Research Article | Open Access

# Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family of -Strictly Pseudocontractive Mapping in Hilbert Spaces

**Academic Editor:**Jong Hae Kim

#### Abstract

We first extend the definition of W_{n} from an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family of -strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

#### 1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of and let be a bifunction. We consider the following equilibrium problem (EP) which is to find such that Denote the set of solutions of 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). Some methods have been proposed to solve the equilibrium problem [1–13].

A mapping is called -Lipschitzian if there exists a positive constant such that is said to be -strongly monotone if there exists a positive constant such that A mapping is said to be -strictly pseudocontractive mapping if there exists a constant such that for all and denotes the set of fixed point of the mapping , that is .

If , then is said to a pseudocontractive mapping, that is, is equivalent to for all .

The class of -strict pseudo-contractive mappings extends the class of nonexpansive mappings (A mapping is said to be nonexpansive if , for all ). That is, is nonexpansive if and only if is a -strict pseudocontractive mapping. Clearly, the class of -strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mapping.

In 2006, Marino and Xu [14] introduced the general iterative method and proved that for a given , the sequence generated by the algorithm where is a self-nonexpansive mapping on , is an -contraction of into itself (i.e., , for all and ), satisfies certain conditions, is strongly positive bounded linear operator on , and converges strongly to fixed point of which is the unique solution to the following variational inequality:

Tian [15] considered the following iterative method, for a nonexpansive mapping : with , where is -Lipschitzian and -strongly monotone operator. The sequence converges strongly to fixed-point in which is the unique solution to the following variational inequality:

For finding a common element of , S. Takahashi and W. Takahashi [16] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial point , define sequences and recursively by They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to , where .

Liu [17] introduced the following scheme: and where is a -strict pseudo-contractive mapping and is a strongly positive bounded linear operator. They proved that under certain appropriate conditions imposed on , and , the sequence converges strongly to , where .

In [18], the concept of mapping had been modified for a countable family of nonexpansive mappings by defining the sequence of -mappings generated by and , proceeding backward Yao et al. [19] using this concept, introduced the following algorithm: and They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to .

Colao and Marino [20] considered the following explicit viscosity scheme where is a strongly positive operator on . Under certain appropriate conditions, the sequences and converge strongly to .

Motivated and inspired by these facts, in this paper, we first extend the definition of from an infinite family of nonexpansive mappings to an infinite family of strictly pseudo-contractive mappings, and then propose the iteration scheme (3.2) for finding an element of , where is an infinite family of -strictly pseudo-contractive mappings of into itself. Finally, the convergence theorem of the iteration scheme is obtained. Our results include Yao et al. [19], Colao and Marino [20] as some special cases.

#### 2. Preliminaries

Throughout this paper, we always assume that is a nonempty closed convex subset of a Hilbert space . We write to indicate that the sequence converges weakly to . implies that converges strongly to . We denote by and the sets of positive integers and real numbers, respectively. For any , there exists a unique nearest point in , denoted by , such that Such a is called the metric projection of onto . It is known that is nonexpansive. Furthermore, for and , It is widely known that satisfies Opials condition [21], that is, for any sequence with , the inequality holds for every with .

In order to solve the equilibrium problem for a bifunction , we assume that satisfies the following conditions:(A1)for all .(A2) is monotone, that is, , for all .(A3), for all .(A4) For each is convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

Lemma 2.1 2.1 (see [22]). *Let be a bifunction from into satisfying (A1), (A2), (A3), and (A4). Then, for any and , there exists such that
**
Furthermore, if , then the following hold:*(1)* is single-valued.*(2)* is firmly nonexpansive, that is,
*(3)*. *(4)* is closed and convex.*

Lemma 2.2 2.2 (see [23]). *Let be a k-strictly pseudo-contractive mapping. Define by for each . Then, as , is nonexpansive mapping such that .*

Lemma 2.3 2.3 (see [24]). *In a Hilbert space , there holds the inequality
*

Lemma 2.4 (see [25]). *Let be a Hilbert space and be a closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then .*

Lemma 2.5 (see [26]). *Let and be bounded sequences in a Banach space E and be a sequence in satisfying the following condition
**
Suppose that and . Then .*

Lemma 2.6 (see [27]). *Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence in , such that*(i)*.*(ii)* or .**Then, .*

Let be an infinite family of -strictly pseudo-contractive mappings of into itself, we define a mapping of into itself as follows, where , and for . We can obtain is a nonexpansive mapping and by Lemma 2.2. Furthermore, we obtain that is a nonexpansive mapping.

*Remark 2.7. *If , and for , then the definition of in (2.9) reduces to the definition of in (1.13).

To establish our results, we need the following technical lemmas.

Lemma 2.8 (see [18]). *Let be a nonempty closed convex subset of a strictly convex Banach space. Let be an infinite family of nonexpansive mappings of into itself and let be a real sequence such that for every . Then, for every and , the limit exists.*

In view of the previous lemma, we will define

Lemma 2.9 (see [18]). *Let be a nonempty closed convex subset of a strictly convex Banach space. Let be an infinite family of nonexpansive mappings of into itself such that and let be a real sequence such that for every . Then, .*

The following lemmas follow from Lemmas 2.2, 2.8, and 2.9.

Lemma 2.10. *Let be a nonempty closed convex subset of a strictly convex Banach space. Let be an infinite family of -strictly pseudo-contractive mappings of into itself such that . Define and and let be a real sequence such that for every . Then, .*

Lemma 2.11 (see [28]). *Let be a nonempty closed convex subset of a Hilbert space. Let be an infinite family of nonexpansive mappings of into itself such that and let be a real sequence such that for every . If is any bounded subset of , then
*

#### 3. Main Results

Let be a real Hilbert space and be a -Lipschitzian and -strongly monotone operator with , , and . Then, for , is a contraction with contractive coefficient and .

In fact, from (1.2) and (1.3), we obtain Thus, is a contraction with contractive coefficient .

Now, we show the strong convergence results for an infinite family -strictly pseudo-contractive mappings in Hilbert space.

Theorem 3.1. *Let be a nonempty closed convex subset of a real Hilbert space and be a bifunction from satisfying (A1)–(A4). Let be a -strictly pseudo-contractive mapping with and be a real sequence such that , . Let be a contraction of into itself with and be -Lipschitzian and -strongly monotone operator on with coefficients , , and . Let be a sequence generated by
**
where and is the sequence defined by (2.9). If , , , and satisfy the following conditions:*(i)*, , ,*(ii)*,
*(iii)*,
,
*(iv)*, , .**Then converges strongly to , where is the unique solution of variational inequality
**
that is, , which is the optimality condition for the minimization problem
**
where is a potential function for (i.e., for ).*

*Proof . *We divide the proof into five steps.*Step 1*. We prove that is bounded.

Noting the conditions (i) and (ii), we may assume, without loss of generality, that . For , we obtain
Take . Since and , then from Lemma 2.1, we know that, for any ,
Furthermore, since and (3.6), we have
Thus, it follows from (3.7) that
By induction, we have
Hence, is bounded and we also obtain that , , , , and are all bounded. Without loss of generality, we can assume that there exists a bounded set such that , , , , , for all .*Step 2*. We show that .

Let . We note that
and then
Therefore,
It follows from (3.2) that

We will estimate . From and , we obtain

Taking in (3.14) and in (3.15), we have

So, from (A2), one has
furthermore,
Since , we assume that there exists a real number such that for all . Thus, we obtain
which means
where .

Next, we estimate . Notice that
From (2.9), we obtain
where is a constant such that , for all .

Substituting (3.20) and (3.22) into (3.21), we obtain
Hence, we have
where .

Furthermore,
It follows from (3.25) that
By the conditions (i), (iii), and (iv), we obtain
Hence, by Lemma 2.5, one has
which implies
*Step 3*. We claim that .

Notice that
It follows from (3.2) that
By the condition (iii), we obtain

First, we show . From (3.2), for all , applying Lemma 2.3 and noting that is convex, we obtain
Since , , we have
which implies
Substituting (3.35) into (3.33), we have
which means
Noticing and , we have

Second, we show . It follows from (3.2) that
This implies that
Noticing , and (3.30), we have
Thus, substituting (3.41) and (3.38) into (3.32), we obtain
Furthermore, (3.42), (3.30), and Lemma 2.11 lead to
*Step 4*. Letting , we show
We know that is a contraction. Indeed, for any , we have
and hence is a contraction due to . Thus, Banach’s Contraction Mapping Principle guarantees that has a unique fixed point, which implies .

Since is bounded in , without loss of generality, we can assume that , it follows from (3.43) that . Since is closed and convex, is weakly closed. Thus we have .

Let us show . For the sake of contradiction, suppose that , that is, . Since , by the Opial condition, we have
It follows (3.43) that
This is a contradiction, which shows that .

Next, we prove that . By (3.2), we obtain
It follows from (A2) that
Replacing by , we have
Since and , it follows from (A4) that for all . Put for all and . Then, we have and then . Hence, from (A1) and (A4), we have
which means . From (A3), we obtain for and then . Therefore, .

Since , it follows from (3.38), (3.42), and Lemma 2.11 that
*Step 5*. Finally we prove that as . In fact, from (3.2) and (3.7), we obtain
which implies
From condition (i) and (3.7), we know that and . we can conclude from Lemma 2.6 that as . This completes the proof of Theorem 3.1.

*Remark 3.2. *If , , and , , for , then Theorem 3.1 reduces to Theorem 3.5 of Yao et al. [19]. Furthermore, we extend the corresponding results of Yao et al. [19] from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings.

*Remark 3.3. *If and , , for , then Theorem 3.1 reduces to Theorem 10 of Colao and Marino [20]. Furthermore, we extend the corresponding results of Colao and Marino [20] from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings, and from a strongly positive bounded linear operator to a -Lipschitzian and -strongly monotone operator .

Theorem 3.4. *Let be a nonempty closed convex subset of a real Hilbert space and be a bifunction from satisfying (A1)–(A4). Let be a nonexpansive mapping with . Let be a contraction of into itself with and be -Lipschitzian and -strongly monotone operator on with coefficients , , and . Let be sequence generated by
**
where . If , and satisfy the following conditions:*(i)*, , ,*(ii)*,
*(iii)*,
,
*(iv)*, , .**Then converges strongly to , where is the unique solution of variational inequality
**
that is, .*

*Proof . *By Theorem 3.1, letting , , and for , we can obtain Theorem 3.4.

#### 4. Numerical Example

Now, we present a numerical example to illustrate our theoretical analysis results obtained in Section 3.

*Example 4.1. *Let , , , , , , , for all , , , , for all , with contraction coefficient , , , for every . Then is the sequence generated by
and , as , where 0 is the unique solution of the minimization problem