Open Access

Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space

Journal of Inequalities and Applications20102010:246237

https://doi.org/10.1155/2010/246237

Received: 26 January 2010

Accepted: 15 April 2010

Published: 24 May 2010

Abstract

We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space . A mapping of into itself is said to be nonexpansive if for all , and a mapping is said to be firmly nonexpansive if for all . Let be a smooth,strictly convex and reflexive Banach space; let be the duality mapping of and a nonempty closed convex subset of . A mapping is said to be nonspreading if
(1.1)
for all , where for all ; see, for instance, Kohsaka and Takahashi [1]. In the case when is a Hilbert space, we know that for all Then a nonspreading mapping in a Hilbert space is defined as follows:
(1.2)

for all Let be the set of fixed points of , and be nonempty; a mapping is said to be quasi-nonexpansive if for all and

Remark 1.1.

In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [1].

Fixed point iterations process for nonexpansive mappings and asymptotically nonexpansive mappings in Banach spaces including Mann and Ishikawa iterations process have been studied extensively by many authors to solve the nonlinear operator equations. In 1953, Mann [2] introduced Mann iterative process defined by
(1.3)

where and satisfies the assumptions and and proved that in case is a Banach space, and is closed, and is continuous, then the convergence of to a point implies that . Recently, Dotson [3] proved that a Mann iteration process was applied to the approximation of fixed points of quasi-nonexpansive mappings in Hilbert space and in uniformly convex and strictly convex Banach spaces.

On the other hand, Kohsaka and Takahashi [1] proved an existence theorem of fixed points for nonspreading mappings in a Banach space. Very recently, Iemoto and Takahashi [4] studied the approximation theorem of common fixed points for a nonexpansive mapping of into itself and a nonspreading mapping of into itself in a Hilbert space. In particular, this result reduces to approximation fixed points of a nonspreading mapping of into itself in a Hilbert space by using iterative scheme
(1.4)
Let be a real-valued function and an equilibrium bifunction, that is, for each . The mixed equilibrium problem is to find such that
(1.5)
Denote the set of solutions of (1.5) by . The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equiliubrium problems, and the equilibrium problems as special cases (see, e.g., Blum and Oettli [5]). In particular, if , this problem reduces to the equilibrium problem, which is to find such that
(1.6)
The set of solutions of (1.6) is denoted by Numerous problems in physics, optimization, and economics reduce to find a solution of (1.6). Let be two monotone bifunctions and is constant. In 2009, Moudafi [6] introduced an alternating algorithm for approximating a solution of the system of equilibrium problems: finding such that
(1.7)
Let be two monotone bifunctions and are two constants. In this paper, we consider the following problem for finding such that
(1.8)
which is called a system of mixed equilibrium problems. In particular, if , then problem (1.8) reduces to finding such that
(1.9)
The system of nonlinear variational inequalities close to these introduced by Verma [7] is also a special case: by taking , , and , where are two nonlinear mappings. In this case, we can reformulate problem (1.7) to finding such that
(1.10)

which is called a general system of variational inequalities where and are two constants. Moreover, if we add up the requirement that , then problem (1.10) reduces to the classical variational inequality .

In 2008,Ceng and Yao[8] considered a new iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.They also proved a strong convergence theorem for the iterative scheme. In the same year, Yao et al. [9] introduced a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. Very recently, Ceng et al. [10] introduced and studied a relaxed extragradient method for finding a common of the set of solution (1.10) for the and -inverse strongly monotones and the set of fixed points of a nonexpansive mapping of into a real Hilbert space . Let , and are given by
(1.11)

Then, they proved that the iterative sequence converges strongly to a common element of the set of fixed points of a nonexpansive mapping and a general system of variational inequalities with inverse-strongly monotone mappings under some parameters controlling conditions.

In this paper, we introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping.

2. Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . For every point , there exists a unique nearest point in , denoted by , such that
(2.1)
is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies
(2.2)
for all . Moreover, is characterized by the following properties: and
(2.3)

for all Further, for all and , if and only if , for all .

A space is said to satisfy Opial's condition if for each sequence in which converges weakly to point , we have
(2.4)

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 (see [11]).

Let be an inner product space. Then for all and with we have
(2.5)

Lemma 2.2 (see [12]).

Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to y, then .

Lemma 2.3 (see [1]).

Let be a Hilbert space and a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then the following are equivalent:

(1)there exists such that is bounded;

(2) is nonempty.

Lemma 2.4 (see [1]).

Let be a Hilbert space and a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then is closed and convex.

Lemma 2.5 (see [4]).

Let be a Hilbert space, a closed convex subset of , and a nonspreading mapping with . Then is demiclosed, that is, and imply

Lemma 2.6 (see [13]).

Let be a closed convex subset of a real Hilbert space and let be a sequence in . Suppose that for all ,
(2.6)

for every . Then, converges strongly to some .

Lemma 2.7 (see [14]).

Assume is a sequence of nonnegative real numbers such that
(2.7)

where is a sequence in and is a sequence in such that

(1) ,

(2) or

Then

For solving the mixed equilibrium problems for an equilibrium bifunction , let us assume that satisfies the following conditions:

(A1) for all ;

(A2) is monotone, that is, for all

(A3)for each , is weakly upper semicontinuous;

(A4)for each , is convex, semicontinuous.

The following lemma appears implicitly in [5, 15].

Lemma 2.8 (see [5]).

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that
(2.8)

The following lemma was also given in [15].

Lemma 2.9 (see [15]).

Assume that satisfies (A1)–(A4). For and , define a mapping as follows:
(2.9)

for all . Then, the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any ,

(3)

(4) is closed and convex.

We note that Lemma 2.9 is equivalent to the following lemma.

Lemma 2.10.

Let a nonempty closed convex subset of a real Hilbert space . Let be an equilibrium bifunction satisfying (A1)–(A4) and let be a lower semicontinuous and convex functional. For and define a mapping as follows.
(2.10)

Then, the following results hold:

(i)for each , ;

(ii) is single-valued;

(iii) is firmly nonexpansive, that is, for any ,
(2.11)

(iv)

(v) is closed and convex.

Proof.

Define , for all . Thus, the bifunction satisfies, (A1)–(A4). Hence, by Lemmas 2.8 and 2.9, we have (i)–(v).

Lemma 2.11 (see [6]).

Let be a closed convex subset of a real Hilbert space . Let and be two mappings from satisfying (A1)–(A4) and let and be defined as in Lemma 2.10 associated to and , respectively. For given is a solution of problem (1.8) if and only if is a fixed point of the mapping defined by
(2.12)

where .

Proof.

By a similar argument as in the proof of Proposition  2.1 in [6], we obtain the desired result.

We note from Lemma 2.11 that the mapping is nonexpansive. Moreover, if is a closed bounded convex subset of , then the solution of problem (1.8) always exists. Throughout this paper, we denote the set of solutions of (1.8) by

3. Main Result

In this section, we prove a weak convergence theorem for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the system of mixed equilibrium problems.

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 2.10 associated to and , respectively. Let be a quasi-nonexpansive mapping of into itself such that Suppose and , are given by
(3.1)

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.8), where

Proof.

Let . Then and Put , and . Since
(3.2)
it follows by Lemma 2.1 that
(3.3)
Hence is a decreasing sequence and therefore exists. This implies that , , , and are bounded. From (3.3), we note that
(3.4)
Since and , we obtain
(3.5)
This implies that . Since and are firmly nonexpansive, it follows that
(3.6)
and so . By the convexity of , we have
(3.7)
This implies that
(3.8)
Since and , we obtain Similarly, we note that
(3.9)
and so . Thus, we have
(3.10)
and hence
(3.11)
It follows from and that Hence
(3.12)
and therefore
(3.13)
Since is a bounded sequence, there exists a subsequence such that converges weakly to . From Lemma 2.5, we have . Let be a mapping which is defined as in Lemma 2.11. Thus, we have
(3.14)
and hence
(3.15)
From and , we get According to Lemmas 2.2 and 2.11, we have . Hence Since and , we obtain . Let be another subsequence of such that converges weakly to . We may show that , suppose not. Since exists for all , it follows by the Opial's condition that
(3.16)
This is a contradiction. Thus, we have . This implies that converges weakly to . Put . Finally, we show that . Now from (2.2) and , we have
(3.17)
Since is nonnegative and decreasing for any it follows by Lemma 2.6 that converges strongly to some and hence
(3.18)

Therefore, .

Corollary 3.2.

Let be a closed convex subset of a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 2.10 associated to and , respectively. Let be a nonspreading mapping of into itself such that Suppose and , , are given by
(3.19)

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.8), where

Setting and in Theorem 3.1, we have following result.

Corollary 3.3.

Let be a closed convex subset of a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 2.10 associated to and , respectively, such that Suppose and , , are given by
(3.20)

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.9), where

Setting in Theorem 3.1, we have the following result.

Corollary 3.4.

Let be a closed convex subset of a real Hilbert space . Let be a bifunction from satisfying (A1)–(A4). Let and let be defined as in Lemma 2.9 associated to . Let be a quasi-nonexpansive mapping of into itself such that Suppose and and are given by
(3.21)

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.7), where

Declarations

Acknowledgments

The authors would like to thank the referee for the insightful comments and suggestions. The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Naresuan University

References

  1. Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Archiv der Mathematik 2008, 91(2):166–177. 10.1007/s00013-008-2545-8MathSciNetView ArticleMATHGoogle Scholar
  2. Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar
  3. Dotson WG Jr.: On the Mann iterative process. Transactions of the American Mathematical Society 1970, 149: 65–73. 10.1090/S0002-9947-1970-0257828-6MathSciNetView ArticleMATHGoogle Scholar
  4. Iemoto S, Takahashi W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2009, 71: 2082–2089. 10.1016/j.na.2009.03.064MathSciNetView ArticleMATHGoogle Scholar
  5. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123–145.MathSciNetMATHGoogle Scholar
  6. Moudafi A: From alternating minimization algorithms and systems of variational inequalities to equilibrium problems. Communications on Applied Nonlinear Analysis 2009, 16(3):31–35.MathSciNetMATHGoogle Scholar
  7. Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities. Computers & Mathematics with Applications 2001, 41(7–8):1025–1031. 10.1016/S0898-1221(00)00336-9MathSciNetView ArticleMATHGoogle Scholar
  8. Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008, 214(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleMATHGoogle Scholar
  9. Yao Y, Liou Y-C, Yao J-C: A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems. Fixed Point Theory and Applications 2008, -15.Google Scholar
  10. Ceng L-C, Wang C-Y, Yao J-C: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Mathematical Methods of Operations Research 2008, 67(3):375–390. 10.1007/s00186-007-0207-4MathSciNetView ArticleMATHGoogle Scholar
  11. Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Computers & Mathematics with Applications 2000, 40(4–5):559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleMATHGoogle Scholar
  12. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar
  13. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003, 118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
  14. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004, 298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
  15. Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997, 78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© S. Plubtieng and K. Sombut. 2010

This article is published under license to BioMed Central Ltd. 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.