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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 141376, 17 pages
http://dx.doi.org/10.1155/2010/141376
Research Article

Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 11 April 2010; Revised 9 July 2010; Accepted 16 July 2010

Academic Editor: Simeon Reich

Copyright © 2010 Prasit Cholamjiak and Suthep Suantai. 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 a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.

1. Introduction

Let be a real Banach space, and let be the dual of . Let be a closed and convex subset of . Let be bifunctions from to , where is the set of real numbers and is an arbitrary index set. The system of equilibrium problems is to find such that If is a singleton, then problem (1.1) reduces to find such that The set of solutions of the equilibrium problem (1.2) is denoted by .

Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element in the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem.

In 2004, Matsushita and Takahashi [2] introduced the following algorithm for a relatively nonexpansive mapping in a Banach space : for any initial point , define the sequence by where is the duality mapping on , is the generalized projection from onto , and is a sequence in . They proved that the sequence converges weakly to fixed point of under some suitable conditions on .

In 2008, Takahashi and Zembayashi [3] introduced the following iterative scheme which is called the shrinking projection method for a relatively nonexpansive mapping and an equilibrium problem in a Banach space : They proved that the sequence converges strongly to under some appropriate conditions.

2. Preliminaries and Lemmas

Let be a real Banach space, and let be the unit sphere of . A Banach space is said to be strictly convex if, for any , It is also said to be uniformly convex if, for each , there exists such that, for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. The function which is called the modulus of convexity of is defined as follows: The space is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . The duality mapping is defined by for all . If is a Hilbert space, then , where is the identity operator. It is also known that, if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subset of (see [4] for more details).

Let be a smooth Banach space. The function is defined by for all . In a Hilbert space , we have for all .

Let be a closed and convex subset of , and let be a mapping from into itself. A point in is said to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping is said to be relatively nonexpansive [68] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [6, 7]. is said to be relatively quasi-nonexpansive if and for all and . It is obvious that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings. The class of relatively quasi-nonexpansive mappings was studied by many authors (see, for example, [912]). Recall that is closed if

The aim of this paper is to introduce a new hybrid projection algorithm for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of closed and relatively quasi-nonexpansive mappings in the frameworks of Banach spaces.

We will need the following lemmas.

Lemma 2.1 (Kamimura and Takahashi [8]). Let be a uniformly convex and smooth Banach space, and let be two sequences of . If and either or is bounded, then as .

Let be a reflexive, strictly convex, and smooth Banach space, and let be a nonempty, closed, and convex subset of . The generalized projection mapping, introduced by Alber [13], is a mapping that assigns to an arbitrary point the minimum point of the function ; that is, , where is the solution of the minimization problem The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the duality mapping (see, for instance, [4, 8, 1315]). In a Hilbert space, is coincident with the metric projection.

Lemma 2.2 (Alber [13], Kamimura and Takahashi [8]). Let be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space , let , and let . Then if and only if

Lemma 2.3 (Alber [13], Kamimura and Takahashi [8]). Let be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space . Then

Lemma 2.4 (Qin et al. [16]). Let be a uniformly convex, smooth Banach space, and let be a closed and convex subset of . Let be a closed and relatively quasi-nonexpansive mapping from into itself. Then is closed and convex.

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

Lemma 2.5 (Blum and Oettli [17]). Let be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to which satisfies conditions (A1)–(A4), and let and . Then there exists such that

Lemma 2.6 (Takahashi and Zembayashi [18]). Let be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to which satisfies conditions (A1)–(A4). For all and , define the mapping as follows: Then, the following statements hold:(1) is single valued;(2) is of firmly nonexpansive type [19]; that is, for all , (3);(4) is closed and convex.

Lemma 2.7 (Takahashi and Zembayashi [18]). Let be a closed and convex subset of a smooth, strictly, and reflexive Banach space , let be a bifunction from to which satisfies conditions (A1)–(A4), and let . Then, for all and ,

3. Strong Convergence Theorems

Theorem 3.1. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty, closed, and convex subset of . Let be bifunctions from to which satisfies conditions (A1)–(A4), and let be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from into itself. Assume that . For any initial point with and , define the sequence as follows: Assume that and for are sequences which satisfy the following conditions:(B1);(B2).
Then the sequence converges strongly to .

Proof. We divide our proof into six steps as follows.
Step 1. for all .
From Lemma 2.4 we know that is closed, and convex for all . From Lemma 2.6(4), we also know that is closed and convex for each . Hence is a nonempty, closed and convex subset of . Clearly is closed and convex. Suppose that is closed and convex for some . For each and , we see that is equivalent to By the construction of the set , we see that Hence is also closed and convex.
It is obvious that . Now, suppose that for some , and let . Then Hence . By induction, we can conclude that for all .
Step 2. exists.
From and , we have From Lemma 2.3 we get that Combining (3.5) and (3.6), we get that exists.
Step 3. is a Cauchy sequence in .
Since for , we obtain from Lemma 2.3 that We see that as , which implies with Lemma 2.1 that as . Therefore is a Cauchy sequence. By the completeness of the space and the closedness of the set , we can assume that as . Moreover, we get that Since , we have for all that Applying Lemma 2.1 to (3.8) and (3.9), we derive This shows that as for all . Since is uniformly norm-to-norm continuous on bounded subsets of , we obtain that
Step 4. .
Denote for any and for all . We note that for all . From (3.4) we observe that Since for all , it follows from (3.12) and Lemma 2.7 that From (3.10) and (3.11), we get that for all . From Lemma 2.1, we have From (3.10) and (3.14), we have and hence, Again, since for all , it follows from (3.12) and Lemma 2.7 that From (3.15) and (3.16), we also have Hence, from (3.15) and (3.18), we get In a similar way, we can verify that for all , for all , for all . Hence, we can conclude that for each and . Observe that then we obtain from (B1) and (3.23) that Since is also uniformly norm-to-norm continuous on bounded subsets, we get that Since is closed for all and , we conclude that .
Step 5. .
From (3.24) and (B2), we have that as . Then, for each , we obtain that From (A2) we have that From (A4) and the fact that for , we get for all . For each and , denote . Then , which implies that . From (A1) and (A4), we obtain that . Thus, . From (A3), we have for all and . Hence .
Step 6. .
From , we have Since , we also have Letting in (3.31), we obtain that From Lemma 2.2 we conclude that . This completes the proof.

Remark 3.2. Theorem 3.1 improves and extends Theorem 3.1 of Takahashi and Zembayashi in [3] in the following senses:(i)from the case of an equilibrium problem to a finite family of equilibrium problems;(ii) from a single relatively nonexpansive mapping to an infinitely countable family of relatively quasi-nonexpansive mappings;(iii)if and for all , then our restriction on is weaker than that of Theorem 3.1 of [3].

Remark 3.3. The iteration (3.1) is a modification of (1.4) in the following ways.(i)We use the composition of mappings in the second step.(ii)We construct the set by using the concept of supremum concerning an infinitely countable family of closed and relatively quasi-nonexpansive mappings . If and for all , then the iteration (3.1) reduces to that of (1.4).
If we take for all in Theorem 3.1, then we have the following corollary.

Corollary 3.4. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty, closed, and convex subset of . Let be bifunctions from to which satisfies conditions (A1)–(A4), and let be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from into itself. Assume that . For any initial point with and , define the sequence as follows: If    for each , then converges strongly to .

4. Applications

In this section, we give several applications of Theorem 3.1 in the framework of Banach spaces and Hilbert spaces.

Let be a nonlinear mapping. The classical variational inequality problem is to find that such that The solutions set of (4.1) is denoted by . For each and , define the mapping as follows:

Theorem 4.1. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty, closed, and convex subset of . Let be continuous and monotone operators from to , and let be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from into itself such that . For any initial point with and , define the sequence as follows: Assume that and for are sequences which satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence converges strongly to .

Proof. Define for all and . First, we see that for each .
Next, we show that satisfy conditions (A1)–(A4).(A1) Consider for all and .(A2) For each and , we observe that By the monotonicity of , we obtain that is monotone. Thus satisfy condition (A2).(A3) For each and , we have by the continuity of that This shows that satisfy condition (A3).(A4)Let and . Then, for each and , we have Thus is convex in the second variable. Let and . Then This shows that is lower semicontinuous in the second variable. Hence satisfy condition (A4). From Theorem 3.1 we obtain the desired result.

If we take for all in Theorem 4.1, then we have the following corollary.

Corollary 4.2. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty, closed, and convex subset of . Let be continuous and monotone operators from to , and let be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from into itself such that . For any initial point with and , define the sequence as follows: If    for each , then converges strongly to .
Let be a real-valued function. The convex minimization problem is to find that such that The solutions set of (4.9) is denoted by . For each and , define the mapping as follows:

Theorem 4.3. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty, closed, and convex subset of . Let be lower semicontinuous and convex functions from to , and let be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from into itself such that . For any initial point with and , define the sequence as follows: Assume that and for are sequences which satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence converges strongly to .

Proof. Define for all and . Then for each , and therefore satisfy conditions (A1) and (A2).
Next, we show that satisfy conditions (A3) and (A4). For each , we have by the lower semicontinuity of that This implies that satisfy condition (A3).
Let and . For each , we have by the convexity of that On the other hand, let and . By the lower semicontinuity of we have Thus satisfy condition (A4). From Theorem 3.1 we also obtain the desired result.

If we take for all in Theorem 4.3, then we have the following corollary.

Corollary 4.4. Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty, closed, and convex subset of . Let be lower semicontinuous and convex functions from to , and let be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from into itself such that . For any initial point with and , define the sequence as follows: If    for each , then converges strongly to .

As a direct consequence of Theorem 3.1, we obtain the following application in a Hilbert space.

Theorem 4.5. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be bifunctions from to which satisfies conditions (A1)–(A4), and let be an infinitely countable family of closed and quasi-nonexpansive mappings from into itself such that . For any initial point with and , define the sequence as follows: where is the metric projection. Assume that and for are sequences which satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence converges strongly to .

Proof. Taking in Theorem 3.1, the result is obtained immediately.

Remark 4.6. Theorem 4.5 improves and extends the main results of [2022] in the following senses:(i)from the case of an equilibrium problem to a finite family of equilibrium problems;(ii) from the class of nonexpansive mappings to the class of an infinitely countable family of quasi-nonexpansive mappings.

Acknowledgments

The authors would like to thank Professor Simeon Reich and the referee for valuable suggestions on improving the manuscript and the Thailand Research Fund for financial support. The first author was supported by the Royal Golden Jubilee Grant PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.

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