Strong Convergence Theorems by Hybrid Methods for Strict Pseudocontractions and Equilibrium Problems

Abstract

Let be N strict pseudocontractions defined on a closed convex subset of a real Hilbert space . Consider the problem of finding a common element of the set of fixed point of these mappings and the set of solutions of an equilibrium problem with the parallel and cyclic algorithms. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly by hybrid methods.

1. Introduction

Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction from to , where is the set of real numbers.

The equilibrium problem for is to find such that

(1.1)

for all . The set of such solutions is denoted by .

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

(1.2)

for all ; see [1]. We denote the set of fixed points of by (i.e., ).

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

(1.3)

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

Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem (1.1); see for instance [2–5]. In particular, Combettes and Hirstoaga [6] proposed several methods for solving the equilibrium problem. On the other hand, Mann [7], Nakajo and Takahashi [8] considered iterative schemes for finding a fixed point of a nonexpansive mapping.

Recently, Acedo and Xu [9] considered the problem of finding a common fixed point of a finite family of strict pseudocontractive mappings by the parallel and cyclic algorithms. Very recently, Liu [3] considered a general iterative method for equilibrium problems and strict pseudocontractions. In this paper, motivated by [3, 5, 9–12], applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the set of fixed points of a finite family of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1) by the hybrid methods.

We will use the notation

(1) for weak convergence and for strong convergence,

(2) denotes the weak -limit set of .

2. Preliminaries

We need some facts and tools in a real Hilbert space which are listed as below.

Lemma 2.1.

Let be a real Hilbert space. There hold the following identities.

(i), for all

(ii), for all , for all

Lemma 2.2 (see [4]).

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

(2.1)

is convex (and closed).

Recall that given a nonempty closed convex subset of a real Hilbert space , for any , there exists a 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 .

Lemma 2.3 (see [4]).

Let be a nonempty closed convex subset of a real Hilbert space . Given and , then if and only if there holds the relation

(2.3)

Lemma 2.4 (see [13]).

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

(2.4)

Then .

Lemma 2.5 (see [9]).

Let be a nonempty closed convex subset of . Let is a sequence in and . Assume

(i)the weak -limit set ,

(ii)for each exists.

Then is weakly convergent to a point in .

Proposition 2.6 (see [9]).

Assume be a nonempty closed 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 of is closed and convex so that the projection is well defined.

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

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

(2.6)

Lemma 2.7 (see [1]).

Let be a -strict pseudocontraction. Define by for each . Then, as is a nonexpansive mapping such that .

For solving the equilibrium problem, let us 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 semicontionuous for each

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.8 (see [14]).

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

(2.7)

Lemma 2.9 (see [6]).

For , define a 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 set of fixed points of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1) in Hilbert spaces.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be an integer. Let, for each be a -strict pseudocontraction for some . Let Assume the set . Assume also 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 generated by the following algorithm:

(3.2)

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

Proof.

The proof is divided into several steps.

Step 1.

Show first that is well defined.

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

Step 2.

Show for all .

Indeed, take , from we have

(3.3)

for all . From Proposition 2.6, Lemma 2.7, and (3.3), we get

(3.4)

So for all . Next we show that for all by induction. 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 .

Step 3.

Show that

(3.6)

Notice that the definition of actually . This together with the fact further implies

(3.7)

Then is bounded and (3.6) holds. From (3.3), (3.4), and Proposition 2.6(i), we also obtain and are bounded.

Step 4.

Show that

(3.8)

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

(3.9)

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

Step 5.

Show that

(3.10)

From , we have

(3.11)

By (3.8), we obtain

(3.12)

For , we have

(3.13)

hence,

(3.14)

Therefore, by the convexity of , we get

(3.15)

Since , we get

(3.16)

It follows that

(3.17)

from (3.12). Observe that we also have . On the other hand, from , we compute

(3.18)

From , (3.17), and , we obtain . It is easy to get

(3.19)

Combining the above results, we obtain From (3.2), we have

(3.20)

It follows from that

Step 6.

Show that

(3.21)

We first show . To see this, we take and assume that as for some subsequence of .

Without loss of generality, we may assume that

(3.22)

It is easily seen that each and . We also have

(3.23)

where Note that by Proposition 2.6, is -strict pseudocontraction and . Since

(3.24)

we obtain by virtue of (3.10) and (3.22)

(3.25)

So by the demiclosedness principle (Proposition 2.6(ii)), it follows that and hence holds.

Next we show take , and assume that as for some subsequence of . From (3.17), we obtain . Since and is closed convex, we get

By we have

(3.26)

From the monotonicity of , we get

(3.27)

hence

(3.28)

From (3.17) and condition (A4), we have

(3.29)

For with and , let . Since and , we obtain and hence . So, we have

(3.30)

Dividing by , we get

(3.31)

Letting and from (A3), we get

(3.32)

for all and . Hence (3.21) holds.

Step 7.

From (3.6) and Lemma 2.4, we conclude that , where .

A very similar result obtained in a way completely different is Theorem of [11].

Theorem 3.2.

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

(3.33)

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

(3.34)

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

Proof.

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

Step 1.

is well defined for all

We show is closed convex for all by induction. For , we have is closed convex. Assume that for some is closed convex, from Lemma 2.2, we have is also closed convex. The assumption holds.

Step 2.

.

Step 3.

for all , where

Step 4.

Step 5.

Step 6.

Step 7.

.

The proof of Steps 2–7 is similar to that of Theorem 3.1.

A very similar result obtained in a way completely different is Theorem of [10].

4. Cyclic Algorithm

Let be a closed 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 convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be an integer. Let, for each be a -strict pseudocontraction for some . Let Assume the set . Given , let , and be sequences generated by the following algorithm:

(4.4)

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

Proof.

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

Step 1.

is well defined for all

Step 2.

.

Step 3.

for all , where

Step 4.

Step 5.

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

Step 6.

Show that

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

(4.5)

Then the demiclosedness principle (Proposition 2.6(ii)) implies that for all . This ensures that .

The proof of is similar to that of Theorem 3.1.

Step 7.

Show that

The strong convergence to of is the consequence of Step 3, Step 5, and Lemma 2.4.

Theorem 4.2.

Let be a nonempty closed convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be an integer. Let, for each be a -strict pseudocontraction for some . Let Assume the set . Given , let , and be sequences generated by the following algorithm:

(4.6)

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

Proof.

The proof of this theorem can consult Step 1 of Theorem 3.2 and Steps 2–7 of Theorem 4.1.