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

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

It is well known that (1.5) is equivalent to

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

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

(1.10)

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

(1.11)

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:

(1.12)

Then, the following statements hold:

(1) is single-valued,

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

(1.13)

(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

(1.14)

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,

(2.1)

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

(2.2)

for all and .

In the sequel, we need the following lemmas.

Lemma 2.1 (see [27, 28]).

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

(2.3)

where satisfies conditions: . If  , then as .

Lemma 2.3 (see [30]).

Assume that is a sequence of nonnegative real numbers such that

(2.4)

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

(2.5)

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 ,

(2.6)

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

(2.7)

Then, for each bounded subset of ,

(2.8)

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

Lemma 2.6 (see [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

(2.9)

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

(2.10)

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

(3.1)

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

(3.2)

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,

(3.3)

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

(3.4)

It follows that

(3.5)

Hence, is bounded and so are and . Observe that

(3.6)

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

(3.7)

It follows that

(3.8)

where . So, we have

(3.9)

which implies that

(3.10)

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

(3.11)

This implies that

(3.12)

where . This shows that

(3.13)

Hence,

(3.14)

From (3.10), we obtain

(3.15)

as . So, we can conclude that

(3.16)

for each . Observing

(3.17)

it follows by (3.16) that

(3.18)

From (3.6) and (3.18), we have

(3.19)

Hence,

(3.20)

as .

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

(3.21)

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

(3.22)

From (A2), we have

(3.23)

Therefore,

(3.24)

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

(3.25)

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

(3.26)

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

(3.27)

This implies that

(3.28)

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

(3.29)

Hence ; consequently, . Further, we see that

(3.30)

So, we have

(3.31)

In particular,

(3.32)

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

(4.1)

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

(4.2)

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

Next, we show that

(4.3)

Since and ,

(4.4)

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

(4.5)

Since satisfies the -condition and , it follows that

(4.6)

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

(4.7)

Hence,

(4.8)

Observe that

(4.9)

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

(4.10)

(4.11)

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

(4.12)

which implies that

(4.13)

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

(4.14)

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

(4.15)

This implies that

(4.16)

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

(4.17)

for each .

Next, we show that

(4.18)

Observing

(4.19)

it follows, by (4.17), that

(4.20)

From (4.9) and (4.20), we have

(4.21)

We see that

(4.22)

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

(4.23)

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

(4.24)

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

(4.25)

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

(4.26)

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 .