About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 489629, 13 pages
http://dx.doi.org/10.1155/2012/489629
Research Article

Strong and Weak Convergence Theorems for Equilibrium Problems and Weak Relatively Uniformly Nonexpansive Multivalued Mappings in Banach Spaces

Department of Foundation, Shandong Yingcai University, Jinan 250104, China

Received 22 August 2012; Accepted 21 September 2012

Academic Editor: Yongfu Su

Copyright © 2012 Zi-Ming Wang. 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

Equilibrium problem and fixed point problem are considered. A general iterative algorithm is introduced for finding a common element of the set of solutions to the equilibrium problem and the common set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings. Furthermore, strong and weak convergence results for the common element in the two sets mentioned above are established in some Banach space.

1. Introduction

Let be a smooth Banach space, and let be a nonempty closed convex subset of . In the sequel, we denote by the family of all nonempty subsets of . We use to denote the Lyapunov functional defined by We know several fundamental properties of as follows: for all . For a sequence and , is bounded if and only if is bounded.

Let be a multivalued mapping. We denote by the set of fixed points of , that is,

For a multivalued mapping , we define an asymptotic fixed point and a strong asymptotic fixed point of as follows.

Definition 1.1 (see [1]). Let be a multivalued mapping.(1)A point in is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and there exists a sequence such that . The set of asymptotic fixed point of will be denoted by . (2)A point in is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to and there exists a sequence such that . The set of strong asymptotic fixed point of will be denoted by .

Definition 1.2 (see [1]). A multivalued mapping is called relatively nonexpansive multivalued mapping (weak relatively nonexpansive multivalued mapping) if the following conditions are satisfied:(1) is nonempty;(2);(3).

Definition 1.3 (see [1]). A multivalued mapping is called relatively uniformly nonexpansive multivalued mapping (weak relatively uniformly nonexpansive multivalued mapping) if the following conditions are satisfied:(1) is nonempty;(2);(3).

Remark 1.4. By comparing condition (2) of Definitions 1.2 and 1.3, one easily draws the following conclusions:(1)the class of relatively nonexpansive multivalued mappings contains the class of relatively uniformly nonexpansive multivalued mappings as a subclass, but the converse may be not true;(2)the class of weak relatively nonexpansive multivalued mappings contains the class of weak relatively uniformly nonexpansive multivalued mappings as a subclass, but the converse may be not true.

For any operator , is held. So we have the following remark.

Remark 1.5. From Definitions 1.2 and 1.3, the following conclusions can easily be drawn: (1)the class of weak relatively nonexpansive multivalued mappings contains the class of relatively nonexpansive multivalued mappings as a subclass, but the converse may be not true;(2)the class of weak relatively uniformly nonexpansive multivalued mappings contains the class of relatively uniformly nonexpansive multivalued mappings as a subclass, but the converse may be not true.

Remark 1.6. The examples of weak relatively uniformly nonexpansive multivalued mapping can be found in Su [1] and Homaeipour and Razani [2].
Let be a real Banach space, and let be the dual space of . Let be a bifunction from to . The equilibrium problem is to find The set of solutions of (1.3) is denoted by . Given a mapping , let for all . Then if and only if for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see [35] for details.

In recent years, iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by many authors, see [2, 69] for details. In 2011, Homaeipour and Razani [2] introduced the concept of relatively nonexpansive multivalued mappings and proved some weak and strong convergence theorems to approximate a fixed point for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space which improved and extended the corresponding results of Matsushita and Takahashi [10]. Very recently, Su [1] not only redefined relatively nonexpansive multivalued mappings, which was different from Homaeipour and Razani [2]'s definition, but also introduced some interesting examples about the multivalued mappings. On the other hand, in 2009, Qin et al. [11] introduced an iterative algorithm for the equilibrium problem (1.3) and relatively nonexpansive mappings. Moreover, they proved a weak convergence theorem for finding a common element of the set of solutions to the equilibrium problem (1.3) and the common set of fixed points of two relatively nonexpansive mappings, which improved and extended the corresponding results of Takahashi and Zembayashi [12].

Motivated and inspired by the above facts, the purpose of this paper will introduce an iterative algorithm for the equilibrium problem (1.3) and two weak relatively uniformly nonexpansive multivalued mappings. Furthermore, a weak convergence theorem will given for finding a common element of the set of solutions to the equilibrium problem (1.3) and the common set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings in some Banach space. Our results improve and extend the corresponding results of Qin et al. [11] and Takahashi and Zembayashi [12].

2. Preliminaries

Let be a real Banach space with norm , and let be the normalized duality mapping from into given by for all , where denotes the dual space of and the generalized duality pairing between and . It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of .

As we all know that if is a nonempty closed convex subset of a Hilbert space , and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces, and, consequently, it is not available in more general Banach spaces. In this connection, Alber [13] introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces. The generalized projection is a map that assigns to an arbitrary point the minimum point of the Lyapunov functional , that is, , where is the solution to the minimization problem; The existence and uniqueness of the operator follow from the properties of the Lyapunov functional and strict monotonicity of the mapping , see, for example, [13, 14]. In Hilbert spaces, . It is obvious from the definition of function that

A Banach space is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in such that and . Let be the unit sphere of . Then the Banach space is said to be smooth provided by which exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

We also need the following lemmas for the proof of our main results.

Lemma 2.1 (see [2]). Let be a strictly convex and smooth Banach space, then if and only if .

Lemma 2.2 (see [2]). Let be a uniformly convex and smooth Banach space and . Then, for all , where is a continuous, strictly increasing, and convex function with .

Lemma 2.3 (see [11]). Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space . Then

Lemma 2.4 (see [11]). Let be a uniformly convex Banach space and be a closed ball of . Then there exists a continuous strictly increasing convex function with such that for all and with .

Lemma 2.5. Let be a strictly convex and smooth Banach space, and let be a closed convex subset of . Suppose is a weak relatively uniformly nonexpansive multivalued mapping. Then, F(T) is closed and convex.

Proof. First, we show that is closed. Let be a sequence in such that as . Since the multivalued operator is uniformly weak relatively nonexpansive, one has for all and for all . Therefore, Applying Lemma 2.1, one gets . Hence . Therefore, .
Next, we show that is convex. To this end, for arbitrary . Putting , we prove that . Let , we have Using Lemma 2.1 again, we also obtain . Hence, , that is, . Therefore, is convex.

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions: for all ; is monotone, that is, for all ; for each , for each is convex and lower semicontinuous.

Lemma 2.6 (see [12]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying , and let and . Then, there exists such that

Lemma 2.7 (see [12]). Let be a closed subset of a strictly convex, uniformly smooth, and reflexive Banach space , and let be a bifunction from to satisfying . For all and , define a mapping as follows: for all . Then, the following hold: (1) is single-valued; (2) is firmly nonexpansive mapping, that is, for all , (3)(4) is closed and convex.

Lemma 2.8 (see [12]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying , and let and and ,

Lemma 2.9 (see [12]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let , and let . Then

3. Main Results

In this section, we prove a weak convergence theorem for finding a common element of the set of solutions for an equilibrium problem and the set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings in a Banach space. Before proving the result, we need the following theorem.

Theorem 3.1. Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying and let be two weak relatively uniformly nonexpansive multivalued mappings such that . Let be a sequence generated by the following manner: where , , and are the duality mapping on . Assume that , , and are three sequences in satisfying the following conditions: (a); (b); (c) for some . Then converges strongly to , where is the generalized projection of onto .

Proof. Let . Putting for all , it is well known that is relatively nonexpansive, one has Therefore, exists. Since is bounded, are bounded.
Define for all . Then, from and (3.2), one gets Since is the generalized projection, from Lemma 2.3, one sees Hence, from (3.3), one has Therefore, is a convergent sequence. Applying (3.3) again, one also obtains that, for all , From and Lemma 2.3, one has and hence Let . From Lemma 2.2, there exists a continuous, strictly increasing, and convex function with such that Therefore, one has Since is a convergent sequence, from the property of , one obtains that is a Cauchy sequence. Since is closed, converges strongly to . This completes the proof of Theorem 3.1.

In the following, we give our weak convergence result in this paper.

Theorem 3.2. Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying and let be two weak relatively uniformly nonexpansive multivalued mappings such that . Let be a sequence generated by the following manner: where , , and are the duality mapping on . Assume that , , and are three sequences in satisfying the following conditions: (a); (b); (c) for some . If is weakly sequentially continuous, then converges weakly to , where .

Proof. In view of the proof Theorem 3.1, one has that , , and are bounded. Let from Lemma 2.4, for , one has It follows that In view of , by taking the limit in (3.14), one sees From the property of , one has Since is also uniformly norm-to-norm continuous on bounded sets, one obtains Similarly, one could obtain Noticing that is bounded, one gets that there exists a subsequence of such that converges weakly to . From (3.17) and (3.18), and , one has
Next, we show that . Let . From Lemma 2.2, there exists a continuous, strictly increasing, and convex function with such that Noticing that and from Lemma 2.8 and (3.13), for , one has Noticing that exists, one gets It follows from the property of that Since is uniformly norm-to-norm continuous on bounded sets, one has From the assumption , one obtains Since , one has By replacing by and from , one sees Taking in the above inequality and from , one has For and , define . Since , one gets , which yields that . It follows from that that is, Let ; from , we obtain for all . This implies that . Therefore, .
On the other hand, let ; from Lemma 2.9 and , we have From Theorem 3.1, one knows that converges strongly to . Since is weakly sequentially continuous, one has as . On the other hand, since is monotone, one has as . Also, since is uniformly convex, one has . This completes the proof of Theorem 3.2.

If we only consider one operator , the following corollary can been established by Theorem 3.2.

Corollary 3.3. Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying , and let be a weak relatively uniformly nonexpansive multivalued mapping such that . Let be a sequence generated by the following manner: where , and is the duality mapping on . Assume that is a sequence in satisfying the following conditions: (a); (b) for some . If is weakly sequentially continuous, then converges weakly to , where .

If and are two relatively uniformly nonexpansive multivalued mappings, from Definitions 1.1 and 1.3, it is easy to know that the class of weak relatively uniformly nonexpansive multivalued mappings contains the class of relatively uniformly nonexpansive multivalued mappings as a subclass. Therefore, the following corollary can be easily obtained by Theorem 3.2.

Corollary 3.4. Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying and let be two relatively uniformly nonexpansive multivalued mappings such that . Let be a sequence generated by the following manner: where , , and are the duality mapping on . Assume that , , and are three sequences in satisfying the following conditions: (a); (b); (c) for some . If is weakly sequentially continuous, then converges weakly to , where .

Remark 3.5. Our results improve Theorem 4.1. of Takahashi and Zembayashi [12] and Theorem 3.5. of Qin et al. [11] in the following senses: (1)from single-valued mappings to multivalued ones; (2)from relatively nonexpansive single-valued mappings (the definition can be found in [1, 11, 12]) to weak relatively uniformly nonexpansive multivalued ones.

Acknowledgment

The author would like to thank the referees for their comments and suggestions.

References

  1. Y. Su, “On the weak relatively nonexpansive multi-valued mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 730619, 7 pages, 2012. View at Publisher · View at Google Scholar
  2. S. Homaeipour and A. Razani, “Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, article 73, 2011.
  3. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. View at Zentralblatt MATH
  4. 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
  5. A. Moudafi, “Second-order differential proximal methods for equilibrium problems,” JIPAM, vol. 4, no. 1, article 18, 2003. View at Zentralblatt MATH
  6. N. Shahzad and H. Zegeye, “Strong convergence results for nonself multimaps in Banach spaces,” Proceedings of the American Mathematical Society, vol. 136, no. 2, pp. 539–548, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. N. Shahzad and H. Zegeye, “On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 838–844, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Y. Song and H. Wang, “Convergence of iterative algorithms for multivalued mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 4, pp. 1547–1556, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. S. Jung, “Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 11, pp. 2345–2354, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 15–50, Dekker, New York, 1996. View at Zentralblatt MATH
  14. W. Takahashi, Nonlinear Functional Analysis, Yokohama, Yokohama, Japan, 2000.