Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces

Abstract

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with the work of S. Takahashi and W. Takahashi (2007) and Iemoto and Takahashi (2009).

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively, and let be a closed convex subset of . Let be bifunction, where is the set of real numbers. The equilibrium problem for is to find such that

(1.1)

The set of solution 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. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); see, for example, [1–9] and the references therein.

A mapping of into itself is said to be nonexpansive if for all , and a mapping is said to be firmly  nonexpansive if for all . Let be a smooth, strictly convex and reflexive Banach space, and let be the duality mapping of and a nonempty closed convex subset of . A mapping is said to be nonspreading if

(1.2)

for all , where for all ; see, for instance, Kohsaka and Takahashi [10]. In the case when is a Hilbert space, we know that for all . Then a nonspreading mapping in a Hilbert space is defined as follows:

(1.3)

for all . Let be the set of fixed points of , and nonempty; a mapping is said to be quasi-nonexpansive if for all and .

Remark 1.1.

In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and that if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [10, 11].

In 1953, Mann [12] introduced the iteration as follows: a sequence defined by

(1.4)

where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence (see [12, 13]). Fourteen years later, Halpern [14] introduced the following iterative scheme for approximating a fixed point of :

(1.5)

for all , where and is a sequence of . Strong convergence of this type iterative sequence has been widely studied: Wittmann [15] discussed such a sequence in a Hilbert space.

On the other hand, Kohsaka and Takahashi [10] proved an existence theorem of fixed point for nonspreading mappings in a Banach space. Recently, Lemoto and Takahashi [16] studied the approximation theorem of common fixed points for a nonexpansive mapping of into itself and a nonspreading mapping of into itself in a Hilbert space. In particular, this result reduces to approximation fixed points of a nonspreading mapping of into itself in a Hilbert space by using iterative scheme

(1.6)

Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, [1, 2, 6, 7, 17–20] and the references therein. In 1997, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Recently, S. Takahashi and W. Takahashi [8] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let be a nonexpansive mapping. In 2008, Plubtieng and Punpaeng [7] introduced a new iterative sequence for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space which is the optimality condition for the minimization problem. Very recently, S. Takahashi and W. Takahashi [9] introduced an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets.

In this paper, motivated by S. Takahashi and W. Takahashi [8] and Lemoto and Takahashi [16], we introduce an iterative sequence and prove a strong convergence theorem for finding solution of equilibrium problems and the set of fixed points of a nonspreading mapping in Hilbert spaces.

2. Preliminaries

Let be a real Hilbert space. When is a sequence in , implies that converges weakly to and means the strong convergence. Let be a nonempty closed convex subset of . For every point , there exists a unique nearest point in ; denote by , such that

(2.1)

is called the metric projection of onto . We know that is nonexpansive. Further, for and ,

(2.2)

Moreover, is characterized by the following properties: and

(2.3)

for all , . We also know that satisfies Opial's condition [21], that is, for any sequence with , the inequality

(2.4)

holds for every with ; see [21, 22] for more details.

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 (see [23]).

Let be an inner product space. Then for all and with , one has

(2.5)

Lemma 2.2 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then the following are equivalent.

(1)There exists such that is bounded;

(2) is nonempty.

Lemma 2.3 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then is closed and convex.

Lemma 2.4.

Let be a real Hilbert space. Then for all ,

(1);

(2).

Lemma 2.5 (see [24]).

Let , and let be sequences of real numbers such that

, for all ,

and .

Then, .

Lemma 2.6 (see [16]).

Let be a Hilbert space, a closed convex subset of , and a nonspreading mapping with . Then is demiclosed, that is, and imply .

Lemma 2.7 (see [16]).

Let be a Hilbert space, a nonempty closed convex subset of a real Hilbert space , and let be a nonspreading mapping of into itself, and let . Then

(2.6)

Lemma 2.8 (see [25]).

Assume is a sequence of nonnegative real numbers such that

(2.7)

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

(1);

(2) or .

Then .

For solving the equilibrium problems for a bifunction , let us assume that satisfies the following conditions:

(A1);

(A2) is monotone, that is, ;

(A3)for each ,  ;

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

The following lemma appears implicitly in [26].

Lemma 2.9 (see [26]).

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

(2.8)

The following lemma was also given in [4].

Lemma 2.10 (see [4]).

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

(2.9)

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 2.11 (see [27]).

Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also consider the sequence of integers defined by

(2.10)

Then is a nondecreasing sequence verifying , and the following properties are satisfied for all :

(2.11)

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 nonspreading mapping and the set of solutions of the equilibrium problems.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunctions from satisfying (A1)–(A4), and let be a nonspreading mapping of into itself such that . Let , and let and be sequences generated by and

(3.1)

for all , where and satisfy

, , ,

, ,

, and .

Then converges strongly to , where .

Proof.

Let . From , we have

(3.2)

for all . Put . We divide the proof into several steps.

Step 1.

We claim that the sequences , , , and are bounded. First, we note that

(3.3)

and so

(3.4)

Putting , we note that for all . In fact, it is obvious that . Assume that for all . Thus, we have

(3.5)

By induction, we obtain that for all . So, is bound. Hence, , , and are also bounded.

Step 2.

Put . We claim that as . We note that

(3.6)

where . On the other hand, from and , we have

(3.7)

(3.8)

for all . Putting in (3.7) and in (3.8), we have

(3.9)

So, from (A2), we note that

(3.10)

and hence

(3.11)

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

(3.12)

and hence

(3.13)

where . So, from (3.6), we note that

(3.14)

By Lemma 2.5, we have

(3.15)

for . We note from that

(3.16)

and hence

(3.17)

Therefore, from the convexity of , we have

(3.18)

and hence

(3.19)

So, we have . Indeed, since , it follows that

(3.20)

Then, we note that

(3.21)

Since, and , it follows that

(3.22)

Step 3.

Put . From , it follows by Lemma 2.7 that

(3.23)

Since , we have . Therefore, by (3.23), we obtain

(3.24)

Step 4.

Putting , we claim that the sequence converges strongly to . Indeed, we discuss two possible cases.

Case 1.

Assume that there exists such that the sequence is a nonincreasing sequence for all . Then we have (for ), and hence exists. Therefore

(3.25)

By (3.22), (3.24), and (3.25), we get

(3.26)

Let be a subsequence of such that

(3.27)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . Since is closed and convex, we note that is weakly closed. So, we have . Since , it follows by Lemma 2.6 that . From (3.27) and the property of metric projection, we have

(3.28)

Finally, we prove that . In fact, since , it follows that

(3.29)

By (3.28) and , we immediately deduce by Lemma 2.8 that .

Case 2.

Assume that for all , there exits such that . Put for all . Thus, it follows that there exists a subsequence of such that for all . Let be a mapping defined by

(3.30)

where . By Lemma 2.11, we note that is a nondecreasing sequence such that as and that the following properties are satisfied by all numbers :

(3.31)

From (3.24), we have

(3.32)

This implies that

(3.33)

Take a subsequence of such that

(3.34)

From the boundedness of , we can assume that . Since is closed and convex, it follows that is weakly closed. So, we have . Since , it follows by Lemma 2.6 that . From (3.34) and the property of metric projection, we have

(3.35)

By the same argument as (3.29) in Case 1, we conclude immediately that, for all ,

(3.36)

which implies that

(3.37)

By (3.35), we have

(3.38)

and hence

(3.39)

Since for all , we have

(3.40)

This completes the proof.

As direct consequences of Theorem 3.1, we obtain corollaries.

Corollary 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunctions from satisfying (A1)–(A4), and let be a firmly nonexpansive mapping of into itself such that . Let , and let and be sequences generated by and

(3.41)

for all , where and satisfy

, , ,

, ,

, and .

Then converges strongly to , where .