrium problems. You may also see [5, 25].

Inspired and motivated by the above works, Chen et al. [7] proposed an algorithm for finding a fixed point of Bregman weak relatively nonexpansive mapping satisfying in a reflexive Banach space as follows:

where satisfy certain conditions and is an error sequence in with as . They proved that, under suitable conditions, the sequence generated by (16) converges strongly to , where is the Bregman projection of onto .

Moreover, in [26], Naraghirad and Yao introduced a hybrid iteration algorithm for finding a common fixed point of infinite family of Bregman weak relatively nonexpansive mapping provided that , in a reflexive Banach space . They proved that, under suitable conditions, their hybrid iteration algorithm converges strongly to .

Remark 6. It is worth mentioning that all the iteration algorithms introduced and used above seem cumbersome and complicated in the sense that at each stage of the iteration computations of the set(s) and/or are required and the next iterate is taken as the Bergman projection of on the intersection of and/or . This seems difficult to do in applications.

It is our purpose in this paper to introduce an iterative algorithm for finding a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. We prove strong convergence theorem for the sequence produced by the method. Our scheme does not require computations of the set or at each stage of iterates. We prove strong convergence theorems for the sequences produced by the method. Our results improve and generalize many known results in the current literature (see, e.g., [7, 23, 24]).

2. Preliminaries

Let be a function. For any and , the right-hand derivative of at in the direction of is defined by . 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 for any . The function is said to be Fréchet differentiable at if this limit is attained uniformly in and is said to be uniformly Fréchet differentiable on a subset of if the limit is attained uniformly for and .

Let . The subdifferential of at is the convex set defined by

The Fenchel conjugate of is the function defined by .

The function is said to be essentially smooth if is both locally bounded and single-valued on its domain. It is called essentially strictly convex, if is locally bounded on its domain and is strictly convex on every convex subset of . is said to be a Legendre, if it is both essentially smooth and essentially strictly convex. When the subdifferential of is single-valued, it coincides with the gradient (see [27]).

For a Legendre function the following properties are known.

(i) is essentially smooth if and only if is essentially strictly convex (see [28, Theorem 5.4]).(ii)One has (see [29]).(iii) is Legendre if and only if is Legendre (see [28, Corollary 5.5]).(iv)If is Legendre, then is a bijection satisfying , , and (see [28, Theorem 5.10]).

When is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is . In this case the gradient of coincides with the generalized duality mapping of ; that is, . In particular, , the identity mapping in Hilbert spaces.

A function on is coercive [30] if the sublevel set of is bounded; equivalently, . A function on is said to be strongly coercive [31] if .

Let , for all , and . Then, a function is said to be uniformly convex on bounded subsets of ([31, pp. 203]) 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 if and only if is uniformly convex on bounded subsets of .

In the sequel, we will need the following lemmas.

Lemma 7 (see [26]). Let be a constant and let be uniformly convex on bounded subsets of . Then,

for all , , , and , with , where is the gauge of uniform convexity of .

Let be a Gâteaux differentiable function. The modulus of total convexity of at dom is the function defined by

The function is called totally convex at if , whenever . The function is called totally convex if it is totally convex at any point and is said to be totally convex on bounded sets if for any nonempty bounded subset of and , where the modulus of total convexity of the function on the set is the function defined by

We know that is totally convex on bounded sets if and only if is uniformly convex on bounded sets (see [32, Theorem 2.10]). The next lemma will be useful in the proof of our main results.

Lemma 8 (see [27]). Let be a proper, lower semicontinuous, and convex function; then, for all , one has

Lemma 9 (see [33]). Let be Gâteaux differentiable on such that is bounded on bounded subsets of . Let and . If is bounded, so is the sequence .

Lemma 10 (see [32]). Let be a nonempty, closed, and convex subset of . Let be a Gâteaux differentiable and totally convex function and let . Then,

(i) if and only if ; (ii).

Lemma 11 (see [34]). Let be uniformly Fréchet differentiable and bounded on bounded sets of ; then is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of .

Let be a Legendre and Gâteaux differentiable function. Following [2, 3], we make use of the function associated with , which is defined by

Then, is nonnegative and

Moreover, by the subdifferential inequality,

, and (see [35]).

3. Main Result

In the sequel, we will need the following lemma.

Lemma 12. Let be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . For and such that one has that

Proof. Since is uniformly Fréchet differentiable function we have by Lemma 11 that is uniformly continuous and hence by Theorem of [31] we get that is uniformly convex on bounded subsets of . This, with (24), (23), and Lemma 7, give that

for . The proof is complete.

We now prove the following theorem.

Theorem 13. Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be a finite family of Bregman weak relatively nonexpansive mappings. Assume that the interior of is nonempty. For let be a sequence generated by

where satisfy , , and . Then, the sequence generated by (28) converges strongly to an element of .

Proof. Let . Then, from (28), Lemma 10, and Lemma 12 we have that

for each , and hence

Therefore, exists and hence by Lemma 9 we get that is bounded. Now, from (2), we also have that

This implies that

Since the interior of is nonempty, there exists and such that , whenever . Thus, from (30), we have that

Therefore, from (32) and (33), we get that

Then, this and (32) imply that

and hence

Since with is arbitrary, we have

So, if , then

We know that converges. So, is a Cauchy sequence. Since is complete there exists such that converges strongly to . Furthermore, since is Legendre there exists such that and hence

Now, since is strongly coercive and uniformly convex on bounded subsets of , is uniformly Fréchet differentiable and bounded on bounded subsets of (see [31, Prop. 3.6.2]). Thus, applying Lemma 11 we get that , as .

Moreover, since is subset of , which is closed and convex, we have that .

Now, we show that . But from (29) we have that

for each . Then, the property of implies that as . From the fact that , for each , is Bregman weak relatively nonexpansive we obtain that , for each , and hence . This completes the proof.

If, in Theorem 13, every for each is Bregman relatively nonexpansive we have for each . Thus, we get the following corollary.

Corollary 14. Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be a finite family of Bregman relatively nonexpansive mappings. Assume that the interior of is nonempty. For let be a sequence generated by (28). Then, the sequence converges strongly to an element of .

If, in Theorem 13, we take , for each , to be continuous Bregman quasi-nonexpansive mappings, then we have for each . Thus, we have the following corollary.

Corollary 15. Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be a finite family of continuous Bregman quasi-nonexpansive mappings. Assume that the interior of is nonempty. For let be a sequence generated by (28). Then, the sequence converges strongly to an element of .

If, in Theorem 13, we take , then we have the following corollary.

Corollary 16. Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be a finite family of Bregman relatively nonexpansive mappings. Assume that the interior of is nonempty. For let be a sequence generated by

where , satisfy and . Then, the sequence generated by (41) converges strongly to an element of .

Remark 17. Our results are new even if the convex function is chosen to be in uniformly smooth and uniformly convex spaces.

Remark 18. Our theorems improve and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem extends Theorem of [23], Theorem of [24], and Theorem of [7] in the sense that either our theorem is applicable to a more general class of a finite family of Bregman weak relatively nonexpansive mappings or our scheme does not require the computation of or for each provided that the interior of is nonempty.

Moreover, we observe that Theorem 3.2 extends Theorem 3.1 of [26] in the sense that our scheme does not require the computations of for each when we consider finite family of Bregman weak relatively nonexpansive mappings and the interior of is nonempty.

Disclosure

The first author undertook this work when he was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, as a regular associate.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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