Abstract and Applied Analysis

Volume 2013 (2013), Article ID 169206, 6 pages

http://dx.doi.org/10.1155/2013/169206

Research Article

Weak and Strong Convergence Theorems for Strictly Pseudononspreading Mappings and Equilibrium Problem in Hilbert Spaces

Yun He Zhao and Shih-sen Chang

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 14 June 2013; Revised 10 August 2013; Accepted 24 August 2013

Academic Editor: Abdelghani Bellouquid

Copyright © 2013 Yun He Zhao and Shih-sen Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to propose an iterative algorithm for equilibrium problem and a class of strictly pseudononspreading mappings which is more general than the class of nonspreading mappings studied recently in Kurokawa and Takahashi (2010). We explored an auxiliary mapping in our theorems and proofs and under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve some recent results announced by some authors.

1. Introduction and Preliminaries

Throughout this paper, we assume that is a real Hilbert space and is a nonempty and closed convex subset of . In the sequel, we denote by “” and “” the strong and weak convergence of , respectively. Denote by the set of fixed points of a mapping .

Definition 1. Let be a mapping.

(1) is said to be nonexpansive, if , .(2) is said to be quasinonexpansive, if is nonempty and

(3) is said to be nonspreading [1, 2], if

It is easy to prove that is nonspreading if and only if

(4) is said to be -strictly pseudononspreading in the terminology of Browder-Petryshyn [3], if there exists such that

Remark 2. (1) If is a nonspreading mapping with , then is quasinonexpansive and is closed and convex.

(2) Clearly every nonspreading mapping is -strictly pseudononspreading with , but the inverse is not true. This can be seen from the following example.

Example 3. Let denote the set of all real numbers. Let be a mapping defined by

It is easy to see that is a -strictly pseudononspreading mapping with , but it is not nonspreading (see, [4]).

Definition 4. (1) Let be a mapping. is said to be demiclosed at 0, if for any sequence with and , we have .

(2) A Banach space is said to have Opial's property, if for any sequence with , we have

It is well known that each Hilbert space processes opial property.

(3) A mapping is said to be semicompact, if for any bounded sequence with , then there exists a subsequence such that converges strongly to some point .

Lemma 5 (see [5]). Let be a uniformly convex Banach space and let be a closed ball with center and radius . For any given sequence and any given number sequence with , , there exists a continuous strictly increasing and convex function with such that for any , the following holds:

Lemma 6. Let be a real Hilbert space, be a nonempty and closed convex subset of , and let be a -strictly pseudononspreading mapping.

(i) If , then it is closed and convex.(ii) is demiclosed at origin.

Lemma 7. Let be a -strictly pseudononspreading mapping with . Denote by , where , then

(i); (ii) the following inequality holds:

(iii) is a quasinonexpansive mapping, that is,

Proof. The conclusion (i) is obvious. Now we prove the conclusion (ii). Since is -strictly pseudononspreading, for any we have

Take in (8), then . Hence, conclusion (iii) is proved.

This completes the proof.

In the sequel, we assume that is a bifunction satisfying the following conditions:

(A1);(A2) is monotone, that is, ;(A3);(A4) for each , is convex and lower semicontinuous.

Recalled that the “so-called” equilibrium problem for a bifunction function   is to find a point , such that

Lemma 8 (see [6, 7]). Let be a nonempty and closed convex subset of a Hilbert space and let be a bi-function satisfying conditions: (A1), (A2), (A3), and (A4). Then, for any and , there exists such that

Furthermore, if for given , we define a mapping by

then the following hold:

(1) is single-valued;(2) is firmly nonexpansive, that is, ;(3), where is the set of solutions of the equilibrium problem (11);(4) is a closed and convex subset of .

Concerning the weak and strong convergence problem for some kinds of iterative algorithms for nonspreading mappings, -strictly pseudononspreading mappings and other kind of nonlinear mappings have been considered in Osilike and Isiogugu [4], Igarashi et al. [8], Iemoto and Takahashi [9], Kurokawa and Takahashi [10], and Kim [11–28]. The purpose of this paper is to propose an iterative algorithm for an infinite family of strictly pseudononspreading mappings and equilibrium problem. Under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve the corresponding results in [4, 8–11].

2. Main Results

Throughout this section, we assume that the following conditions are satisfied.

(1) is a real Hilbert spaces, is a nonempty and close convex subset of . (2) For each , is a -strictly pseudononspreading mapping with . For given , denoted by , for each , it follows from (8) that

(3) is a bifunction satisfying the conditions (A1)–(A4). Then it follows from Lemma 8 that the mapping defined by (13) is single valued, , (where is the solution set of the equilibrium problem (11)), and is a closed and convex subset of .

We are now in a position to give the following result.

Theorem 9. Let , , , , , , and be the same as above. Let and be the sequences defined by

where and satisfy the following conditions:

(a), for each ; (b) for each ,  ; (c) and . (I) If , then both and converge weakly to some point ; (II) in addition, if there exists some positive integer such that is semicompact, then both and converge strongly to .

Proof. First, we prove the conclusion (I). The proof is divided into three steps.

Step  1. We prove that the sequences , , , and all are bounded, and for each the limits , exist and

In fact, it follows from Lemma 8 that , , and

Since , by Lemma 7(i), . Hence, it follows from (17) and (9) that

This implies that for each , the limits and exist. And so and are bounded and (16) holds.

Furthermore, by (9), it is easy to see that for each , and are also bounded.

Step  2. Next we prove that for each the following holds:

In fact, by Lemma 5 for any positive integer and , we have

This shows that

Since is a continuous and strictly increasing function with . By condition (b), it yields that

Therefore, we have

On the other hand, it follows from Lemma 8 that and for each

This shows that

In view of (20) and (25)

that is,

In view of (27), (22), (14), and noting that is bounded, we have

Therefore, we have

The conclusion is proved.

Step  3. Next we prove that the weak-accumulation point set of the sequence is a singleton and .

In fact, for any , their exists a subsequence such that . It follows from (27) that . Since , from (15) and condition (A2) we have

Since and , it follows from condition (A4) that

For any , letting , then . By condition (A1) and (A4), we have

This implies that . Letting , by condition (A3) we have

This shows that is a solution to the equilibrium (11), that is, .

On the other hand, by Lemma 6, for each , is demiclosed at . In view of (19), we know that . Due to the arbitrariness of , we have .

Now we prove that is a singleton. Suppose to the contrary that there exist with . Therefore, there exist subsequences and in such that and . Since , by (16), the limits and exist. By using the opial property of , we have

This is a contradiction. Therefore, is a singleton. Without loss of generality, we can assume that and . By using (15) and (19), we have .

This completes the proof of the conclusion (I).

Next we prove the conclusion (II).

Without loss of generality, we can assume that is semicompact. From (19) we have that

Therefore, there exists a subsequence of such that . Since , we have and so . By virtue of (16), we have

This completes the proof of Theorem 9.

Taking and in Theorem 9, we have , Therefore, the following theorem can be obtained from Theorem 9 immediately.

Theorem 10. Let , , and be the same as in Theorem 9. Let be the sequences defined by

where satisfies the following conditions:

(a), for each ;(b) for each ,  .(I) If , then both converge weakly to some point ;(II) in addition, if there exists some positive integer such that is semicompact, then converge strongly to .

Remark 11. Theorems 9 and 10 improve and extend the corresponding recent results of [4, 8–11].

Acknowledgments

The authors would like to express their thanks to the referees for their helpful comments and advices. This work was supported by the Natural Science Foundation of Yunnan Province (Grant no. 2011FB074).

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