Abstract and Applied Analysis

Volume 2014 (2014), Article ID 436069, 27 pages

http://dx.doi.org/10.1155/2014/436069

Research Article

Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

Lu-Chuan Ceng ¹ and Juei-Ling Ho ²

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

²Department of Finance, Tainan University of Technology, No. 529, Zhongzheng Road, YongKang District, Tainan 71002, Taiwan

Received 15 October 2013; Revised 25 November 2013; Accepted 25 November 2013; Published 4 February 2014

Academic Editor: Erdal Karapınar

Copyright © 2014 Lu-Chuan Ceng and Juei-Ling Ho. 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 two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm ; let be a nonempty closed convex subset of and let be the metric projection of onto . Let be a self-mapping on . We denote by the set of fixed points of and by the set of all real numbers.

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that

We denote the set of solutions of GMEP (1) by . The GMEP (1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. The GMEP is further considered and studied in [2–5].

Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set

(assuming the existence of minimizers). We denote by the set of minimizers of CMP (2). The gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection as follows:

or more generally,

where, in both (3) and , the initial guess is taken from arbitrarily and the parameters or are positive real numbers. The convergence of algorithms (3) and depends on the behavior of the gradient .

Since the Lipschitz continuity of the gradient implies that it is actually -inverse strongly monotone (ism) [6], its complement can be an averaged mapping (i.e., it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping). Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Recently, Xu [7] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.

Assume that the CMP (2) is consistent and the gradient is -Lipschitz continuous with . Let be a -contraction with . Xu [7] introduced the following hybrid GPA:

where and . It was proven that under appropriate conditions the sequence converges in norm to a minimizer of CMP (2); see [7, Theorem 5.2].

It is worth emphasizing that the regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem

where is the regularization parameter and again is convex with -Lipschitz continuous gradient . In [7], Xu introduced another hybrid GPA with regularization

where (i) for all ; (ii) (and ) as ; (iii) ; and (iv) as . It was proven that converges strongly to the minimum-norm solution of CMP (2); see [7, Theorem 6.1]. Very recently, the hybrid GPA with regularization is extended to develop new extragradient methods with regularization in Ceng et al. [8, 9] for finding a common solution of the split feasibility problem (SFP) and the fixed point problem of a nonexpansive mapping in a real Hilbert space.

On the other hand, consider the following variational inequality problem (VIP): find a such that

The solution set of VIP (7) is denoted by .

The VIP (7) was first discussed by Lions [10] and now is well known; there are a lot of different approaches towards solving VIP (7) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (7) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [11–14]. It is well known that if is a strongly monotone and Lipschitz-continuous mapping on , then VIP (7) has a unique solution. Not only the existence and uniqueness of solutions are important topics in the study of VIP (7), but also how to actually find a solution of VIP (7) is important.

Motivated by the idea of Korpelevič's extragradient method [15], Nadezhkina and Takahashi [16] introduced an extragradient iterative scheme:

where is a monotone, -Lipschitz continuous mapping, is a nonexpansive mapping, and for some and for some . They proved the weak convergence of to an element of . Recently, inspired by Nadezhkina and Takahashi's iterative scheme [16], Zeng and Yao [17] introduced another iterative scheme for finding an element of and derived the weak convergence result. Furthermore, by combining the CQ method and extragradient method, Nadezhkina and Takahashi [18] introduced an iterative process:

They proved the strong convergence of to an element of under appropriate conditions. Later on, Ceng and Yao [19] introduced an extragradient-like approximation method which is based on the above extragradient method and viscosity approximation method and derived a strong convergence result as well. Next, recall some concepts. 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.

Recall that 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 obvious that if is -inverse-strongly monotone, then is monotone and -Lipschitz continuous.

Definition 1. Let be a nonempty subset of a normed space and let be a self-mapping on .

(i) is asymptotically nonexpansive (cf. [20]) if there exists a sequence of positive numbers satisfying the property and

(ii) is asymptotically nonexpansive in the intermediate sense [21] provided is uniformly continuous and

(iii) is uniformly Lipschitzian if there exists a constant such that

It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [20] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [20] as follows.

Theorem GK (see [20, Theorem ]). If is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point in .

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [21]. Recently, Kim and Xu [22] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as follows.

Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exists 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 .

Recently, Sahu et al. [23] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 3. 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 (17) reduces to the relation

Whenever for all in (18), then is an asymptotically -strict pseudocontractive mapping with sequence . In 2009, Sahu et al. [23] derived the weak and strong convergence of the modified Mann iteration process for the class of asymptotically -strictly pseudocontractive mappings in the intermediate sense with sequence . More precisely, they established the following theorems.

Theorem SXY1. Let be a nonempty closed convex subset of a real Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that and . Assume that is a sequence in such that and . Let be a sequence in generated by the modified Mann iteration process:

Then converges weakly to an element of .

Theorem SXY2. Let be a nonempty closed convex subset of a real Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let be a sequence in such that for all . Let be the sequence in generated by the following (CQ) algorithm:

where and . Then converges strongly to .

Subsequently, the iterative algorithms in Theorems SXY1 and SXY2 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 in a real Hilbert space; see, for example, [24, 25].

On the other hand, Yao et al. [26] introduced two 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. Then they obtained some weak and strong convergence theorems for the proposed iterative algorithms. Very recently, motivated by Yao et al. [26], Cai and Bu [3] introduced two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then they proved some strong and weak convergence theorems for the proposed iterative algorithms under appropriate conditions.

In this paper, inspired by the above facts, we introduce two iterative algorithms by hybrid extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite GMEPs, the set of solutions of finite VIPs for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then we prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions. For recent related results, see, for example, [7, 24, 27–31] and ther references therein.

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 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,

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 4. For given and :

(i) ,   for all  ; (ii) ,  for all  ; (iii) ,  for all  .

Consequently, is nonexpansive and monotone.

If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that, for all and ,

So if , then is a nonexpansive mapping from to .

Definition 5. A mapping is said to be

(a) nonexpansive if

(b) firmly nonexpansive if is nonexpansive, or equivalently, if is -inverse strongly monotone (-ism),

alternatively, is firmly nonexpansive if and only if can be expressed as

where is nonexpansive; projections are firmly nonexpansive.

It can be easily seen that if is nonexpansive, then is monotone. It is also easy to see that a 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 6. A mapping is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping; that is,

where and is nonexpansive. More precisely, when the last equality holds, we say that is -averaged. Thus, firmly nonexpansive mappings (in particular, projections) are -averaged maps.

Proposition 7 (see [32]). Let   be a given mapping.

(i) is nonexpansive if and only if the complement is -ism. (ii) If is -ism, then for is -ism. (iii) is averaged if and only if the complement is -ism for some . Indeed, for is -averaged if and only if is -ism.

Proposition 8 (see [32, 33]). Let be given operators.

(i) If for some and if is averaged and is nonexpansive, then is averaged. (ii) is firmly nonexpansive if and only if the complement is firmly nonexpansive. (iii) If for some and if is firmly nonexpansive and is nonexpansive, then is averaged. (iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where . (v) If the mappings are averaged and have a common fixed point, then

The notation denotes the set of all fixed points of the mapping ; that is, = .

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

Lemma 9. Let be a real inner product space. Then there holds the following inequality:

Lemma 10. Let be a bounded sequence on a reflexive Banach space . If , then .

Lemma 11. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 4 (i)) implies

Lemma 12. Let be a real Hilbert space. Then the following hold:

(i) for all ; (ii) for all and for all ; (iii) If is a sequence in such that , it follows that

Lemma 13 ([23, 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).

Lemma 14 ([23, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and let be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then

for all and .

Lemma 15 ([23, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and let be 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 16 (demiclosedness principle [23, Proposition 3.1]). Let be a nonempty closed convex subset of a Hilbert space and let be 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 17 ([23, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and let be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.

Remark 18. Lemmas 16 and 17 give some basic properties of an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Moreover, Lemma 16 extends the demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [22], Górnicki [34], Marino and Xu [35], and Xu [36].

To prove a weak convergence theorem by a modified extragradient method with regularization for the CMP (2) and the fixed point problem of an asymptotically -strict pseudocontractive mapping in the intermediate sense, we need the following lemma due to Osilike et al. [37].

Lemma 19 (see [37, 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 .

Corollary 20 (see [38, page 303]). Let and be two sequences of nonnegative real numbers satisfying the inequality

If converges, then exists.

Recall that a Banach space is said to satisfy the Opial condition [39] if for any given sequence which converges weakly to an element , there holds the inequality

It is well known in [39] that every Hilbert space satisfies the Opial condition.

Lemma 21 (see [24, 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 .

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for for all implies . Let be a monotone, -Lipschitz continuous mapping, and let be the normal cone to at ; that is, . Define

It is known that in this case is maximal monotone, and if and only if ; see [40].

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:

(A1) for all ; (A2) is monotone, that is, for any ; (A3) is upper-hemicontinuous, that is, for each ,

(A4) is convex and lower semicontinuous for each ; (B1) for each and , there exists a bounded subset and such that for any ,

(B2) is a bounded set.

Lemma 22 (see [41]). Assume that satisfies (A1)–(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:

for all . Then the following hold:

(1) for each ; (2) is single-valued; (3) is firmly nonexpansive; that is, for any ,

(4) ; (5) is closed and convex.

Lemma 23 (see [42]). 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 prove a strong convergence theorem for a hybrid extragradient iterative algorithm with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm with regularization is based on the extragradient method, shrinking projection method, Mann-type iterative method, and hybrid gradient projection algorithm (GPA) with regularization.

Theorem 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where , . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in and let be 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 hold:

(i) ; (ii) for some ; (iii) ,for all ; (iv) ,for all .

Then converge strongly to provided either (B1) or (B2) holds.

Proof. First of all, one can show that is -averaged for each , where

which shows that is nonexpansive. Furthermore, for with , we have

Without loss of generality, we may assume that

Consequently, it follows that for each integer , is -averaged with

This immediately implies that is nonexpansive for all .

We divide 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 13 we know that is convex for every .

Next we show that for all . Put

for all and ,

for all and , and , where is the identity mapping on . Then we have that and . Suppose that for some . Take arbitrarily. Then from (23) and Lemma 22 we have

Similarly, we have

Combining (52) and (53), we have

Also, it follows from (44) that

Note that for every . Then, by Proposition 4(ii), we have

Further, by Proposition 4(i), we have

So from (54) and (55), we obtain

By Lemma 12 and (58), we have

It follows from (59) and that

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

Step 2. We prove that 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 , we obtain

which implies

It follows from that and hence

From (64) and , we have

Note that

Since and (66), we obtain

Step 3. We prove that , and as .

Indeed, from (58) and (59) it follows that

Next we prove that

For , it follows from (23) that

By (52), (53), (58), (69), and (71), we obtain

which implies that

Since , , , , and , we conclude from (68) and the boundedness of that

By Lemmas 12 and 22, we have

which implies that

Utilizing (72) and (76), we have

which implies

Since , , , , , and , we conclude from (68) and (74) and the boundedness of that (70) holds. Hence we obtain

Next we show that , . It follows from (23) that

Utilizing (54), (72), and (80), we get

which implies

Since , , , , , and , we conclude from (68) and the boundedness of that

By Proposition 4 and Lemma 12, we obtain

which implies

Utilizing (81) and (85), we get

which implies

Since , , , , , and , we conclude from (68) and (83) and the boundedness of , and that

From (88) we get

By (79) and (89), we have

From (64) and (90), we have

By (68), (79), and (89), we get

On the other hand, utilizing (58) and (59) we have

which yields

Since , , and , we conclude from (92) and the boundedness of , and that

Also, utilizing the similar arguments to those of (58), we obtain

which together with (59) leads to

So we have

Since , , and , we conclude from (92) and the boundedness of , and that

Utilizing (92)–(99), we get

Since , we get

which together with (100) implies

In addition, observe that

From (68), (79), and (100), it immediately follows that

Moreover, note that

From (91), (92), and (100), it immediately follows that

Meantime, it is clear that

From (102) and (106) and Lemma 14, we obtain

Furthermore, we note that

From (102) and (108) and the uniform continuity of , we have

Step 4. Finally we prove that as .

Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (70), (88)–(90), and (104), we have that ,    ,    ,   ,  where . Since is uniformly continuous, by (110) we get for any . Hence from Lemma 16, we obtain . Next we prove that . Let

where . Let . Since and , we have

On the other hand, from and , we have

and hence

Therefore, we have

From (88) and since is continuous, we obtain that . From , , for all and (88), we have

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

By (A2), we have

Let for all and . This implies that . Then we have

By (70), we have as . Furthermore, by the monotonicity of , we obtain . Then by (A4), we obtain

Utilizing (A1), (A4), and (120), we obtain

and hence

Letting , we have, for each ,

This implies that and hence .

Further, let us show that . As a matter of fact, since and (due to (95) and (99)), we have  and . Let

where is the normal cone to at . We have already mentioned that in this case the mapping is maximal monotone, and if and only if ; see [40] for more details. Let be the graph of and let . Then, we have and hence . So we have for all . On the other hand, from and , we have

and hence

Therefore, from for all and , we have

Note that , and (due to the -Lipschitz continuity of ). Thus, we obtain as . Since is maximal monotone, we have and hence . Clearly, . Consequently, . This shows that . From (61) and Lemma 23 we infer that as . This completes the proof.

Corollary 25 (i.e., [3, Theorem 3.1]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and let be -inverse strongly monotone and -inverse-strongly monotone, respectively, where , . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Assume that is nonempty and bounded. Let and be sequences in such that and . Pick any and set , . Let be a sequence generated by the following algorithm:

where , , , , , . Assume that either (B1) or (B2) holds. Then converge strongly to .

Proof. In Theorem 24, put and for all . Then and . In this case, we obtain from (44) that

Thus, the iterative scheme (44) reduces to (128). Since for all and is bounded, we know that , and

It is easy to see that all the conditions of Theorem 24 are satisfied. Therefore, in terms of Theorem 24, we derive the desired result.

Corollary 26. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in , and let be 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 hold:

(i) ; (ii) for some ; (iii) , ; (iv) .

Then converge strongly to provided either (B1) or (B2) holds.

Corollary 27. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in , and let be 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 hold:

(i) ; (ii) for some ; (iii) ; (iv) .

Then converge strongly to provided either (B1) or (B2) holds.

Corollary 28. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be a uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence such that . Assume that is nonempty and bounded. Let be a sequence in , and let be 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 hold:

(i) ; (ii) for some ; (iii) ; (iv) .

Then converge strongly to provided either (B1) or (B2) holds.

Remark 29. Theorem 24 extends, improves, supplements, and develops Yao et al. [26, Theorem 3.1] in the following aspects.

(i) Theorem 24 generalizes and extends [26, Theorem 3.1] from the asymptotically -strict pseudocontractive mapping to the asymptotically -strict pseudocontractive mapping in the intermediate sense and from one mixed equilibrium problem to finite GMEPs. (ii) We add finite VIPs and the CMP (2) in our algorithm such that it can be applied to find a common element of the set of solutions of finite GMEPs, the set of solutions of finite VIPs for inverse strongly monotone mappings, the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense, and the CMP (2) for a convex functional with -Lipschitz continuous gradient . (iii) Theorem 24 also removes the condition (ii) in [26, Theorem 3.1].

4. Weak Convergence Theorem

In this section, we prove a new weak convergence theorem by a modified extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm with regularization is based on the extragradient method, Mann-type iterative method, and hybrid gradient projection algorithm (GPA) with regularization.

Theorem 30. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where , . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let be a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:

where , , , and . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:

(i) and ; (ii) for some .

Then

(a) converges weakly to an element ; (b) converges weakly to provided is bounded.

Proof. First of all, again one can show that is -averaged for each , where

This shows that is nonexpansive. Furthermore, for with , without loss of generality, we may assume that

Consequently, it follows that for each integer , is -averaged with

This immediately implies that is nonexpansive for all .

Next let us show that exists for any . Put

for all , , and , where is the identity mapping on . Then we have that and . Take arbitrarily. Similarly to the proof of Theorem 24, we obtain that

We observe that

It follows from (142) and (147) and that

From and condition (i) we have

So, applying Lemma 19 to (148), we deduce that exists. This implies that is bounded and hence , and are also bounded. In addition, by Lemma 12 and (59) we obtain from that

Thus, it is easy to see from that

Since exists, , , and the sequence is bounded, we obtain

Taking into consideration , we get , which together with (152) leads to

Combining (140), (142), (143), and (147), we have

which yields

Since , , , , , and , we conclude from (152) and the boundedness of , and that

Combining (140), (142), (144), and (147), we have

which implies

Since , , , , and , we conclude from (152) and (156) and the boundedness of , and that

From (159), we have

Combining (140), (142), (145), and (147), we have

where , which implies

Since , , , , , and , we conclude from (152) and the boundedness of , and that

Combining (140), (142), (146), and (147), we have

which implies

Since , , , , , and , we conclude from (152) and (163) and the boundedness of , and that

By (166), we have

From (160) and (167), we have

By (152) and (168), we obtain

Furthermore, combining (139), (140), (142), and (147), we have

which yields

Since , , , and , we obtain from (152) and the boundedness of , and that

Also, combining (139), (140), , and (147), we have

which leads to

Since , , and , we obtain from (152) and the boundedness of , , , and that

Hence, combining (172) and (175), we get

We note that

From and (169), we have

On the other hand, we observe that

By (153), (168), and (176), we have

We note that

From (178), (180), Lemma 14, and the uniform continuity of , we obtain

Since is bounded, there exists a subsequence of which converges weakly to . From (168) and (176), we have that . From (182) and the uniform continuity of , we have for any . So, from Lemma 16, we have . Similarly to the arguments in the proof of Theorem 24, we can derive . 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 contraction. So we have . This shows that . Therefore, by Lemma 10, we know that .

Finally, we claim that provided is bounded. Put . Since , we have . By (148) and Lemma 21, we have which converges strongly to some . Since converges weakly to , we have

Therefore, we obtain . This completes the proof.

Corollary 31 (i.e., [3, Theorem 4.1]). Let be a nonempty closed convex subset of a real Hilbert space . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where , . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let and be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:

where , , , and . Assume that and , and that either (B1) or (B2) holds. Then converges weakly to .

Proof. In Theorem 30, put and for all . Then and . In this case, we obtain from (134) that

Thus, the iterative scheme (134) reduces to (95). It is easy to see that all the conditions of Theorem 24 are satisfied. In terms of Theorem 24, we have that converges weakly to an element . Now, put . Since , we have . Taking into account that for all , we conclude from (148) that

By Lemma 21, we have that converges strongly to some . Since converges weakly to , we have

Therefore, we obtain . This completes the proof.

Corollary 32. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let is a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:

where , , and . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:

(i) and ; (ii) for some .

Then

(a) converges weakly to an element ; (b) converges weakly to provided is bounded.

Corollary 33. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let be a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:

where , . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:

(i) and ; (ii) for some .

Then

(a) converges weakly to an element ; (b) converges weakly to provided is bounded.

Corollary 34. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be the uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence . Assume that is nonempty. Let be a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm:

where . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:

(i) and ; (ii) for some .

Then

(a) converges weakly to an element ; (b) converges weakly to provided is bounded.

Finally, we provide an example to illustrate Corollary 34.

Example 35. 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 be a convex functional with -Lipschitz continuous gradient , for instance, putting , where . Then is -Lipschitz continuous with (due to ). Put and for all . Then it is clear that is a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, for instance, putting and . Then we can take . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence , for instance, putting . Then and for all (due to the nonexpansivity of ). Thus, we know that . Take , and . Let be a sequence in such that , and let be sequences in such that and . Pick any . In this case, the algorithm (191) reduces to the following algorithm:

So it follows that

Since , we deduce that there exists an integer such that

Therefore, from (194) we obtain that for all

This shows that

(a) converges to the unique point in ; (b) converges to .

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), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). This research was partially supported by a grant from the NSC (101-2115-M-165 -001) as well.

References

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, 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.

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.

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.

J.-B. Baillon and G. Haddad, “Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977.

H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360–378, 2011.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012.

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

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.

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.

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.

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.

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.

K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

R. E. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169–179, 1993.

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.

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 J.-C. Yao, “Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense,” Acta Applicandae Mathematicae, vol. 115, no. 2, pp. 167–191, 2011.

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

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.

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.

L.-C. Ceng and J.-C. Yao, “Approximate proximal methods in vector optimization,” European Journal of Operational Research, vol. 183, no. 1, pp. 1–19, 2007.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012.

C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.

P. L. Combettes, “Solving monotone inclusions via compositions of nonexpansive averaged operators,” Optimization, vol. 53, no. 5-6, pp. 475–504, 2004.

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.

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.

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.

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.

K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.

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

R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.

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.

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.