Journal of Applied Mathematics

Volume 2014 (2014), Article ID 580686, 9 pages

http://dx.doi.org/10.1155/2014/580686

Research Article

Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

Mohammed Ali Alghamdi,¹ Naseer Shahzad,¹ and Habtu Zegeye²

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana

Received 9 May 2014; Accepted 24 June 2014; Published 21 July 2014

Academic Editor: Giuseppe Marino

Copyright © 2014 Mohammed Ali Alghamdi et al. 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 study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

1. Introduction

Throughout this paper, is a real reflexive Banach space with the dual space . The norm and the dual pair between and are denoted by and , respectively. We also assume that is a proper, lower semicontinuous, and convex function. Denote the domain of by ; that is, . Let . The subdifferential of at is the convex set defined by

The Fenchel conjugate of is the function defined by .

Let 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 and hence coincides with , the value of the gradient of at . The function is said to be Gâteaux differentiable if it is Gâteaux differentiable for any . Furthermore, is said to be Fréchet differentiable at if this limit is attained uniformly in and it is called uniformly Fréchet differentiable on a subset of if the limit is attained uniformly for and .

Let be a Gâteaux differentiable function. The function defined by

is called the Bregman distance with respect to [1].

When is a smooth Banach space, setting for all , we have , for all , where is the normalized duality mapping from into , and hence reduces to , for all , which is the Lyapunov function introduced by Alber [2]. If , a Hilbert space, is identity mapping and hence becomes , for all , .

Let be a nonempty and convex subset of and let be a mapping. is said to be nonexpansive if for all , and is said to be quasinonexpansive if and , for all and , where stands for the fixed point set of ; that is, . A point is said to be an asymptotic fixed point of (see [3]) if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by .

A mapping with is called

(i)quasi-Bregman nonexpansive [4] if

(ii)Bregman relatively nonexpansive [4] if

(iii)Bregman firmly nonexpansive [5] if, for all ,

or, equivalently,

Iterative methods for approximating fixed points of nonexpansive, quasinonexpansive mappings and their generalizations have been studied by various authors (see, e.g., [6–16] and the references therein) in Hilbert spaces. But extending this theory to Banach spaces encountered some difficulties because the useful examples of nonexpansive operators in Hilbert spaces are no longer nonexpansive in Banach spaces (e.g., the resolvent , for , of a monotone mapping and the metric projection onto a nonempty, closed, and convex subset of ). To overcome these difficulties, Bregman [1] discovered techniques with the use of Bregman distance function instead of norm in the process of designing and analyzing feasibility and optimization problems. This opened a growing area of research for solving variational inequalities and approximating solutions or fixed points of nonlinear mappings (see, e.g., [1, 17–21] and the references therein).

The existence and approximation of fixed points of Bregman firmly nonexpansive mappings were studied in [5]. It is also known that if is Bregman firmly nonexpansive and is Legendre function which is bounded, uniformly Frêchet differentiable, and totally convex on bounded subsets of , then and is closed and convex (see [5]). It also follows that every Bregman firmly nonexpansive mapping is Bregman relatively nonexpansive and hence quasi-Bregman nonexpansive mapping.

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

Remark 1. If is a smooth Banach space and for all , then the Bregman projection reduces to the generalized projection (see, e.g., [2]) which is defined by

where .

Very recently, by using Bregman projection, Reich and Sabach [4] introduced an algorithm for finding a common zero of many finitely maximal monotone mappings satisfying in a reflexive Banach space as follows:

where . Under suitable conditions, they proved that if, for each , we have that and the sequences of errors satisfy , then the sequence converges strongly to .

In [22], Reich and Sabach proposed an algorithm for finding a common fixed point of many finitely Bregman firmly nonexpansive mappings satisfying in a reflexive Banach space as follows:

They proved that, under suitable conditions, the sequence generated by (10), converges strongly to and applied it to the solution of convex feasibility and equilibrium problems.

Remark 2. But it is worth mentioning that the iteration processes (9) and (10) seem difficult in the sense that, at each stage of iteration, the set(s) and/or are/is computed and the next iterate is taken as the Bregman projection of onto the intersection of and/or . This seems difficult to do in applications.

In this paper, we investigate an iterative scheme for finding a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in reflexive Banach spaces. We prove strong convergence theorems for the sequences produced by the method. Furthermore, we apply our method to prove strong convergence theorems for finding a solution of a finite family of equilibrium problems and for finding a common zero of a finite family of maximal monotone mappings. Our results improve and generalize many known results in the current literature (see, e.g., [4, 23])

2. Preliminaries

Let be a Gâteaux differentiable function. 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 [24]).

We note that if is a reflexive Banach space, then we have the following.

(i) is essentially smooth if and only if is essentially strictly convex (see [25, Theorem 5.4]).(ii) (see [26]).(iii) is Legendre if and only if is Legendre (see [25, Corollary 5.5]).(iv)If is Legendre, then is a bijection satisfying , , and (see [25, 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.

Let be a Banach space and let , for all and . Then a function is said to be uniformly convex on bounded subsets of [27, page 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 [27, page 207] if , for all , where is defined by

for all .

In the sequel, we will need the following lemmas.

Lemma 3 (see [28]). Let be a Banach space, let be a constant, and let be a uniformly convex on bounded subsets of . Then

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

Lemma 4 (see [24]). Let be a proper, lower semicontinuous, and convex function; then is a proper, lower semicontinuous, and convex function. Thus, for all , we have

Lemma 5 (see [29]). Let be a Gâteaux differentiable on such that is bounded on bounded subsets of . Let and . If is bounded, so is the sequence .

A function on is coercive [30] if the sublevel set of is bounded; equivalently, . A function on is said to be strongly coercive [27] if .

Lemma 6 (see [27]). Let be a reflexive Banach space and let be 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 Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of ;(3), is strongly coercive and uniformly convex on bounded subsets of .

Lemma 7 (see [5]). Let be a Legendre function. Let be a nonempty closed convex subset of and let be a quasi-Bregman nonexpansive mapping. Then is closed and convex.

Let be a 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 [18, Theorem 2.10]). The following lemmas will be useful in the proof of our main results.

Lemma 8 (see [31]). 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,

Lemma 9 (see [27, 32]). Let be a strongly coercive and uniformly convex on bounded subsets of ; then is bounded on bounded sets and uniformly Fréchet differentiable on bounded subsets of .

Lemma 10 (see [33]). 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 .

Lemma 11 (see [18]). 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).

Let be a Legendre and Gâteaux differentiable function. Following [2, 34], we make use of the function associated with , which is defined by

Then is nonnegative and

Moreover, by the subdifferential inequality,

where and (see [35]).

Lemma 12 (see [36]). Let be a sequence of nonnegative real numbers satisfying the following relation:

where and satisfying the following conditions: , and . Then, .

Lemma 13 (see [37]). 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

We now prove the following theorem.

Theorem 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 quasi-Bregman nonexpansive mappings such that , for . Assume that is nonempty. For let be a sequence generated by

where and satisfying , , and . Then, converges strongly to .

Proof. Lemma 7 ensures that each , for , and are closed and convex. Thus, is well defined. Let . Then, from (23), Lemmas 11 and 4, and property of we get that

Moreover, from (23), (18), and (19) we get that

Since is uniformly Fréchet differentiable function we have that is uniformly smooth and hence by Theorem of [27] we get that is uniformly convex. This, with Lemmas 3 and (24), gives that

for each . Thus, by induction,

which implies that is bounded. Now, let . Then we note that . Since is strongly coercive, uniformly convex, uniformly Fréchet differentiable, and bounded on bounded subsets of by Lemmas 10, 9, and 6 we have that and are bounded on bounded sets and hence and are bounded. In addition, using (19), (20), and property of we obtain that

Furthermore, from (26) and (29) we have that

Now, we consider two cases.

Case 1. Suppose that there exists such that is nonincreasing for all . In this situation, is convergent. Then, from (30) we have that

which implies, by the property of , that

Now, since is a strongly coercive and uniformly convex on bounded subsets of , is uniformly Fréchet differentiable on bounded subsets of (see [27, Proposition ]) and is Legendre; we have by Lemma 10 that

for each .

Furthermore, Lemma 11, property of , and the fact that , as , imply that

and hence by Lemma 8 we get that

Since is bounded and is reflexive, we choose a subsequence of such that and . Then, from (36) we get that

Thus, from (34) and the fact that we obtain that , for each and hence .

Therefore, by Lemma 11, we immediately obtain that . It follows from Lemma 12 and (31) that , as . Consequently, by Lemma 8 we obtain that .

Case 2. Suppose that there exists a subsequence of such that

for all . Then, by Lemma 13, there exists a nondecreasing sequence such that , and , for all . Then from (30) and the fact that we obtain that

for each . Thus, following the method of proof of Case 1, we obtain that , , , as , and hence we obtain that

Then from (31) we have that

Now, since , inequality (41) implies that

In particular, since , we get

Then, from (40) we obtain , as . This together with (41) gives , as . But , for all ; thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to and the proof is complete.

If, in Theorem 14, we consider a single quasi-Bregman nonexpansive mapping, we get the following corollary.

Corollary 15. Let be a 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 quasi-Bregman nonexpansive mapping such that . Let be a sequence generated by

where and satisfying and . Then converges strongly to .

If, in Theorem 14, we assume that each , for , is Bregman relatively nonexpansive mapping, then we have that for each . Thus, we get 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 , for , be a finite family of Bregman relatively nonexpansive mappings. Assume that is nonempty. For let be a sequence generated by

where and satisfying , , and . Then, converges strongly to .

If, in Theorem 14, we assume that each is Bregman firmly nonexpansive, then we get the following corollary.

Corollary 17. 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 firmly nonexpansive mappings with . Let be a sequence generated by

where and satisfying , , and . Then converges strongly to .

4. Applications

4.1. Equilibrium Problems

Let be a coercive Legendre function. Let be a nonempty, closed, and convex subset of . Let be a bifunction with the following conditions (see example [38]).

(B1), for all ;(B2) is monotone; that is, , for all ;(B3)for all , ;(B4)for each , is convex and lower semicontinuous.

The equilibrium problem corresponding to is to find such that

The set of solutions of (47) is denoted by EP. Equilibrium problem is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, and so forth; see [20, 30, 38–43].

The resolvent of a bifunction [39] is the operator , defined by

We know the following lemma in [23].

Lemma 18. Let be a coercive Legendre function. Let be a nonempty, closed, and convex subset of . If the bifunction satisfies conditions (B1)–(B4), then

(1) is single-valued and ;(2) is a Bregman firmly nonexpansive mapping;(3) is a closed and convex subset of and ;(4)For all and for all , we have

In addition, if is uniformly Fréchet differentiable and bounded on bounded subsets of , then by Reich and Sabach [5] we have that . Thus, considering , for , in Corollary 17, we get the following result.

Theorem 19. 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 bifunctions which satisfy the conditions (B1)–(B4) and is nonempty. For let be a sequence generated by

where , , and satisfy , , and . Then, converges strongly to .

4.2. Zeroes of Maximal Monotone Operators

In this section we present an algorithm for finding a common zero of a finite family of maximal monotone mappings.

Let be a mapping with range and domain . Then, is said to be monotone if, for any , we have

It is said to be maximal monotone if is monotone and the graph of is not a proper subset of the graph of any other monotone mappings, where graph of is given by . It is known that if is maximal monotone, then the set is closed and convex.

Let be maximal monotone mapping and is Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of . The resolvent of with respect to is the mapping defined by

It is known that is single-valued, Bregman firmly nonexpansive and (see [17]). Furthermore, the result by Reich and Sabach [5] implies that . If, in Theorem 14, we assume that , then we obtain an algorithm for finding a common zero of a finite family of maximal monotone mappings.

Theorem 20. Let be a 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 maximal monotone mappings with . Then, for each , the sequence is defined by

where , , and satisfy , , and . Then, converges strongly to .

For a common point of a solution of equilibrium problem and a zero of maximal monotone mapping we have the following corollary.

Corollary 21. 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 bifunction which satisfy the conditions (B1)–(B4) and let be a maximal monotone mapping. Assume that is nonempty. For let be a sequence generated by

where and satisfying , , and . Then, converges strongly to .

Remark 22. Corollary 17 and Corollary 4.3 improve Theorem 1 of Reich and Sabach [23] and Theorem 4.2 of Reich and Sabach [4], respectively, in the sense that at each stage the computation of or is not required.

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 170/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, they would like to thank the anonymous reviewers for their valuable comments and suggestions.

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