About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 619762, 9 pages
http://dx.doi.org/10.1155/2013/619762
Research Article

Strong Convergence Iterative Algorithms for Equilibrium Problems and Fixed Point Problems in Banach Spaces

1Department of Applied Mathematics and Physics, North China Electric Power University, Baoding 071003, China
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 1 February 2013; Accepted 9 March 2013

Academic Editor: Yisheng Song

Copyright © 2013 Shenghua Wang and Shin Min Kang. 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 the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.

1. Introduction

Let be a real reflexive Banach space with norm and the dual space of equipped with the inducted norm . Throughout this paper, is a proper, lower semicontinuous, and convex function and the Fenchel conjugate of is the function defined by

We denote by the domain of , that is, the set .

Let be a nonempty, closed, and convex subset of and a nonlinear mapping. The fixed points set of is denoted by

Recall that a mapping is said to be nonexpansive if, for each ,

Nakajo-Takahashi [1] introduced the following hybrid method which is the so-called CQ-method for a nonexpansive mapping in a Hilbert space : where and is the metric projection from onto a closed and convex subset of . They proved that the sequence generated by (4) converges strongly to a fixed point of under suitable conditions.

Takahashi et al. [2] introduced a new hybrid iterative scheme called the shrinking projection method for a nonexpansive mapping in a Hilbert space as follows: where , and they proved that the sequence generated by (5) converges strongly to a fixed point of under suitable conditions.

In 2010, Reich and Sabach [3] introduced the following two hybrid iterative schemes for Bregman strongly nonexpansive mappings in a reflexive Banach space : where is the Bregman projection with respect to from onto a closed and convex subset of . They proved that the sequence generated by both (6) and (7) converges strongly to a common fixed point of .

The construction of fixed points for Bregman-type mappings via iterative processes has been investigated in, for example, [48].

In this paper, we design a new hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove some strong convergence theorems. Our results extend the recent one of Reich and Sabach [3].

2. Preliminaries

Let be a real Banach space. For any and , we define the right-hand derivative of at in the direction 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 int dom . The function is said to be Fréchet differentiable at if this limit is attained uniformly in . Finally, is said to be uniformly Fréchet differentiable on a subset of if the limit is attained uniformly for and .

Let be a reflexive Banach space. The Legendre function is defined from a general Banach space into (see [9]). According to [9], the function is Legendre if and only if it satisfies the following conditions(L1) The interior of the domain of (denoted by ) is nonempty; is Gâteaux differentiable on and dom .(L2) The interior of the domain (denoted by int dom) is nonempty; is Gâteaux differentiable on int dom and dom .

Since is reflexive, we always have (see [10]). This fact, when combined with conditions (L1) and (L2), implies the following equalities:

Also, conditions (L1) and (L2), in conjunction with [9], imply that the functions and are strictly convex on the interior of their respective domains. Several interesting examples of the Legendre functions are presented in [9, 11]. Especially, the functions with are Legendre, where the Banach space is smooth and strictly convex and, in particular, a Hilbert space.

Let be a convex and Gâteaux differentiable function. The function defined as is called the Bregman distance with respect to [12].

By the definition, we know the following property (the three point identity): for any and ,

Recall that the Bregman projection [13] of onto the nonempty, closed, and convex subset of is the necessarily unique vector satisfying

Let be a convex and Gâteaux differentiable function. The function is said to be totally convex at if its modulus of total convexity at , that is, the function defined by is positive whenever . The function is said to be totally convex when it is totally convex at every point . In addition, the function is said to be totally convex on bounded sets if is positive for any nonempty bounded subset of and , where the modulus of total convexity of the function on the set is the function defined by

Some examples of the totally convex functions can be found in [14, 15].

Recall that the function is said to be sequentially consistent [15] if, for any two sequences and in such that the first is bounded,

Let be a nonempty, closed, and convex subset of and a bifunction that satisfies the following conditions: (C1) for all ; (C2) is monotone, that is, for all ; (C3) for all ; (C4) for all is convex and lower semicontinuous.

The equilibrium problem with respect to is as follows: find such that

The set of all solutions of (16) is denoted by . The resolvent of a bifunction [16] is the operator denoted by For any , there exists such that ; see [3].

Let be a convex subset of and a mapping. A point in the closure of is said to be an asymptotic fixed point of [17, 18] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . The mapping is called Bregman quasi-nonexpansive if and is said to be Bregman (quasi)-strongly nonexpansive [6] with respcet to a nonempty if for all and , and if whenever is bounded, , and it follows that The mapping is called Bregman firmly nonexpansive if for all .

Next, we introduce a new mapping that is called Bregman asymptotically quasinonexpansive mapping which is a natural extension of Bregman quasinonexpansive mapping introduced by Reich and Sabach [3]. The mapping is said to be Bregman asymptotically quasi-nonexpansive if there exists a sequence satisfying such that, for every , Obviously, every Bregman quasinonexpansive mapping is a Bregman asymptotically quasi-nonexpansive one with .

Let be a Banach space and a nonempty subset of . The mapping is said to be uniformly asymptotically regular on if The mapping is said to be closed if, for any sequence in such that and , .

The following is an important result which will be used in the next section.

Lemma 1. Let be a reflexive Banach space and a Gâteaux differentiable and Legendre function which is totally convex on bounded sets. Let be a nonempty, closed and convex subset of and a closed Bergman asymptotically quasi-nonexpansive mapping with the sequence such that as . Then is closed and convex.

Proof. The closedness of comes directly from the closedness of . Now, for arbitrary , , put . We prove that . Indeed, from the definition of , we see that This implies that . It follows from Lemma 3 below that that is, as . In view the closedness of , we can obtain the desired conclusion. This completes the proof.

Finally, we state some lemmas that will used in the proof of main results in next section.

Lemma 2 (see [7]). If is uniformly Fréchet differentiable and bounded on bounded subsets of , then is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of .

Lemma 3 (see [14]). The function is totally convex on bounded sets if and only if it is sequentially consistent.

Lemma 4 (see [15]). Suppose that is Gâteaux differentiable and totally convex on . Let and a nonempty, closed, and convex subset of . If , then the following conditions are equivalent.(i) The vector is the Bregman projection of onto with respect to .(ii) The vector is the unique solution of the variational inequality. (iii) The vector is the unique solution of the inequality

Lemma 5 (see [6]). Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is also bounded.

Lemma 6 (see [3]). Let be a coercive i.e., and Gâteaux differentiable function. Let be a closed and convex subset of . If the bifunction satisfies conditions , then (1) is single-valued; (2) is a Bregman firmly nonexpansive mapping; (3)the set of fixed points of is the solution set of the equilibrium problem, that is, ; (4) is a closed and convex subset of ; (5)for all and , one has

3. Main Results

Now, we give our main theorems.

Theorem 7. Let be a reflexive Banach space and 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 a countable family of closed Bregman asymptotically quasi-nonexpansive mappings with the sequences such that for every . Let and suppose that . Let be a bifunction satisfying conditions (C1)–(C4). Assume that each is uniformly asymptotically regular and is nonempty and bounded. Let be a real sequence in with for every and for every . Let be a sequence generated by the following manner: where for each . Then, defined by (30) converges strongly to as .

Proof. First, we prove that the sequence is well defined. Note that is that is, This shows that is closed and convex for every . From the definition of , it is easy to see that is closed and convex for every . For every and , Lemma 6 shows that and for any and . Hence,
Since for every , we have This shows that for every . Thus for every . Further, we have for every . Thus the sequence is well defined.
From , by Lemma 4(iii) we have for any . Hence the sequence is bounded. Therefore by Lemma 5 the sequence is bounded.
On the other hand, in view of and , from Lemma 4(iii) we have that is, Therefore the sequence is increasing, and since it is also bounded, exists. By the construction of , we have that and for any positive integer . It follows that Letting in (39), we see that . It follows from Lemma 3 that as . Hence, is a Cauchy sequence. Since is a Banach space and is closed and convex, we can assume that By taking in (39), we see that Lemma 3 implies that Since , we have Then (41) implies that Note that and , we have for every . It follows from Lemma 3 that for every . Note that Combining (42) with (46), we see that for every . This means that the sequence is bounded. Since is uniformly Fréchet differentiable, it follows from Lemma 2 that Since is uniformly Fréchet differentiable, it is also uniformly continuous (see [19, Theorem 1.8, p.13]) and therefore From the definition of the Bregman distance, we obtain that for every for any . Since every sequence is bounded, is also bounded for every . Now from (48)–(50), we have for any and for every .
In view of , by Lemma 6 (5) we have Note that is bounded and as . It follows from (52) that for every . Lemma 3 shows that Note that . From (48) and (55) we get for every . Note that
It follows from (40) and (56) that for every . On the other hand, we have Since every is uniformly asymptotically regular and (58), we obtain that, for every , that is, as . From the closedness of , we see that for every . Thus .
Next we prove that for every . Since is uniformly Fréchet differentiable, is uniformly continuous. Thus, by (55) we have Since , we have We have from (C2) that Letting , we have from (61) and (C4) that For with and , let . Since and , we have and hence . So, from (C1) we have Dividing by , we have Letting , from (C3) we have Therefore, . Thus .
Finally, we show that . Since for every , by Lemma 4(ii) we arrive at Taking the limit as in (68), we obtain that and hence by Lemma 4(ii). This completes the proof.

Corollary 8. Let be a reflexive Banach space and 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 a closed Bregman asymptotically quasi-nonexpansive mapping with the sequence such that . Let be a bifunction satisfying conditions (C1)–(C4). Assume that is uniformly asymptotically regular and is nonempty and bounded. Let be a sequence generated by the following manner: where for each . Then, defined by (70) converges strongly to as .

Since every Bregman quasi-nonexpansive mapping is Bregman quasi-asymptotically nonexpansive, we have the following results.

Corollary 9. Let be a reflexive Banach space and 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 . Let be a countable family of closed Bregman quasi-nonexpansive mappings and a bifunction satisfying conditions (C1)–(C4). Assume that . Let be a real sequence in with and for every . Let be a sequence generated by the following manner: Then, defined by (71) converges strongly to as .

Corollary 10. Let be a reflexive Banach space and 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 . Let be a closed Bregman quasi-nonexpansive mapping and a bifunction satisfying conditions (C1)–(C4). Assume that . Let be a sequence generated by the following manner: Then, defined by (72) converges strongly to as

Remark 11. Set for each and and for each and . Then and . Also, for every . Hence, and satisfy the conditions of Theorem 7.

Remark 12. It needs to notice that Corollaries 9 and 10 still hold if we replace the closedness of the mappings with .

In the equilibrium problem, the bifunction is usually required to satisfy conditions (C1)–(C4). But, if the condition (C3) is replaced with the following condition: for every fixed is continuous, then we have the following result:

Lemma 13. Let be a coercive i.e., and Gâteaux differentiable function. Let be a closed and convex subset of . If the bifunction satisfies conditions , , , and , then the mapping defined by (2.2) is closed.

Proof. Let converge to and to . To end the conclusion, we need to prove that . Indeed, for each , Lemma 6 shows that there exists a unique such that , that is,
Since is uniformly Fréchet differentiable, is uniformly continuous. So, taking the limit as in (73), by using we get which implies that . This completes the proof.

If the bifunction satisfies conditions (C1), (C2), , and (C4) instead of (C1)–(C4), then we have a simple method to prove that in the proof of Theorem 7. Indeed, from the proof of Theorem 7, we see that

Note that as . This shows that as for every . It follows from the closedness of that . Lemma 6 shows that .

Remark 14. Obviously, the proof process of is simple if we replace condition (C3) with which is such that is closed. In fact, although condition is stronger than (C3), it is not easier to verify condition (C3) than to verify the condition . Hence, from this viewpoint, the condition is acceptable.

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant Number: 13MS109) and the HeBei Education 4 Department (Grant Number: 936101101).

References

  1. K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Reich and S. Sabach, “Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 73, no. 1, pp. 122–135, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. H. H. Bauschke and P. L. Combettes, “Construction of best Bregman approximations in reflexive Banach spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 12, pp. 3757–3766, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. F. Kohsaka and W. Takahashi, “Proximal point algorithms with Bregman functions in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 3, pp. 505–523, 2005. View at Zentralblatt MATH · View at MathSciNet
  6. S. Reich and S. Sabach, “Two strong convergence theorems for a proximal method in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 31, no. 1–3, pp. 22–44, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Reich and S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 10, no. 3, pp. 471–485, 2009. View at Zentralblatt MATH · View at MathSciNet
  8. S. Reich and S. Sabach, “A projection method for solving nonlinear problems in reflexive Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 9, no. 1, pp. 101–116, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces,” Communications in Contemporary Mathematics, vol. 3, no. 4, pp. 615–647, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, NY, USA, 2000. View at MathSciNet
  11. H. H. Bauschke and J. M. Borwein, “Legendre functions and the method of random Bregman projections,” Journal of Convex Analysis, vol. 4, no. 1, pp. 27–67, 1997. View at Zentralblatt MATH · View at MathSciNet
  12. Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,” Journal of Optimization Theory and Applications, vol. 34, no. 3, pp. 321–353, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. M. Bregman, “A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, pp. 620–631, 1967. View at MathSciNet
  14. D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  15. D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,” Abstract and Applied Analysis, Article ID 84919, 39 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. View at Zentralblatt MATH · View at MathSciNet
  17. Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178, pp. 313–318, Marcel Dekker, New York, NY, USA, 1996. View at Zentralblatt MATH · View at MathSciNet
  19. A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, UK, 1993. View at MathSciNet