Convergence Theorems Concerning Hybrid Methods for Strict Pseudocontractions and Systems of Equilibrium Problems

Abstract

Let be strict pseudo-contractions defined on a closed and convex subset of a real Hilbert space . We consider the problem of finding a common element of fixed point set of these mappings and the solution set of a system of equilibrium problems by parallel and cyclic algorithms. In this paper, new iterative schemes are proposed for solving this problem. Furthermore, we prove that these schemes converge strongly by hybrid methods. The results presented in this paper improve and extend some well-known results in the literature.

1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty, closed, and convex subset of .

Let be a countable family of bifunctions from to , where is the set of real numbers. Combettes and Hirstoaga [1] considered the following system of equilibrium problems:

(1.1)

where is an arbitrary index set. If is a singleton, then problem (1.1) becomes the following equilibrium problem:

(1.2)

The solution set of (1.2) is denoted by .

A mapping of is said to be a -strict pseudocontraction if there exists a constant such that

(1.3)

for all ; see [2]. We denote the fixed point set of by , that is, .

Note that the class of strict pseudocontractions properly includes the class of nonexpansive mappings which are mapping on such that

(1.4)

for all . That is, is nonexpansive if and only if is a 0-strict pseudocontraction.

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance, [1, 3, 4] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.1); related work can also be found in [5–8].

Recently, Acedo and Xu [9] considered the problem of finding a common fixed point of a finite family of strict pseudo-contractive mappings by the parallel and cyclic algorithms. Very recently, Duan and Zhao [10] considered new hybrid methods for equilibrium problems and strict pseudocontractions. In this paper, motivated by [5, 8–12], applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the fixed point set of a finite family of strict pseudocontractions and the solution set of the system of equilibrium problems (1.1) by the hybrid methods.

We will use the following notations:

(1) for the weak convergence and for the strong convergence,

(2) denotes the weak -limit set of .

2. Preliminaries

We will use the facts and tools in a real Hilbert space which are listed below.

Lemma 2.1.

Let be a real Hilbert space. Then the following identities hold:

(i)

(ii)

Lemma 2.2 (see [6]).

Let be a real Hilbert space. Given a nonempty, closed, and convex subset , points , and a real number , then the set

(2.1)

is convex (and closed).

Recall that given a nonempty, closed, and convex subset of a real Hilbert space , for any , there exists the unique nearest point in, denoted by , such that

(2.2)

for all . Such a is called the metric (or the nearest point) projection of onto . As we all know if and only if there holds the relation

(2.3)

Lemma 2.3 (see [13]).

Let be a nonempty, closed, and convex subset of . Let be a sequence in and . Let . Suppose that is such that and satisfies the following condition:

(2.4)

Then .

Proposition 2.4 (see [9]).

Let be a nonempty, closed, and convex subset of a real Hilbert space .

(i)If is a -strict pseudocontraction, then satisfies the Lipschitz condition

(2.5)

(ii)If is a -strict pseudocontraction, then the mapping is demiclosed (at 0). That is, if is a sequence in such that and , then .

(iii)If is a -strict pseudocontraction, then the fixed point set of is closed and convex. Therefore the projection is well defined.

(iv)Given an integer , assume that, for each , is a -strict pseudocontraction for some . Assume that is a positive sequence such that . Then is a -strict pseudocontraction, with

(v)Let and be given as in item (iv). Suppose that has a common fixed point. Then

(2.6)

Lemma 2.5 (see [2]).

Let be a -strict pseudocontraction. Define by for any . Then, for any , is a nonexpansive mapping with .

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

(A1) for all

(A2) is monotone, that is, for any

(A3)for each

(A4) is convex and lower semicontinuous for each

Lemma 2.6 (see [3]).

Let be a nonempty, closed, and convex subset of , let be bifunction from to which satisfies conditions (A1)–(A4), and let and . Then there exists such that

(2.7)

Lemma 2.7 (see [1]).

For , define the mapping as follows:

(2.8)

for all . Then, the following statements hold:

(i) is single valued;

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

(2.9)

(iii);

(iv) is closed and convex.

3. Parallel Algorithm

In this section, we apply the hybrid methods to the parallel algorithm for finding a common element of the fixed point set of strict pseudocontractions and the solution set of the problem (1.1) in Hilbert spaces.

Theorem 3.1.

Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be bifunctions from to which satisfies conditions (A1)–(A4). Let, for each , be a -strict pseudocontraction for some . Let Assume that . Assume also that is a finite sequence of positive numbers such that for all and for all . Let the mapping be defined by

(3.1)

Given , let , and , be sequences which are generated by the following algorithm:

(3.2)

where for some , for some , and satisfies for all . Then, converge strongly to .

Proof.

Denote for every and for all . Therefore . The proof is divided into six steps.

Step 1.

The sequence is well defined.

It is obvious that is closed and is closed and convex for every . From Lemma 2.2, we also get that is convex.

Take , since for each , is nonexpansive, , and , we have

(3.3)

for all . From Proposition 2.4, Lemma 2.5, and (3.3), we get

(3.4)

So for all . Thus Next we will show by induction that for all . For , we have . Assume that for some . Since , we obtain

(3.5)

As by induction assumption, the inequality holds, in particular, for all . This together with the definition of implies that . Hence holds for all Thus , and therefore the sequence is well defined.

Step 2.

(3.6)

From the definition of we imply that . This together with the fact that further implies that

(3.7)

Then is bounded and (3.6) holds. From (3.3), (3.4), and Proposition 2.4 (i), we also obtain that , ,and are bounded.

Step 3.

The following limit holds:

(3.8)

From and , we get . This together with Lemma 2.1 (i) implies that

(3.9)

Then , that is, the sequence is nondecreasing. Since is bounded, exists. Then (3.8) holds.

Step 4.

The following limit holds:

(3.10)

From , we have

(3.11)

By (3.6), we obtain

(3.12)

Next we will show that

(3.13)

Indeed, for , it follows from the firm nonexpansivity of that for each we have

(3.14)

Thus we get

(3.15)

which implies that, for each

(3.16)

Therefore, by the convexity of and Lemma 2.5, we get

(3.17)

It follows that

(3.18)

Since , we get from (3.12) that (3.13) holds; then we have

(3.19)

Observe that we also have as . On the other hand, from , we observe that

(3.20)

From , (3.19), and , we obtain as . It is easy to see that

(3.21)

Combining the above arguments and (3.2), we have

(3.22)

Now, it follows from that as .

Step 5.

The following implication holds:

(3.23)

We first show that . To this end, we take and assume that as for some subsequence of .Without loss of generality, we may assume that

(3.24)

It is easily seen that each and . We also have

(3.25)

where Note that, by Proposition 2.4, is a -strict pseudocontraction and . Since

(3.26)

we obtain by virtue of (3.10) and (3.24) that

(3.27)

So by the demiclosedness principle (Proposition 2.4 (ii)), it follows that , and hence .

Next we will show that . Indeed, by Lemma 2.6, we have that, for each

(3.28)

From (A2), we get

(3.29)

Hence,

(3.30)

From (3.13), we obtain that as for each (especially, ). Together with (3.13) and (A4) we have, for each that

(3.31)

For any and , let . Since and , we obtain that , and hence . So, we have

(3.32)

Dividing by t, we get, for each that

(3.33)

Letting and from (A3), we get

(3.34)

for all and for each that is, Hence (3.23) holds.

Step 6.

From (3.6) and Lemma 2.3, we conclude that , where .

Remark 3.2.

In 2007, Acedo and Xu studied the following CQ method [9]:

(3.35)

In this paper, we first turn the strict pseudocontraction into nonexpansive mapping then replace with a more simple form in the iterative algorithm.

Remark 3.3.

If , , and , we can obtain [14, Theorem ].

Remark 3.4.

If , , , and and we use to replace , we can get the result that has been studied by Tada and Takahashi in [8] for nonexpansive mappings. If , , , and , we can get [7, Theorem ].

Theorem 3.5.

Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be bifunctions from to which satisfies conditions (A1)–(A4). Let, for each , be a -strict pseudocontraction for some . Let Assume that . Assume also that is a finite sequence of positive numbers such that for all and for all . Let the mapping be defined by

(3.36)

Given , let {}, {}, and {} be sequences which are generated by the following algorithm:

(3.37)

where for some , for some , and, satisfies for all . Then, converge strongly to .

Proof.

The proof of this theorem is similar to that of Theorem 3.1.

Step 1.

The sequence is well defined. We will show by induction that is closed and convex for all . For , we have which is closed and convex. Assume that for some is closed and convex; from Lemma 2.2, we have that is also closed and convex; The proof of is similar to the one in Step 1 of Theorem 3.1.

Step 2.

for all , where

Step 3.

as

Step 4.

as

Step 5.

Step 6.

.

The proof of Step 2–Step 6 is similar to that of Theorem 3.1.

Remark 3.6.

If , we can obtain the two corresponding theorems in [10].

4. Cyclic Algorithm

Let be a closed, and convex subset of a Hilbert space , and let be -strict pseudocontractions on such that the common fixed point set

(4.1)

Let and let be a sequence in . The cyclic algorithm generates a sequence in the following way:

(4.2)

In general, is defined by

(4.3)

where , with .

Theorem 4.1.

Let be a nonempty, closed, and convex subset of a real Hilbert space and let be bifunctions from to which satisfies conditions (A1)–(A4). Let, for each , be a -strict pseudocontraction for some . Let Assume that . Given , let {}, {}, and {} be sequences which are generated by the following algorithm:

(4.4)

where for some , for some , and satisfies for all . Then, converge strongly to .

Proof.

The proof of this theorem is similar to that of Theorem 3.1. The main points are the following.

Step 1.

The sequence is well defined.

Step 2.

for all , where

Step 3.

Step 4.

To prove the above steps, one simply replaces with in the proof of Theorem 3.1.

Step 5.

Indeed, let and for some subsequence of . We may assume that for all . Since, by , we also have for all , we deduce that

(4.5)

Then the demiclosedness principle implies that for all . This ensures that .The Proof of is similar to that of Theorem 3.1.

Step 6.

The sequence converges strongly to

The strong convergence to of is a consequence of Step 2, Step 5, and Lemma 2.3.

Theorem 4.2.

Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be bifunctions from to which satisfies conditions (A1)–(A4). Let, for each , be a -strict pseudocontraction for some . Let Assume that . Given , let {}, {}, and {} be sequences whic are generated by the following algorithm:

(4.6)

where for some , for some , and satisfies for all . Then, converge strongly to .

Proof.

The proof of this theorem is similar to that of Step 1 in Theorem 3.5 and Step 2–Step 6 in Theorem 4.1.

Remark 4.3.

If , we can obtain the two corresponding theorems in [10].