Abstract and Applied AnalysisVolume 2012 (2012), Article ID 316276, 39 pagesdoi:10.1155/2012/316276

Research Article

A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

Pongsakorn Sunthrayuth1,2 and Poom Kumam1,2

¹Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 30 August 2011; Accepted 1 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Pongsakorn Sunthrayuth and Poom Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

1. Introduction

Throughout this paper, we denoted by and the set of all positive integers and all positive real numbers, respectively. We always assume that is a real Hilbert space with inner product and norm , respectively, and is a nonempty closed convex subset of . Let be a real-valued function, be an equilibrium bifunction, and be two nonlinear mappings. The generalized mixed equilibrium problem with perturbed mapping is to find such that The set of solutions of the problem (1.1) is denoted by . As special cases of the problem (1.1), we have the following. (1)If , then the problem (1.1) reduces to the generalized mixed equilibrium problem of finding such that which was introduced and studied by Peng and Yao [1]. The set of solutions of the problem (1.2) is denoted by . (2)If , then the problem (1.1) reduces to the mixed equilibrium problem of finding such that which was consider by Ceng and Yao [2]. The set of solutions of the problem (1.3) is denoted by . (3)If , then the problem (1.1) reduces to the generalized equilibrium problem of finding such that which was consider by S. Takahashi and W. Takahashi [3]. The set of solutions of the problem (1.4) is denoted by .(4)If , then the problem (1.1) reduces to the equilibrium problem of finding such that The set of solutions of the problem (1.5) is denoted by .(5)If , then the problem (1.1) reduces to the classical variational inequality problem of finding such that The set of solutions of the problem (1.6) is denoted by . It is known that is a solution of the problem (1.6) if and only if is a fixed point of the mapping , where is a constant and is the identity mapping.

The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [4, 5]). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.2). Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature (see, e.g., [6–34]).

Let be a strongly positive bounded linear operator on ; that is, there exists a constant such that

Recall that a mapping is said to be contractive if there exists a constant such that

A mapping is said to be (1)nonexpansive if (2)firmly nonexpansive if (3)-strictly pseudocontractive if there exists a constant such that We denote by the set of fixed points of , that is, .

Recall the following definitions of a nonlinear mapping ; the following is mentioned.

Definition 1.1. The nonlinear mapping is said to be (i)monotone if (ii)-strongly monotone if there exists a constant such that (iii)-inverse-strongly monotone if there exists a constant such that

Let be a set-valued mapping. The set defined by is said to be the domain of . The set defined by is said to be the range of . The set defined by is said to be the graph of .

Recall that is said to be monotone if

is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, is maximal monotone if for all . For a maximal monotone operator on and , we may define the single-valued resolvent . It is known that is firmly nonexpansive , where denotes the fixed point set of .

We discuss the following algorithms for solving the solutions of variational inequality problems and fixed point problems for a nonexpansive mapping (see, e.g., [29, 35–43]).

In 2010, Chantarangsi et al. [44] introduced a new viscosity hybrid steepest descent method for solving the generalized mixed equilibrium problems (1.2), variational inequality problems, and fixed point problems of nonexpansive mappings in a real Hilbert space. More precisely, they proved the following theorem.

Theorem CCK [see [44]]Let be a nonempty closed and convex subset of a real Hilbert space . Let , be two bifunctions satisfying condition (H1)–(H5), let , be -inverse-strongly monotone mapping and -inverse-strongly monotone mapping, respectively, and let be a nonexpansive mapping. Let be an -Lipschitz continuous and relaxed cocoercive mapping, a contraction mapping with coefficient , and a strongly positive linear bounded self-adjoint operator with coefficient and . Suppose that . Let , , , and be generated by where , , , , , and are three sequences in satisfying the following conditions: (C1) and , (C2), (C3) and , (C4) and , (C5), and . Then, converges strongly to .

Very recently, Yu and Liang [45] proved the following convergence theorem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and inverse strongly monotone mapping in a real Hilbert space.

Theorem YL [see [45]]Let be a real Hilbert space and a nonempty close and convex subset of . Let and be two maximal monotone operators such that and , respectively. Let be a -strict pseudocontraction mapping, an -inverse-strongly monotone mapping, and an -inverse-strongly monotone mapping. Assume that . Let be a sequence generated by where is a fixed element, , is a sequence in is a sequence in , and , , , and are sequences in satisfying the following conditions: (C1) and , (C2), (C3) and , (C4) and , (C5) and . Then, the sequence converges strongly to .

On the other hand, the following optimization problem has been studied extensively by many authors: where are infinitely many closed convex subsets of such that , , is a real number, is a strongly positive linear bounded operator on , and is a potential function for (i.e., for all ). This kind of optimization problem has been studied extensively by many authors (see, e.g., [5, 46–52]) for when and , where is a given point in .

The following questions naturally arise in connection with above the results.

Question 1. Could we weaken the control conditions of Theorems CCK and YL in and ?

Question 2. Can Theorem YL be extended to finding a common element of the set of solutions of a system generalized mixed equilibrium problems and the set of common fixed points of infinite family of nonexpansive mappings?

The purpose of this paper is to give the affirmative answers to these questions mentioned above. Motivated by the iterative process (1.16) and (1.17), we introduce a new iterative algorithm (3.2) below, for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Then, we prove the strong convergence theorem of these iterative process in a real Hilbert space. The results presented in the paper improve and extend the recent ones announced by many others.

2. Preliminaries

Definition 2.1 (see [53]). Let be a nonempty convex subset of a real Hilbert space . Let , , be mappings of into itself. For each , let , where and . For every , we define the mapping as follows: Such a mapping is nonexpansive from into itself, and it is called -mapping generated by and .

Lemma 2.2 (see [53]). Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself with and let , where , , , and for all . For all , let and be -mappings generated by and and and , respectively. Then, (i) is nonexpansive and ,  for all , (ii)for all and for all positive integer , the exists,(iii)the mapping defined by is a nonexpansive mapping such that , and it is called the -mapping generated by and , (iv) if is any bounded subset of C, then

Lemma 2.3 (see [54]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all integers and . Then, .

Lemma 2.4 (see [55]). Let be a real Hilbert space. Then, the following inequalities hold: (i),(ii).

Lemma 2.5 (see [56]). Let be a nonempty closed convex subset of a real Hilbert space , a mapping, and a maximal monotone mapping. Then,

Lemma 2.6 (see [57]). Let be a real Hilbert space and let be a maximal monotone operator on . For and , define the resolvent . Then, the following holds: for all and .

For solving the equilibrium problem for bifunction , let us assume that satisfies the following conditions: (H1) for all , (H2) is monotone; that is, for all ,(H3) for each , is concave and upper semicontinuous, (H4) for each , is convex, (H5) for each , is lower semicontinuous.

Definition 2.7. A differentiable function on a convex set is called (i)convex [2] ifwhere is the Fréchet differentiable of at ; (ii)strongly convex [2] 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 .

Let be an equilibrium bifunction satisfying the conditions (H1)–(H5). Let be any given positive number. For a given point , consider the auxiliary mixed equilibrium problem to finding such that where is the Fréchet differentiable of at . Let be the mapping such that for each , is the set of solutions of , that is, Then, the following conclusion holds.

Lemma 2.8 (see [58]). Let be a nonempty closed convex subset of a real Hilbert space , and let be a lower semicontinuous and convex functional. Let be a bifunction satisfying the conditions (H1)–(H5). Assume that (i) is strongly convex with constant and the function is weakly upper semicontinuous for each ;(ii) for each , there exist a bounded subset and such that for all , Then, the following holds: (a) is single-valued mapping; (b) is nonexpansive if is Lipschitz continuous with constant and where for ; (c); (d) is closed and convex.

In particular, whenever is a bifunction satisfying the conditions (H1)–(H5) and , for all , then is firmly nonexpansive; that is, for any , In this case, is rewritten as . If, in addition, , then is rewritten as (see [59, Lemma 2.1] for more details).

Remark 2.9. We remark that Lemma 2.8 is not a consequence of [2, Lemma 3.1] because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .

Lemma 2.10. Let , and be as in Lemma 2.8. Then, the following holds: for all and .

Proof. By similar argument as in the proof of Proposition 1 in [58], for all and , let and ; we have Let in (2.14) and in (2.15); we have Adding up the last two inequalities and from the monotonicity of , we obtain that It follows that We derive from (2.18) that Hence, we obtain that

The following lemma can be found in [60, 61] (see also [62, Lemma 2.2]).

Lemma 2.11. Let be a nonempty closed convex subset of a real Hilbert space and a proper lower semicontinuous differentiable convex function differentiable convex function. If is a solution to the minimization problem then In particular, if solves the optimization problem then where is a potential function for .

Lemma 2.12 (see [49]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i), (ii) or . Then, .

3. Main Results

Now, we give our main results in this paper.

Theorem 3.1. Let be a nonempty closed convex subset of a Hilbert space . Let be two lower semicontinuous and convex functionals, and let be two bifunctions satisfying conditions (H1)–(H5). Let be -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, and -inverse-strongly monotone mapping, respectively, and let be -inverse-strongly monotone mapping and -inverse-strongly monotone mapping, respectively. Let be an infinite family of nonexpansive mappings, and let , where , , , and for all . For all , let and be -mappings generated by and and and , respectively. Let be two maximal monotone operators such that and , respectively. Assume that Let be a contraction mapping with a coefficient , and let be a strongly positive linear bounded operator with a coefficient . Let and be two constants such that . Let be a sequence defined by and where , , , , , , and . Assume that the following conditions are satisfied: (C1) for all , is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each , (C2)for all and for each , there exist a bounded subset and such that, for all , (C3), (C4) and , (C5), (C6) and , (C7) and , (C8) and , C9 and . Then, the sequence defined by (3.2) converges strongly to , provided is firmly nonexpansive, where solves the following optimization problem:

Proof. By the conditions (C4) and (C5), we may assume, without loss of generality, that for all . Since is a linear bounded self-adjoint operator on , we have Observe that This shows that is positive. It follows that First, we show that is bounded. Take . Since and by Lemma 2.4  (i), we have In a similar way, we can get It follows from (3.8), and (3.9) that Setting . Since , we have In a similar way, we can get It follows from (3.10), (3.11) and (3.12) that Since , it follows from (3.13) that By induction, we have Hence, is bounded, so are , , and .Next, we show that . Since and , we have In a similar way, we can get Substitution (3.16) into (3.17), we obtain where .On the other hand, notice from Lemma 2.6 that where is an appropriate constant such that .In a similar way, we can get from Lemma 2.6 that where is an appropriate constant such that . It follows from (3.18) and (3.19) that where .Define the sequence by . Then, , , we note that It follows that Now, we estimate . By definition of , for all and for all , we have By the condition (C3), we obtain that It follows from (3.21) that It follows from (3.23) and (3.26) that Note that and . Since is strongly monotone and Lipschitz continuous, then, from Lemma 2.10, we obtain that In a similar way, we can get Consequently, it follows from (3.25), (3.27), (3.28), (3.29), and the conditions (C4)–(C9) that Hence, by Lemma 2.3, we have Moreover, we have Since it follows that It follows from (3.32) and the conditions (C4) and (C5) that Next, we show that From (3.8), (3.9), (3.11), and (3.12), we have On the other hand, from Lemma 2.4   (ii), we have where It follows from the condition (C4) and (3.35) that Substituting (3.37) into (3.38), we have which in turn implies that It follows from the conditions (C4)–(C9), (3.32), and (3.40) that Next, we show that . Since is firmly nonexpansive, we have which in turn implies that Substituting (3.45) into (3.38), we have which in turn implies that Since , and from (3.32), we obtain that In a similar way, we can get It follows from (3.38) and (3.49) that which in turn implies that Since and from (3.32), we obtain that In addition, from the firmly nonexpansivity of , we have which in turn implies that It follows from (3.38) and (3.54) that which in turn implies that Since , , and from (3.32), we obtain that In a similar way, we can get It follows from (3.38) and (3.58) that which in turn implies that Since , , and from (3.32), we obtain that Notice that It follows from (3.35), (3.48), (3.52), (3.57), and (3.61) that Moreover, we note that where is any bounded subset of . From Lemma 2.2 and (3.63), we obtain that Next, we show that where is a solution of optimization problem (3.4). To show this, we choose a subsequence of such that Since is bounded, there exists a subsequence of such that as . On the other hand, we note that It follows from (3.48), (3.52), (3.57), and (3.61) that Now, we show that .(i) First, we show that . By Lemma 2.2, we have . Suppose the contrary, , from (3.65) and Opial's condition (see [63]), we have which is a contradiction. So, we obtain .(ii) Now, we show that . From , we know that From (H2), we have Now, replacing by in (3.72), we have Put for all and . Then, from (3.73), we have Since , we have and from Lipschitz continuity of , we have . Further, from the monotonicity of , we have Since , and , we have . Then, from (H4), (H5), the weakly lower semicontinuous of , and , we obtain that From (H1), (H4), and (3.76), we also have and hence Letting , we have, for all , This implies that . In a similar way, we can get .(iii) Now, we show that . In fact, notice that Let . Since is monotone, we have It follows from the condition (C8) and (3.48) that This implies that , that is . In a similar way, we can get . Therefore . Now, from Lemma 2.11, (3.67), and (3.69), we obtain that Finally, we show that as . Again, from Lemma 2.4  , we compute It follows that where . Put and Then, (3.85) reduces to formula It follows from the condition (C4) and (3.83) that and Hence, by Lemma 2.12, we obtain that as . This completes the proof.

Remark 3.2. The control condition and other on (C6)–(C9) are replaced by the strictly weaker conditions: as shown in the next example.

Example 3.3. (a) If , then .(b) The converse of is not true.

Proof. (a) Since and , there exists a constant such that for all . We observe that Hence, we obtain that . (b) Let , then . We see that but Then, converse of (a) is not true. Hence, (b) is proved.

Remark 3.4. Theorem 3.1 improves and extends [45, Theorem 3.3] in the following respects. (1)The problem of finding the common element is more general and more complex than the one of finding the common element in [44, Theorem 3.3].(2)In [44, Algorithm 3.13], the function is chosen as for .(3)The conditions (C3) and (C4) of Chantarangsi et al. [44, Theorem 3.3] are replaced by the strictly weaker conditions.

Remark 3.5. Theorem 3.1 improves and extends [45, Theorem 2.1] in the following respects. (1)Theorem 3.1 extended [64, Theorem 2.1] to finding common element of the set of solutions of a system of generalized mixed equilibrium problem, and the set of infinite family of nonexpansive mapping involves strongly positive linear bounded operator and the optimization problem.(2)The conditions (C3) and (C4) of Yu and Liang [45, Theorem 2.1] are replaced by the strictly weaker conditions.

4. Some Applications

Let be a real Hilbert space and a proper convex lower semicontinuous function. Then, the subdifferential of is defined as follows: From Rockafellar [65], we know that is maximal monotone. Let be a closed and convex subset of , and let be the indicator function of , that is, Since is a proper lower semicontinuous convex function on , the subdifferential of is a maximal monotone operator.

Lemma 4.1 (see [57]). Let be a nonempty closed convex subset of a real Hilbert space , let be the metric projection from onto , and let be the subdifferential of , where is as defined in (4.2) and . Then,

Now, we consider the existence of solution of the variational inequality (1.6).

Theorem 4.2. Let be a nonempty closed convex subset of a Hilbert space . Let be two lower semicontinuous and convex functionals, and let be two bifunctions satisfying conditions (H1)–(H5). Let be -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, and -inverse-strongly monotone mapping, respectively, and let be -inverse-strongly monotone mapping and -inverse-strongly monotone mapping, respectively. Let be an infinite family of nonexpansive mappings, and let , where , , and for all . For all , let and be -mappings generated by and and and , respectively. Assume that Let be a contraction mapping with a coefficient , and let be a strongly positive linear bounded operator with a coefficient . Let and be two constants such that . Let be a sequence defined by and where , and and . Assume the following conditions are satisfied: (C1) for all , is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each , (C2) for all and for each , there exist a bounded subset and such that, for any , (C3), (C4) and , (C5), (C6) and , (C7) and , (C8) and ,(C9) and . Then, the sequence defined by (4.5) converges strongly to , provided is firmly nonexpansive, where solves the following optimization problem:

Proof. Now, we show that and , respectively. Set in Theorem 3.1. Notice that In a similar way, we can get From Lemma 4.1, we can conclude the desired conclusion immediately. This completes the proof.

Let be -strict pseudocontraction mapping. Setting , we see that, for all , On the other hand, we note that For all , we obtain that Then, is -inverse-strongly monotone mapping.

Theorem 4.3. Let be a nonempty closed convex subset of a Hilbert space . Let be two lower semicontinuous and convex functionals, and let be two bifunctions satisfying conditions (H1)–(H5). Let be -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, -inverse-strongly monotone mapping, and -inverse-strongly monotone mapping, respectively, and let be -inverse-strongly monotone mapping and -inverse-strongly monotone mapping, respectively. Let be an infinite family of nonexpansive mappings, and let , where , , and for all . For all , let and be -mappings generated by and and and , respectively. Let be a -pseudocontraction mapping for all . Assume that Let be a contraction mapping with a coefficient , and let be a strongly positive linear bounded operator with a coefficient . Let and be two constants such that . Let be a sequence defined by and where , , , , and and . Assume the conditions (C1)– (C9) in Theorem 4.2. Then, the sequence defined by (4.14) converges strongly to ,  provided is firmly nonexpansive, where solves the following optimization problem:

Proof. Taking , we see that is strict pseudocontraction mapping with and for . From Theorem 4.2, we can conclude the desired conclusion easily. This completes the proof.

Acknowledgments

This research was supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand (under the Project no. RG-1-53-03-2). P. Sunthrayuth was partially supported by the Centre of Excellence in Mathematics for the Ph.D. Program at KMUTT. Moreover, the authors are greatly indebted to the reviewers for their extremely constructive comments and valuable suggestions leading to the revised version.

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 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.

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.

E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

C. Jaiboon and P. Kumam, “A general iterative method for addressing mixed equilibrium problems and optimization problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1180–1202, 2010.

L.-C. Ceng, A. R. Khan, Q. H. Ansari, and J.-C. Yao, “Viscosity approximation methods for strongly positive and monotone operators,” Fixed Point Theory, vol. 10, no. 1, pp. 35–72, 2009.

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.

H.-Y. Hu and L.-C. Ceng, “A general system of generalized nonlinear mixed composite-type equilibria in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 927–959, 2011.

A. Moudafi and M. Théra, “Proximal and dynamical approaches to equilibrium problems,” in Ill-Posed Variational Problems and Regularization Techniques, vol. 477 of Lecture Notes in Economics and Mathematical Systems, pp. 187–201, Springer, Berlin, Germany, 1999.

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.

M. A. Noor, “General variational inequalities and nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 810–822, 2007.

P. Kumam and P. Katchang, “A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings,” Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 475–486, 2009.

S. Plubtieng and P. Kumam, “Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 614–621, 2009.

X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.

S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages, 2010.

S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi-ø-asymptotically nonexpansive mappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010.

S. Saewan and P. Kumam, “A new modified block iterative algorithm for uniformly quasi-ø-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2011, 35 pages, 2011.

S. Saewan and P. Kumam, “A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces,” Computers and Mathematics with Applications, vol. 62, pp. 1723–1735, 2011.

S. Saewan and P. Kumam, “Strong convergence theorems for countable families of uniformly quasi-ø-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Abstract and Applied Analysis, vol. 2011, Article ID 701675, 2011.

S. Saewan and P. Kumam, “The shrinking projection method for solving generalized equilibrium problem and common fixed points for asymptotically quasi-ø-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2011, 9 pages, 2011.

A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.

W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.

P. Kumam, W. Kumam, and P. Junlouchai, “Generalized systems of variational inequalities and projection methods for inverse-strongly monotone mappings,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 976505, 2011.

P. Kumam, “A relaxed extragradient approximation method of two inverse-strongly monotone mappings for a general system of variational inequalities, fixed point and equilibrium problems,” Bulletin of the Iranian Mathematical Society, vol. 36, no. 1, pp. 227–250, 2010.

P. Kumam and C. Jaiboon, “Approximation of common solutions to system of mixed equilibrium problems, variational inequality problem, and strict pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, Article ID 347204, 30 pages, 2011.

P. Katchang and P. Kumam, “Convergence of iterative algorithm for finding common solution of fixed points and general system of variational inequalities for two accretive operators,” Thai Journal of Mathematics, vol. 9, no. 2, pp. 343–360, 2011.

P. Katchang and P. Kumam, “A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems,” Journal of Inequalities and Applications, vol. 2010, Article ID 370197, 25 pages, 2010.

P. Katchang, T. Jitpeera, and P. Kumam, “Strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 838–852, 2010.

N. Petrot, R. Wangkeeree, and P. Kumam, “A viscosity approximation method of common solutions for quasi variational inclusion and fixed point problems,” Fixed Point Theory, vol. 12, no. 1, pp. 165–178, 2011.

S. Saewan and P. Kumam, “A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages, 2010.

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.

S. Saewan, P. Kumam, and K. Wattanawitoon, “Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 734126, 25 pages, 2010.

J. Zhao and S. He, “A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 670–680, 2009.

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.

Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, no. 79, 2011.

Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 4, pp. 635–641, 2010.

Y. Yao, Y.-C. Liou, and C.-P. Chen, “Algorithms construction for nonexpansive mappings and inversestrongly monotone mappings,” Taiwanese Journal of Mathematics, vol. 15, pp. 1979–1998, 2011.

Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical & Computer Modelling. In press

P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 139–150, 2010.

T. Jitpeera and P. Kumam, “A general iterative algorithm for generalized mixed equilibrium problems and variational inclusions approach to variational inequalities,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 619813, 25 pages, 2011.

T. Jitpeera and P. Kumam, “An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 71–91, 2010.

T. Jitpeera and P. Kumam, “A composite iterative method for generalized mixed equilibrium problems and variational inequality problems,” Journal of Computational Analysis and Applications, vol. 13, no. 2, pp. 345–361, 2011.

T. Jitpeera and P. Kumam, “A new hybrid algorithm for a system of mixed equilibrium problems, fixed point problems for nonexpansive semigroup, and variational inclusion problem,” Fixed Point Theory and Applications, Article ID 217407, 27 pages, 2011.

W. Chantarangsi, C. Jaiboon, and P. Kumam, “A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 390972, 39 pages, 2010.

L. Yu and M. Liang, “Convergence of iterative sequences for fixed point and variational inclusion problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 368137, 15 pages, 2011.

H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.

P. L. Combettes, “Hilbertian convex feasibility problem: convergence of projection methods,” Applied Mathematics and Optimization, vol. 35, no. 3, pp. 311–330, 1997.

F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998.

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

T. Jitpeera, U. Witthayarat, and P. Kumam, “Hybrid algorithms of common solutions of generalized mixed equilibrium problems and the common variational inequality problems with applications,” Fixed Point Theory and Applications, vol. 2011, Article ID 971479, 28 pages, 2011.

T. Jitpeera and P. Kumam, “Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 648617, 26 pages, 2011.

R. Wangkeeree, N. Petrot, P. Kumam, and C. Jaiboon, “Convergence theorem for mixed equilibrium problems and variational inequality problems for relaxed cocoercive mappings,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 425–449, 2011.

A. Kangtunyakarn, “A new iterative algorithm for the set of fixed-point problems of nonexpansive mappings and the set of equilibrium problem and variational inequality problem,” Abstract and Applied Analysis, vol. 2011, Article ID 562689, 24 pages, 2011.

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

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

K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 471–489, 2007.

S. Takahashi, W. Takahashi, and M. Toyoda, “Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 147, no. 1, pp. 27–41, 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. In press

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.

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

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.

J. S. Jung, “Iterative algorithms with some control conditions for quadratic optimizations,” Panamerican Mathematical Journal, vol. 16, no. 4, pp. 13–25, 2006.

Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

L. Yu and M. Liang, “Convergence of iterative sequences for fixed point and variational inclusion problems,” Fixed Point Theory and Applications, Article ID 368137, 15 pages, 2011.

R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970.