Open Access

An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions

Fixed Point Theory and Applications20092009:794178

DOI: 10.1155/2009/794178

Received: 5 August 2008

Accepted: 4 January 2009

Published: 10 February 2009

Abstract

We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

1. Introduction

Let be a real Hilbert space with inner product and induced norm and let be a nonempty-closed convex subset of . Let be a function and let be a bifunction from to such that , where is the set of real numbers and . Flores-Bazán [1] introduced the following generalized equilibrium problem:
(1.1)

The set of solutions of (1.1) is denoted by . Flores-Bazán [1] provided some characterizations of the nonemptiness of the solution set for problem (1.1) in reflexive Banach spaces in the quasiconvex case. Bigi et al. [2] studied a dual problem associated with the problem (1.1) with .

Let . Here denotes the indicator function of the set ; that is, if and otherwise. Then the problem (1.1) becomes the following equilibrium problem:
(1.2)

The set of solutions of (1.2) is denoted by . The problem (1.2) includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem. For more detail, please see [35] and the references therein.

If for all , where is a function, then the problem (1.1) becomes a problem of finding which is a solution of the following minimization problem:
(1.3)

The set of solutions of (1.3) is denoted by .

If is replaced by a real-valued function , the problem (1.1) reduces to the following mixed equilibrium problem introduced by Ceng and Yao [6]:
(1.4)
Recall that a mapping is said to be a -strict pseudocontraction [7] if there exists such that
(1.5)

where denotes the identity operator on . When , is said to be nonexpansive. Note that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive mappings. We denote the set of fixed points of by .

Ceng and Yao [6], Yao et al. [8], and Peng and Yao [9, 10] introduced some iterative schemes for finding a common element of the set of solutions of the mixed equilibrium problem (1.4) and the set of common fixed points of a family of finitely (infinitely) nonexpansive mappings (strict pseudocontractions) in a Hilbert space and obtained some strong convergence theorems(weak convergence theorems). Some methods have been proposed to solve the problem (1.2); see, for instance, [35, 1118] and the references therein. Recently, S. Takahashi and W. Takahashi [12] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al. [13] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an -inverse strongly monotone mapping in a Hilbert space. Tada and Takahashi [14] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem. Ceng et al. [15] introduced an iterative algorithm for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a strict pseudocontraction mapping. Chang et al. [16] introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of problem (1.2), and the set of solutions of a variational inequality problem for an -inverse strongly monotone mapping. Colao et al. [17] introduced an iterative method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space and proved the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. To the best of our knowledge, there is not any algorithms for solving problem (1.1).

On the other hand, Marino and Xu [19] and Zhou [20] introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. Acedo and Xu [21] introduced some parallel and cyclic algorithms for finding a common fixed point of a family of finite strict pseudocontraction mappings and obtained both weak and strong convergence theorems for the sequences generated by the iterative schemes.

In the present paper, we introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

2. Preliminaries

Let be a real Hilbert space with inner product and norm . Let be a nonempty-closed convex subset of . Let symbols and denote strong and weak convergences, respectively. In a real Hilbert space , it is well known that
(2.1)

for all and .

For any , there exists a unique nearest point in , denoted by , such that for all . The mapping is called the metric projection of onto . We know that is a nonexpansive mapping from onto . It is also known that and
(2.2)

for all and .

For each , we denote by the convex hull of . A multivalued mapping is said to be a KKM map if, for every finite subset ,

We will use the following results in the sequel.

Lemma 2.1 (see [22]).

Let be a nonempty subset of a Hausdorff topological vector space and let be a KKM map. If is closed for all and is compact for at least one , then

For solving the generalized equilibrium problem, let us give the following assumptions for the bifunction , and the set :

(A1) for all ;

(A2) is monotone, that is, for any ;

(A3) for each , is weakly upper semicontinuous;

(A4)for each is convex;

(A5)for each is lower semicontinuous;

(B1)For each and , there exist a bounded subset and such that for any ,
(2.3)

(B2) is a bounded set.

Lemma 2.2.

Let be a nonempty-closed convex subset of . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function such that . For and define a mapping as follows:
(2.4)

for all . Assume that either (B1) or (B2) holds. Then, the following conclusions hold:

(1)for each , ;

(2) is single-valued;

(3) is firmly nonexpansive, that is, for any
(2.5)

(4)

(5) is closed and convex.

Proof.

Let be any given point in . For each , we define
(2.6)
Note that for each , is nonempty since and for each , . We will prove that is a KKM map on . Suppose that there exists a finite subset of and for all with such that for each . Then we have
(2.7)
for each . By (A4) and the convexity of , we have
(2.8)
which is a contradiction. Hence, is a KKM map on . Note that (the weak closure of ) is a weakly closed subset of for each . Moreover, if (B2) holds, then is also weakly compact for each . If (B1) holds, then for , there exists a bounded subset and such that for any ,
(2.9)
This shows that
(2.10)

Hence, is weakly compact. Thus, in both cases, we can use Lemma 2.1 and have .

Next, we will prove that for each ; that is, is weakly closed. Let and let be a sequence in such that . Then,
(2.11)
Since is weakly lower semicontinuous, we can show that
(2.12)
It follows from (A3) and the weak lower semicontinuity of that
(2.13)

This implies that . Hence, is weakly closed. Hence, . Hence, from the arbitrariness of , we conclude that , .

We observe that . So by similar argument with that in the proof of Lemma 2.3 in [9], we can easily show that is single-valued and is a firmly nonexpansive-type map. Next, we claim that . Indeed, we have the following:
(2.14)

At last, we claim that is a closed convex. Indeed, Since is firmly nonexpansive, is also nonexpansive. By [23, Proposition 5.3], we know that is closed and convex.

Remark 2.3.

It is easy to see that Lemma 2.2 is a generalization of [9, Lemma 2.3].

Lemma 2.4 (see [24, 25]).

Assume that is a sequence of nonnegative real numbers such that
(2.15)
where is a sequence in and is a sequence such that
(2.16)

Then,

Lemma 2.5.

In a real Hilbert space , there holds the following inequality:
(2.17)

for all

3. Strong Convergence Theorems

In this section, we show a strong convergence of an iterative algorithm based on both viscosity approximation method and parallel method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions in a Hilbert space.

We need the following assumptions for the parameters and :

(C1) and ;

(C2) ;

(C3) for some and ;

(C4) and ;

(C5) for all .

Theorem 3.1.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5), and let be a proper lower semicontinuous and convex function such that . Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Assume that either (B1) or (B2) holds. Let be a contraction of into itself and let , and be sequences generated by
(3.1)

for every , where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Proof.

We show that is a contraction of into itself. In fact, there exists such that for all . So, we have
(3.2)

for all Since is complete, there exists a unique element such that .

Let and let be a sequence of mappings defined as in Lemma 2.2. From , we have
(3.3)
We define a mapping by
(3.4)
By [21, Proposition 2.6], we know that is an -strict pseudocontraction and . It follows from (3.3), and such that
(3.5)
Put It is obvious that Suppose From (3.3), (3.5), and , we have
(3.6)

for every Therefore, is bounded. From (3.3) and (3.5), we also obtain that and are bounded.

Following [26], define by
(3.7)
As shown in [26], each is a nonexpansive mapping on . Set , we have
(3.8)
On the other hand, from and , we have
(3.9)
(3.10)
Putting in (3.9) and in (3.10), we have
(3.11)
So, from the monotonicity of , we get
(3.12)
hence
(3.13)
Without loss of generality, let us assume that there exists a real number such that for all Then,
(3.14)
hence
(3.15)

where

It follows from (3.8) and (3.15) that
(3.16)
Define a sequence such that
(3.17)
Then, we have
(3.18)
From (3.18) and (3.16), we have
(3.19)
It follows from (C1)–(C5) that
(3.20)
Hence, by [27, Lemma 2.2], we have . Consequently,
(3.21)
Since , we have
(3.22)
thus
(3.23)

It follows from (C1) and (C2) that .

Since for , it follows from (3.5) and (3.3) that
(3.24)
from which it follows that
(3.25)
It follows from (C1)–(C3) and that
(3.26)
For , we have from Lemma 2.2,
(3.27)
Hence,
(3.28)
By (3.24) and (3.28), we have
(3.29)
Hence,
(3.30)

It follows from (C1), (C2), and that .

Next, we show that
(3.31)
where . To show this inequality, we can choose a subsequence of such that
(3.32)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From , we obtain that . From , we also obtain that . Since and is closed and convex, we obtain .

We first show that To see this, we observe that we may assume (by passing to a further subsequence if necessary) (as ) for It is easy to see that and . We also have
(3.33)
where . Note that by [21, Proposition 2.6], is an -strict pseudocontraction and . Since
(3.34)
it follows from (3.26) and that
(3.35)

So by the demiclosedness principle [21, Proposition 2.6(ii)], it follows that .

We now show By we know that
(3.36)
It follows from (A2) that
(3.37)
Hence,
(3.38)
It follows from (A4), (A5) and the weakly lower semicontinuity of , and that
(3.39)
For with and , let Since and , we obtain and hence . So by (A4) and the convexity of , we have
(3.40)
Dividing by , we get
(3.41)
Letting , it follows from (A3) and the weakly lower semicontinuity of that
(3.42)
for all . Observe that if , then holds. Moreover, hence . This implies Therefore, we have
(3.43)

Finally, we show that , where .

From Lemma 2.5, we have
(3.44)
thus
(3.45)

It follows from (C1), (3.43), (3.45), and Lemma 2.4 that . From and , we have and . The proof is now complete.

Theorem 3.2.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5), and let be a proper lower semicontinuous and convex function such that . Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Assume that either (B1) or (B2) holds. Let be an arbitrary point in and let , and be sequences generated by
(3.46)

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Proof.

Let for all , by Theorem 3.1, we obtain the desired result.

4. Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence theorems. Now, give some examples as follows: for , let , by Theorems 3.1 and 3.2, respectively, we have the following results.

Theorem 4.1.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5), and let be a proper lower semicontinuous and convex function such that . Let be an -strict pseudocontraction for some such that . Assume that either (B1) or (B2) holds. Let be a contraction of into itself and let , and be sequences generated by
(4.1)

for every where , , and are sequences of numbers satisfying the conditions (C1)–(C4). Then, , and converge strongly to .

Theorem 4.2.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5), and let be a proper lower semicontinuous and convex function such that . Let be an -strict pseudocontraction for some such that . Assume that either (B1) or (B2) holds. Let be an arbitrary point in and let , and be sequences generated by
(4.2)

for every where , , and are sequences of numbers satisfying the conditions (C1)–(C4). Then, , and converge strongly to .

We need the following two assumptions.

(B3)For each and , there exist a bounded subset and such that for any ,
(4.3)
(B4)For each and , there exist a bounded subset and such that for any ,
(4.4)

Let , by Theorems 3.1 and 3.2, respectively, we obtain the following results.

Theorem 4.3.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5). Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Assume that either (B3) or (B2) holds. Let be a contraction of into itself, and let , and be sequences generated by
(4.5)

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Theorem 4.4.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5). Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Assume that either (B3) or (B2) holds. Let be an arbitrary point in and let , and be sequences generated by
(4.6)

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Let for all , by Theorems 3.1 and 3.2, respectively, we obtain the following results.

Theorem 4.5.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a lower semicontinuous and convex function, and let be a proper lower semicontinuous and convex function such that . Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Assume that either (B4) or (B2) holds. Let be a contraction of into itself, and let , and be sequences generated by
(4.7)

for every , where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Theorem 4.6.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be a lower semicontinuous and convex function, and let be a proper lower semicontinuous and convex function such that . Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Assume that either (B4) or (B2) holds. Let be an arbitrary point in and let , and be sequences generated by
(4.8)

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Let , and let for all . Then . By Theorems 3.1 and 3.2, we obtain the following results.

Theorem 4.7.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Let be a contraction of into itself, and let and be sequences generated by
(4.9)

for every , where and are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then, and converge strongly to .

Theorem 4.8.

Let be a nonempty-closed convex subset of a real Hilbert space . Let be an integer. For each , let be an -strict pseudocontraction for some such that . Assume for each , is a finite sequence of positive numbers such that for all and for all . Let . Let be an arbitrary point in and let and be sequences generated by
(4.10)

for every , where and are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then, and converge strongly to .

Remark 4.9.
  1. (1)

    Since the nonexpansive mappings have been replaced by the strict pseudocontractions, Theorems 3.1, 3.2, 4.1 and 4.2 extend and improve [6, Theorem 3.1], [8, Theorem 3.5], [9, Theorems 4.1 and 4.2], [18, Theorem 4.1], and the main results in [911, 1316].

     
  2. (2)

    Since the weak convergence has been replaced by strong convergence, Theorems 3.1, 3.2, 4.1−4.4 extend and improve [12, Theorem 3.1], [10, Corollary 4.1].

     
  3. (3)

    Theorems 4.7 and 4.8 are strong convergence theorems for strict pseudocontractions without constraints and hence they improve the corresponding results in [19, 21]. Theorems 3.1 and 3.2 also improve [10, Corollary 3.1].

     

Declarations

Acknowledgments

The first author was supported by the National Natural Science Foundation of China (Grants No. 10771228 and No. 10831009), the Science and Technology Research Project of Chinese Ministry of Education (Grant no. 206123), the Education Committee project Research Foundation of Chongqing Normal University (Grant no. KJ070816); the second and third authors were partially supported by the Grants NSC97-2221-E-230-017 and NSC96-2628-E-110-014-MY3, respectively.

Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Chongqing Normal University
(2)
Department of Information Management, Cheng Shiu University
(3)
Department of Applied Mathematics, National Sun Yat-sen University

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© Jian-Wen Peng et al. 2009

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