Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping

Abstract

The purpose of this paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method.

1. Introduction

Let be a closed convex subset of a real Hilbert space , and let be a bifunction. Recall that the equilibrium problem for a bifunction is to find such that

(1.1)

The set of solutions of (1.1) is denoted by . Given a mapping , let for all . Then, if and only if for all ; that is, is a solution of the variational inequality. Let be a nonlinear mapping. The variational inequality problem is to find a such that

(1.2)

for all . The set of solutions of the variational inequality is denoted by . Now, we consider the following generalized equilibrium problem:

(1.3)

The set of is denoted by , that is,

(1.4)

In the case of , is denoted by . In the case of , is also denoted by . Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of (1.3); see, for instance, [1–3].

A mapping of into is called inverse strongly monotone mapping, see [4], if there exists a positive real number such that

(1.5)

for all . The following definition is well known.

Definition 1.1.

A mapping is said to be a -strict pseudocontraction if there exists such that

(1.6)

A mapping is called nonexpansive if

(1.7)

for all .

We know that -strict pseudocontraction includes a class of nonexpansive mappings. If , is said to be a pseudocontractivemapping. is strong pseudocontraction if there exists a positive constant such that is pseudocontraction. In a real Hilbert space , (1.6) is equivalent to

(1.8)

is pseudocontraction if and only if

(1.9)

Then, is strong pseudocontraction if there exists positive constant

(1.10)

The class of -strict pseudocontractions falls into the one between classes of nonexpansive mappings, and the pseudocontraction mappings, and the class of strong pseudocontraction mappings is independent of the class of -strict pseudocontraction.

We denote by the set of fixed points of . If is bounded, closed, and convex, and is a nonexpansive mapping of into itself, then is nonempty; for instance, see [5]. Browder and Petryshyn [6] show that if a -strict pseudocontraction has a fixed point in , then starting with an initial , the sequence generated by the recursive formula:

(1.11)

where is a constant such that , converges weakly to a fixed point of . Marino and Xu [7] have extended Browder and Petryshyns above-mentioned result by proving that the sequence generated by the following Manns algorithm [8]:

(1.12)

converges weakly to a fixed point of provided the control sequence satisfies the conditions that for all and . In 1974, S. Ishikawa proved the following strong convergence theorem of pseudocontractive mapping.

Theorem 1.2 (see [9]).

Let be a convex compact subset of a Hilbert space , and let be a Lipschitzian pseudocontractive mapping. For any , suppose that the sequence is defined by

(1.13)

where , are two real sequences in satisfying

(i),

(ii),

(iii).

Then converges strongly to a fixed point of .

In order to prove a strong convergence theorem of Mann algorithm (1.12) associated with strictly pseudocontractive mapping, in 2006, Marino and Xu [7] proved the following theorem for strict pseudocontractive mapping in Hilbert space by using method.

Theorem 1.3 (see [7]).

Let be a closed convex subset of a Hilbert space . Let be a -strict pseudocontraction for some , and assume that the fixed point set of is nonempty. Let be the sequence generated by the following algorithm:

(1.14)

Assume that the control sequence is chosen so that for all . Then converges strongly to . Very recently, in 2010, [10] established the hybrid algorithm for Lipschitz pseudocontractive mapping as follows:

(1.15)

Under suitable conditions of and , they proved that the sequence defined by (1.15) converges strongly to .

Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15]. The motivation of (1.14), (1.15), and the research in this direction, we prove the strong convergence theorem for finding solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.

2. Preliminaries

In order to prove our main results, we need the following lemmas. Let be closed convex subset of a real Hilbert space , and let be the metric projection of onto ; that is, for , satisfies the property

(2.1)

The following characterizes the projection .

Lemma 2.1 (see [5]).

Given that and , then if and only if the following inequality holds:

(2.2)

The following lemma is well known.

Lemma 2.2.

Let be Hilbert space, and let be a nonempty closed convex subset of . Let be -strictly pseudocontractive, then the fixed point set of is closed and convex so that the projection is well defined.

Lemma 2.3 ((demiclosedness principle) (see [16]).

If is a -strict pseudocontraction on closed convex subset of a real Hilbert space , then is demiclosed at any point .

To solve the equilibrium problem for a bifunction , assume that satisfies the following conditions:

() for all ,

() is monotone, that is, ,

()for all ,

(2.3)

()for all is convex and lower semicontinuous.

The following lemma appears implicitly in [1].

Lemma 2.4 (see [1]).

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

(2.4)

for all .

Lemma 2.5 (see [11]).

Assume that satisfies . For and , define a mapping as follows:

(2.5)

for all . Then, the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is,

(2.6)

(3);

(4) is closed and convex.

Lemma 2.6 (see [17]).

Let be a closed convex subset of . Let be a sequence in and . Let ; if is such that and satisfy the condition

(2.7)

then , as .

Lemma 2.7 (see [7]).

For a real Hilbert space H, the following identities hold: if is a sequence in H weak convergence to , then

(2.8)

for all .

3. Main Result

Theorem 3.1.

Let be a nonempty closed convex subset of a Hilbert space . Let and be bifunctions from into satisfying , respectively. Let be an α-inverse strongly monotone mapping, and let be a inverse strongly monotone mapping. Let be a strict pseudocontraction mapping with . Let be a sequence generated by and

(3.1)

where is sequence in , , and satisfy the following conditions:

(i),

(ii).

Then converges strongly to .

Proof.

First, we show that is nonexpansive. Let . Since is inverse strongly monotone mapping and , we have

(3.2)

Thus is nonexpansive, so are , , and . Since

(3.3)

then we have

(3.4)

By Lemma 2.5, we have . By the same argument as above, we conclude that .

Let . Then and . Hence

(3.5)

Again by Lemma 2.5, we have . By nonexpansiveness of and , we have

(3.6)

By (3.6), we have

(3.7)

Next, we show that is closed and convex for every . It is obvious that is closed. In fact, we know that, for ,

(3.8)

So, we have that for all and , it follows that

(3.9)

Then, we have that is convex. By Lemmas 2.5 and 2.2, we conclude that is closed and convex. This implies that is well defined. Next, we show that for every .

Taking , we have

(3.10)

It follows that . Then, we have , . Since , for every , we have

(3.11)

In particular, we have

(3.12)

By (3.11), we have that is bounded, so are . Since and , we have

(3.13)

It is implied that

(3.14)

Hence, we have that exists. Since

(3.15)

it is implied that

(3.16)

Since , we have

(3.17)

And by (3.16), we have

(3.18)

Since

(3.19)

by (3.16) and (3.18), we have

(3.20)

Next, we show that

(3.21)

Let , by (3.10) and (3.7), we have

(3.22)

Since , , we have

(3.23)

Since , , we have

(3.24)

Substituting (3.23) and (3.24) into (3.22),

(3.25)

It is implied that

(3.26)

By (3.20) and condition , we have

(3.27)

By using the same method as (3.27), we have

(3.28)

By Lemma 2.5 and firm nonexpansiveness of , we have

(3.29)

By (3.29), it is implied that

(3.30)

Again, by Lemma 2.5 and firm nonexpansiveness of , we have

(3.31)

By (3.31), it is implied that

(3.32)

Substituting (3.30) and (3.32) into (3.22), we have

(3.33)

which implies that

(3.34)

and by (3.27), (3.28), (3.20), and conditions , , we have

(3.35)

By using the same method as (3.35), we have

(3.36)

Since

(3.37)

from (3.35), (3.36), and condition , we have

(3.38)

By (3.20) and (3.38), we have

(3.39)

Since

(3.40)

from (3.39) and condition , we have

(3.41)

Let be the set of all weaks -limit of . We will show that . Since is bounded, then . Letting , there exists a subsequence of converging to . By (3.35), we have as . Since , for any , we have

(3.42)

From , we have

(3.43)

This implies that

(3.44)

Put for all and . Then, we have . So, from (3.44), we have

(3.45)

Since , we have . Further, from monotonicity of , we have . So, we have

(3.46)

From , , and (3.46), we also have

(3.47)

Thus

(3.48)

Letting , we have, for each ,

(3.49)

This implies that

(3.50)

From (3.36), we have . Since , for any , we have

(3.51)

By using the same method as (3.50), we have

(3.52)

Since and (3.38), we have . By Lemma 2.3, is demiclosed at zero, and by (3.41), we have

(3.53)

From (3.50), (3.52), and (3.53), we have . Hence . Therefore, by (3.12) and Lemma 2.6, we have that converges strongly to . The proof is completed.

4. Applications

By using our main result, we have the following results in Hilbert spaces.

Theorem 4.1.

Let be a nonempty closed convex subset of a Hilbert space . Let and be bifunctions from into satisfying , respectively. Let be a -strict pseudocontraction mapping with . Let be a sequence generated by and

(4.1)

where is sequence in , , and satisfy the following conditions:

(i),

(ii).

Then converges strongly to .

Proof.

Putting in Theorem 3.1, we have the desired conclusions.

Theorem 4.2.

Let be a nonempty closed convex subset of a Hilbert space . Let be bifunctions from into satisfying , respectively. Let be an α-inverse strongly monotone mapping, and let be a sequence generated by and

(4.2)

where is sequence in , , and , for all . Then converges strongly to .

Proof.

Putting , , and , for all , in Theorem 3.1, we have the desired conclusions.