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Research Article | Open Access

Volume 2012 |Article ID 416476 | https://doi.org/10.1155/2012/416476

Haitao Che, Meixia Li, Xintian Pan, "Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family of -Strictly Pseudocontractive Mapping in Hilbert Spaces", Journal of Applied Mathematics, vol. 2012, Article ID 416476, 23 pages, 2012. https://doi.org/10.1155/2012/416476

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

Revised08 Jun 2012
Accepted24 Jun 2012
Published31 Jul 2012

#### Abstract

We first extend the definition of Wn 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 .

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  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  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  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  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 , 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.  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  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. , Colao and Marino  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 , 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 ). 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 ). 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 ). In a Hilbert space , there holds the inequality

Lemma 2.4 (see ). 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 ). 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 ). 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 ). 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 ). 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 ). 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. . Furthermore, we extend the corresponding results of Yao et al.  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 . Furthermore, we extend the corresponding results of Colao and Marino  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