Convergence Theorems on Generalized Equilibrium Problems and Fixed Point Problems with Applications

Abstract

The purpose of this work is to introduce an iterative method for finding a common element of a solution set of a generalized equilibrium problem, of a solution set solutions of a variational inequality problem and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces.

1. Introduction and Preliminaries

Let be a real Hilbert space, a nonempty closed and convex subset of and a nonlinear mapping. Recall the following definitions.

(a)The mapping is said to be monotone if

(1.1)

(b) is said to be -strongly monotone if there exists a constant such that

(1.2)

(c) is said to be -inverse-strongly monotone if there exists a constant such that

(1.3)

The classical variational inequality problem is to find such that

(1.4)

In this paper, we use to denote the solution set of the problem (1.4). One can easily see that the variational inequality problem is equivalent to a fixed point problem. is a solution to the problem (1.4) if and only if is a fixed point of the mapping , where is a constant and is the identity mapping.

Let be a nonlinear mapping. In this paper, we use to denote the fixed point set of . Recall the following definitions.

(d)The mapping is said to be nonexpansive if

(1.5)

(e) is strictly pseudocontractive with a constant if

(1.6)

For such a case, is called a -strict pseudocontraction.

(f) is said to be pseudocontractive if

(1.7)

Clearly, the class of strict pseudocontractions falls into the one between a class of nonexpansive mappings and a class of pseudocontractions.

Recently, many authors considered the problem of finding a common element of the solution set of the variational inequality (1.4) and of fixed point set of a nonexpansive mapping in Hilbert spaces; see, for examples, [1–5] and the references therein.

In 2005, Iiduka and Takahashi [2] obtained the following theorem in a real Hilbert space.

Theorem 1 IT.

Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that . Suppose and is given by

(1.8)

for every where is a sequence in and is a sequence in . If and are chosen so that for some with ,

(1.9)

then converges strongly to

Let be an inverse-strongly monotone mapping, and a bifunction of into , where is the set of real numbers. We consider the following equilibrium problem:

(1.10)

In this paper, the set of such is denoted by , that is,

(1.11)

In the case of , the zero mapping, the problem (1.10) is reduced to

(1.12)

In this paper, we use to denote the solution set of the problem (1.12), which was studied by many others; see, for examples, [1, 3, 6–23] and the reference therein. In the case of , the problem (1.10) is reduced to the classical variational inequality (1.4). The problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instances, [15, 24].

To study the problems (1.10) and (1.12), we may assume that the bifunction satisfies the following conditions:

(A1) for all ;

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

(A3)for each ,

(1.13)

(A4)for each , is convex and weakly lower semicontinuous.

Recently, S. Takahashi and W. Takahashi [21] considered the problem (1.12) by introducing an iterative method in a Hilbert space. To be more precise, they proved the following theorem.

Theorem 1 TT1.

Let be a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A4), and let be a nonexpansive mapping of into such that . Let be a contraction of into itself, and let and be sequences generated by and

(1.14)

where and satisfy

(1.15)

Then and converge strongly to where

Very recently, S. Takahashi and W. Takahashi [22] further considered the problem (1.10). Strong convergence theorems of common elements are established. More precisely, they obtained the following result.

Theorem 2 TT2.

Let be a closed convex subset of a real Hilbert space and let be a bifunction satisfying (A1), (A2), (A3) and (A4). Let be an -inverse-strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that . Let and and let and be sequences generated by

(1.16)

where , , and satisfy

(1.17)

Then, converges strongly to

In this paper, motivated by Theorem IT, Theorem TT1, and Theorem TT2, we introduce a general iterative method for the problem of finding a common element of a solution set of a generalized equilibrium problem (1.10), of a solution set of a variational inequality problem (1.4), and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

In order to prove our main results, we need the following lemmas.

The following lemmas can be found in [11, 24].

Lemma 1.1.

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

(1.18)

Further, define a mapping by

(1.19)

for all and Then, the following hold.

(1) is single-valued;

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

(1.20)

(3);

(4) is closed and convex.

Lemma 1.2 (see [25]).

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

Lemma 1.3 (see [26]).

Let and be bounded sequences in a Banach space and let be a sequence in with

(1.21)

Suppose that for all integers and

(1.22)

Then

The following lemma can be deduced from Bruck [8].

Lemma 1.4.

Let be a closed convex subset of a strictly convex Banach space . Let , and be three nonexpansive mappings on . Suppose is nonempty. Let , and be three constant in Then the mapping on defined by

(1.23)

for is well defined, nonexpansive, and holds.

Lemma 1.5 (see [6]).

Let be a real Hilbert space, a nonempty closed and convex subset of and a nonexpansive mapping. Then is demiclosed at zero.

Lemma 1.6 (see [14]).

Let be a real Hilbert space, a nonempty closed and convex subset of and a -strict pseudocontraction. Then is closed and convex.

Lemma 1.7 (see [27]).

Assume that is a sequence of nonnegative real numbers such that

(1.24)

where is a sequence in and is a sequence such that

(a)

(b) or

Then

2. Main Results

Theorem 2.1.

Let be a nonempty closed and convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be an -inverse-strongly monotone mapping of into and a -inverse-strongly monotone mapping of into . Let be a -strict pseudocontraction with a fixed point. Assume that . Let be a sequence in generated by

(2.1)

where is a fixed element in , , , , , and are sequences in , is sequence in , and . Assume that the above control sequences satisfy the following restrictions

(R1)

(R2), ;

(R3)

(R4) and .

Then the sequence defined by the iterative process (2.1) converges strongly to .

Proof.

The proof is divided into six steps.

Step 1.

Show that is well defined.

From Lemma 1.6, we see that is closed and convex. On the other hand, we see that the mapping , where , is nonexpansive. Indeed, for any , we have that

(2.2)

This shows that is nonexpansive mapping. Similarly, we can prove that , where is nonexpansive. It follows that , is closed and convex. From Lemma 1.1, we see that . Since is nonexpansive, we obtain that is closed and convex. This shows that is well defined.

Step 2.

Show that is bounded.

Put for each In view of Lemma 1.2 and (R4), we obtain that is nonexpansive and . Letting , we obtain that

(2.3)

Note that for each It follows that Putting

(2.4)

we have that

(2.5)

It follows that

(2.6)

This shows that the sequence is bounded. Note that

(2.7)

This proves that the sequences and are bounded, too.

Step 3.

Show that as

Note that

(2.8)

where is an appropriate constant such that . On the other hand, we have

(2.9)

It follows from (2.1) and (2.9) that

(2.10)

where is an appropriate constant such that

(2.11)

Put , for each , that is,

(2.12)

Now, we compute Notice that

(2.13)

It follows that

(2.14)

Substituting (2.10) into (2.14), we arrive at

(2.15)

It follows from the restrictions (R2)–(R4) that

(2.16)

From Lemma 1.3, we obtain that

(2.17)

From (2.12), we see that

(2.18)

In view of (2.17), we get that

(2.19)

Step 4.

Show that as

From the iterative process (2.1), we have

(2.20)

This implies that

(2.21)

It follows from the restrictions (R2) and (R3) that we arrive at

(2.22)

Step 5.

Show that where

To show that, we can choose a sequence of such that

(2.23)

Since is bounded, we see that there exists a subsequence of which converges weakly to . Without loss of generality, we may assume that .

Next, we show that . In fact, define a mapping by

(2.24)

where . Note that is nonexpansive and . From Lemma 1.4, we see that is a nonexpansive mapping such that

(2.25)

On the other hand, we have

(2.26)

It follows from the condition (R4) that Note that

(2.27)

From (2.22), we arrive at

(2.28)

It follows from Lemma 1.5 that

(2.29)

Thanks to (2.23), we arrive at

(2.30)

Step 6.

Show that as

Notice that

(2.31)

which yields that

(2.32)

In view of the restrictions (R2) and (2.30), we from Lemma 1.7 can conclude the desired conclusion easily. This completes the proof.

As corollaries of Theorem 2.1, we have the following results.

Corollary 2.2.

Let be a nonempty closed and convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be a -inverse-strongly monotone mapping of into . Let be a -strict pseudocontraction with a fixed point. Assume that . Let be a sequence in generated by

(2.33)

where is a fixed element in , , , , , and are sequences in , is sequence in , and . Assume that the above control sequences satisfy the following restrictions:

(R1)

(R2), ;

(R3)

(R4) and for all .

Then the sequence defined by the iterative process (2.1) converges strongly to .

Proof.

In Theorem 2.1, put , the zero mapping. Then for any , we see that the the following inequality holds.

(2.34)

Then, we can obtain the desired conclusion easily from Theorem 2.1. This completes the proof.

Corollary 2.3.

Let be a nonempty closed and convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be a -strict pseudocontraction of into and a -strict pseudocontraction of into . Let be a -strict pseudocontraction with a fixed point. Assume that . Let be a sequence in generated by

(2.35)

where is a fixed element in , , , , , and are sequences in , is sequence in , and . Assume that the above control sequences satisfy the following restrictions:

(R1)

(R2), ;

(R3)

(R4) and .

Then the sequence defined by the above iterative process converges strongly to .

Proof.

Put and . Then, we see that is -inverse-strongly monotone and is -inverse-strongly monotone; see [7]. We have and

(2.36)

It is easy to obtain the desired conclusion from Theorem 2.1.

Remark 2.4.

If is a contractive mapping and we replace by in the recursion formula (2.1), we can obtain the so-called viscosity iteration method. We note that all theorems and corollaries of this paper carry over trivially to the so-called viscosity iteration method; see [28] for more details.