Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications

Abstract

We introduce new iterative algorithms by hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally, we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others.

1. Introduction

Throughout this paper, let be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . A mapping is called nonexpansive if , for all . The set of fixed points of denoted by ; that is, . If is bounded, closed, and convex and is a nonexpansive mapping of into itself, then ; see, for instance, [1]. Let be a bifunction of into , where is the set of real numbers, a mapping, and a real-valued function. The generalized mixed equilibrium problem is for finding such that

(1.1)

The set of solutions of (1.1) is denoted by ; that is,

(1.2)

The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problems as special cases [2–7].

In particular, if , the problem (1.1) is reduced into the mixed equilibrium problem [8] for finding such that

(1.3)

The set of solutions of (1.3) is denoted by .

If , (1.1) is reduced into the generalized equilibrium problem [9] for finding such that

(1.4)

The set of solutions of (1.4) is denoted by , which this problem was studied by S. Takahashi and W. Takahashi [10].

If and , then the generalized mixed equilibrium problem (1.1) becomes the following equilibrium problem which is to find such that

(1.5)

The set of solutions of (1.5) is denoted by . Many problems in applied sciences, such as numerous problems in physics, optimization, and economics reduce into finding a solution of (1.5). Some methods have been proposed to solve the generalized mixed equilibrium problems, equilibrium problems, and fixed point problems ([2, 6, 11–29]) and references therein. If and , then the generalized mixed equilibrium problem (1.1) becomes the following variational inequality problem, denoted by , is to find such that

(1.6)

The variational inequality problem has been extensively studied in the literature. See, for example [30, 31] and the references therein. A mapping of into is called monotone if

(1.7)

is called an -inverse-strongly  monotone if there exists a positive real number such that

(1.8)

In 2008, Takahashi et al. [32] introduced an iterative method for finding the set of fixed point by Hybrid method in Hilbert spaces. Starting with , , define sequence , as follows:

(1.9)

where is a metric projection of onto and is a nonexpansive mapping of into itself. They proved that if the sequence of parameters satisfies appropriate conditions, then generated by (1.9) converges strongly to . In 2009, Kumam [20] introduced an iterative method for finding a common element of the set of common fixed points of nonexpansive mapping, the set of solutions of a variational inequality problem, and the set of solutions of an equilibrium problem in Hilbert spaces. Starting with an arbitrary , , define sequence , as follows:

(1.10)

where is a nonexpansive mapping of into itself and is a -inverse-strongly monotone mapping of into . He proved that if the sequences , , and of parameters satisfies appropriate conditions, then generated by (1.10) converges strongly to . In 2010, Kangtunyakarn [33] introduced a new method for a common of generalized equilibrium problems, common of variational inequality problems, and fixed point problems by using -mapping generated by a finite family of nonexpansive mappings and real numbers in Hilbert spaces. Starting with an arbitrary , , in , define the sequences , as follows:

(1.11)

where is the -mapping and , are , -inverse-strongly monotone mappings of into , respectively. He proved that if the sequences , , , , , , and of parameters satisfies appropriate conditions, then generated by (1.11) converges strongly to .

Recently, Shehu [34] motivated Chantarangsi et al. [35] who studied the problem of approximating a common element of the set of fixed points of an infinite family of nonexpansive mapping, the set of common solutions of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert spaces.

In this paper, motivated by the above results, we present a new hybrid iterative scheme for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces. Then, we prove strong convergence theorems under some mild conditions. Finally, we give some applications of our results. The results presented in this paper generalize, extend, and improve the results of Takahashi et al. [32], Kumam [20], Kangtunyakarn [33], and many 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 converges weakly to , and means converges strongly to . 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)

Moreover, is characterized by the following properties: ,

(2.4)

(2.5)

for all , . It is also known that satisfies the Opial condition; for any sequence with , the inequality

(2.6)

holds for every with .

It is obvious that any -inverse-strongly monotone mapping is -Lipschitz monotone and continuous mapping. We also have that for all and ,

(2.7)

So, if , then is a nonexpansive mapping of into .

For solving the generalized mixed equilibrium problem, let us assume that the bifunction , the nonlinear mapping is continuous monotone, is convex, and lower semicontinuous satisfies the following conditions:

(A1) for all ,

(A2) is monotone; that is, for any ,

(A3) is upper-hemicontinuous; that is, for each ,

(2.8)

(A4) is convex and lower semicontinuous for each ,

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

(2.9)

(B2) is a bounded set.

The following lemma appears implicitly in [2]. We need the following lemmas for proving our main result.

Lemma 2.1 (see [2]).

Let be a nonempty closed convex subset of , and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

(2.10)

The following lemma was also given in [36].

Lemma 2.2 (see [36]).

Assume that satisfies (A1)–(A4). For and , define a mapping as follows:

(2.11)

for all . Then, the following hold:

(1) is single-valued,

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

(3),

(4) is closed and convex.

Lemma 2.3 (see [37]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction mapping satisfies (A1)–(A4), and let be convex and lower semicontinuous such that . Assume that either (B1) or (B2) holds. For and , there exists such that

(2.12)

Define a mapping as follows:

(2.13)

for all . Then, the following hold:

(1) is single-valued,

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

(3),

(4) is closed and convex.

Lemma 2.4 (see [38]).

Assume that is a sequence of nonnegative real numbers such that

(2.14)

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

(1),

(2) or .

Then, .

3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces.

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)–(A4), and let be a proper lower semicontinuous and convex function. Let , , , and be , , , and -inverse-strongly monotone mapping of into , respectively. Let be an infinite nonexpansive mapping such that . Assume that either (B1) or (B2) holds. Let be a sequence generated by , , and

(3.1)

for every , where , and satisfying the following conditions:

(i),

(ii),

(iii),

(iv),

(v),

(vi).

Then, converges strongly to .

Proof.

Let , then , , , and . By nonexpansiveness of , and , we have

(3.2)

Since both and are nonexpansive for each and (2.7), we have

(3.3)

Therefore, we obtain and .

Next, we will divide the proof into four steps.

Step 1.

We show that is well defined. Let , then is closed and convex for each . Suppose that is closed convex for some . Then, by definition of , we know that is closed convex for . Hence, is closed convex for and for each . This implies that is closed convex for . Moreover, we show that . For , . For , let . Then,

(3.4)

which shows that , for all , for all . So, , for all , for all . Therefore, it follows that , for all . This implies that is well defined.

Step 2.

We claim that and .

From , we get

(3.5)

for each . Since , we have

(3.6)

Hence, for , we obtain

(3.7)

It follows that

(3.8)

From and , we have

(3.9)

For , we compute

(3.10)

and then

(3.11)

Thus, the sequence is a bounded and nondecreasing sequence, so exists. That is, there exists such that

(3.12)

Hence, is bounded and so are , , , , , , , , and for , and . From (3.9), we get

(3.13)

By (3.12), we obtain

(3.14)

Since , we have

(3.15)

By (iii) and (3.14), we get

(3.16)

It follows that

(3.17)

By (3.14) and (3.16), we have

(3.18)

Step 3.

We claim that the following statements hold:

(S1),

(S2),

(S3).

For (3.2), we note that

(3.19)

Since , , we have

(3.20)

By condition (iii) and (3.18), by using the same method with (3.20). Hence, from (3.19), since , , we have

(3.21)

By condition (iii) and (3.18), then we have . On the other hand, we compute

(3.22)

and hence,

(3.23)

By using the same method as (3.23), we also have

(3.24)

Furthermore, we observe that

(3.25)

By condition (i)–(iv), (3.18), and , then we get

(3.26)

Therefore, we have

(3.27)

Similar to (3.26), from (3.25) by conditions (i)–(iv), (3.18), , and , we get

(3.28)

Therefore, we have

(3.29)

From (3.1), (3.3), we have

(3.30)

Furthermore, we observe that

(3.31)

Since , , we have

(3.32)

By conditions (i)–(v), (3.18), , and , then . By using the same method with (3.32). Hence, from (3.31), and since , , we have

(3.33)

By conditions (i)–(iv), (vi), (3.18), , and , then . From (3.1), we have

(3.34)

Assume that and . By nonexpansiveness of and , we also have

(3.35)

It follows that

(3.36)

Similar to (3.36), we obtain

(3.37)

Substituting (3.36), (3.37) into (3.34), we have

(3.38)

By (3.38), we have

(3.39)

It follows that

(3.40)

By conditions (iii)–(vi), (3.18), and , we get

(3.41)

By using (3.41), we can prove that

(3.42)

Applying (3.27) and (3.41), we also have

(3.43)

From (3.29) and (3.42), we obtain

(3.44)

Since and , we have

(3.45)

By (3.43) and (3.44), we obtain

(3.46)

By condition (iii), we have , which implies that

(3.47)

From (3.18) and , we have

(3.48)

Since

(3.49)

By (3.46) and (3.48), we have that , for all .

Step 4.

We show that .

First, we show that . Assume that . Since and , we have that . By and , , from Opial's condition, we have

(3.50)

which is a contradiction. Thus, we obtain .

Next, we show that . Since , we have for any that

(3.51)

From (A2), we also have

(3.52)

For with and , let . Since and , we have . Then, we have

(3.53)

Since , we have . Further, from an inverse-strongly monotonicity of , we have . So, from (A4) and the weak lower semicontinuity of and , we have at the limit

(3.54)

as . From (A1), (A4), and (3.54), we also get

(3.55)

Letting , we have, for each ,

(3.56)

This implies that . By the same arguments, we can show that .

Lastly, by the same proof of [39, Theorem 3.1, pages 346-347], we can show that and . Therefore, ; that is, .

Noting that since , by(2.4), we have

(3.57)

Since and by the continuity of inner product, we obtain from the above inequality that

(3.58)

By (2.4), again, 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)–(A4), and let be a proper lower semicontinuous and convex function. Let , , , and be , , , and -inverse-strongly monotone mapping of into , respectively. Let be nonexpansive mapping such that . Assume that either (B1) or (B2) holds. Let be a sequence generated by , and

(3.59)

for every , where , , and satisfy the following conditions:

(i),

(ii),

(iii),

(iv),

(v),

(vi).

Then, converges strongly to .

Proof.

Taking for , to be nonexpansive mappings, in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

A mapping is said to be a -strict pseudocontraction [40] if there exists a constant such that

(3.60)

where denotes the identity operator on .

Lemma 3.3 (see [41]).

Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strict pseudocontraction. Define by for each . Then, as )   is nonexpansive such that .

Using Theorem 3.1, we obtain the following result.

Theorem 3.4.

Let be a nonempty closed convex subset of a real Hilbert Space . Let , be a bifunction of into real numbers satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function. Let , , , and be , , , and -inverse-strongly monotone mapping of into , respectively. Let be a finite family of -psuedocontractions such that . Define a mapping by . Let be the -mappings generated by and . Assume that either (B1) or (B2) holds. Let be a sequence generated by , , , and

(3.61)

for every , where , , and satisfy the following conditions:

(i),

(ii),

(iii),

(iv),

(v),

(vi).

Then, converges strongly to .

Proof.

From Theorem 3.1, is a finite family of -strict pseudocontraction. By Lemma 3.3, we have that is nonexpansive mappings. The conclusion of Theorem  3.4 can be obtained from Theorem 3.1 immediately.

4. Some Applications

4.1. Complementarity Problem

Let be a nonempty closed and convex cone in , and let be an operator of into . We define the polar of in to be the set

(4.1)

Then, the element is called a solution of the complementarity problem if

(4.2)

The set of solution of the complementarity problem is denoted by , . We will assume that , satisfies the following conditions:

(E1), is a , -inverse-strongly monotone mapping, respectively,

(E2).

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

(4.3)

(B2) is a bounded set.

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)–(A4), and let be a proper lower semicontinuous and convex function. Let , , , and be , , , and -inverse-strongly monotone mapping of into , respectively. Let be infinite nonexpansive mapping such that . Assume that either (B1) or (B2) holds. Let be a sequence generated by , , , and

(4.4)

for every , where , , and satisfy the following conditions:

(i),

(ii),

(iii),

(iv),

(v),

(vi).

Then, converges strongly to .

Proof.

Using Lemma  7.1.1 of [42], we have that and . Hence, by Corollary 4.1, we can conclude the desired conclusion easily. This completes the proof.

4.2. Optimization Problem

In this section, we study a kind of multiobjective 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:

(4.5)

where is a convex and lower semicontinuous functional, and define as a closed convex subset of a real Hilbert space . We denote the set of solutions of (4.5) by and . Let be a bifunction defined by . We consider the equilibrium problem, it is obvious that , . Therefore, from Theorem 3.1, we obtained the following corollary.

Corollary 4.2.

Let be a nonempty closed convex subset of a real Hilbert Space . Let be a bifunction of into real numbers satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function. Let , , , and be , , , and -inverse-strongly monotone mapping of into , respectively. Let be an infinite nonexpansive mapping such that . Assume that either (B1) or (B2) holds. Let be a sequence generated by , , , , and

(4.6)

for every , where , , and satisfy the following conditions:

(i),

(ii),

(iii),

(iv).

Then, converges strongly to .

Proof.

From Theorem 3.1, put , , and . The conclusion of Corollary 4.2 can be obtained from Theorem 3.1 immediately.

4.3. Minimization Problem

In this section, we study the problem for finding a minimizer of a continuously Frèchet differentiable convex functional in a Hilbert space.

First, we use the following lemma in our result.

Lemma 4.3 (see [43]).

Let be a Banach space, let be a continuously Frèchet differentiable convex functional on , and let be the gradient of . If is -Lipschitz continuous, then is an -inverse-strongly monotone.

Let , be functionals on which satisfies the following conditions:

(C1)let, , be a continuously Frèchet differentiable convex functional on , and let, , be , -Lipschitz continuous, respectively,

(C2) and .

Corollary 4.4.

Let be a real Hilbert Space. Let , be a bifunction of into real numbers satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function. Let , be , -inverse-strongly monotone mapping of into , respectively. Let be infinite nonexpansive mappings. Let , be functionals on which satisfies the conditions (C1) and (C2). Suppose that . Assume that either (B1) or (B2) holds. Let be a sequence generated by , , , , and

(4.7)

for every , where , , and satisfying the following conditions:

(i),

(ii),

(iii),

(iv),

(v),

(vi).

Then, converges strongly to .

Proof.

We know form condition (C1) and Lemma 4.3 that , is a , -inverse-strongly monotone operator from in to itself, respectively. The conclusion of Corollary 4.4 can be obtained from Theorem 3.1 immediately.