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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 859598, 16 pages
http://dx.doi.org/10.1155/2012/859598
Research Article

Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems

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 14 June 2012; Accepted 25 July 2012

Academic Editor: Yonghong Yao

Copyright © 2012 H. Zegeye and N. Shahzad. 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 introduce an iterative process which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces.

1. Introduction

Let be a real Banach space with dual . The function , defined by is studied by Alber [1] and Reich [2], where is the normalized duality mapping from to defined by , where denotes the generalized duality pairing. It is well known that is smooth if and only if is single valued and if is uniformly smooth then is uniformly continuous on bounded subsets of . We note that in a Hilbert space ,   is the identity operator.

Let be a nonempty closed convex subset of a Hilbert space . It is well known that the metric projection of onto ,  , is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this direction, Alber [1] introduced a generalized projection operator in a Banach space which is an analogue of metric projection in Hilbert spaces.

Let be a nonempty closed and convex subset of a reflexive, strictly convex and smooth Banach space . The generalized projection mapping, introduced by Alber [1], is a mapping , that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem Let be a nonempty closed convex subset of a Banach space . Let be a single-valued mapping. An element is called a fixed point of if . The set of fixed points of is denoted by . A point in is said to be an asymptotic fixed point of (see [2]) if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . is said to be nonexpansive if for each and is called relatively nonexpansive if (A1) ; (A2) for and and (A3) .

Let be a nonempty closed convex subset of a Banach space and let and denote the family of nonempty subsets and nonempty closed bounded subsets of , respectively. Let be the Hausdorff metric on defined by for all , where is the distance from the point to the subset .

Let be a multivalued mapping. is said to be a nonexpansive if , for . An element is called a fixed point of , if , where . A point called an asymptotic fixed point of , if there exists a sequence in which converges weakly to such that . is said to be relatively nonexpansive if (B1) ; (B2) for , , and (B3) , where is the set of asymptotic fixed points of .

We remark that the class of relatively nonexpansive single-valued mappings is contained in a class of relatively nonexpansive multi-valued mappings. An example of relatively nonexpansive multi-valued mapping by Homaeipour and Razani [3] is given below.

Example 1.1. Let , , and . Let be defined by It is shown in [3] that is relatively nonexpansive multi-valued mapping which is not nonexpansive.

The study of fixed points for multi-valued nonexpansive mappings in relation to Hausdorff metric was introduced by Markin [4] (see also [5]). Since then a lot of activity in this area and fixed point theory for multi-valued nonexpansive mappings has been developed which has some nontrivial applications in pure and applied sciences including control theory, convex optimization, differential inclusion, and economics (see, e.g., [6] and references therein). Later, Lim [7] established the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces.

It is well known that the normal Mann’s iterative [8] algorithm has only weak convergence in an infinite-dimensional Hilbert space even for nonexpansive single-valued mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann’s iteration algorithm, the so called hybrid projection iteration method is such a modification. The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [9] in 1968. For 40 years, (HPIA) has received rapid developments. For details, the readers are referred to papers [1012] and the references therein.

In 2003, Nakajo and Takahashi [12] proposed the following modification of the Mann iteration method for a nonexpansive single-valued mapping in a Hilbert space : where is a closed convex subset of , denotes the metric projection from onto . They proved that if the sequence is bounded above from one then the sequence generated by (1.5) converges strongly to .

In spaces more general than Hilbert spaces, Matsushita and Takahashi [11] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive single-valued mapping in a Banach space : They proved the following convergence theorem.

Theorem MT. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , let be a relatively nonexpansive single-valued mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.6), where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .

Let be a bifunction, where is the set of real numbers. The equilibrium problem for is The solution set of (1.7) is denoted by .

If , where is a monotone mapping, then the problem (1.7) reduces to the system of variational inequality problem That is, the problem (1.8) is a special case of (1.7). The set of solutions of inequality (1.8) is denoted by .

For solving the equilibrium problem for a bifunction , we assume that satisfies the following conditions:(A1) , for all ,(A2) is monotone, that is,  , for all ,(A3) for each , ,(A4) for each ,   is convex and lower semicontinuous.

Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive single-valued mapping and the set of solutions of an equilibrium problems in the frame work of Hilbert spaces and Banach spaces, respectively: see, for instance, [2, 1321] and the references therein.

In [22], Kumam introduced the following iterative scheme in a Hilbert space: for finding a common element of the set of fixed point of nonexpansive single-valued mapping and set of solution of equilibrium problems.

In the case that is a Banach space, Takahashi and Zembayashi [16] introduced the following iterative scheme which is called the shrinking projection method: where is the duality mapping on , is the generalized projection from onto and is relatively nonexpansive single-valued mapping. They proved that the sequence converges strongly to a common element of the set of fixed point of relatively nonexpansive single-valued mapping and set of solution of equilibrium problem under appropriate conditions.

We remark that the computation of in (1.9) and (1.10) is not simple because of the involvement of computation of from for each .

More recently, Homaeipour and Razani [3] studied the following iterative scheme for a fixed point of relatively nonexpansive multi-valued mapping in uniformly convex and uniformly smooth Banach space : where for all and . They proved that if is weakly sequentially continuous then the sequence converges weakly to a fixed point of . Furthermore, it is shown that the scheme converges strongly to a fixed point of if interior of is nonempty.

But it is worth mentioning that the convergence of the scheme is either weak or it requires that the interior of   is nonempty.

In this paper, motivated by Kumam [22], Takahashi and Zembayashi [16], and Homaeipour and Razani [3], we construct an iterative scheme which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces. Our scheme does not involve computation of and , for each , and the requirement that the interior of is nonempty is dispensed with. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

2. Preliminaries

Let be a normed linear space with  . The modulus of smoothness of is the function defined by The space is said to be smooth if  ,  for all   and is called uniformly smooth if and only if .

The modulus of convexity of is the function defined by is called uniformly convex if and only if , for every .

In the sequel, we will need the following results.

Lemma 2.1   (see [1]). Let be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space and let . Then for all ,

We make use of the function , defined by studied by Alber [1]. That is, for all and . We know the following lemma.

Lemma 2.2 (see [1]). Let be reflexive strictly convex and smooth Banach space with as its dual. Then for all and .

Lemma 2.3 (see [1]). Let be a convex subset of a real smooth Banach space . Let . Then if and only if

Lemma 2.4 (see [23]). 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 , such that , and , for .

Lemma 2.5 (see [24]). Let be a real smooth and uniformly convex Banach space and let and be two sequences of . If either or is bounded and as , then , as .

Proposition 2.6 (see [3]). Let be a strictly convex and smooth Banach space and be a nonempty closed convex subset of . Let be a relatively nonexpansive multi-valued mapping. Then is closed and convex.

Lemma 2.7 (see [16]). Let be a nonempty, closed and convex subset of a uniformly smooth, strictly convex and reflexive real Banach space . Let be a bifunction from to which satisfies conditions (A1)–(A4). For and , define the mapping as follows: Then the following statements hold:(1) is single-valued,(2),(3), for ,(4) is closed and convex.

Lemma 2.8 (see [25]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

Lemma 2.9 (see [26]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , and . Then, .

3. Main Result

Let be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive real Banach space with dual . Let be a bifunction. For the rest of this paper, is a mapping defined as follows. For , let be given by where , for some .

Theorem 3.1. Let be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space . Let , be a bifunction which satisfies conditions (A1)–(A4). Let , for , be a finite family of relatively nonexpansive multi-valued mappings. Assume that is nonempty. Let be a sequence generated by where such that , , ,  for , satisfying , for each . Then converges strongly to an element of .

Proof. Since is nonempty closed and convex, put . Now from (3.2), Lemma 2.7(3) and property of , we get that Now, from (3.2), Lemma 2.7(3), relatively nonexpansiveness of , property of and (3.3), we have that where . Thus, by induction, which implies that is bounded and hence and are bounded. Now let . Then we have that . Using Lemma 2.2 and property of , we obtain that
Furthermore, from (3.2), Lemma 2.4, relatively nonexpansiveness of , for each , Lemma 2.7(3), and (3.6) we have that and hence where , for all . Note that satisfies and .

Now, we consider two cases.

Case 1. Suppose that there exists such that is nonincreasing for all . In this situation, is then convergent. Then from (3.7), we have that and hence Lemma 2.5 implies that Moreover, from (3.7), we have that , as , which implies by the property of that , as , for each , and hence, since uniformly continuous on bounded sets, we obtain that Furthermore, by Lemma 2.1, property of and the fact that , as , imply that and hence Therefore, from (3.9), (3.10), and (3.12), we obtain that as , for each .

Let be a subsequence of such that   and . Then, from (3.12), (3.13), and the uniform continuity of , we get that Now, we show that . But, from the definition of and (A2) we note that Letting , we have from (3.15) and (A4) that , for all . Now, for and , let . Since and , we have and hence . So, from the convexity of the equilibrium bifunction on the second variable , we have and hence . Now, letting and condition (A3), we obtain that , for all , and hence .

Next, we show that . But, since each satisfies condition (B3) we obtain from (3.13) and (3.15) that , for each , and hence . Thus, from the above discussions we obtain that . Therefore, by Lemma 2.3, we immediately obtain that . It follows from (3.8) and Lemma 2.9 that , as . Consequently, by Lemma 2.5.

Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 2.8, there exist a nondecreasing sequence such that , and , for all . Now, from (3.7) and the fact that , we have as . Thus, using the same proof of Case 1, we obtain that and , as , for each and hence Then from (3.8), we have that Since , (3.21) implies that In particular, since , we get Then, from (3.20), we obtain that , as . This together with (3.21) 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 3.1, we assume that , for continuous monotone mapping, then we obtain the following corollary.

Corollary 3.2. Let be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space . Let be a continuous monotone mapping. Let , for , be a finite family of relatively nonexpansive multi-valued mappings. Assume that is nonempty. Let be a sequence generated by where such that , , , for , satisfying , for each . Then converges strongly to an element of .

Proof. Let . Since is monotone and continuous, we get that a bifunction satisfies conditions (A1)–(A4). Thus, the conclusion follows from Theorem 3.1.

If in Theorem 3.1, we assume that , then we get the following theorem.

Corollary 3.3. Let be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space . Let , be a bifunction which satisfies conditions (A1)–(A4). Let be a relatively nonexpansive multi-valued mapping. Assume that is nonempty. Let be a sequence generated by where such that , , , for each . Then converges strongly to an element of .

Proof. The proof follows from Theorem 3.1 with .

If in Theorem 3.1, we assume that , we get the following corollary.

Corollary 3.4. Let be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space . Let , for , be a finite family of relatively nonexpansive multi-valued mappings. Assume that is nonempty. Let be a sequence generated by where such that , , , for satisfying , for each . Then converges strongly to an element of .

If in Theorem 3.1, we assume that each ,   is single valued, we get the following corollary.

Corollary 3.5. Let be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space . Let , be a bifunction which satisfies conditions (A1)–(A4). Let , for , be a finite family of relatively nonexpansive single-valued mappings. Assume that is nonempty. Let be a sequence generated by where such that , , , for , satisfying , for each . Then converges strongly to an element of .

If , a real Hilbert space, then is uniformly convex and uniformly smooth real Banach space. In this case, , identity map on and , projection mapping from onto . Thus, the following corollary holds.

Corollary 3.6. Let be a nonempty, closed, and convex subset of a Hilbert space . Let , be a bifunction which satisfies conditions (A1)–(A4). Let , for , be a finite family of relatively nonexpansive multi-valued mappings. Assume that is nonempty. Let be a sequence generated by where such that , , , for , satisfying , for each . Then converges strongly to an element of .

Remark 3.7. (1) Theorem 3.1 improves and extends the corresponding results of Kumanm [22] and Takahashi and Zembayashi [16] in the sense that either our scheme does not require computation of , for each , or the space considered is more general.
(2) Theorem 3.1 improves the corresponding results of Homaeipour and Razani [3] in the sense that our convergence is strong and the requirement that the interior of is nonempty is dispensed with.

Acknowledgment

The authors thank the reviewers for their useful comments.

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