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# Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions

*Fixed Point Theory and Applications***volume 2011**, Article number: 941090 (2011)

## Abstract

We introduce a new iterative algorithm for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions.

## 1. Introduction

Let be a real Hilbert space with the inner product and inducted norm . Let be a nonempty, closed, and convex subset of . Let be a family of bifunctions, and let be a family of nonlinear mappings, where is an arbitrary index set. The system of generalized equilibrium problems is to find such that

If is a singleton, then (1.1) reduces to find such that

The solutions set of (1.2) is denoted by . If , then the solutions set of (1.2) is denoted by , and if , then the solutions set of (1.2) is denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and the Nash equilibrium problem in noncooperative games; see also [1, 2]. Some methods have been constructed to solve the system of equilibrium problems (see, e.g., [3–7]). Recall that a mapping is said to be

(1)*monotone* if

(2)*α-inverse-strongly monotone* if there exists a constant such that

It is easy to see that if is *α*-inverse-strongly monotone, then is monotone and -Lipschitz.

For solving the equilibrium problem, let us assume that satisfies the following conditions:

(A1) for all ,

(A2) is monotone, that is, for all ,

(A3)for each , ,

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

Throughout this paper, we denote the fixed points set of a nonlinear mapping by . Recall that is said to be a *-strict pseudocontraction* if there exists a constant such that

It is well known that (1.5) is equivalent to

It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. It is also known that every -strict pseudocontraction is -Lipschitz; see [8].

In 1953, Mann [9] introduced the iteration as follows: a sequence defined by and

where . If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence defined by (1.7) converges weakly to a fixed point of (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [10]).

In 1967, Browder and Petryshyn [11] introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann iterative algorithm (1.7) with a constant sequence for all . Recently, Marino and Xu [8] and Zhou [12] extended the results of Browder and Petryshyn [11] to Mann's iteration process (1.7). Since 1967, the construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors (see, e.g., [13–22]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping, a bifunction, and let be an inverse-strongly monotone mapping.

In 2008, Moudafi [23] introduced an iterative method for approximating a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem as follows: a sequence defined by and

where and . He proved that the sequence generated by (1.8) converges weakly to an element in under suitable conditions.

Due to the weak convergence, recently, S. Takahashi and W. Takahashi [24] introduced another modification iterative method of (1.8) for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space. To be more precise, they proved the following theorem.

Theorem 1.1 (see [24]).

Let be a closed convex subset of a real Hilbert space , and let be a bifunction satisfying (A1)–(A4). Let be an *α*-inverse-strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Let and , and let and be sequences generated by

where , and satisfy

(i) and ,

(ii),

(iii),

(iv).

Then, converges strongly to .

Recently, Yao et al. [25] introduced a new modified Mann iterative algorithm which is different from those in the literature for a nonexpansive mapping in a real Hilbert space. To be more precise, they proved the following theorem.

Theorem 1.2 (see [25]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping such that . Let , and let be two real sequences in . For given arbitrarily, let the sequence , , be generated iteratively by

Suppose that the following conditions are satisfied:

(i) and ,

(ii),

then, the sequence generated by (1.10) strongly converges to a fixed point of .

We know the following crucial lemmas concerning the equilibrium problem in Hilbert spaces.

Lemma 1.3 (see [1]).

Let be a nonempty, closed, and convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4). Let and . Then, there exists such that

Lemma 1.4 (see [26]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4). For and , define the mapping as follows:

Then, the following statements hold:

(1) is single-valued,

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

(3),

(4) is closed and convex.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let for each . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let be a countable family of -strict pseudocontractions. For each , denote the mapping by , where is the mapping defined as in Lemma 1.4.

Motivated and inspired by Marino and Xu [8], Moudafi [23], S. Takahashi and W. Takahashi [24], and Yao et al. [25], we consider the following iteration: and

where , and .

In this paper, we first prove a path convergence result for a nonexpansive mapping and a system of generalized equilibrium problems. Then, we prove a strong convergence theorem of the iteration process (1.14) for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the main results obtained by Yao et al. [25] in several aspects.

## 2. Preliminaries

Let be a nonempty, closed, and convex subset of a real Hilbert space . For each , there exists a unique nearest point in , denoted by , such that . is called the metric projection of onto . It is also known that for and , is equivalent to for all . Furthermore,

for all , . In a real Hilbert space, we also know that

for all and .

In the sequel, we need the following lemmas.

Let be a real uniformly convex Banach space, and let be a nonempty, closed, and convex subset of , and let be a nonexpansive mapping such that , then is demiclosed at zero.

Lemma 2.2 (see [29]).

Let and be two sequences in a Banach space such that

where satisfies conditions: . If , then as .

Lemma 2.3 (see [30]).

Assume that is a sequence of nonnegative real numbers such that

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

(a) ; (b) or .Then, .

Lemma 2.4 (see [31]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let the mapping be *α*-inverse-strongly monotone, and let be a constant. Then, we have

for all . In particular, if , then is nonexpansive.

To deal with a family of mappings, the following conditions are introduced: let be a subset of a real Hilbert space , and let be a family of mappings of such that . Then, is said to satisfy the *-condition* [32] if for each bounded subset of ,

Lemma 2.5 (see [32]).

Let be a nonempty and closed subset of a Hilbert space , and let be a family of mappings of into itself which satisfies the -condition. Then, for each , converges strongly to a point in . Moreover, let the mapping be defined by

Then, for each bounded subset of ,

The following results can be found in [33, 34].

Let be a closed, and convex subset of a Hilbert space . Suppose that is a family of -strictly pseudocontractive mappings from into with and is a real sequence in such that . Then, the following conclusions hold:

(1) is a -strictly pseudocontractive mapping,

(2).

Lemma 2.7 (see [34]).

Let be a closed and convex subset of a Hilbert space . Suppose that is a countable family of -strictly pseudocontractive mappings of into itself with . For each , define by

where is a family of nonnegative numbers satisfying

(i) for all ,

(ii) for all ,

(iii).

Then,

(1)Each is a -strictly pseudocontractive mapping.

(2) satisfies -condition.

(3)If is defined by

then and .

In the sequel, we will write satisfies the -condition if satisfies the -condition and is defined by Lemma 2.5 with .

## 3. Path Convergence Results

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping. Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let . For each , we denote the mapping by

where is the mapping defined as in Lemma 1.4. For each , we define the mapping as follows:

By Lemmas 1.4(2) and 2.4, we know that and are nonexpansive for each . So, the mapping is also nonexpansive for each . Moreover, we can check easily that is a contraction. Then, the Banach contraction principle ensures that there exists a unique fixed point of in , that is,

Theorem 3.1.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping. Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let . For each , let the mapping be defined by (3.1). Assume that . For each , let the net be generated by (3.3). Then, as , the net converges strongly to an element in .

Proof.

First, we show that is bounded. For each , let and . From (3.3), we have for each that

It follows that

Hence, is bounded and so are and . Observe that

as since is bounded.

Next, we show that as . Denote for any and . We note that for each . From Lemma 2.4, we have for each and that

It follows that

where . So, we have

which implies that

for each . Since is firmly nonexpansive for each , we have for each and that

This implies that

where . This shows that

Hence,

From (3.10), we obtain

as . So, we can conclude that

for each . Observing

it follows by (3.16) that

From (3.6) and (3.18), we have

Hence,

as .

Next, we show that is relatively norm compact as . Let be a sequence such that as . Put . From (3.20), we obtain

Since is bounded, without loss of generality, we may assume that converges weakly to . Applying Lemma 2.1 to (3.21), we can conclude that .

Next, we show that . Note that for each . Hence, for each and , we obtain

From (A2), we have

Therefore,

For each and , put . Then, we have . From (3.24), we get that

We note that , as , and is a family of monotone mappings. It follows from (3.25) that

So, by (A1), (A4) and (3.26), we have for each and that

This implies that

Letting in (3.28), it follows from (A3) that

Hence ; consequently, . Further, we see that

So, we have

In particular,

Since , we have as . By using the same argument as in the proof of Theorem 3.1 of [25], we can show that as . This completes the proof.

## 4. Strong Convergence Results

Theorem 4.1.

Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings and let be a countable family of -strict pseudocontractions for some such that . Assume that , , and for each satisfy the following conditions:

(i) and ,

(ii).

Suppose that satisfies the -condition. Then, generated by (1.14) converges strongly to an element in .

Proof.

For each , define by , . Then, , since . Moreover, we know that satisfies the -condition, since satisfies the -condition. By Lemma 2.5, we can define the mapping by for . Hence, , , since for . Further, we know that is nonexpansive for each . Indeed, for each and , we have

Hence, is nonexpansive for each and so is .

Next, we show that is bounded. Denote for any and . We note that . From (1.14), we have for each that

Hence, by induction, is bounded and so are and .

Next, we show that

Since and ,

Set , . So, we have from (1.14) and (4.4) that

Since satisfies the -condition and , it follows that

So, by Lemma 2.2 and (ii), we obtain

Hence,

Observe that

as . Similar to the proof of Theorem 3.1, we obtain for each that

for some and . Then, from (4.10), we have

which implies that

So, from (4.8), (i), (ii) and for each , we have

for each . Similarly, from (4.11), we have

This implies that

From (i), (ii), (4.8), and (4.14), it follows that

for each .

Next, we show that

Observing

it follows, by (4.17), that

From (4.9) and (4.20), we have

We see that

So, by (4.7), (4.21), and Lemma 2.5, we have

Let the net be defined by (3.3). By Theorem 3.1, we have as . Moreover, by proving in the same manner as in Theorem 3.2 of [25], we can show that

Finally, we show that as . From (1.14), we have

By (i) and (4.24), it follows from Lemma 2.3 that . This completes the proof.

As a direct consequence of Lemmas 2.6 and 2.7 and Theorem 4.1, we obtain the following result.

Theorem 4.2.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let be a sequence of -strict pseudocontractions of into itself such that and . Assume that and for each . Define the sequence by and

where and are real sequences in which satisfy (i)-(ii) of Theorem 4.1 and is a real sequence which satisfies (i)–(iii) of Lemma 2.7. Then, converges strongly to an element in .

Remark 4.3.

Theorems 4.1 and 4.2 extend the main results in [25] from a nonexpansive mapping to an infinite family of strict pseudocontractions and a system of generalized equilibrium problems.

Remark 4.4.

If we take and for each , then Theorems 3.1, 4.1, and 4.2 can be applied to a system of equilibrium problems and to a system of variational inequality problems, respectively.

Remark 4.5.

Let be an infinite family of nonexpansive mappings of into itself, and let be real numbers such that for all . Moreover, let and be the -mappings [35] generated by and and and . Then, we know from [7, 35] that satisfies the -condition. Therefore, in Theorem 4.1, the mapping can be also replaced by .

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## Acknowledgments

The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund. The first author is supported by the Royal Golden Jubilee Grant no. PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.

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### Keywords

- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space
- Common Fixed Point
- Convex Banach Space