A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces

Abstract

We introduce a viscosity approximation method for finding a common element of the set of solutions for an equilibrium problem involving a bifunction defined on a closed, convex subset and the set of fixed points for a nonexpansive semigroup on another one in Hilbert's spaces.

1. Introduction

Let be a nonempty, closed, and convex subset of a real Hilbert space . Denote the metric projection from onto by . Let be a nonexpansive mapping on , that is, and , for all . We use to denote the set of fixed points of , that is, .

Let be a nonexpansive semigroup on a closed convex subset , that is,

(1)for each , is a nonexpansive mapping on ,

(2) for all ,

(3) for all ,

(4)for each , the mapping from into is continuous.

Denote by . We know [1, 2] that is a closed, convex subset in and if is bounded.

The equilibrium problem is for a bifunction defined on to find such that

(1.1)

Assume that the bifunction satisfies the following set of standard properties:

(A1), for all ,

(A2) for all ,

(A3)for every , is weakly lower semicontinuous and convex,

(A4), for all .

Denote the set of solutions of (1.1) by . We also know [3] that is a closed convex subset in .

The problem studied in this paper is formulated as follows. Let and be closed convex subsets in . Let be a bifunction satisfying conditions (A1)–(A4) with replaced by and let be a nonexpansive semigroup on . Find an element

(1.2)

where and denote the set of solutions of an equilibrium problem involving by a bifunction on and the fixed point set of a nonexpansive semigroup on a closed convex subset , respectively.

In the case that , , , and , a nonexpansive mapping on , for all , (1.2) is the fixed point problem of a nonexpansive mapping. In 2000, Moudafi [4] proved the following strong convergence theorem.

Theorem 1.1.

Let be a nonempty, closed, convex subset of a Hilbert space and let be a nonexpansive mapping on such that . Let be a contraction on and let be a sequence generated by: and

(1.3)

where satisfies

(1.4)

Then, converges strongly to , where .

Such a method for approximation of fixed points is called the viscosity approximation method. It has been developed by Chen and Song [5] to find , the set of fixed points for a semigroup on . They proposed the following algorithm: and

(1.5)

where , is a contraction, and are sequences of positive real numbers satisfying the conditions: , , and as .

Recently, Yao and Noor [6] proposed a new viscosity approximation method

(1.6)

where , , and are in , , for finding , when satisfies the uniformly asymptotically regularity condition

(1.7)

uniformly in , and is any bounded subset of . Further, Plubtieng and Pupaeng in [7] studied the following algorithm:

(1.8)

where and are in satisfying the following conditions: , , , and is a positive divergent real sequence.

There were some methods proposed to solve equilibrium problem (1.1); see for instance [8–12]. In particular, Combettes and Histoaga [3] proposed several methods for solving the equilibrium problem.

In 2007, S. Takahashi and W. Takahashi [13] combinated the Moudafi's method with the Combettes and Histoaga's result in [3] to find an element . They proved the following strong convergence theorem.

Theorem 1.2.

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

(1.9)

where and satisfy

(1.10)

Then, and converge strongly to , where .

Very recently, Ceng and Wong in [14] combined algorithm (1.6) with the result in [3] to propose the following procudure:

(1.11)

for finding an element in the case that under the uniformly asymptotic regularity condition on the nonexpansive semigroup on .

In this paper, motivated by the above results, to solve (1.2), we introduce the following algorithm:

(1.12)

where is a contraction on , that is, and , for all , ,

(1.13)

for all , , , and be the sequences in (0,1), and , are the sequences in satisfy the following conditions:

(i),

(ii), ,

(iii),

(iv) with bounded ,

(v) and .

The strong convergence of (1.12)-(1.13) and its corollaries are showed in the next section.

2. Main Results

We formulate the following facts needed in the proof of our results.

Lemma 2.1.

Let be a real Hilbert space . There holds the following identity:

(2.1)

Lemma 2.2 (see [15]).

Let be a nonempty, closed, convex subset of a real Hilbert space . For any , there exists a unique such that , for all , and if and only if for all .

Lemma 2.3 (see [16]).

Let be a sequence of nonnegative real numbers satisfying the following condition:

(2.2)

where and are sequences of real numbers such that , , and . Then, .

Lemma 2.4 (see [9]).

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

(2.3)

Lemma 2.5 (see [9]).

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

(2.4)

Then, the following statements hold:

(i) is single-valued,

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

(2.5)

(iii),

(iv) is closed and convex.

Lemma 2.6 (see [17]).

Let be a nonempty bounded closed convex subset in a real Hilbert space and let be a nonexpansive semigroup on . Then, for any ,

(2.6)

Lemma 2.7 (Demiclosedness Principle [18]).

If is a closed convex subset of , is a nonexpansive mapping on , is a sequence in such that and , then .

Lemma 2.8 (see [19]).

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

Now, we are in a position to prove the following result.

Theorem 2.9.

Let and be two nonempty, closed, convex subsets in a real Hilbert space . Let be a bifunction from to satisfying conditions (A1)–(A4) with replaced by , let be a nonexpansive semigroup on such that and let be a contraction of into itself. Then, and generated by (1.12)-(1.13) converge strongly to , where .

Proof.

Let . Then, is a contraction of into itself. In fact, from for all and the nonexpansive property of for a closed convex subset in , it implies that

(2.7)

Hence, is a contraction of into itself. Since is complete, there exists a unique element such that . Such a is an element of , because .

By Lemma 2.4, and are well defined. For each , by putting and using (ii) and (iii) in Lemma 2.5, we have that

(2.8)

Put . Clearly, . Suppose that . Then, we have, from the nonexpansive property of , condition (i) and (2.8), that

(2.9)

So, for all and hence is bounded. Therefore, , , and are also bounded.

Next, we show that as . For this purpose, we define a sequence by

(2.10)

Then, we observe that

(2.11)

and, hence,

(2.12)

Now, we estimate the value by using and . We have from (2.4) that

(2.13)

(2.14)

Putting in (2.13) and in (2.14), adding the one to the other obtained result and using (A2), we obtain that

(2.15)

and, hence,

(2.16)

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

(2.17)

and, hence,

(2.18)

On the other hand,

(2.19)

So, we get from (2.10), (2.12), (2.18), (2.19), and the nonexpansive property of that

(2.20)

So,

(2.21)

and by Lemma 2.8, we have

(2.22)

Consequently, it follows from (2.10) and condition (iii) that

(2.23)

By (2.18), (2.23), and

(2.24)

we also obtain

(2.25)

We have, for every , from (iii) in Lemma 2.5, that

(2.26)

and, hence,

(2.27)

Therefore, from the convexity of and condition (i), we have

(2.28)

and, hence,

(2.29)

Without loss of generality, we assume that for all . Then, for sufficiently large ,

(2.30)

So, we have

(2.31)

Further, since , by condition (i), (2.19) and

(2.32)

we obtain that

(2.33)

Then, from (2.25), (2.33) and the conditions on and , it implies that

(2.34)

and so

(2.35)

Since

(2.36)

we obtain from (2.31) that

(2.37)

Next, we show that

(2.38)

We choose a subsequence of the sequence such that

(2.39)

As is bounded, there exists a subsequence of the sequence which converges weakly to . From (2.37), we also have that converges weakly to . Since and and , are two closed convex subsets in , we have that .

First, we prove that . From (2.4) it follows that

(2.40)

and, hence, by using condition (A2), we get

(2.41)

Therefore,

(2.42)

This together with condition (A3) and (2.31) imply that

(2.43)

So, for all . It means that .

Next we show that . Since , we have

(2.44)

and, hence, from (2.31) it follows that

(2.45)

Thus, (2.37) together with (2.45) imply

(2.46)

Therefore, also converges weakly to , as .

On the other hand, for each , we have that

(2.47)

Let . Since , we have from (2.33) that

(2.48)

So, is a nonempty bounded closed convex subset. It is easy to verify that is a nonexpansive semigroup on . By Lemma 2.6, we get

(2.49)

for every fixed , and hence, by (2.45)–(2.47), we obtain

(2.50)

for each . By Lemma 2.7, for all , because for any mapping . It means that . Therefore, . Since , we have from Lemma 2.2 that

(2.51)

So, (2.38) is proved. Further, since , by using Lemma 2.1, we have that

(2.52)

This with (2.8) implies that

(2.53)

where

(2.54)

Using Lemma 2.3, we get

(2.55)

From (2.33) it follows that as . This completes the proof.

Remarks 2.

• (a) Note that the following parameters , , , for any fixed number , and with for all satisfy all conditions in Theorem 2.9.

• (b) If for all and , then we have the following corollary.

Corollary 2.10.

Let be a nonempty, closed, convex subsets in a real Hilbert space . Let be a bifunction from to satisfying conditions (A1)–(A4), let be a nonexpansive mapping on such that and let be a contraction of into itself. Let and be sequences generated by and

(2.56)

where , , , and satisfy conditions (i)–(v). Then, and converge strongly to , where .

Proof.

From the proof of the theorem, in (2.12).

• (c) In the case that , a closed convex subset in , for all , we have the following result.

Corollary 2.11.

Let be a nonempty, closed, convex subsets in a real Hilbert space . Let be a nonexpansive semigroup on such that and let be a contraction of into itself. Let and be sequences generated by and

(2.57)

where is defined by (1.13) for all and , , , and satisfy conditions (i)–(v). Then, the sequences and converge strongly to , where .

Proof.

By Lemma 2.2, if and only if

(2.58)

Clearly, in addition, if is a contraction of into itself and , then we obtain the algoritm

(2.59)

where is defined by (1.13) and , , , and satisfy conditions (i)–(v). This algorithm is different from Yao and Noor's algorithm (1.6), in which for all . It likes completely the Plubtieng and Punpaeng's algorithm (1.8), but converges under a new condition on .