Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems

Abstract

We introduce an iterative scheme by the viscosity iterative method for finding a common element of the solution set of an equilibrium problem, the solution set of the variational inequality, and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space. Then we prove our main result under some suitable conditions.

1. Introduction

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

(1.1)

The solution set of (1.1) is denoted by .

Let be a mapping. The classical variational inequality, denoted by , is to find such that

(1.2)

The variational inequality has been extensively studied in the literature (see, e.g., [1–3]). The mapping is called -inverse-strongly monotone if

(1.3)

where is a positive real number.

A mapping is called strictly pseudocontractive if there exists with such that

(1.4)

It is easy to know that is ()-inverse-strongly-monotone. If , then is nonexpansive. We denote by the fixed points set of .

In 2003, for , Takahashi and Toyoda [4] introduced the following iterative scheme:

(1.5)

where is a sequence in , is an -inverse-strongly monotone mapping, is a sequence in , and is the metric projection. They proved that if , then converges weakly to some

Recently, S. Takahashi and W. Takahashi [5] introduced an iterative scheme for finding a common element of the solution set of (1.1) and the fixed points set of a nonexpansive mapping in a Hilbert space. If is bifunction which satisfies the following conditions:

() for all

() is monotone, that is, for all

() for each

() for each is convex and lower semicontinuous,

then they proved the following strong convergence theorem.

Theorem A (see [5]).

Let be a closed and convex subset of a real Hilbert space . Let be a bifunction which satisfies conditions .

Let be a nonexpansive mapping such that and let be a contraction; that is, there is a constant such that

(1.6)

and let and be sequences generated by and

(1.7)

where and satisfy , and

Then, and converge strongly to where

Let be a sequence of nonexpansive mappings of into itself and a sequence of nonnegative numbers in . For each , define a mapping of into itself as follows:

(1.8)

Such a mapping is called the -mapping generated by and (see [6]).

In this paper, we introduced a new iterative scheme generated by and find such that

(1.9)

where and are sequences in and are sequences in , is a fixed contractive mapping with contractive coefficient , is an -inverse-strongly monotone mapping of to , is a bifunction which satisfies conditions , and is generated by (1.8). Then we proved that the sequences and converge strongly to , where .

2. Preliminaries

Let be a real Hilbert space and let be a closed and convex subset of is the metric projection from onto , that is, for any , for all It is easy to see that is nonexpansive and

(2.1)

If is an -inverse-strongly monotone mapping of to , then it is obvious that is ()-Lipschitz continuous. We also have that for all and ,

(2.2)

So, if , then is nonexpansive.

Lemma 2.1 (see [7]).

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

Lemma 2.2 (see [8]).

Assume that is a sequence of nonnegative real numbers such that

(2.3)

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

(2.4)

Then

Lemma 2.3 (see [9]).

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

(2.5)

Lemma 2.4 (see [9]).

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

(2.6)

Then, the following holds:

(i) is single-valued;

(ii) is firmly nonexpansive, that is,

(2.7)

(iii)

(iv) is closed and convex.

Lemma 2.5 (Opial's theorem [10]).

Each Hilbert space satisfies Opial's condition; that is, for any sequence with , the inequality

(2.8)

holds for each with

Let be a sequence of nonexpansive self-mappings on , where is a nonempty, closed and convex subset of a real Hilbert space . Given a sequence in , one defines a sequence of self-mappings on generated by (1.8). Then one has the following results.

Lemma 2.6 (see [6]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and is a sequence in for some . Then, for every and the limit exists.

Remark 2.7.

It can be shown from Lemma 2.6 that if is a nonempty and bounded subset of , then for there exists such that for all .

Remark 2.8.

Using Lemma 2.6, we can define a mapping as follows:

(2.9)

for all Such a is called the -mapping generated by and Since is nonexpansive, is also nonexpansive. Indeed, observe that for each ,

(2.10)

Let be a bounded sequence in and . Then, it is clear from Remark 2.7 that for there exists such that for all

(2.11)

This implies that

Lemma 2.9 (see [6]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and is a sequence in for some . Then, .

3. Strong Convergence Theorem

Theorem 3.1.

Let be a Hilbert space. Let be a nonempty, closed, and convex subset of . Let be a bifunction which satisfies conditions , an -inverse-strongly monotone mapping of to , a contraction of into itself, and a sequence of nonexpansive self-mappings on such that . Suppose that , and are sequences in , and and are sequences in which satisfies the following conditions:

(i)

(ii)

(iii)

(iv)

(v).

Then and generated by (1.9) converge strongly to , where .

Proof.

Let . It follows from Lemma 2.4 and (1.9) that , and hence,

(3.1)

for all . Let . Since is nonexpansive and , we have

(3.2)

(3.3)

Thus,

(3.4)

Hence is bounded. So , and are also bounded.

Next, we claim that Indeed, assume that where , . Then,

(3.5)

(3.6)

Using (1.8) and the nonexpansivity of , we deduce that

(3.7)

for some constant On the other hand, from and , we obtain

(3.8)

(3.9)

Setting in (3.8) and in (3.9), we get

(3.10)

From , we have

(3.11)

and hence

(3.12)

Without loss of generality, we may assume that there exists a real number such that for all Then

(3.13)

and hence

(3.14)

where . It follows from (3.5), (3.6), (3.7), and (3.14) that

(3.15)

Therefore, .

Since and , hence,

(3.16)

Lemma 2.1 yields that . Consequently,

For , we obtain

(3.17)

and hence

(3.18)

This together with (3.2) yields that

(3.19)

and hence,

(3.20)

So (note that and . Since

(3.21)

we obtain and hence . Thus,

Let Then is a contraction of into itself. In fact, there exists such that for all . So

(3.22)

for all So is a contraction by Banach contraction principle [11]. Since is a complete space, there exists a unique element such that .

Next we show that

(3.23)

where To show this inequality, we choose a subsequence of such that

(3.24)

Since is bounded, there exists a subsequence of which converges weakly to some , that is, . From , we obtain that . Now we will show that . First, we will show . From we have

(3.25)

By , we also have

(3.26)

and hence

(3.27)

Since and , it follows from that for all . For any and , let Since and , then we have and hence . This together with and yields that

(3.28)

and thus . From , we have for all and hence . Now, we show that . Indeed, we assume that ; from Opial's condition, we have

(3.29)

This is a contradiction. Thus, we obtain that . Finally, by the same argument as in the proof of [3, Theorem 3.1], we can show that . Hence Hence,

(3.30)

Now we show that

From (1.9), we have

(3.31)

and hence,

(3.32)

Using (3.23) and Lemma 2.2, we conclude that converges strongly to Consequently, converges strongly to This completes the proof.

Using Theorem 3.1, we prove the following theorem.

Theorem 3.2.

Let , and be given as in Theorem 3.1 and let be an -strictly pseudocontractive mapping such that . Suppose that and Let and be the sequences and find such that

(3.33)

where , and are given as in Theorem 3.1. Then and converge strongly to , where .

Proof.

Put . Then is ()-inverse-strongly-monotone. We have and put . So by Theorem 3.1 we obtain the desired result.