Abstract and Applied Analysis

Volume 2014 (2014), Article ID 540381, 24 pages

http://dx.doi.org/10.1155/2014/540381

Research Article

Iterative Algorithms for Systems of Generalized Equilibrium Problems with the Constraints of Variational Inclusion and Fixed Point Problems

Lu-Chuan Ceng,¹ Abdul Latif,² and Abdullah E. Al-Mazrooei²

¹Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

²Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 6 December 2013; Accepted 11 January 2014; Published 12 March 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 Lu-Chuan Ceng et al. 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 and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.

1. Introduction

Let be a real Hilbert space with inner product and norm , be a nonempty closed convex subset of and be the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called strongly positive on if there exists a constant such that

A mapping is called -Lipschitz continuous if there exists a constant such that

In particular, if then is called a nonexpansive mapping; if then is called a contraction.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP) [1] which is to find a point such that

The solution set of VIP (3) is denoted by .

Let be a real-valued function, be a nonlinear mapping and be a bifunction. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that

We denote the set of solutions of GMEP (4) by . The system of equilibrium problems or generalized equilibrium problems is a tool to study Nash eequilibrium problems, see for example [3–8]. In fact, the GMEP (4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see for example, [9–15]. Here we also consider a system of two generalized equilibrium problem that could be usefull to study the Two players game problem, see [16].

Throughout this paper, it is assumed as in [2] that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (H5), where

(H1) for all ;(H2) is monotone, that is, for any ;(H3) is upper-hemicontinuous, that is, for each ,

(H4) is convex and lower semicontinuous for each ;(H5)for each and , there exists a bounded subset and such that for any ,

Given a positive number . Let is the solution set of the auxiliary mixed equilibrium problem, that is, for each ,

In particular, whenever , is rewritten as .

Let be two bifunctions, and be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find such that

where and are two constants. It is introduced and studied in [17]. Whenever , the SGEP reduces to a system of variational inequalities, which is considered and studied in [18]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.

In 2010, Ceng and Yao [17] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [17]). Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse-strongly monotone for . Let for . Then, is a solution of (8) if and only if is a fixed point of the mapping defined by where . Here, we denote the fixed point set of by .

Let be an infinite family of nonexpansive mappings on and be a sequence of nonnegative numbers in . For any , define a mapping on as follows:

Such a mapping is called the -mapping generated by and .

In 2011, for the case where , Yao et al. [14] proposed the following hybrid iterative algorithm

where be a contraction, is differentiable and strongly convex, and are given, for finding a common element of the set and the fixed point set of an infinite family of nonexpansive mappings on . They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (10) to a point under some appropriate conditions. This point also solves the following optimization problem:

where is the potential function of .

Let be a contraction and be a strongly positive bounded linear operator on . Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)–(H4), and that are inverse-strongly monotone. Let the mapping be defined as in Proposition CY. Very recently, Ceng et al. [11] introduced the following hybrid extragradient-like iterative algorithm

for finding a common solution of GMEP (4), SGEP (8) and the fixed point problem of an infinite family of nonexpansive mappings on , where , and are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (11) to a point under some suitable conditions. This point also solves the following optimization problem:

where is the potential function of .

On the other hand, let be a single-valued mapping of into and be a set-valued mapping with . Consider the following variational inclusion: find a point such that

We denote by the solution set of the variational inclusion (12). In particular, if , then . If , then problem (12) becomes the inclusion problem introduced by Rockafellar [19]. It is known that problem (12) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, and so forth. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows:

where is a positive number.

In this paper, we will introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems. Such solution also solves an optimization problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement and generalization of the previously known results in this area.

2. Preliminaries

Throughout this paper, we assume that is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence , that is,

Definition 1. A mapping is called

(i) monotone if

(ii) -strongly monotone if there exists a constant such that

(iii) -inverse-strongly monotone if there exists a constant such that

It is easy to see that the projection is -ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition 2. A differentiable function is called:

(i) convex, if

where is the Frechet derivative of at ;

(ii) strongly convex, if there exists a constant such that

It is easy to see that if is a differentiable strongly convex function with constant then is strongly monotone with constant .

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property

Some important properties of projections are gathered in the following proposition.

Proposition 3. For given and :

(i);(ii), ;(iii), . (This implies that is nonexpansive and monotone.)

By using the technique of [20], we can readily obtain the following elementary result.

Proposition 4 (see [11, Lemma 1 and Proposition 1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)–(H4). Assume that

(i) is strongly convex with constant and the function is weakly upper semicontinuous for each ;(ii)for each and , there exists a bounded subset and such that for any ,

Then the following hold:(a) for each ;(b) is single-valued;(c) is nonexpansive if is Lipschitz continuous with constant and

where for ;(d) for all and

(e) ;(f) is closed and convex.

In particular, whenever is a bifunction satisfying the conditions (H1)–(H4) and , then that is, for any ,

( is firmly nonexpansive) and

In this case, is rewritten as . If, in addition, , then is rewritten as .

Remark 5. Suppose is strongly convex with constant and is Lipschitz continuous with constant . Then is -strongly monotone and -Lipschitz continuous with positive constants . Utilizing Proposition 4 (d) we can show that for all and ,

We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 6. Let be a real inner product space. Then there holds the following inequality

Lemma 7. Let be a real Hilbert space. Then the following hold:

(a) for all ;(b) for all and with ;(c)If is a sequence in such that , it follows that

We have the following crucial lemmas concerning the -mappings defined by (9).

Lemma 8 (see [21, Lemma 3.2]). Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and the limit exists, where is defined by (9).

Remark 9 (see [22, Remark 3.1]). It can be known from Lemma 8 that if is a nonempty bounded subset of , then for there exists such that for all

Remark 10 (see [22, Remark 3.2]). Utilizing Lemma 8, we define a mapping as follows:

Such a is called the -mapping generated by and . Since is nonexpansive, is also nonexpansive. Indeed, observe that for each

If is a bounded sequence in , then we put . Hence, it is clear from Remark 5 that for an arbitrary there exists such that for all

This implies that

Lemma 11 (see [21, Lemma 3.3]). Let be a sequence of nonexpansive self-mappings on such that , and let be a sequence in for some . Then, .

Lemma 12 (see [23, Demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .

Lemma 13. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3 (i)) implies

Lemma 14 (see [24]). Let and be bounded sequences in a real Banach space and be a sequence in with . Suppose

Then, .

Lemma 15 (see [25]). Assume that is a sequence of nonnegative real numbers such that

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

(i);(ii) or .

Then .

Recall that a set-valued mapping is called monotone if for all and imply

A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for for every implies . Next we provide an example to illustrate the concept of maximal monotone mapping.

Let be a monotone, -Lipschitz-continuous mapping and let be the normal cone to at , that is,

Define

Then, is maximal monotone and if and only if ; see [19].

Assume that is a maximal monotone mapping. Let . In terms of Huang [26] (see also [27]), there holds the following property for the resolvent operator .

Lemma 16. is single-valued and firmly nonexpansive, that is,

Consequently, is nonexpansive and monotone.

Lemma 17 (see [28]). Let be a maximal monotone mapping with . Then for any given is a solution of problem (12) if and only if satisfies

Lemma 18 (see [27]). Let be a maximal monotone mapping with and let be a strongly monotone, continuous and single-valued mapping. Then for each , the equation has a unique solution for .

Lemma 19 (see [28]). Let be a maximal monotone mapping with and be a monotone, continuous and single-valued mapping. Then for each . In this case, is maximal monotone.

Lemma 20 (see [29]). Let be a nonempty closed convex subset of a real Hilbert space , and be a proper lower semicontinuous differentiable convex function. If is a solution the minimization problem

then,

In particular, if solves (OP), then

3. Main Results

In this section, we introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.

Theorem 21. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that for any ,

(iii), and ;(iv), and satisfies

(v).

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Proof. Since and , we may assume, without loss of generality, that . Since is a -strongly positive bounded linear operator on , we know that

Observe that

that is, is positive. It follows that

Put

for all , and , where is the identity mapping on . Then we have that .

We divide the rest of the proof into several steps.

Step 1. We show that is bounded. Indeed, take arbitrarily. Since , is -inverse strongly monotone and , we have, for any ,

Since and is -inverse strongly monotone, where , by Lemma 16 we deduce that for each

Combining (52) and (53), we have

Since , is -inverse-strongly monotone for , and for , we deduce that, for any ,

(This shows that is nonexpansive.) Thus, from (54), we get

Set . Then from (47) we have

By induction, we get

Therefore, is bounded and so are the sequences and .

Step 2. We show that as .

Indeed, define

Then from the definition of , we obtain

It follows that

From (9), since and are all nonexpansive, we have

where is a constant such that

On the other hand, we estimate . Taking into account that and , we may assume, without loss of generality, that and . Utilizing Remark 5 and Lemma 16, we have

where for some .

Note that

Since is nonexpansive, from (62), (64) and (66) it follows that

Utilizing (61), and (68), we have

where for some . Since and , we deduce from (69) that

Since for all , by Lemma 14 we obtain from that

which immediately yields

Step 3. as .

Indeed, utilizing Lemmas 6 and 7 (b) we obtain from (47) and (54) that

which leads to

Since and , we deduce from the boundedness of and that

Also, by Lemma 7 (b) we deduce from (47) and (54) that

From (73) and (76) we get

which immediately implies that

Since and , we deduce from the boundedness of and that

So, it follows that

Step 4.  and as .

Indeed, for , we find that

From (47), (53) and (81), we obtain

which together with (73), implies that

So, it follows that

Since and , from the boundedness of and we get

Furthermore, from the firm nonexpansivity of , we have

which implies that

From (47) and (87), we have

which together with (73), implies that

So, it follows that

Since and , from (85) and the boundedness of and we get

Next we show that ,   . Observe that

From (47) and (92), we have

which together with (73), implies that

So, it follows that

Since ,   ,   and , from the boundedness of and we get

By Lemmas 7 (a) and 16, we obtain

which implies that

From (47) and (98), we have

which together with (73), implies that

So, it follows that

Since and , from (96) and the boundedness of and we get

From (102) we get

By (91) and (103), we have

On the other hand, for simplicity, we write and for all . Then

We now show that , that is, . As a matter of fact, for , it follows from (47), (54) and (55) that

which together with (73), implies that

So, it follows that

Since ,   ,    and , from the boundedness of and we get

Also, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from and (54)-(55) that

Thus, we have

Consequently, from (47), (54), (55) and (111) it follows that

which together with (73), implies that

So, it follows that

Since and , from (109) and the boundedness of and we get

Furthermore, from (47), (54) and (112) it follows that

which together with (73), implies that

So, it follows that

Since and , from (109) and the boundedness of and we get

Note that

Hence from (116) and (120) we get

which together with (79), implies that

Also, observe that

From (122), Remark 10 and the boundedness of we immediately obtain

Step 5. We show that

where is a solution of .

Indeed, we note that is a -strongly positive bounded linear operator and is an -Lipschitzian mapping with . It is clear that

Hence we deduce that is -strongly monotone. In the meantime, it is easy to see that is -Lipschitzian with constant . Thus, there exists a unique solution in to the VIP

Equivalently, solves (due to Lemma 20).

First, we observe that there exists a subsequence of such that

Since is bounded, there exists a subsequence of which converges weakly to some . Without loss of generality, we may assume that . From (91) and (102)–(104), we have that and , where . By (122) and (125) we have that and as . Utilizing the similar arguments to those of (55), we know that is nonexpansive. Hence, by Lemma 12 we obtain and (due to Lemma 11). Next, we prove that . As a matter of fact, since is -inverse strongly monotone, is a monotone and Lipschitz continuous mapping. It follows from Lemma 19 that is maximal monotone. Let , that is, . Again, since , we have

that is,

In terms of the monotonicity of , we get

and hence

In particular,

Since (due to (102)) and (due to the Lipschitz continuity of ), we conclude from and that

It follows from the maximal monotonicity of that , that is, . Therefore, .

Next, we show that . In fact, from , we know that

From (H2) it follows that

Replacing by , we have

Put for all and . Then, from (138) we have

Since as , we deduce from the Lipschitz continuity of and that and as . Further, from the monotonicity of , we have . So, from (H4), the weakly lower semicontinuity of and , we have

From (H1), (H4) and (140) we also have

and hence

Letting , we have, for each ,

This implies that . Therefore, . This shows that . Consequently, from (128) and (129) we have

Step 6. Finally, we show that as .

Indeed, from (47) and (54), we have

In terms of Lemma 6 we have

which leads to

where and . It is easy to see that and . Hence by Lemma 15, we infer that the sequence converges strongly to . This completes the proof.

Corollary 22. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that . Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;

(ii) for each , there exist a bounded subset and such that for any ,

(iii) , and ;

(iv) , and satisfies ;

(v) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Proof. In Theorem 21, putting and , we get and . Utilizing Theorem 21 we derive the desired result.

Corollary 23. Let be a nonempty closed convex subset of a real Hilbert space . Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively, for and . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;

(ii) for each , there exist a bounded subset and such that for any ,

(iii) , and ;

(iv) ,   ,   ,  , and satisfies ;

(v) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Corollary 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively, for . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;

(ii) for each , there exist a bounded subset and such that for any ,

(iii) , and ;

(iv) , , and satisfies ;

(v) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Corollary 25. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;

(ii) for each , there exist a bounded subset and such that for any ,

(iii) , and ;

(iv) ,  ,  ,  , and is a bounded sequence in satisfying ;

(v) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Proof. In Theorem 21, for all ,   is equivalent to

Put . Then it follows that

Observe that for all

So, whenever for some , we obtain the desired result by using Theorem 21.

Let be a -strictly pseudocontractive mapping. For recent convergence result for strictly pseudocontractive mappings, we refer to [16]. Putting , we know that for all

Note that

Hence we have for all

Consequently, if is a -strictly pseudocontractive mapping, then the mapping is -inverse strongly monotone.

Corollary 26. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -strictly pseudocontractive, -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where and is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;

(ii) for each , there exist a bounded subset and such that for any ,

(iii) , and ;

(iv) ,  ,  ,  , and satisfies

(v) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Proof. Since is a -strictly pseudocontractive mapping, the mapping is -inverse strongly monotone. In this case, put . Moreover, we obtain that

So, from Theorem 21, we obtain the desired result.

Corollary 27. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;

(ii) for each , there exist a bounded subset and such that for any ,

(iii) , and ;

(iv) ,  ,  ,  , and satisfies

(v) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Proof. Put for all integers and all . Then, the desired result follows from Theorem 21.

Corollary 28. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let and be two bifunctions from to satisfying (H1)–(H4). Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a sequence of nonexpansive mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Let be the -mapping defined by (9). Assume that where is defined as in Proposition CY. Let and be three sequences in . Assume that:

(i), and ;

(ii) ,  ,  , and satisfies

(iii) .

Given arbitrarily, then the sequence generated iteratively by

converges strongly to which solves the following optimization problem provided is firmly nonexpansive:

where is the potential function of .

Proof. Put ,   for all , for all and . Take for all . Then we get in Theorem 21 and the conclusion follows.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support. This research was partially supported to first author by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). The authors are thankful to the reviewers for their valuable suggestions/comments with appreciation.

References

J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaire, Dunod, Paris, France, 1969.

J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008.

Q. H. Ansari and J.-C. Yao, “Systems of generalized variational inequalities and their applications,” Applicable Analysis, vol. 76, no. 3-4, pp. 203–217, 2000.

Q. H. Ansari, S. Schaible, and J.-C. Yao, “The system of generalized vector equilibrium problems with applications,” Journal of Global Optimization, vol. 22, no. 1–4, pp. 3–16, 2002.

L.-J. Lin and Q. H. Ansari, “Collective fixed points and maximal elements with applications to abstract economies,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 455–472, 2004.

S. Al-Homidan, Q. H. Ansari, and S. Schaible, “Existence of solutions of systems of generalized implicit vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 134, no. 3, pp. 515–531, 2007.

S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “Collectively fixed point and maximal element theorems in topological semilattice spaces,” Applicable Analysis, vol. 90, no. 6, pp. 865–888, 2011.

S. Al-Homidan and Q. H. Ansari, “Fixed point theorems on product topological semilattice spaces, generalized abstract economies and systems of generalized vector quasi-equilibrium problems,” Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 307–330, 2011.

G. Cai and S. Q. Bu, “Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 247, pp. 34–52, 2013.

L. C. Ceng, H.-Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1341–1367, 2011.

L.-C. Ceng, Q. H. Ansari, and S. Schaible, “Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems,” Journal of Global Optimization, vol. 53, no. 1, pp. 69–96, 2012.

L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 2012, article 92, 2012.

L.-C. Ceng and A. Petruşel, “Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem,” Taiwanese Journal of Mathematics, vol. 16, no. 2, pp. 445–478, 2012.

Y. Yao, Y.-C. Liou, and J.-C. Yao, “New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems,” Optimization, vol. 60, no. 3, pp. 395–412, 2011.

Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.

L.-C. Ceng, Q. H. Ansari, S. Schaible, and J.-C. Yao, “Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory, vol. 12, no. 2, pp. 293–308, 2011.

L.-C. Ceng and J.-C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1922–1937, 2010.

L.-C. Ceng, C.-Y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008.

R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.

L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

J. G. O'Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to convex feasibility problems in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 9, pp. 2022–2042, 2006.

Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 64363, 12 pages, 2007.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990.

T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.

N.-J. Huang, “A new completely general class of variational inclusions with noncompact valued mappings,” Computers & Mathematics with Applications, vol. 35, no. 10, pp. 9–14, 1998.

L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “Characterization of H-monotone operators with applications to variational inclusions,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 329–337, 2005.

L.-C. Ceng, Q. H. Ansari, M. M. Wong, and J.-C. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012.

Y. Yao, M. A. Noor, S. Zainab, and Y. C. Liou, “Mixed equilibrium problems and optimization problems,” Journal of Mathematical Analysis and Applications, vol. 354, pp. 319–329, 2009.