Strong Convergence Theorems for an Infinite Family of Equilibrium Problems and Fixed Point Problems for an Infinite Family of Asymptotically Strict Pseudocontractions

Abstract

We prove a strong convergence theorem for an infinite family of asymptotically strict pseudo-contractions and an infinite family of equilibrium problems in a Hilbert space. Our proof is simple and different from those of others, and the main results extend and improve those of many others.

1. Introduction

Let be a closed convex subset of a Hilbert space . Let be a mapping and if there exists an element such that , then is called a fixed point of . The set of fixed points of is denoted by . Recall that

(1) is called nonexpansive if

(1.1)

(2) is called asymptotically nonexpansive [1] if there exists a sequence with such that

(1.2)

(3) is called to be a -strict pseudo-contraction [2] if there exists a constant with such that

(1.3)

(4) is called an asymptotically-strict pseudo-contraction [3, 4] if there exists a constant with and a sequence with such that

(1.4)

It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudo-contraction and every -strict pseudo-contraction is an asymptotically -strict pseudo-contraction with for all . Moreover, every asymptotically -strict pseudo-contraction with sequence is uniformly -Lispchitzian, where and the fixed point set of asymptotically -strict pseudo-contraction is closed and convex; see [3, Proposition 2.6].

Let be a bifunction from to , where is the set of real numbers. The equilibrium problem for is to find such that for all . The set of such solutions is denoted by .

In 2007, S. Takahashi and W. Takahashi [5] first introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem which is connected with Combettes and Hirstoaga's result [6] and Wittmann's result [7]. More precisely, they gave the following theorem.

Theorem 1.1 (see [5]).

Let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the following assumptions:

(A1) for all ;

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

(A3)for all ,

(1.5)

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

Let be a nonexpansive mapping such that , be a contraction and , be the sequences generated by

(1.6)

where and satisfy the following conditions:

(1.7)

Then, the sequences and converge strongly to , where .

In [8], Tada and Takahashi proposed a hybrid algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem and proved the following strong convergence theorem.

Theorem 1.2 (see [8]).

Let be a nonempty closed convex subset of a Hilbert space . Let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping of into such that . Let and be sequences generated by and

(1.8)

where for some and satisfies . Then converges strongly to .

Many methods have been proposed to solve the equilibrium problems and fixed point problems; see [9–13].

Recently, Kim and Xu [3] proposed a hybrid algorithm for finding a fixed point of an asymptotically -strict pseudo-contraction and proved a strong convergence theorem in a Hilbert space.

Theorem 1.3 (see [3]).

Let be a closed convex subset of a Hilbert space . Let be an asymptotically -strict pseudo-contraction for some . Assume that is nonempty and bounded. Let be the sequence generated by the following algorithm:

(1.9)

where

(1.10)

Assume that the control sequence is chosen such that . Then converges strongly to .

In this paper, motivated by [3, 8], we propose a new algorithm for finding a common element of the set of fixed points of an infinite family of asymptotically strict pseudo-contractions and the set of solutions of an infinite family of equilibrium problems and prove a strong convergence theorem. Our proof is simple and different from those of others, and the main results extend and improve those Kim and Xu [3], Tada and Takahashi [8], and many others.

2. Preliminaries

Let be a Hilbert space, and let be a nonempty closed convex subset of . It is well known that, for all and ,

(2.1)

and hence

(2.2)

which implies that

(2.3)

for all and with .

For any , there exists a unique nearest point in , denoted by , such that

(2.4)

Let denote the identity operator of , and let be a sequence in a Hilbert space and . Throughout the rest of the paper, denotes the strong convergence of to .

We need the following lemmas for our main results in this paper.

Lemma 2.1 (see [14]).

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

(2.5)

Lemma 2.2 (see [6]).

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

(2.6)

Then the following hold:

(1) is single-valued,

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

(2.7)

(3), and

(4) is closed and convex.

3. Main Results

Now, we are ready to give our main results.

Lemma 3.1.

Let be a nonempty closed convex subset of a Hilbert space . Let be an asymptotically -strict pseudo-contraction with sequence such that . Assume that and define a mapping for each . Then the following hold:

(3.1)

Proof.

For all , we have

(3.2)

By this result, for all and , we have

(3.3)

and hence

(3.4)

This completes the proof.

Lemma 3.2.

Let be a nonempty closed subset of a Hilbert space . Let be an asymptotically -strict pseudo-contraction with sequence satisfying as . Let be a sequence in such that and as . Then as .

Proof.

The proof method of this lemma is mainly from [15, Lemma 2.7]. Since is an asymptotically -strict pseudo-contraction, we obtain from [3, Proposition 2.6] that

(3.5)

where . Note that , which implies that , and observe that

(3.6)

Since is uniformly Lipschitzian, is uniformly continuous. So we have

(3.7)

It follows from and as that . This completes the proof.

Let be a Hilbert space, and, let be a nonempty closed and convex subset of . Let be a countable family of bifunctions from to satisfying (A1)–(A4) and let be a real number sequence in with . Define

(3.8)

Lemma 2.2 shows that every () is a firmly nonexpansive mapping and hence nonexpansive and .

Theorem 3.3.

Let be a nonempty closed convex subset of a Hilbert space . Let be an infinite family of asymptotically -strict pseudocontractions with the sequence satisfying as for each and for each and . Let be a countable family of bifunctions from to satisfying (A1)–(A4). Assume that is nonempty and bounded. Set and . Assume that is a strictly decreasing sequence in for some , is a strictly decreasing sequence in , is a sequence in with for each , and is a sequence in with . The sequence is generated by and

(3.9)

where is defined by (3.8) and

(3.10)

Then converges strongly to .

Proof.

We show first that the sequence is well defined. Obviously, is closed for all . Since

(3.11)

is equivalent to

(3.12)

is convex for all . So is also closed and convex for all .

For each and , put . Let . Note that , is strictly decreasing and each is firmly nonexpansive. Hence we have

(3.13)

Since and is strictly decreasing, by (3.13) and Lemma 3.1, we have

(3.14)

So we have and hence for all . This shows that for all . This implies that the sequence is well defined.

Since is a nonempty closed convex subset of , there exists a unique such that

(3.15)

From , we have

(3.16)

Since , we have

(3.17)

Therefore, is bounded. From (3.13) and (3.14), and are also bounded.

From and , one sees that for all . It follows that

(3.18)

Since is bounded, the sequence is bounded and nondecreasing. So there exists such that

(3.19)

Since , and , we have

(3.20)

So we get

(3.21)

Since , we obtain

(3.22)

that is,

(3.23)

Now, for each , from (3.23) we get

(3.24)

This implies that there exists an element such that as .

Next we show that and .

From , we have

(3.25)

By (3.10) and (3.23), we obtain

(3.26)

For , we have, from Lemma 2.2,

(3.27)

and hence

(3.28)

Therefore

(3.29)

By (3.29) and Lemma 3.1, we have

(3.30)

and hence

(3.31)

This shows that

(3.32)

Since with , , is strictly decreasing and , we get

(3.33)

Let for each . Then as . Hence, from (3.33), one has

(3.34)

From (3.26) and (3.34), we obtain

(3.35)

Noting that

(3.36)

we have

(3.37)

By Lemma 3.1, we have

(3.38)

Therefore, combining this inequality with (3.37), we get

(3.39)

and hence (noting that for each )

(3.40)

From (3.26), (3.35) and we have

(3.41)

From the definition of and (3.41), we have (noting that )

(3.42)

We next show (3.42) implies that

(3.43)

As a matter of fact, from (3.23) and (3.34) we have

(3.44)

Now, (3.42), (3.44), and Lemma 3.2 imply (3.43).

Since each is uniformly continuous and as , one get for each and hence .

Now we show .

Since every is nonexpansive, from (3.33) and , we have and hence . Lemma 2.2 shows that .

Finally, we prove that . From , one sees

(3.45)

Since for all , one arrives at

(3.46)

Taking the limit for above inequality, we get

(3.47)

Hence . This completes the proof.

As direct consequences of Theorem 3.3, we can obtain the following corollaries.

Corollary 3.4.

Let be a nonempty closed convex subset of a Hilbert space . Let be a countable family of bifunctions from to satisfying (A1)–(A4). Assume that is nonempty and bounded. Let be a sequence in with . Set . The sequence is generated by and

(3.48)

where is defined by (3.8) and is a strictly decreasing sequence in . Then converges strongly to .

Proof.

Putting for all and for all in Theorem 3.3, we obtain Corollary 3.4.

Corollary 3.5.

Let be a nonempty closed subset of a Hilbert space . Let be an asymptotically -strict pseudo-contraction with sequence satisfying as and . Let and be sequences generated by and

(3.49)

where , with , and with . Then converges strongly to .

Proof.

Put for all and set for all in Theorem 3.3. By Lemma 2.2, we have for each . Hence, by Theorem 3.3, we obtain Corollary 3.5.

Remark 3.6.

Our algorithms are of interest because the sequence in Theorem 3.3 is very different from the known manner. The proof is simple and different from those of others. The main results extend and improve those of Kim and Xu [3], Tada and Takahashi [8], and many others.

Remark 3.7.

Put , , , , , , , , for all and all , , and . Then these control sequences satisfy all the conditions of Theorem 3.3.