A Viscosity of Cesàro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems

Abstract

We introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a -inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesàro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang (2009), Peng and Yao (2009), Shimizu and Takahashi (1997), and some authors.

1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm are denoted by and , respectively and let be a nonempty closed convex subset of . A mapping is called 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 a on if there exists a constant and such that

Let be a proper extended real-valued function and be a bifunction of into , where is the set of real numbers. Ceng and Yao [1] considered the following for finding such that

(1.1)

The set of solutions of (1.1) is denoted by . We see that x is a solution of problem (1.1) implies that . If , then the mixed equilibrium problem (1.1) becomes the following equilibrium problem is to find such that

(1.2)

The set of solutions of (1.2) is denoted by . 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 (1.2). Some methods have been proposed to solve the equilibrium problem (see [2–14]).

Let be a mapping. The    denoted by , is to find such that

(1.3)

for all The variational inequality problem has been extensively studied in the literature. See, for example, [15, 16] and the references therein. A mapping of into is called monotone if

(1.4)

for all . B is called -- if there exists a positive real number such that for all

(1.5)

Let be a strongly positive linear bounded operator on : that is, there is a constant with property

(1.6)

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :

(1.7)

where is strongly positive linear bounded operator and is a potential function for (i.e., for ). Moreover, it is shown in [17] that the sequence defined by the scheme

(1.8)

converges strongly to .

In 1997, Shimizu and Takahashi [18] originally studied the convergence of an iteration process for a family of nonexpansive mappings in the framework of a real Hilbert space. They restate the sequence as follows:

(1.9)

where and are all elements of and is an appropriate in They proved that converges strongly to an element of fixed point of which is the nearest to

In 2007, Plubtieng and Punpaeng [19] proposed the following iterative algorithm:

(1.10)

They proved that if the sequence and of parameters satisfy appropriate condition, then the sequences and both converge to the unique solution of the variational inequality

(1.11)

which is the optimality condition for the minimization problem

(1.12)

where is a potential function for (i.e., for ).

In 2008, Peng and Yao [20] introduced an iterative algorithm based on extragradient method which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. The sequences generated by ,

(1.13)

for all , where is W-mapping. They proved the strong convergence theorems under some mind conditions.

In this paper, motivated by the above results and the iterative schemes considered in [9, 18–20], we introduce a new iterative process below based on viscosity and Cesàro mean approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for a -inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we prove strong convergence theorems which are connected with [5, 21–24]. We extend and improve the corresponding results of Kumam and Katchang [9], Peng and Yao [20], Shimizu and Takahashi [18] and some authors.

2. Preliminaries

Let be a real Hilbert space with inner product and norm and let be a nonempty closed convex subset of . Then

(2.1)

(2.2)

for all and . For every point , there exists a unique in , denoted by , such that

(2.3)

is called the metric projection of onto It is well known that is a nonexpansive mapping of onto and satisfies

(2.4)

for every Moreover, is characterized by the following properties: and

(2.5)

(2.6)

for all . Let be a monotone mapping of C into H. In the context of the variational inequality problem the characterization of projection (2.5) implies the following:

(2.7)

It is also known that H satisfies the Opial condition [25], that is, for any sequence with , the inequality

(2.8)

holds for every with .

A set-valued mapping is called if for all , and imply . A monotone mapping is 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 . Let be a monotone mapping of into and let be the to at , that is, and define

(2.9)

Then is the maximal monotone and if and only if ; see [26].

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction and a proper extended real-valued function satisfies the following conditions:

(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 ,

(2.10)

(B2)C is a bounded set.

We need the following lemmas for proving our main results.

Lemma 2.1 (Peng and Yao [20]).

Let be a nonempty closed convex subset of H. Let be a bifunction satisfies (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:

(2.11)

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 [27]).

Assume is a sequence of nonnegative real numbers such that

(2.12)

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

(1),

(2) or

Then

Lemma 2.3 (Osilike and Igbokwe [28]).

Let be an inner product space. Then for all and with we have

(2.13)

Lemma 2.4 (Suzuki [29]).

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

Lemma 2.5 (Marino and Xu [17]).

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

Lemma 2.6 (Bruck [30]).

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and a nonexpansive mapping. For each and the Cesàro means then

3. Main Results

In this section, we show 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 mixed equilibrium problem and the set of solutions of a variational inequality problem for a -inverse-strongly monotone mapping in a real Hilbert space by using the viscosity of Cesàro mean approximation method.

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 semicontinuos and convex function. Let be a nonexpansive mappings for all , such that . Let be a contraction of into itself with coefficient and let be a -inverse-strongly monotone mapping of into Let be a strongly positive bounded linear self-adjoint on with coefficient and . Assume that either or holds. Let , and be sequences generated by , and

(3.1)

where , and , and satisfy the following conditions:

(i) and

(ii)

(iii)

(iv), and

(v) and

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

(3.2)

Proof.

Now, we have (see [9, page 479]). Since and is a -inverse-strongly monotone mapping. For any , we have

(3.3)

It follows that , hence is nonexpansive.

Let be a sequence of mapping defined as in Lemma 2.1 and , for all we have

(3.4)

By the fact that and are nonexpansive and , we get

(3.5)

Let ; it follows that

(3.6)

which implies that is nonexpansive. Since we have , for all By (3.5) and (3.6), we have

(3.7)

Hence is bounded and also , and are bounded.

Next, we show that Observing that dom and dom , we get

(3.8)

(3.9)

Take in (3.8) and in (3.9), by using condition (A2); it follows that

(3.10)

Thus . Without loss of generality, let us assume that there exists a nonnegative real number such that , for all . Then, we have

(3.11)

and hence

(3.12)

where . On the other hand, let ; it follows from the definition of that

(3.13)

We compute that

(3.14)

Let it follows that

(3.15)

and hence

(3.16)

Therefore,

(3.17)

It follows from and the conditions (i)–(v), that

(3.18)

From Lemma 2.4 and (3.18), we obtain and also

(3.19)

Next, we show that as For we obtain

(3.20)

and hence

(3.21)

Since and from Lemma 2.3 and (3.21), we obtain

(3.22)

Then, we have

(3.23)

By (i) and (iv), imply that

(3.24)

Since we obtain

(3.25)

Next, we show that Indeed, observe that

(3.26)

and then

(3.27)

Since (i) and (iv), we get

Next, we show that where From Lemma 2.3 and (3.3), we obtain

(3.28)

It follows that

(3.29)

Since and as we obtain as Using (2.1), we have

(3.30)

so, we obtain

(3.31)

and hence

(3.32)

which implies that

(3.33)

Since , as and the condition (i)–(iii), we have as At the same time, we note that

(3.34)

since , we have as Consequently, we observe that

(3.35)

It is easy to see that is a contradiction of into itself. Hence is complete, there exists a unique fixed point , such that

Next, we show that

(3.36)

Indeed, we can choose a subsequence of such that

(3.37)

Since is bounded, there exists a subsequence of which converge weakly to Without loss of generality, we can assume that From , we obtain

Let us show Since , we obtain

(3.38)

From (A2), we also have

(3.39)

and hence

(3.40)

From and we get . Since thus that from (A4) and the weakly lower semicontinuity of that , for all For with and let . Since and , we have and hence So, from (A1), (A4) and the convexity of , we have

(3.41)

Dividing by 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

(3.42)

which is a contradiction. Thus, we obtain

Now, let us show that Let be a set-valued mapping is defined by

(3.43)

where is the normal cone to at We have is maximal monotone and if and only if Let hence and we have On the other hand, from we have

(3.44)

that is

(3.45)

Therefor, we have

(3.46)

Noting that as and is -inverse-strongly monotone, hence from (3.46), we obtain as . Since is maximal monotone, we have , and hence . Therefore,

Since , we have

(3.47)

Finally, we show that converge strongly to we obtain that

(3.48)

where By (3.47), (i) and (iii), we get Applying Lemma 2.2 to (3.48) 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). Let be a contraction of into itself with coefficient and let be a nonexpansive mapping such that . Let be a -inverse-strongly monotone mapping of into Let , and be sequences generated by , and

(3.49)

where , for all satisfy the condition (i), (iii)–(v). Then, converges strongly to where

Proof.

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

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 nonexpansive mapping such that . Let be a -inverse-strongly monotone mapping of into Let , and be sequences generated by , and

(3.50)

where , for all satisfy the condition (i), (iii)–(v). Then, converges strongly to where

Proof.

Taking and , for all in Corollary 3.2, we can conclude the desired conclusion easily.

4. Applications to Optimization Problem

Let be a real Hilbert space, a nonempty closed convex subset of , a strongly positive linear bounded operator on with a constant , and be a nonexpansive mapping. In this section we will utilize the results present in section main results to study the following optimization problem:

(4.1)

where is the set of fixed points of in and is a potential function for (i.e., for ). We have the following theorem.

Theorem 4.1.

Let be a nonempty closed convex subset of a real Hilbert Space . Let be a nonexpansive mappings and let be a contraction of into itself with coefficient . Let be a strongly positive linear bounded operator on with coefficient and . Let satisfying condition (i) and (iii) in Theorem 3.1. If is a nonempty compact subset of then for each there is a unique such that

(4.2)

and the sequence converges strongly to some point , which solves the following optimization problem (4.1).

Proof.

Taking , in Corollary 3.2, we get and we also have Hence the sequence converges strongly to some point which is the unique solution of the following variational inequality:

(4.3)

Since is nonexpansive, then is convex. Again by the assumption that is compact, therefore, it is a compact and convex subset of , and

(4.4)

is a continuous mapping. By virtue of the well-know Weierstrass theorem, there exists a point which is a minimal point of optimization problem (4.1). As is know to all, (4.3) is the optimality necessary condition (see Xu [31]) for the optimization problem (4.1). Then, we also have

(4.5)

Since is the unique solution of (4.3), therefor, . This complete the proof of Theorem 4.1.