Open Access

A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems

Journal of Inequalities and Applications20102010:370197

https://doi.org/10.1155/2010/370197

Received: 27 October 2009

Accepted: 16 March 2010

Published: 31 May 2010

Abstract

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.

1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm being denoted by and , respectively, denoting the family of all subsets of and leting be a closed convex subset of . mapping is called nonexpansive if for all . We use to denote the set of fixed points of , that is, . It is assumed throughout the paper that is a nonexpansive mapping such that . Recall that a self-mapping is contraction on if there exists a constant and such that . Let be a strongly positive bounded linear operator on : that is, there is a constant with property
(1.1)
Let be a single-valued nonlinear mapping and let be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point such that
(1.2)

where is the zero vector in . The set of solutions of problem (1.2) is denoted by .

If , where is a proper convex lower semi-continuous function and is the subdifferential of , then the variational inclusion problem (1.2) is equivalent to find such that
(1.3)

which is called the mixed quasivariational inequality (see [1]).

If , where is a nonempty closed convex subset of and is the indicator function of , that is,
(1.4)
then the variational inclusion problem (1.2) is equivalent to find such that
(1.5)

This problem is called Hartman-Stampacchia variational problem (see [24]).

A set-valued mapping is called monotone if, for all , and imply that . A monotone mapping is maximal if the graph of 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 every implies that .

Let the set-valued mapping be a maximal monotone. We define the resolvent operator that is associate with and as follows:
(1.6)

where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone (see [5, 6]).

Let be a proper extended real-valued function and let be a bifunction of into , where is the set of real numbers. Ceng and Yao [7] considered the following mixed equilibrium problem for finding such that
(1.7)
The set of solutions of (1.7) is denoted by . We see that x is a solution of problem (1.7) implying that . If , then the mixed equilibrium problem (1.7) becomes the following equilibrium problem that is to find such that
(1.8)

The set of solutions of (1.8) is denoted by EP(F). Given a mapping , let for all . Then if and only if for all that is, is a solution of the variational inequality. The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.8). Some methods have been proposed to solve the equilibrium problem (see [821]).

In 2008, Zhang et al. [6] introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The iterative scheme is and:
(1.9)
for all . They proved the strong convergence theorem under some mind conditions. In the same year, Peng et al. [22] introduced an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse-strongly monotone mapping, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert spaces. The sequence is generated as follows:
(1.10)

They proved that if and satisfy appropriate conditions, then converges strongly to , where .

In 2009, Plubtieng and Sriprad [23] introduced an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mapping, the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse-strongly monotone mappings, and the set of solutions of an equilibrium problem in Hilbert spaces. Starting with an arbitrary , define the sequences , and by
(1.11)
where is an inverse-strongly monotone mapping and is a bounded linear operator on . They proved that if the sequences and of parameters satisfy appropriate conditions, then is generated by (1.11) converging strongly to the unique solution of the variational inequality
(1.12)
which is the optimality condition for the minimization problem
(1.13)

where is a potential function for , that is, , for all .

Let , where , be a family of finitely nonexpansive mappings. Let the mapping be defined by
(1.14)

where . Such a mapping is called the W-mapping generated by and . Nonexpansivity of each ensures the nonexpansivity of . Moreover, in [24, Lemma ], it is shown that .

The concept of -mappings was introduced in [25, 26]. It is now one of the main tools in studying convergence of iterative methods for approaching common fixed points of nonlinear mappings; more recent progresses can be found in [24, 27, 28] and the references cited therein.

Following from -mappings, Peng and Yao [29] introduced iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings, and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The sequence is generated by
(1.15)

where is monotone and Lipschitz continuous mapping and is an inverse-strongly monotone mapping. They proved the weak convergence theorem if the sequences and of parameters satisfy appropriate conditions.

In this paper, motivated by the above results and the iterative schemes considered by Zhang et al. in [6], Peng et al. in [22], Peng and Yao in [29], and Plubtieng and Sriprad in [23], we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Zhang et al. [6], Peng et al. [22], Peng and Yao [29], Plubtieng and Sriprad [23], and some authors.

2. Preliminaries

Let be a real Hilbert space with norm and inner product and let be a closed convex subset of . When is a sequence in , means that converges weakly to and means the strong convergence. In a real Hilbert space , we have
(2.1)
for all and . For every point , there exists a unique nearest point in , denoted by , such that
(2.2)
is called the metric projection of onto It is well known that is a nonexpansive mapping of onto and satisfies
(2.3)
for every Moreover, is characterized by the following properties:
(2.4)

for all .

Recall that a mapping of into itself is called -inverse-strongly monotone if there exists a positive real number such that
(2.5)
for all It is obvious that any -inverse-strongly monotone mapping is -Lipschitz monotone and continuous mapping. We also have that, for all and
(2.6)

So, if then is a nonexpansive mapping from into itself.

It is also known that H satisfies the Opial condition [30], that is, for any sequence with , the inequality
(2.7)

holds for every with .

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, , and the set C.

(A1) for all

(A2) is monotone, that is, for all

(A3)For each

(A4)For each is convex and lower semicontinuous.

(A5)For each is weakly upper semicontinuous.

(B1)For each and , there exist a bounded subset and such that for any ,
(2.8)

(B2)C is a bounded set.

We need the following lemmas for proving our main result.

Lemma 2.1 (Peng and Yao [31]).

Let C be a nonempty closed convex subset of H. Let be a bifunction satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:
(2.9)

for all . Then, the following hold.

(1)For each .

(2) is single valued.

(3) is firmly nonexpansive, that is, for any

(4)

(5) is closed and convex.

Lemma 2.2 (Xu [32]).

Assume that is a sequence of nonnegative real numbers such that
(2.10)

where is a sequence in and is a sequence in such that

(1)

(2) or

Then

Lemma 2.3 (Osilike and Igbokwe [33]).

Let be an inner product space. Then for all and with one has
(2.11)

Lemma 2.4 (Colao et al. [28]).

Let be a nonempty convex subset of a Banach space. Let be a family of finitely nonexpansive mappings of into itself and let be sequences in such that . Moreover for every integer , let and be the -mappings generated by and and and , respectively. Then for every , it follows that
(2.12)

Lemma 2.5 (Suzuki [34]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose that for all integers and Then,

Lemma 2.6 (Marino and Xu [35]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 2.7 (Brézis [5]).

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

Remark 2.8.

Lemma 2.7 implies that is closed and convex if is a maximal monotone mapping and is a Lipschitz continuous mapping.

Lemma 2.9 (Zhang et al. [6]).

is a solution of variational inclusion (1.2) if and only if for all that is,
(2.13)

3. Main Result

In this section, we prove 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 in a real Hilbert space.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert Space . Let be a bifunction of into real numbers satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Let be a contraction of into itself with coefficient . Let be an -inverse-strongly monotone mapping of into itself, be a maximal monotone mapping, and let be a strongly bounded linear operator on with coefficient and . Let be a family of finitely nonexpansive mappings of into such that and let be the -mapping generated by and . Assume that either (B1) or (B2) holds. Let be a sequence generated by and
(3.1)

for every , where and satisfy the following conditions:

(i) and ,

(ii) ,

(iii) ,

(iv) for all .

Then converges strongly to , where , which is the unique solution of the variational inequality
(3.2)

Proof.

Since , we may assume, without loss of generality, that for all . We assume that . Since is linear bounded self-adjoint operator on , we have
(3.3)
Observe that
(3.4)
this shows that is positive. It follows that
(3.5)
Let , let be a sequence of mappings defined as in Lemma 2.1, and let . For any , we have
(3.6)
Since , we have . From and being nonexpansive, then we have
(3.7)
for all . It follows that
(3.8)
for every . It follows by mathematical induction that
(3.9)

Therefore, is bounded. We also obtain that and , are all bounded.

Next, we show that Observing that and , we get
(3.10)
(3.11)
Take in (3.10) and in (3.11); by using condition (A2), we obtain
(3.12)
Thus . Without loss of generality, let us assume that there exists a real number such that for all . Then, we have
(3.13)
and hence
(3.14)

where .

On the other hand, again since and are nonexpansive, we obtain
(3.15)
It follows from (3.14) and (3.15) that
(3.16)
Define the sequence by for all . Then, observe that
(3.17)
It follows that
(3.18)
From the definition of , since and are nonexpansive, we have
(3.19)

where is an approximate constant such that , .

Since for all and , we compute
(3.20)
It follows that
(3.21)
Substituting (3.21) into (3.19) yields that
(3.22)
and hence
(3.23)
Combining (3.16) and (3.23), we obtain
(3.24)
So
(3.25)
Conditions (i)–(iv) imply that
(3.26)
Hence, by Lemma 2.5, we have
(3.27)
Consequently,
(3.28)
From (ii), (3.14), (3.16), and (3.28), we also have and as We note that
(3.29)
It follows that
(3.30)
From (i), (iii), and (3.28), we obtain
(3.31)
Next, we shall show that . For any , since is firmly nonexpansive, we have
(3.32)
It follows that
(3.33)
Therefore, we have
(3.34)
Then, we obtain
(3.35)
By (i), (iii), and (3.28) imply that
(3.36)
Since we have
(3.37)
We note that, by (3.34), nonexpansiveness of and the inverse-strong monotonicity of imply that
(3.38)
It follows from (i), (iii), and (3.28) that
(3.39)
which implies that
(3.40)
On the other hand, since is firmly nonexpansive, we have
(3.41)
which yields that
(3.42)
From (3.34) and (3.42), we obtain
(3.43)
Hence, we get
(3.44)
By (i), (iii), (3.28), and (3.40), we have
(3.45)
Similarly, we can prove that
(3.46)
By the same idea in (3.42), (3.45) and using (3.46), then we obtain that
(3.47)
From
(3.48)
hence
(3.49)
and also
(3.50)
Observe that is a contraction of into itself. Indeed, for all , we have
(3.51)
Since is complete, there exists a unique fixed point such that Next, we show that
(3.52)
Indeed, we can choose a subsequence of such that
(3.53)
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From we obtain . Let us show that Since , we have
(3.54)
From (A2), we also have
(3.55)
and hence
(3.56)
From and we get . Since it follows by (A4) and the weakly lower semicontinuity of that
(3.57)
For with and let Since and we have and hence So, from (A1), (A4), and the convexity of , we have
(3.58)

Dividing by t, we get . From (A3) and the weakly lower semicontinuity of , we have for all and hence

Next, we show that . Assume that . Since and , from Opial's condition, we have
(3.59)

which is a contradiction. Thus, we obtain .

Next, we show that . In fact, having as -inverse-strongly monotone, implies that is -Lipschitz continuous monotone mapping and that domain of is equal to . It follows from Lemma 2.7 that is a maximal monotone. Let , that is, . Since , we have , that is,
(3.60)
With being a maximal monotone, we have
(3.61)
and so
(3.62)
It follows from , , and that
(3.63)

It follows from the maximal monotonicity of that , that is, . This implies that .

Since , it follows that
(3.64)
By (3.49), (3.50), and the last inequality, we have
(3.65)
Finally, we show that converges strongly to . Indeed, from (3.1), we have
(3.66)

where . By (3.65), we get . Hence by Lemma 2.2 to (3.66), we conclude that . This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2.

Let be a nonempty closed convex subset of a real Hilbert Space . Let be a bifunction of into real numbers satisfying (A1)–(A5) and let be a contraction of into itself with coefficient . Let be an -inverse-strongly monotone mapping of into itself and let be a maximal monotone mapping such that . Let be a sequence generated by and
(3.67)

for every , where and satisfy the conditions (i)–(iii) in Theorem 3.1. Then converges strongly to

Proof.

Taking for , and , in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 3.3.

Let be a nonempty closed convex subset of a real Hilbert Space . Let be a bifunction of into real numbers satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Let be a contraction of into itself with coefficient . Let be an -inverse-strongly monotone mapping of into and let be a strongly bounded linear operator on with coefficient and . Let be a family of finitely nonexpansive mappings of into such that and let be the -mapping generated by and . Assume that either (B1) or (B2) holds. Let be a sequence generated by and
(3.68)
For every . where and satisfy the condition (i)–(iv) in Theorem 3.1. Then converges strongly to which is the unique solution of the variational inequality
(3.69)

Equivalently, one has

Proof.

From Theorem 3.1 put ; then . So we have and . The conclusion of Corollary 3.3 can be obtained from Theorem 3.1 immediately.

4. Application

In this section, we study a kind of optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
(4.1)

where is a convex and lower semicontinuous functional defined convex subset of a Hilbert space . We denote by the set of solutions of (4.1). Let be a bifunction defined by . We consider the equilibrium problem (1.8); it is obvious that . Therefore, from Theorem 3.1, we give the following corollary.

Corollary 4.1.

Let be a nonempty closed convex subset of a real Hilbert Space . Let be a bifunction of into real numbers satisfying (A1)–(A5) and let be a lower semicontinuous and convex function. Let be a contraction of into itself with coefficient . Let be an -inverse-strongly monotone mapping of into itself, let be a maximal monotone mapping, and let be a strongly bounded linear operator on with coefficient and . Let be a family of finitely nonexpansive mappings of into such that and let be the -mapping generated by and . Let be a sequence generated by and
(4.2)

for every , where and satisfy the following conditions:

(i) and .

(ii) .

(iii) .

(iv) for all .

Then converges strongly to , where , which is the unique solution of the variational inequality
(4.3)

Proof.

From Theorem 3.1 put and . The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.

Declarations

Acknowledgments

The authors would like to thank the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. Mr. Phayap Katchang was supported by King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT for Ph.D. Program at KMUTT. Moreover, the authors are also very grateful to Professor Yeol Je Cho and Professor Jong Kyu Kim for the hospitality.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, (KMUTT)
(2)
Centre of Excellence in Mathematics, CHE

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© P. Katchang and P. Kumam 2010

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