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`Abstract and Applied AnalysisVolume 2014 (2014), Article ID 584395, 8 pageshttp://dx.doi.org/10.1155/2014/584395`
Research Article

## Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces

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 21 February 2014; Accepted 7 May 2014; Published 27 May 2014

#### Abstract

We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces. Furthermore, we apply our method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our theorems complement some recent results that have been proved for this important class of nonlinear mappings.

#### 1. Introduction

In this paper, without other specifications, let be a real reflexive Banach space and as its dual, let be the set of real numbers, and let be a nonempty, closed, and convex subset . Let be a proper convex and lower semicontinuous function. Denote the domain of by ; that is, . The Fenchel conjugate of is the function defined by . is called cofinite if . For any and , the right-hand derivative of at in the direction of is defined by .

The function is called Gâteaux differentiable at if exists for any . In this case, coincides with , the value of the gradient of at . The function is called Gâteaux differentiable if it is Gâteaux differentiable for any int(dom). 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 .

The function is said to be bounded if it maps bounded subsets of into bounded sets. We note that if is uniformly Fréchet differentiable and bounded, then is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of (Proposition 2.1, [1]) and is uniformly Fréchet on bounded subsets of (see [2]) and hence is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of .

Let be a Gâteaux differentiable function. The function defined by is called the Bregman distance with respect to [3].

A Bregman projection [4] of onto the nonempty closed and convex set is the unique vector satisfying

Remark 1. If is a smooth and strictly convex Banach space and for all , then we have that for all , where is the normalized duality mapping from into , and hence(i) reduces to , for all , which is the Lyapunov function introduced by Alber [5] and(ii) reduces to the generalized projection (see, e.g., [5]) which is defined by If , a Hilbert space, is the identity mapping and hence the Bregman distance becomes , for , and the Bregman projection reduces to the metric projection of onto .

Let be a nonlinear mapping. Denote by the set of fixed points of . A mapping is said to be nonexpansive if , for all , and is called quasinonexpansive if , for all and . A point is called an asymptotic fixed point of (see [6]) if contains a sequence which converges weakly to such that . We denote by the set of asymptotic fixed points of .

A mapping is called(i)left quasi-Bregman nonexpansive [7] if and (ii)left Bregman relatively nonexpansive [7] if and (iii)left Bregman strongly nonexpansive (see [8, 9]), with respect to nonempty , if and, if, whenever is bounded, and it follows that (iv)left Bregman firmly nonexpansive [10] if and for all , or, equivalently, If is left Bregman firmly nonexpansive and is Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of , then it is known in [10] that and is closed and convex (see [10]). It follows that every left Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to a nonempty set .

Existence and approximation of fixed points of nonexpansive and quasinonexpansive mappings have been intensively studied for almost fifty years or so by various authors (see e.g., [1124] and the references therein) in Hilbert spaces. But most of the methods failed to give the same conclusion in Banach spaces more general than Hilbert spaces. One of the reasons is that a nonexpansive mapping in Hilbert spaces may not be nonexpansive in Banach spaces (e.g., the resolvent of a maximal monotone mapping and the metric projection onto a nonempty, closed, and convex subset of ).

To overcome this problem, researchers use the distance function introduced by Bregman [4] instead of norm which opened a growing area of research in designing and analyzing iterative techniques for solving variational inequalities, approximating equilibria, computing fixed points of nonlinear mappings, and approximating solutions of convex feasibility problems (see, e.g., [4, 2528] and the references therein).

In [29], Reich and Sabach proposed the following algorithm for finding a common fixed point of finitely many left Bregman firmly nonexpansive self-mappings on satisfying . For let the sequence be defined by They proved that, under some suitable conditions, the sequence generated by (11) converges strongly to a point in and applied it to the solution of convex feasibility and equilibrium problems.

Very recently, by using Bregman projection, Reich and Sabach [9] proposed an algorithm for finding a common fixed point of finitely many left Bregman strongly nonexpansive mappings satisfying in a reflexive Banach space as follows: Under some suitable conditions, they proved that the sequence generated by (12) converges strongly to a point in and applied it to the solution of convex feasibility and equilibrium problems.

The above results naturally bring us to the following: a natural question arises whether we can establish analogous results for right Bregman strongly nonexpansive mappings or not.

A mapping is called(i)right quasi-Bregman nonexpansive [30] if and (ii)right Bregman relatively nonexpansive [30] if and (iii)right Bregman strongly nonexpansive (see [8, 9]), with respect to nonempty , if and if, whenever is bounded, and it follows that (iv)right Bregman firmly nonexpansive [10] if and for all , or, equivalently,

Remark 2. It is shown in [10] that if is right Bregman firmly nonexpansive, then and hence it is right Bregman relatively nonexpansive mapping provided that the Legendre function is uniformly Fréchet differentiable and bounded on bounded sets of .

The class of right Bregman firmly nonexpansive mappings associated with the Bregman distance induced by a convex function was introduced and studied by Martin-Marques et al. [30]. Examples of right Bregman firmly nonexpansive mappings are given in [30]. If is a nonempty and closed subset of , where is a Legendre and Fréchet differentiable function, and is a right Bregman strongly nonexpansive mapping, it is proved that is closed (see [30]). In addition, they have shown that this class of mappings is closed under composition and convex combination and proved weak convergence of the Picard iterative method to a fixed point of a mapping under suitable conditions (see [31]). However, Picard iteration process has only weak convergence.

In this paper, it is our purpose to introduce an iterative scheme which converges strongly to a common fixed point of a finite family of right Bregman strongly nonexpansive mappings. As a consequence, we use our results to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our results complements the recent results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on .

#### 2. Preliminaries

Let be a convex and Gâteaux differentiable function. The modulus of total convexity of at 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 [27], Theorem 2.10).

The function is called essentially smooth, if is both locally bounded and single-valued on its domain and 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 Legendre, if it is both essentially smooth and essentially strictly convex. Since is reflexive, we know that (see [34]), is essentially smooth if and only if is essentially strictly convex (see [35], Theorem 5.4), and is Legendre if and only if is Legendre (see [35], Corollary 5.5); if is Legendre, then is a bijection satisfying , ran , and ran (see [35], Theorem 5.10). From now on, we assume that the convex function is Legendre.

If is a smooth and strictly convex Banach space, then an important and interesting 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.

In the sequel, we shall use the following lemmas.

Lemma 3 (see [31]). Let be a bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . For each , let be a right Bregman strongly nonexpansive mapping with respect to , and let . If is nonempty, then is also right Bregman strongly nonexpansive and .

Lemma 4 (see [30]). Let be a Fréchet differentiable function. Let be a nonempty closed convex subset of and let be a right quasi-Bregman nonexpansive mapping. Then is closed.

Lemma 5 (see [36]). The function is totally convex on bounded subsets of if and only if for any two sequences and in and , respectively, such that the first one is bounded and

Lemma 6 (see [27]). 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 7 (see [37]). If is a proper, lower semicontinuous, and convex function, then is a proper, lower semicontinuous and convex function. Thus, for all , we have

Lemma 8 (see [31]). Let be admissible and totally bounded at a point . Let . If is bounded, then so is the sequence .

Let be a Gâteaux differentiable function. Following [3, 5], we make use of the function associated with , which is defined by

Then, is nonnegative and

Moreover, by the subdifferential inequality,

for all and (see [38]).

Lemma 9 (see [39]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , , and . Then, .

Lemma 10 (see [40]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists an increasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, is the largest number in the set such that the condition holds.

#### 3. Main Results

Theorem 11. Let be a cofinite 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 right Bregman strongly nonexpansive mappings such that , for each . Assume that is nonempty. For , let be a sequence generated by where , satisfy and . Then, converges strongly to a point in .

Proof. Note that from Lemma 3 we have and is right Bregman strongly nonexpansive mapping. Let . Then, using (29), the convexity of , and property of we get that Thus, by induction we obtain that which implies that and hence are bounded. Thus, from Lemma 8 we get that and are bounded. Now, let . Then, iteration process (29) becomes where , a conjugate of . Since and are uniformly continuous on bounded subsets of and , respectively, we get that and are bounded and by Section 6 of Martin-Marquez et al. [31] we have that is left Bergman strongly nonexpansive with respect to . In addition, by Proposition 3.3 of [30] we have that is closed and convex. Let . Now, from (32), (25), (26), and Lemma 6 we obtain that Now, we consider two cases.

Case  1. Suppose that there exists such that is decreasing for all . Then, we get that is convergent and hence In addition, from (32) and Lemma 7 we have that Following from (35), (34), and the fact that , as , we get that This with the fact that is left Bregman strongly nonexpansive implies that Then, by Lemma 5 we obtain that Now, since is reflexive and is bounded, there exists a subsequence of such that Thus, from (39), (38), the fact that is left Bregman strongly nonexpansive mapping with , and Lemma 6 we get that and Therefore, it follows from (33), (41), and Lemma 9 that as . Consequently, by Lemma 5 we obtain that and hence .

Case  2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 10, there exist a nondecreasing sequence such that and for all . Thus, we get that This implies that as . Now, following the method in Case 1 we obtain that Now, from (33) we have that But (43) and (46) imply that and noting that , we get that Thus, using (45) we get that and hence from (46) we have that as . But , for all , implies that and hence by Lemma 5 we obtain that and . Therefore, from the above two cases, we can conclude that converges strongly to and the proof is complete.

Remark 12. We note that the sequence in Theorem 11 converges strongly to a point such that , where .

If, in Theorem 11, we consider a single right Bregman strongly nonexpansive mapping, we get the following corollary.

Corollary 13. Let be a reflexive Banach space and let be a cofinite 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 right Bregman strongly nonexpansive mapping such that . For let be a sequence generated by where satisfy and . Then, converges strongly to some in .

If, in Theorem 11, we assume that each is right Bregman firmly nonexpansive, then we have that is right Bregman firmly nonexpansive with (see [10]) and hence it is right Bregman strongly nonexpansive mapping. Thus, we have the following.

Corollary 14. Let be a cofinite 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 right Bregman firmly nonexpansive mappings with . For let be a sequence generated by where , satisfying and . Then, converges strongly to in .

#### 4. Applications

##### 4.1. Zeroes of Maximal Mappings

Let be a maximal monotone mapping. Recently, many authors studied zero points of monotone mappings using different methods (see e.g., [13, 25, 28, 30, 31, 38]). In this section we use Halpern’s type scheme to find common zeros of a finite family of maximal monotone set-valued mappings.

Definition 15 (see [31]). Let be an admissible function and let be a set-valued mapping such that . The conjugate resolvent of with respect to , or the conjugate -resolvent, is the operator defined by

Remark 16. If, in addition, is monotone and is strictly convex, then it is shown in [31] that is right Bregman firmly nonexpansive and . Moreover, we know that if is Legendre, bounded, and uniformly continuous on bounded subsets of , then, for every right Bregman firmly nonexpansive operator ,   (see [10]). Thus, under these assumptions on and , the operator is right Bregman strongly nonexpansive mapping.

We shall need the following lemma.

Lemma 17 (see [30]). Let be a strictly convex, cofinite, and admissible function, and let be a set-valued monotone mapping. Then is maximal monotone if and only if .

Theorem 18. Let be a cofinite function such that is uniformly Fréchet differentiable and totally convex on bounded subsets of . Let , , be maximal monotone mappings such that . For each , consider the sequence generated iteratively by where , satisfy , and . Then, converges strongly to in , and .

Proof. From Lemma 17 we know that each , is a mapping from into itself, since Remark 16 guarantees that each is right Bregman strongly nonexpansive mapping with respect to . Now the result follows immediately from Theorem 11 applied to .

##### 4.2. Convex Feasibility Problems

The convex feasibility problem (CFP) is finding an element , where for , are nonempty, closed, and convex subsets of . Let . The right Bregman projection [30] onto is the operator defined by If is Legendre and uniformly continuous on bounded subsets of and is weakly sequentially continuous, then the right Bregman projection is right Bregman strongly nonexpansive mapping with (see [30]). Therefore, if we take for each , then we get an algorithm for solving convex feasibility problems. More precisely, we have the following result.

Theorem 19. Let be a cofinite function which is bounded, uniformly continuous, and totally convex on bounded subsets of . Assume that is weakly sequentially continuous. Let be nonempty, closed, and convex subsets of such that . For each , let the sequence be generated iteratively by where , satisfy , and . Then, converges strongly to in .

Remark 20. Theorem 11 complements the results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on , where is nonempty, closed, and convex subset of .

#### Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support.

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