Abstract and Applied Analysis

Volume 2014 (2014), Article ID 513678, 25 pages

http://dx.doi.org/10.1155/2014/513678

Research Article

Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria

Lu-Chuan Ceng,¹ Cheng-Wen Liao,² Chin-Tzong Pang,³ Ching-Feng Wen,⁴ and Zhao-Rong Kong⁵

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

²Department of Food and Beverage Management, Vanung University, Chung-Li 320061, Taiwan

³Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan

⁴Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

⁵Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 20 January 2014; Accepted 12 February 2014; Published 25 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 first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

1. Introduction

Let be a real Hilbert space with inner product and norm , a nonempty closed convex subset of , and 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): find a point such that

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

The VIP (3) was first discussed by Lions [1] and now is well known; there are a lot of different approaches towards solving VIP (3) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (3) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [2–5]. It is well known that if is strongly monotone and Lipschitz-continuous mapping on , then VIP (3) has a unique solution. Not only are the existence and uniqueness of solutions important topics in the study of VIP (3), but also how to actually find a solution of VIP (3) is important. Up to now, there have been many iterative algorithms in the literature, for finding approximate solutions of VIP (3) and its extended versions; see, for example, [6–11].

In 1976, Korpelevič [12] proposed an iterative algorithm for solving the VIP (3) in Euclidean space :

with a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [10, 11, 13–23] and references therein, to name but a few.

Let be a real-valued function,   a nonlinear mapping, and a bifunction. In 2008, Peng and Yao [18] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that

We denote the set of solutions of GMEP (5) by . The GMEP (5) 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, [20, 23–28].

If , then GMEP (5) reduces to the generalized equilibrium problem (GEP) which is to find such that

It is introduced and studied by S. Takahashi and W. Takahashi [29]. The set of solutions of GEP is denoted by .

If , then GMEP (5) reduces to the mixed equilibrium problem (MEP) which is to find such that

It is considered and studied in [30–32]. The set of solutions of MEP is denoted by .

If , , then GMEP (5) reduces to the equilibrium problem (EP) which is to find such that

It is considered and studied in [33, 34]. The set of solutions of EP is denoted by . It is worth mentioning that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.

Throughout this paper, we assume as in [18] 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 be the solution set of the auxiliary mixed equilibrium problem; that is, for each ,

In particular, whenever , , is rewritten as .

Let be two bifunctions and 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 [19]. Whenever , the SGEP reduces to a system of variational inequalities, which is considered and studied in [13]. It is worth mentioning that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.

In 2010, Ceng and Yao [19] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [19]). Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse strongly monotone for . Let for . Then is a solution of SGEP (12) if and only if is a fixed point of the mapping defined by , where . Here, one denotes the fixed point set of by .

Let be an infinite family of nonexpansive mappings on and 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. [25] proposed the following hybrid iterative algorithm:

where is 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 (14) 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 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. [20] introduced the following hybrid extragradient-like iterative algorithm:

for finding a common solution of GMEP (5), SGEP (12), 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 (16) 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 nonempty subset of a normed space . A mapping is called uniformly Lipschitzian if there exists a constant such that

Recently, Kim and Xu [35] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as below.

Definition 1. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exist a constant and a sequence in with such that

They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with . Subsequently, Sahu et al. [36] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that

Put . Then , , and (13) reduces to the relation

Whenever for all in (21) then is an asymptotically -strict pseudocontractive mapping with sequence . In 2009, Sahu et al. [36] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . More precisely, they first established one weak convergence theorem for the following iterative scheme:

where , , and , and then obtained another strong convergence theorem for the following iterative scheme:

where , , and . Subsequently, the above iterative schemes are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense; see, for example, [10, 22].

In 2009, Yao et al. [30] proposed and analyzed iterative algorithms for finding a common element of the set of fixed points of an asymptotically -strict pseudocontraction and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Very recently, motivated by Yao et al. [30], Cai and Bu [26] introduced and analyzed the following iterative algorithm by the hybrid shrinking projection method:

for finding a common element of the set of solutions of finitely many generalized mixed equilibrium problems, the set of solutions of finitely many variational inequalities for inverse strong monotone mappings , and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense (provided that is nonempty and bounded), where , , , , , . It was proven in [26] that under appropriate conditions converge strongly to .

Motivated and inspired by the above facts, we first introduce and analyze one iterative algorithm by using a composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

2. Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We use the notations and to indicate the weak convergence of to and the strong convergence of to , respectively. Moreover, we use to denote the weak -limit set of ; that is,

Definition 3. 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 -inverse strongly monotone. The inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

Definition 4. A differentiable function is called

(i)convex if

where is the Fréchet 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 5. For given and ,

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

By using the technique of [32], we can readily obtain the following elementary result.

Proposition 6 (see [20, 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.

Remark 7. In Proposition 6, whenever is a bifunction satisfying the conditions (H1)–(H4) and , , we have, for any ,

( is firmly nonexpansive) and

In this case, is rewritten as . If, in addition, , then is rewritten as ; see [19, Lemma 2.1] for more details.

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

Lemma 8. Let be a real inner product space. Then the following inequality holds:

Lemma 9. 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 (13).

Lemma 10 (see [37, 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 (13).

Lemma 11 (see [37, 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 [38, 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 5(i)) implies

Lemma 14 (see [36, Lemma 2.5]). Let be a real Hilbert space. Given a nonempty closed convex subset of and points and given also a real number , the set

is convex (and closed).

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 . 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 [39].

Lemma 15 (see [36, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then

for all and .

Lemma 16 (see [36, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as . Then as .

Lemma 17 (see Demiclosedness principle [36, Proposition 3.1]). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 18 (see [36, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.

Remark 19. Lemmas 17 and 18 give some basic properties of an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Moreover, Lemma 17 extends the Demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [35], Górnicki [40], Xu [41], and Marino and Xu [42].

Lemma 20 (see [43, page 80]). Let , , and be sequences of nonnegative real numbers satisfying the inequality

If and , then exists. If, in addition, has a subsequence which converges to zero, then .

Recall that a Banach space is said to satisfy the Opial condition [38] if, for any given sequence which converges weakly to an element , there holds the inequality

It is well known in [38] that every Hilbert space satisfies the Opial condition.

Lemma 21 (see [22, Proposition 3.1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence in . Suppose that

where and are sequences of nonnegative real numbers such that and . Then converges strongly in .

Lemma 22 (see [44]). Let be a closed convex subset of a real Hilbert space . Let be a sequence in and . Let . If is such that and satisfies the condition

then as .

3. Strong Convergence Theorem

In this section, we will introduce and analyze one iterative algorithm by using a composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions we will prove strong convergence of the proposed algorithm.

Theorem 23. 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 let be a lower semicontinuous and convex functional. Let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse strongly monotone, respectively, where and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive mappings on and a sequence in for some . Let be a -strongly positive bounded linear operator with . Let be the -mapping defined by (13). Assume that is nonempty and bounded where is defined as in Proposition CY. Let be a sequence in and , , , and sequences in such that , and . Pick any and set , . Let be a sequence generated by the following algorithm:

where , , , , and , . Assume that the following conditions are satisfied:

(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 exists a bounded subset and such that, for any ,

(iii) and .

Then converges strongly to provided that is firmly nonexpansive.

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

Taking into account that for all , we have

that is, is positive. It follows that

Put

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

We divide the rest of the proof into several steps.

Step 1. We show that is well defined. It is obvious that is closed and convex. As the defining inequality in is equivalent to the inequality

by Lemma 14 we know that is convex for every .

First of all, let us show that for all . Suppose that for some . Take arbitrarily. Since , is -inverse strongly monotone, and , we have, for any ,

Since , , and is -inverse strongly monotone, where , , by Proposition 5(iii) we deduce that for each

Combining (56) and (57), we have

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

(This shows that is nonexpansive.) Also, from (49), (53), (58), and (59) it follows that

which hence yields

By Lemma 9(b), we deduce from (49) and (61) that

So, from (49) and (62) we get

where and . Hence . This implies that for all . Therefore, is well defined.

Step 2. We prove that , , and as .

Indeed, let . From and , we obtain

This implies that is bounded and hence , , , , and are also bounded. Since and , we have

Therefore exists. From , by Proposition 5(ii) we obtain

which implies

It follows from that and hence

From (67) and , we have

Since and , we have

which immediately leads to

Also, utilizing Lemmas 8 and 9(b) we obtain from (49), (58), (59), and (62) that

and hence

So, it follows that

Since , and , it follows from (69) and the boundedness of , and that

Note that

Hence, it follows from (75) and that

Note that

Thus, we deduce from (71) and (77) that

Since and , we have

which, together with (79), yields

Step 3. We prove that , , , , and as .

Indeed, from (57), (59), and it follows that

Next let us show that

For , we find that

Combining (82) and (84), we obtain

which immediately implies that

Since and and are bounded sequences, it follows from (77) that

Furthermore, from the firm nonexpansivity of , we have

which leads to

From (82) and (89), we have

which hence implies that

Since and , , and are bounded sequences, it follows from (77) and (87) that (83) holds.

Next we show that , . As a matter of fact, observe that

Combining (59), (82), and (92), we have

which together with , , implies that

Since and and are bounded sequences, it follows from (77) that

By Proposition 5(iii) and Lemma 9(a), we obtain

which implies

Combining (59), (82), and (97), we have

So, we conclude that

Since and , and are bounded, from (77) and (95) we get

From (100) we get

Taking into account that , we conclude from (83) and (101) that

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 (58), (59), and (82) that

which immediately yields

Since and and are bounded, from (77) we get

Also, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from , and (59) that

Thus, we have

Consequently, from (58), (104), and (108) it follows that

which hence leads to

Since and , and are bounded sequences, we conclude from (77) and (106) that

Furthermore, from (58), (104), and (109) it follows that

which hence yields

Since and , and are bounded sequences, we conclude from (77) and (106) that

Note that

Hence from (112) and (115) we get

then by (75), (102), and (117), we have

Also, observe that

From (118), [45, Remark 3.2], and the boundedness of we immediately obtain

In addition, from (67) and (77), we have

We note that

From (81), (121), and Lemma 15, we obtain

In the meantime, we note that

From (81), (123), and the uniform continuity of , we have

Step 4. We prove that as .

Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (102), (83), (100), and (77), we have that , and , where . Since is uniformly continuous, by (125) we get for any . Hence from Lemma 17, we obtain . In the meantime, utilizing Lemma 12, we deduce from , (117), and (120) that and (due to Lemma 11). Next, we prove that . As a matter of fact, let

where . Let . Since and , we have

On the other hand, from and , we have

and hence

Therefore we have

From (100) and since is uniformly continuous, we obtain that . From , , and (100), we have

Since is maximal monotone, we have and hence , , which implies .

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 (134) 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 (136) we also have

and hence

Letting , we have, for each ,

This implies that . Consequently, . This shows that . From (64) and Lemma 22 we infer that as . This completes the proof.

Corollary 24. Choose in Theorem 23. For any , , and , the iterative scheme (49) reduces to the following iterative one:

where , , , , and . Then converges strongly to provided that is firmly nonexpansive.

Corollary 25. Choose and the identity operator of in Theorem 23. For any , , and , the iterative scheme (49) reduces to the following iterative one:

where , , , , and . Then converges strongly to provided that is firmly nonexpansive.

Proof. In Theorem 23, putting and the identity operator of , we have . In this case, we get

So, the iterative scheme (49) reduces to the iterative one (141). Utilizing Theorem 23, we derive the desired result.

Remark 26. Theorem 23 extends, improves, supplements, and develops Ceng et al.’s [20, Theorem 1] in the following aspects.

(i) The problem of finding a point

in Theorem 23 is very different from the problem of finding a point

in Ceng et al.’s [20, Theorem 1]. There is no doubt that our problem of finding a point is more general and more subtle than the problem of finding a point in [20, Theorem 1].

(ii) The iterative scheme in [20, Theorem 1] is extended to develop the iterative scheme in Theorem 23 by the virtue of Mann-type iterative method and the shrinking projection method. The iterative scheme in Theorem 23 is more advantageous and more flexible than the iterative scheme in [20, Theorem 1] because it involves solving four problems: the GMEP (5), the SGEP (12), finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings.

(iii) The iterative scheme in Theorem 23 is very different from the iterative scheme in [20, Theorem 1] because the iterative scheme in Theorem 23 involves Mann-type iterative method and the shrinking projection method. The proof of [20, Theorem 1] makes use of Lemma 12 (i.e., Demiclosedness principle for a nonexpansive mapping) but no use of Lemma 17 (i.e., Demiclosedness principle for an asymptotically strict pseudocontractive mapping in the intermediate sense). However, the proof of Theorem 23 depends on not only Lemma 12 but also Lemma 17 because there is an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings appearing in the problem of Theorem 23.

(iv) The proof of Theorem 23 combines Cai and Bu convergence analysis for Mann-type iterative method and the shrinking projection method to solve finitely many GMEPs, finitely many VIPs, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense (see [26, Theorem 3.1]) and Ceng et al.'s convergence analysis for hybrid extragradient-like iterative algorithm (see [20, Theorem 3.1]), where for a -strongly positive bounded linear operator . Because in iterative scheme (49) the composite shrinking projection method involves a -strongly positive bounded linear operator with and infinitely many nonexpansive mappings, the properties of the -mappings and and the operator play a key role in the proof of Theorem 23.

(v) Theorem 23 extends Ceng et al.’s [20, Theorem 1] from the fixed point problem of infinitely many nonexpansive mappings to the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings and generalizes Ceng et al.’s [20, Theorem 1] to the setting of finitely many variational inequalities. The proof of Theorem 23 depends on the properties of the -strongly positive bounded linear operator with , the result on the -mappings and (i.e., for any bounded sequence ) (see [45, Remark 3.2]), and the properties of asymptotically strict pseudocontractive mapping in the intermediate sense (see Lemmas 15–18).

Remark 27. Theorem 23 extends, improves, supplements, and develops Yao et al.’s [30, Theorem 3.1] in the following aspects.

(i)Theorem 23 generalizes and extends [30, Theorem 3.1] from the asymptotically -strict pseudocontractive mapping to the asymptotically -strict pseudocontractive mapping in the intermediate sense and from the MEP to the GMEP and generalizes [30, Theorem 3.1] to the setting of SGEP.(ii)We add finitely many variational inequalities and infinitely many nonexpansive mappings in our algorithm such that it can be applied to find a common solution of the GMEP (5), the SGEP (12), finitely many variational inequalities for inverse strongly monotone mappings, and the common fixed point problem of an asymptotically -strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings .

4. Weak Convergence Theorem

In this section, we will propose and analyze another iterative algorithm (involving no shrinking projection method) for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Moreover, under mild conditions we will prove weak convergence of the proposed algorithm.

Theorem 28. Let , , and be the same notations as in Theorem 23, where and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with the sequence such that and such that . Assume that is nonempty where is defined as in Proposition CY. Let be a sequence in and let , and be sequences in such that , and . Pick any and let be a sequence generated by the following algorithm:

where , , and . Assume that the conditions (i)–(iii) are satisfied. Then converges weakly to provided that is firmly nonexpansive.

Proof. As , and , we may assume, without loss of generality, that , and for all . First, let us show that exists for any . Put

for all , and , where is the identity mapping on . Then we get . Take arbitrarily. Repeating the same arguments as in the proof of Theorem 23, we can obtain that

Utilizing (145) and (152), we obtain

Since , and , by Lemma 21 we have that exists. Thus is bounded and so are the sequences , and .

Also, utilizing Lemmas 8 and 9(b), we obtain from (145), (148), (149), and (152) that

and hence

So, it follows that

Since , , and , it follows from the existence of and the boundedness of , and that

Note that

Hence, it follows from (158) and that

In the meantime, from (152) and (155) it follows that

which, together with , leads to

Consequently, from , , , and the existence of , we get

Since , from (163) we have

Note that

Hence from (160) and (164) we have

Repeating the same arguments as those of Step 3 in the proof of Theorem 23, we can obtain that , , , , and , as .

Since is bounded, there exists a subsequence of which converges weakly to . It is easy to see that , and , where . Since is uniformly continuous and as , we get for any . Hence from Lemma 17, we obtain . In the meantime, utilizing Lemma 12, we deduce from , , and that and (due to Lemma 11). Repeating the same arguments as those of Step 4 in the proof of Theorem 23, we can conclude that and . Consequently, . This shows that .

Next let us show that is a single-point set. As a matter of fact, let be another subsequence of such that . Then we get . If , from the Opial condition, we have

This attains a contradiction. So we have . Put . Since , we have . By Lemma 21, we have that converges strongly to some . Since converges weakly to , we have

Therefore we obtain . This completes the proof.

Corollary 29. Choose in Theorem 28. For any the iterative scheme (145) reduces to the following iterative one:

where and for and . Then converges weakly to provided that is firmly nonexpansive.

Corollary 30. Choose and the identity operator of in Theorem 28. For any the iterative scheme (145) reduces to the following iterative one:

where and for . Then converges weakly to provided that is firmly nonexpansive.

In the following, we provide a numerical example to illustrate how Corollary 30 works.

Example 31. Let with inner product and norm which are defined by

for all with and . Let . Clearly, is a nonempty closed convex subset of a real Hilbert space . Let ,  , , , and , . Then , , and  are three bifunctions from to satisfying (H1)–(H4) and is a lower semicontinuous and convex functional. Let be a -strongly positive bounded linear operator with , let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse strongly monotone, respectively, for , and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that such that is nonempty, for instance, putting

It is easy to see that , that is -inverse strongly monotone with , that is a -strongly positive bounded linear operator, that , and are -inverse strongly monotone, and that is a nonexpansive mapping, that is, a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequences () and (). Moreover, it is clear that , and . Hence, . In this case, from iterative scheme (170) in Corollary 30, we obtain that, for any given ,

Whenever and satisfying , we have

Since , we immediately get

This shows that converges to the unique element of .

Conflict of Interests

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

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), the Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and the Ph.D. Program Foundation of the Ministry of Education of China (20123127110002). This work was supported partly by the National Science Council of Taiwan.

References

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.

J. T. Oden, Quantitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer, New York, NY, USA, 1985.

L. C. Zeng, “Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 187, no. 2, pp. 352–360, 1994.

L. C. Zeng, “Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 201, no. 1, pp. 180–194, 1996.

Y. Yao and J. C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007.

L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Iterative methods for triple hierarchical variational inequalities in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 151, no. 3, pp. 489–512, 2011.

L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010.

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.

G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.

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.

L. C. Zeng and J. C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006.

L. C. Ceng and J. C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007.

N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006.

N. Nadezhkina and W. Takahashi, “Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings,” SIAM Journal on Optimization, vol. 16, no. 4, pp. 1230–1241, 2006.

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.

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, 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, N. Hadjisavvas, and N. C. Wong, “Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems,” Journal of Global Optimization, vol. 46, no. 4, pp. 635–646, 2010.

S. Huang, “Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and variational inequality problems,” Optimization, vol. 60, no. 6, pp. 739–754, 2011.

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.

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.

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.

G. Cai and S. 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.

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, 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, 19 pages, 2012.

S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.

Y. Yao, H. Zhou, and Y. C. Liou, “Weak and strong convergence theorems for an asymptotically -strict pseudo-contraction and a mixed equilibrium problem,” Journal of the Korean Mathematical Society, vol. 46, no. 3, pp. 561–576, 2009.

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, no. 1, pp. 319–329, 2009.

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.

L. C. Ceng and J. C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 729–741, 2008.

L. C. Ceng, A. Petruşel, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.

T. H. Kim and H. K. Xu, “Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2828–2836, 2008.

D. R. Sahu, H. K. Xu, and J. C. Yao, “Asymptotically strict pseudocontractive mappings in the intermediate sense,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3502–3511, 2009.

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.

K. Goebel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, UK, 1990.

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

J. Górnicki, “Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 2, pp. 249–252, 1989.

H. K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1139–1146, 1991.

G. Marino and H. K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.

M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88, 2002.

C. Martinez-Yanes and H. K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 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.