Open Access

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 Applications20112011:217407

https://doi.org/10.1155/2011/217407

Received: 14 December 2010

Accepted: 15 January 2011

Published: 23 January 2011

Abstract

The purpose of this paper is to consider a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a -strict pseudocontraction, the set of solutions of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion problem. Strong convergence of the sequences generated by the proposed iterative scheme is obtained. The results presented in this paper extend and improve some well-known results in the literature.

1. Introduction

Throughout this paper, we assume that be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively. Let be a countable family of bifunctions from to , where is the set of real numbers and is an arbitrary index set. Let be a proper extended real-valued function. The system of mixed equilibrium problems is to find such that
(1.1)
The set of solutions of (1.1) is denoted by , that is,
(1.2)
If is a singleton, the problem (1.1) reduces to find the following mixed equilibrium problem (see also the work of Flores-Bazán in [1]). For finding such that,
(1.3)
the set of solutions of (1.3) is denoted by . Combettes and Hirstoaga [2] introduced the following system of equilibrium problems. For finding such that,
(1.4)
the set of solutions of (1.4) is denoted by , that is,
(1.5)
If is a singleton, the problem (1.4) becomes the following equilibrium problem. For finding such that
(1.6)

The set of solution of (1.6) is denoted by EP .

The equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the (mixed) equilibrium problems as special cases (see, e.g., [38]). Some methods have been proposed to solve the equilibrium problem, see, for instance, [917].

Recall that, a mapping is said to be nonexpansive if
(1.7)

We denote the set of fixed points of by , that is .

Definition 1.1.

A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions:

(1) , for all ;

(2) , for all ;

(3) , for all and ;

(4)for all , is continuous.

We denoted by the set of all common fixed points of , that is, . It is know that is closed and convex.

Let be a single-valued nonlinear mapping and be a set-valued mapping. The variational inclusion problem is to find such that
(1.8)

where is the zero vecter in . The set of solutions of problem (1.8) is denoted by . A set-valued mapping is called monotone if for all and imply . A monotone mapping is maximal if its graph of 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 imply .

Definition 1.2.

A mapping is said to be a -Lipschitz continous if there exists a constant such that
(1.9)

Definition 1.3.

A mapping is said to be a -inverse-strongly monotone if there exists a constant with the property
(1.10)

Remark 1.4.

It is obvious that any -inverse-strongly monotone mappings is monotone and -Lipschitz continuous. It is easy to see that for any constant is in , then the mapping is nonexpansive, where is the identity mapping on .

Definition 1.5.

Let is called Lipschitz continuous, if there exists a constant such that
(1.11)

Let be a differentiable functional on a convex set , which is called:

(1) -convex [18] if
(1.12)

where is the Fréchet derivative of at ;

(2) -strongly convex [19] if there exists a constant such that
(1.13)

In particular, if for all , then is said to be strongly convex.

Definition 1.6.

Let be a set-valued maximal monotone mapping, then the single-valued mapping defined by
(1.14)

is called the resolvent operator associated with , where is any positive number and is the identity mapping. The following characterizes the resolvent operator.

(R1)The resolvent operator is single-valued and nonexpansive for all , that is,
(1.15)
(R2)The resolvent operator is 1-inverse-strongly monotone; see [20], that is,
(1.16)
(R3) The solution of problem (1.8) is a fixed point of the operator for all ; see also [21], that is,
(1.17)

(R4) If , then the mapping is nonexpansive.

(R5) is closed and convex.

In 2007, Takahashi et al. [22] proved the following strong convergence theorem for a nonexpansive mapping by using the shrinking projection method in mathematical programming. For and , they define a sequence as follows:
(1.18)

where . They proved that the sequence generated by (1.18) converges weakly to , where .

In 2008, S. Takahashi and W. Takahashi [23] introduced the following iterative scheme for finding a common element of the set of solution of generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. They proved the strong convergence theorems under certain appropriate conditions imposed on parameters. Next, Zhang et al. [24] introduced the following new iterative scheme for finding a common element of the set of solution to the problem (1.8) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary , define a sequence by
(1.19)

where is the resolvent operator associated with and a positive number and is a sequence in the interval . Peng et al. [25] introduced the iterative scheme by the viscosity approximation method for finding a common element of the set of solutions to the problem (1.8), the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in a Hilbert space.

In 2009, Saeidi [26] introduced a more general iterative algorithm for finding a common element of the set of solution for a system of equilibrium problems and the set of common fixed points for a finite family of nonexpansive mappings and a nonexpansive semigroup. In 2010, Katchang and Kumam [27] obtained a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of a mixed equilibrium problem and the set of solutions of a variational inclusion problem for an inverse-strongly monotone mapping. Let be -mapping (defined by (2.8)), be a contraction mapping and be inverse-strongly monotone mappings. Let be the resolvent operator associated with and a positive number . Starting with arbitrary initial , defined a sequence by
(1.20)

They proved that under certain appropriate conditions imposed on , and , the sequence generated by (1.20) converges strongly to , where . Later, Kumam et al. [28] proved a strongly convergence theorem of the iterative sequence generated by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasinonexpansive mappings, and the set of solutions of variational inclusion problems.

Liu et al. [29] introduced a hybrid iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inverse-strongly monotone mappings. Recently, Jitpeera and Kumam [30] considered a shrinking projection method of finding the common element of the set of common fixed points for a finite family of a -strict pseudocontraction, the set of solutions of a systems of equilibrium problems and the set of solutions of variational inclusions. Then, they proved strong convergence theorems of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Very recently, Hao [18] introduced a general iterative method for finding a common element of solution set of quasi variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings.

In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of a -strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.

2. Preliminaries

Let be a real Hilbert space and be a nonempty closed convex subset of . Recall that the (nearest point) projection from onto assigns to each the unique point in satisfying the property .

The following characterizes the projection . We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1.

For a given , , , for all .

It is well known that is a firmly nonexpansive mapping of onto and satisfies
(2.1)
Moreover, is characterized by the following properties: and for all ,
(2.2)

Lemma 2.2 (see [20]).

Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.

Lemma 2.3 (see [31]).

Let be a closed convex subset of . Let be a bounded sequence in . Assume that

(1)the weak -limit set ,

(2)for each , exists.

Then is weakly convergent to a point in .

Lemma 2.4 (see [32]).

Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality , hold for each with .

Lemma 2.5 (see [33]).

Each Hilbert space , satisfies the Kadec-Klee property, that is, for any sequence with and together imply .

For solving the system of mixed equilibrium problem, let us assume that function , satisfies the following conditions:

is monotone, that is, , for all ;

for each fixed , is convex and upper semicontinuous;

for each fixed , is convex.

Lemma 2.6 (see [34]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex functional from to . Let be a bifunction from to satisfying (H1)–(H3). Assume that

(i) is Lipschitz continuous with constant such that;

(a) , for all ,

(b) is affine in the first variable,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology,

(ii) is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology;

(iii)for each , there exist a bounded subset and such that for any ,
(2.3)
For given , Let be the mapping defined by:
(2.4)

for all . Then the following hold

(1) is single-valued;

(2) is nonexpansive if is Lipschitz continuous with constant such that ;

(3) ;

(4) is closed and convex.

Lemma 2.7 (see [35]).

Let be a -strict pseudocontraction, then

(1)the fixed point set of is closed convex so that the projection is well defined;

(2)define a mapping by
(2.5)

If , then is a nonexpansive mapping such that .

A family of mappings is called a family of uniformly -strict pseudocontractions, if there exists a constant such that
(2.6)
Let be a countable family of uniformly -strict pseudocontractions. Let be the sequence of nonexpansive mappings defined by (2.5), that is,
(2.7)
Let be a sequence of nonexpansive mappings of into itself defined by (2.7) and let be a sequence of nonnegative numbers in . For each , define a mapping of into itself as follows:
(2.8)

Such a mapping is nonexpansive from to and it is called the -mapping generated by and . For each , let the mapping be defined by (2.8). Then we can have the following crucial conclusions concerning .

Lemma 2.8 (see [36]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, let be real numbers such that for every . Then, for every and , exists.

Using this lemma, one can define a mapping and as follows and
(2.9)
Such a mapping is called the -mapping. Since is nonexpansive and , is also nonexpansive. Indeed, observe that for each such that
(2.10)

Lemma 2.9 (see [36]).

Let be a nonempty closed convex subset of a Hilbert space , be a countable family of nonexpansive mappings with , be a real sequence such that , for all . Then .

Lemma 2.10 (see [37]).

Let be a nonempty closed convex subset of a Hilbert space , be a countable family of nonexpansive mappings with , be a real sequence such that , for all . If is any bounded subset of , then
(2.11)

Lemma 2.11 (see [38]).

Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on , then for any ,
(2.12)

Lemma 2.12 (see [39]).

Let C be a nonempty bounded closed convex subset of H, be a sequence in C and be a nonexpansive semigroup on C. If the following conditions are satisfied:

(1) ;

(2) , then .

3. Main Results

In this section, we will introduce an iterative scheme by using a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of -strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem in a real Hilbert space.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space , let , be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly -strict pseudocontractions, be the countable family of nonexpansive mappings defined by , for all , for all , , be the -mapping defined by (2.8) and be a mapping defined by (2.9) with . Let be -inverse-strongly monotone mappings and be maximal monotone mappings such that
(3.1)
Let , , which are constants. Let , , , , and be sequences generated by , , , and
(3.2)

where , is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:

is -Lipschitz continuous with constant such that

(a) , for all ,

(b) is affine,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;

is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;

for each and for all , there exist a bounded subset and such that for any ,
(3.3)

, for some ;

, for some ;

, for some ;

, for each .

Then, and converge strongly to .

Proof.

Pick any . Taking for and for all . From the definition of is nonexpansive for each , then also and , we note that . If follows that
(3.4)

Next, we will divide the proof into eight steps.

Step 1.

We first show by induction that for each .

Taking , we get that . Since are nonexpansive. From the assumption, we see that . Suppose for some . For any , we have
(3.5)
(3.6)
which yields
(3.7)
Applying (3.5) and (3.6), we get
(3.8)

Hence . This implies that for each .

Step 2.

Next, we show that is well defined and is closed and convex for any .

It is obvious that is closed and convex. Suppose that is closed and convex for some . Now, we show that is closed and convex for some . For any , we obtain
(3.9)
is equivalent to
(3.10)

Thus is closed and convex. Then, is closed and convex for any . This implies that is well-defined.

Step 3.

Next, we show that is bounded and exists. From , we have
(3.11)
for each . Using , we also have
(3.12)
So, for , we observe that
(3.13)
This implies that
(3.14)
Hence, we get is bounded. It follows by (3.5)–(3.7), that , and are also bounded. From , and , we obtain
(3.15)
It follows that, we have for each
(3.16)
It follows that
(3.17)
Thus, since the sequence is a bounded and nondecreasing sequence, so exists, that is
(3.18)

Step 4.

Next, we show that and .

Applying (3.15), we get
(3.19)
Thus, by (3.18), we obtain
(3.20)
On the other hand, from , which implies that
(3.21)
It follows by (3.21), we also have
(3.22)
By (3.20), we obtain
(3.23)

Step 5.

Next, we show that
(3.24)
for every . Indeed, for , note that is the firmly nonexpansie, so we have
(3.25)
Thus, we get
(3.26)
It follows that
(3.27)
By (3.5), (3.6), (3.7), and (3.27), we have for each
(3.28)
Consequently, we have
(3.29)
Equation (3.23) implies that for every
(3.30)

Step 6.

Next, we show that and , where .

For any given , , and . Since and are nonexpansive, we have
(3.31)
Similarly, we can show that
(3.32)
Observe that
(3.33)
Substituting (3.31) into (3.33) and using conditions (C4) and (C5), we have
(3.34)
It follows that
(3.35)
By (3.23), we obtain
(3.36)
Since the resolvent operator is 1-inverse-strongly monotone, we obtain
(3.37)
which yields
(3.38)
Similarly, we can obtain
(3.39)
Substituting (3.38) into (3.33), and using condition (C4) and (C5), we have
(3.40)
It follows that
(3.41)
By (3.23) and (3.36), we get
(3.42)
From (3.8) and (C4), we also have
(3.43)
Since , we obtain (3.23), we have
(3.44)
Since is a bounded sequence in , from Lemma 2.11 for all , we have
(3.45)
From (3.44) and (3.45), we get
(3.46)
So, we have
(3.47)

Step 7.

Next, we show that .

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .
  1. (1)

    First, we prove that . Indeed, from Lemma 2.12 and (3.47), we get , that is, , for all .

     
  2. (2)
    We show that , where , for all and . Assume that , then there exists a positive integer such that and so . Hence for any , , that is, . This together with , for all shows , for all , therefore we have , for all . It follows from the Opial's condition and (3.44) that
    (3.48)
     
which is a contradiction. Thus, we get .
  1. (3)
    We prove that . Since , and , we have
    (3.49)
     
It follows that
(3.50)
for all . From (3.30) and by conditions (C1)(c) and (C2), we get
(3.51)

By the assumption and by condition (H1), we know that the function and the mapping both are convex and lower semicontinuous, hence they are weakly lower semicontinuous.

These together with and , we have
(3.52)
Then, we obtain
(3.53)
Therefore .
  1. (4)

    Lastly, we prove that .

     
We observe that is an -Lipschitz monotone mapping and . From Lemma 2.2, we know that is maximal monotone. Let that is, . Since , we have
(3.54)
that is,
(3.55)
By virtue of the maximal monotonicity of , we have
(3.56)
and so
(3.57)
By (3.42), and is inverse-strongly monotone, we obtain that and it follows that
(3.58)

It follows from the maximal monotonicity of that , that is, . Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . In similar way, we can obtain , hence .

Step 8.

Finally, we show that and , where .

Since is nonempty closed convex subset of , there exists a unique such that . Since and , we have
(3.59)

for all . From (3.59) and is bounded, so .

By the weakly lower semicontinuous of the norm, we have
(3.60)
However, since , we have
(3.61)
Using (3.59) and (3.60), we obtain . Thus and . So, we have
(3.62)
Thus, we obtain that
(3.63)
From , we obtain . Using the Kadec-Klee property, we obtain that
(3.64)

and hence in norm. Finally, noticing . We also conclude that in norm. This completes the proof.

Theorem 3.2.

Let be a nonempty closed convex subset of a real Hilbert space , let be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly -strict pseudocontractions, be the countable family of nonexpansive mappings defined by , for all , for all , be the -mapping defined by (2.8) and be a mapping defined by (2.9) with . Let be -inverse-strongly monotone mapping. Such that
(3.65)
Let , , which are constants. Let , , , , and be sequences generated by , , , and
(3.66)

where is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:

is -Lipschitz continuous with constant such that

(a) , for all ,

(b) is affine,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;

is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;

for each and for all , there exist a bounded subset and such that for any ,
(3.67)

, for some ;

, for some ;

, for some ;

, for each .

Then, and converge strongly to .

Proof.

In Theorem 3.1, take , where is the indicator function of , that is,
(3.68)
for . Then (1.8) is equivalent to variational inequality problem, that is, to find such that
(3.69)
Again, since , for , then
(3.70)
So, we have
(3.71)

Hence, we can obtain the desired conclusion from Theorem 3.1 immediately.

Next, we consider another class of important mappings.

Definition 3.3.

A mapping is called strictly pseudocontraction if there exists a constant such that
(3.72)
If , then is nonexpansive. In this case, we say that is a -strictly pseudocontraction. Putting . Then, we have
(3.73)
Observe that
(3.74)
Hence, we obtain
(3.75)

Then, is -inverse-strongly monotone mapping.

Now, we obtain the following result.

Theorem 3.4.

Let be a nonempty closed convex subset of a real Hilbert space , let , be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly -strict pseudocontractions, be the countable family of nonexpansive mappings defined by , for all , for all , , be the -mapping defined by (2.8) and be a mapping defined by (2.9) with . Let be -inverse-strongly monotone mapping and be -strictly pseudocontraction mapping of into for some , such that
(3.76)
Let , , which are constants. Let , , , and be sequences generated by , , , and
(3.77)

where , is the mapping defined by (2.4) and be a sequence in for all . Assume the following conditions are satisfied:

is -Lipschitz continuous with constant such that

(a) , for all ,

(b) is affine,

(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;

is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;

for each and for all , there exist a bounded subset and such that for any ,
(3.78)

, for some ;

, for some ;

, for some ;

, for each

Then, and converge strongly to .

Proof.

Taking and , then we see that is -inverse-strongly monotone mapping, respectively. We have and . So, we have
(3.79)

By using Theorem 3.2, it is easy to obtain the desired conclusion.

Declarations

Acknowledgments

The authors would like to thank the Faculty of Science, King Monkut's University of Technology Thonburi for its financial support. Moreover, P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5380044.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT)

References

  1. Flores-Bazán F: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case. SIAM Journal on Optimization 2000,11(3):675–690.MATHMathSciNetView ArticleGoogle Scholar
  2. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MATHMathSciNetGoogle Scholar
  3. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MATHMathSciNetGoogle Scholar
  4. Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. Journal of Optimization Theory and Applications 2003,117(2):245–266. 10.1023/A:1023627606067MATHMathSciNetView ArticleGoogle Scholar
  5. Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with application to noncoercive hemivariational inequalities. Journal of Optimization Theory and Applications 2004,121(3):571–596.MATHMathSciNetView ArticleGoogle Scholar
  6. Konnov IV, Schaible S, Yao JC: Combined relaxation method for mixed equilibrium problems. Journal of Optimization Theory and Applications 2005,126(2):309–322. 10.1007/s10957-005-4716-0MATHMathSciNetView ArticleGoogle Scholar
  7. Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Lecture Notes in Econcmics and Mathematical Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.Google Scholar
  8. Zeng L-C, Wu S-Y, Yao J-C: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese Journal of Mathematics 2006,10(6):1497–1514.MATHMathSciNetGoogle Scholar
  9. Jitpeera T, Kumam P: An extra gradient 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: Theory & Applications 2010,1(1):71–91.MathSciNetGoogle Scholar
  10. Cianciaruso F, Marino G, Muglia L, Yao Y: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory and Applications 2010, 2010:-19.Google Scholar
  11. Cholamjiak P, Suantai S: A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings. Fixed Point Theory and Applications 2009, 2009:-20.Google Scholar
  12. Jaiboon C, Kumam P: Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities. Journal of Inequalities and Applications 2010, 2010:-43.Google Scholar
  13. Jaiboon C, Chantarangsi W, Kumam P: A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings. Nonlinear Analysis. Hybrid Systems 2010,4(1):199–215. 10.1016/j.nahs.2009.09.009MATHMathSciNetView ArticleGoogle Scholar
  14. Kumam P, Jaiboon C: A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems. Nonlinear Analysis. Hybrid Systems 2009,3(4):510–530. 10.1016/j.nahs.2009.04.001MATHMathSciNetView ArticleGoogle Scholar
  15. Shehu Y: Fixed point solutions of variational inequality and generalized equilibrium problems with applications. Annali dell'Universita di Ferrara 2010,56(2):345–368. 10.1007/s11565-010-0102-4MATHMathSciNetView ArticleGoogle Scholar
  16. Yao Y, Liou Y-C, Yao J-C: A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems. Fixed Point Theory and Applications 2008, 2008:-15.Google Scholar
  17. Yao Y, Liou Y-C, Wu Y-J: An extragradient method for mixed equilibrium problems and fixed point problems. Fixed Point Theory and Applications 2009, 2009:-15.Google Scholar
  18. Hao Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications. Applied Mathematics and Computation 2010,217(7):3000–3010. 10.1016/j.amc.2010.08.033MATHMathSciNetView ArticleGoogle Scholar
  19. Ansari QH, Yao JC: Iterative schemes for solving mixed variational-like inequalities. Journal of Optimization Theory and Applications 2001,108(3):527–541. 10.1023/A:1017531323904MATHMathSciNetView ArticleGoogle Scholar
  20. Brézis H: Opérateur maximaux monotones. In Mathematics Studies. Volume 5. North-Holland, Amsterdam, The Netherlands; 1973.Google Scholar
  21. Lemaire B: Which fixed point does the iteration method select? In Recent Advances in Optimization, Lecture Notes in Economics and Mathematical Systems. Volume 452. Springer, Berlin, Germany; 1997:154–167. 10.1007/978-3-642-59073-3_11Google Scholar
  22. Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062MATHMathSciNetView ArticleGoogle Scholar
  23. Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis. Theory, Methods & Applications 2008,69(3):1025–1033. 10.1016/j.na.2008.02.042MATHMathSciNetView ArticleGoogle Scholar
  24. Zhang S-S, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Applied Mathematics and Mechanics. English Edition 2008,29(5):571–581. 10.1007/s10483-008-0502-yMATHMathSciNetView ArticleGoogle Scholar
  25. Peng J-W, Wang Y, Shyu DS, Yao J-C: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. Journal of Inequalities and Applications 2008, 2008:-15.Google Scholar
  26. Saeidi S: Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings. Nonlinear Analysis. Theory, Methods & Applications 2009,70(12):4195–4208. 10.1016/j.na.2008.09.009MATHMathSciNetView ArticleGoogle Scholar
  27. Katchang P, Kumam P: A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems. Journal of Inequalities and Applications 2010, 2010:-25.Google Scholar
  28. Kumam W, Jaiboon C, Kumam P, Singta A: A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings. Journal of Inequalities and Applications 2010, 2010:-25.Google Scholar
  29. Liu M, Chang SS, Zuo P: An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansive semigroup. Journal of Inequalities and Applications 2010, 2010:-23.Google Scholar
  30. Jitpeera T, Kumam P: A new hybrid algorithm for a system of equilibrium problems and variational inclusion. Annali dell'Universita di Ferrara. In pressGoogle Scholar
  31. Acedo GL, Xu H-K: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis. Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036MATHMathSciNetView ArticleGoogle Scholar
  32. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MATHMathSciNetView ArticleGoogle Scholar
  33. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
  34. Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022MATHMathSciNetView ArticleGoogle Scholar
  35. Zhou Haiyun: Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis. Theory, Methods & Applications 2008,69(2):456–462. 10.1016/j.na.2007.05.032MATHMathSciNetView ArticleGoogle Scholar
  36. Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.MATHMathSciNetGoogle Scholar
  37. Chang SS: Variational Inequalities and Related Problems. Chongqing Publishing House, China; 2007.Google Scholar
  38. Shimizu T, Takahashi W: Strong convergence to common fixed points of families of nonexpansive mappings. Journal of Mathematical Analysis and Applications 1997,211(1):71–83. 10.1006/jmaa.1997.5398MATHMathSciNetView ArticleGoogle Scholar
  39. Tan K-K, Xu HK: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1992,114(2):399–404. 10.1090/S0002-9939-1992-1068133-2MATHMathSciNetView ArticleGoogle Scholar

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© Thanyarat Jitpeera and Poom Kumam. 2011

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