Abstract and Applied Analysis

Volume 2013 (2013), Article ID 589282, 11 pages

http://dx.doi.org/10.1155/2013/589282

Research Article

An Iterative Shrinking Metric -Projection Method for Finding a Common Fixed Point of a Closed and Quasi-Strict-Pseudocontraction and a Countable Family of Firmly Nonexpansive Mappings and Applications in Hilbert Spaces

Kasamsuk Ungchittrakool1,2 and Duangkamon Kumtaeng¹

¹ Department of Mathematics, Faculty of Science, Naresuan University,

99 Moo 9, Phitsanulok-Nakhon Sawan Road,

Tha Pho, Mueang, Phitsanulok 65000, Thailand

²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 29 August 2013; Accepted 18 October 2013

Academic Editor: Shawn X. Wang

Copyright © 2013 Kasamsuk Ungchittrakool and Duangkamon Kumtaeng. 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 create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for finding a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to find a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results.

1. Introduction

It is well known that the metric projection operators in Hilbert spaces and Banach spaces play an important role in various fields of mathematics such as functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problem (see, e.g., [1, 2]). In 1994, Alber [3] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by [3] and proved some properties of the generalized -projection operator. Fan et al. [5] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.

Let be a real Hilbert space; a mapping with domain and range in is called firmly nonexpansive if

nonexpansive if

Throughout this paper, stands for an identity mapping. The mapping is said to be a strict pseudocontraction if there exists a constant such that

In this case, may be called a -strict pseudocontraction. We use to denote the set of fixed points of (i.e. . is said to be a quasi-strict pseudocontraction if the set of fixed point is nonempty and if there exists a constant such that

Construction of fixed points of nonexpansive mappings via Mann’s algorithm [6] has extensively been investigated in the literature; see, for example, [6–10] and references therein. However, we note that Mann’s iterations have only weak convergence even in a Hilbert space (see, e.g., [11]). Nakajo and Takahashi [12] modified the Mann iteration method so that strong convergence is guaranteed, later well known as a hybrid projection iteration method. Since then, the hybrid method has received rapid developments. For the details, the readers are referred to papers [13–26] and the references therein.

On the other hand, for a real Banach space and the dual , let be a nonempty closed convex subset of . Let be a bifunction, let be a real-valued function, and let be a nonlinear mapping. The generalized mixed equilibrium problem is to find such that

The solution set of (5) is denoted by ; that is,

If , the problem (5) reduces to the mixed equilibrium problem for , denoted by , which is to find that such that

If , the problem (5) reduces to the mixed variational inequality of Browder type, denoted by , which is to find that such that

If and , the problem (5) reduces to the equilibrium problem for (for short, ), denoted by , which is to find that such that

If and , the problem (5) reduces to the minimization problem for , denoted by , which is to find that such that

The previous formulation, (8), was shown in [27] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem, and the optimization problem, which can also be written in the form of (9). However, (5) is very general; it covers the problems mentioned above as special cases.

In 2007, S. T. Takahashi and W. T. Takahashi [28] and Tada and Takahashi [29, 30] proved weak and strong convergence theorems for finding a common element of the set of solutions of the equilibrium problem (9) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Takahashi et al. [22] studied a strong convergence theorem by the hybrid method for a family of nonexpansive mappings in Hilbert spaces as follows: , , and , and let

where , for all , and is a sequence of nonexpansive mappings of into itself such that . They proved that if satisfies some appropriate conditions, then converges strongly to .

Motivated by Takahashi et al. [22], Takahashi and Zembayashi [31] (see also [32]) introduced and proved a hybrid projection algorithm for solving equilibrium problems and fixed point problems of a relatively nonexpansive mapping within the framework of a uniformly smooth and uniformly convex Banach space.

In 2011, Saewan and Kumam [33] introduced a new hybrid projection method based on the modified Mann iterative scheme by the generalized -projection operator for a countable family of relatively quasi-nonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Later, they [34] also studied the new hybrid Ishikawa iteration process by the generalized -projection operator for finding a common element of the fixed point set for two countable families of weak relatively nonexpansive mappings and the set of solutions of the system of generalized Ky Fan inequalities in a uniformly convex and uniformly smooth Banach space.

Recently, Li et al. [35] have studied the following hybrid iterative scheme for a relatively nonexpansive mapping by using the generalized -projection operator in Banach spaces:

Under some appropriate assumptions, they obtained strong convergence theorems in Banach spaces.

Motivated and inspired by the work mentioned above, in this paper, we are interested to study our theorems within the framework of a real Hibert space, and we create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for find a common fixed point of an quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to finding a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results.

2. Preliminaries

In this section, some definitions are provided, and some relevant lemmas which are useful to prove in the following section are collected. Most of them are known, and others are not hard to find or understand their proofs. Throughout this paper, we will use the notation for weak convergence and for strong convergence.

Lemma 1 (see Takahashi [36]). Let be a sequence of real numbers. Then, if and only if, for any subsequence of , there exists a subsequence of such that .

Definition 2 (see [37–40]). Let be a nonempty, closed, and convex subset of a Banach space , and let be a sequence of mappings of into itself such that . Then, is said to satisfy the -condition if, for each bounded sequence ,

implies that , where is the set of all weak cluster points of (i.e., ).

Let be a real Hilbert space, and let be nonempty, closed, and convex subset of . Let be a functional defined as follows (see Li et al. [35] (see also [4])):

where , , is positive number, and is proper, convex, and lower semicontinuous. From the definitions of and , it is easy to see the following properties:

(i) is convex and continuous with respect to when is fixed; (ii) is convex and lower semicontinuous with respect to when is fixed.

Definition 3 (see Li et al. [35] (see also [4])). Let be a real Hilbert space, and let be nonempty, closed, and convex subset of . We say that is a metric -projection operator if

Lemma 4 (see Li et al. [35, Lemma 3.1(ii)]). Let be a real Hilbert space, and let . Then, for every , if and only if

Lemma 5 (see Li et al. [35, Lemma 3.2]). Let be a real Hilbert space, let be a nonempty, closed, and convex subset of , and let , . Then,

Lemma 6 (see Deimling [41]). Let be a real Hilbert space, and be a lower semicontinuous convex functional. Then, there exist and such that

Due to the properties of , we have the motivation and ideas to create a new type of mappings which is general and covers a quasi-strict pseud-contraction as follows.

Definition 7. Let be a real Hilbert space, and a mapping with domain and range in is called quasi-strict -pseudocontraction if the set of fixed points is nonempty and if there exists a constant such that, for each ,

It is obvious from the previous definition that (19) is equivalent to

Definition 8. A mapping is said to be closed if, for any sequence with and , .

Example 9. Let be a mapping defined by , for all . Then, it is easy to see that . Moreover, it is found that

for all , where , , and . Furthermore, if such that , then we have and . This means that is closed and quasi-strict -pseudocontraction.

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

(A1), for all ; (A2) is monotone; that is, ;(A3) for each , ;(A4) for each , is convex and lower semicontinuous.

Lemma 10 (see [27]). Let be a nonempty closed convex subset of , and let be a bifunction of into satisfying (A1)–(A4). Let 0 and . Then, there exists such that

Lemma 11 (see [32]). Let be a closed convex subset of and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:

for all . Then, the following hold:

(i) is single-valued; (ii) is firmly nonexpansive-type mapping; that is, for any , ;(iii); (iv) is closed and convex;(v), for all .

Lemma 12 (see [42]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Hilbert space . Let be a continuous and monotone mapping, let be a lower semicontinuous and convex function, and let be a bifunction from to satisfying (A1)–(A4). For and , then, there exists such that

Define a mapping as follows:

for all . Then, the following conclusions hold:

(1) is single-valued; (2) is firmly nonexpansive type; that is, for all , ;(3); (4) is closed and convex;(5), for all .

Lemma 13. Let be a real Hilbert space, and let be a mapping. Then, the following are equivalent:

(i) is firmly nonexpansive i.e.,, for all ;(ii), for all .

Proof. For each , we notice that

The proof is complete.

The following lemma is important since it provides the significant inequality related to quasi-strict -pseudocontractions and firmly nonexpansive mappings within the framework of Hilbert spaces.

Lemma 14. Let be a nonempty, closed, convex subset of real Hilbert spaces . Let be a quasi-strict -pseudocontraction, and let be a firmly nonexpansive mapping such that . Then,

for all and .

Proof. Let and . By the quasi-strict -pseudocontractility of , we have that

It follows from Lemma 13 and (28) that

and, then,

This completes the proof.

3. Main Result

In this section, some available properties of a quasi-strict -pseudocontraction are used to prove that the set of fixed points is closed and convex. An iterative shrinking metric -projection method is provided in order to find a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings.

Lemma 15. Let be a nonempty, closed, convex subset of a real Hilbert space , and let be a quasi-strict -pseudocontraction. Then, the fixed point set of is closed and convex.

Proof. Let be a sequence in such that as . It follows from (28) that

Taking on both sides of (31), so we have that

This means that .

We next show that is convex. For arbitrary and , we let . By the definition of , we have that

By (28), it is easy to see that (33) are equivalent to

respectively. Multiplying by and on both sides of (34) and (35), respectively, and adding the two inequalities, we have that

Hence, . This complete the proof.

Theorem 16. Let be a real Hilbert space, a nonempty, closed, be convex subset of , let be a closed and quasi-strict -pseudocontraction from into itself, and let be a countable family of firmly nonexpansive mappings from into itself which satisfies the -condition such that . Define a sequence in by the following algorithm:

Then, the sequence converges strongly to .

Proof. The proof is divided into six steps.

Step 1. Show that is closed and convex, for all .

For , is closed and convex. Assume that is closed and convex for some . For , we have that

It is not hard to see that the continuity and linearity of and together with the lower semicontinuity and convexity of allow to be closed and convex. Then, for all , is closed and convex.

Step 2. Show that .

It is obvious that . Suppose that for some . For any , we have , and by Lemma 14 we obtain that

This means that . By mathematical induction, for all . Therefore, .

Step 3. Show that is bounded and that the exists.

Since is a convex and lower semicontinuous mapping, applying Lemma 6, we see that there exist and such that

It follows that

Since , it follows from (41) that

for each . This implies that and are bounded. By the fact that and Lemma 5, we obtain that

Since , is nondecreasing. Therefore, the limit of exists.

Step 4. Show that as , where .

Let . From the boundedness of , there exists such that

Writing , it is easy to see that , where . Note that

On the other hand, since , so , and, then, by weakly lower semicontinuity of and , we obtain that

Combining (46) and (47), we obtain that

It follows from (45) and (48), that

and, then,

Next, we consider that

Combining (46) and (51), we obtain that

and, then,

Note that

Then, we have that

Combining (47) and (55), we obtain that

and, then,

By Lemma 1, this implies that

On the other hand, we note that

It follows from (44) and (53) that

Therefore, as . This implies by Lemma 1 that

Thus,

It is easy to show that , for all . Hence, . Since , so, by Lemma 4, we have that

Letting , so we obtain that

which implies that .

Step 5. Show that .

Firstly, we prove that and are bounded. Indeed, taking and then by (28), we have that

By a simple calculation, we get that

where and . So we have that

for all . Therefore, is bounded. Notice that, for each ,

for all . Therefore, is also bounded. Moreover, we note that

Thus, by the fact that and by (58), we obtain that

This means that

For this reason, we have that

Next, we have that

That is, as . It follows from the closed mapping of , that ; thus, .

On the other hand, let us consider that

It follows from the -condition of and (62), that . Therefore, .

Step 6. Show that .

Notice by Step 2 that ; so we have that , and then by Step 5 it yields that

This shows that . It follows from the uniqueness . Then, converges strongly to . This completes the proof.

4. Deduced Theorems and Applications

In this section, some applications of the main theorem are provided in order to find some common solutions of problems in a Hilbert space.

If , then , for all , and , for all ; then, by Theorem 16, we obtain the following corollary.

Corollary 17. Let , , be the same as in Theorem 16, and be a closed and quasi-strict -pseudocontraction from into itself such that . Define a sequence in by the following algorithm:

Then, the sequence converges strongly to .

If is a constant function, saying that , then , and it is not hard to see that coincides with a quasi-strict pseudocontraction. Thus, by Theorem 16, we obtain the following corollary.

Corollary 18. Let , , be the same as in Theorem 16, and let be a closed and quasi-strict pseudocontraction from into itself such that . Define a sequence in by the following algorithm:

Then, the sequence converges strongly to .

Let be a firmly nonexpansive mapping and for all . Then, it is not hard to verify that a family satisfies -condition. Therefore, if , then Corollary 18 reduces to the following corollary.

Corollary 19. Let , , be the same as in Corollary 18, and let be a firmly nonexpansive mapping from into itself such that . Define a sequence in by the following algorithm:

Then, the sequence converges strongly to .

Finally, we provide some applications of the main theorem to find a common solution of fixed point problems of a closed and quasi-strict -pseudocontraction and generalized mixed equilibrium problems via an iterative shrinking metric -projection method within the framework of Hilbert spaces.

Theorem 20. Let , , be the same as in Theorem 16, let be a bifunction from to satisfying (A1)–(A4), let be a lower semicontinuous and convex function, and let be a continuous and monotone mapping such that . Define a sequence in by the following algorithm:

where and , for all , with . Then, converges strongly to .

Proof. By Lemma 12 it is not hard to see that . From the definition of , it is easy to see that , and by (62) we have that . Next, we will show that a countable family satisfies the -condition. It is sufficient to show that . It follows from (61) and (71) that as . Define by , for all . It is not hard to verify that satisfies conditions (A1)–(A4). By (A2), we have that

By using (A4) and , we obtain, that , for all . For and , let . So, from (A1) and (A4), we have that

Dividing by , we have that

From (A3), we have that , for all , and, hence, , so . Therefore, satisfies -condition. Applying Theorem 16, we conclude that .

Acknowledgment

The authors would like to thank the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand, for supporting this paper.

References

Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.

J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005.

Ya. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar, Ariel, Israel, vol. 1 of Functional Differential Equation, pp. 1–21, 1994.

K.-Q. Wu and N.-J. Huang, “The generalised -projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006.

J. Fan, X. Liu, and J. Li, “Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 11, pp. 3997–4007, 2009.

W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, vol. 20, no. 1, pp. 103–120, 2004.

S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.

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.

R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992.

A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975.

K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.

A. Jarernsuk and K. Ungchittrakool, “Strong convergence by a hybrid algorithm for solving equilibrium problem and fixed point problem of a Lipschitz pseudo-contraction in Hilbert spaces,” Thai Journal of Mathematics, vol. 10, no. 1, pp. 181–194, 2012.

S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

K. Ungchittrakool, “Strong convergence by a hybrid algorithm for finding a common fixed point of Lipschitz pseudocontraction and strict pseudocontraction in Hilbert spaces,” Abstract and Applied Analysis, vol. 2011, Article ID 530683, 14 pages, 2011.

K. Ungchittrakool and A. Jarernsuk, “Strong convergence by a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2021, article 147, 14 pages, 2012.

K. Ungchittrakool, “An iterative shrinking projection method for solving fixed point problems of closed and ϕ-quasi-strict pseudocontractions along with generalized mixed equilibrium problems in Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 536283, 20 pages, 2012.

K. Ungchittrakool, “A strong convergence theorem for a common fixed points of two sequences of strictly pseudocontractive mappings in Hilbert spaces and applications,” Abstract and Applied Analysis, vol. 2010, Article ID 876819, 17 pages, 2010.

S. Plubtieng and K. Ungchittrakool, “Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 6, pp. 2896–2908, 2010.

S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 431–450, 2007.

S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2, pp. 103–115, 2007.

W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.

C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 64, no. 11, pp. 2400–2411, 2006.

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.

S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 67, no. 7, pp. 2306–2315, 2007.

T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 64, no. 5, pp. 1140–1152, 2006.

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

S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.

A. Tada and W. Takahashi, “Strong convergence theorem for an equilibrium problem and a nonexpansive mapping,” in Nonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka, Eds., pp. 609–617, Yokohama Publishers, Yokohama, Japan, 2007.

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 K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008.

W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 1, pp. 45–57, 2009.

S. Saewan and P. Kumam, “A modified Mann iterative scheme by generalized -projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2011, article 104, 21 pages, 2011.

S. Saewan and P. Kumam, “A generalized -projection method for countable families of weak relatively nonexpansive mappings and the system of generalized Ky Fan inequalities,” Journal of Global Optimization, vol. 56, no. 2, pp. 623–645, 2013.

X. Li, N.-j. Huang, and D. O'Regan, “Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1322–1331, 2010.

W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009.

H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,” Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264, 2001.

U. Kamraksa and R. Wangkeeree, “Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, article 11, 12 pages, 2011.

K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 339–360, 2006.

K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 11–34, 2007.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics and Mechanics. English Edition, vol. 30, no. 9, pp. 1105–1112, 2009.