Journal of Applied Mathematics

Volume 2012 (2012), Article ID 136134, 12 pages

doi:10.1155/2012/136134

Research Article

Strong Convergence Theorems for a Countable Family of Total Quasi--Asymptotically Nonexpansive Nonself Mappings

Liang-cai Zhao¹ and Shih-sen Chang²

¹College of Mathematics, Yibin University, Yibin 644000, China

²College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 3 June 2012; Revised 22 July 2012; Accepted 23 July 2012

Academic Editor: Naseer Shahzad

Copyright © 2012 Liang-cai 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 introduce a class of total quasi-ϕ-asymptotically nonexpansive-nonself mappings and to study the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results announced by some authors recently.

1. Introduction

Throughout this paper, we assume that is a real Banach space, is a nonempty closed and convex subset of , is the dual space of , and is the normalized duality mapping defined by

Recall that a Banach space is said to be strictly convex if for all with . is said to be uniformly convex, if for each , there exists such that for all with . is said to be smooth, if the limit

exists for all . And is said to be uniformly smooth, if the above limit is exists uniformly for .

In the sequel, we shall denote the fixed point set of a mapping by . When is a sequence in , then () will denote strong (weak) convergence of the sequence to .

A mapping is said to be nonexpansive, if

A mapping is said to be asymptotically nonexpansive if there exists a sequence such that

Recall that a subset of is said to be retract of , if there exists a continuous mapping such that , for all .

It is well known that every nonempty closed and convex subset of a uniformly convex Banach space is a retract of . A mapping is said to be a retraction, if . It follows that if a mapping is a retraction, then for all in the range of . A mapping is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from to .

In the sequel, we assume that is a smooth, strictly convex, and reflexive Banach space and is a nonempty closed convex subset of . Throughout this paper we assume that is the Lyapunov function which is defined by

It is obvious from the definition of that

Following Alber [1], the generalized projection is defined by

Lemma 1.1 (see [1]). Let be a smooth, strictly convex, and reflexive Banach space and be a nonempty closed convex subset of . Then the following conclusions hold:

(1) for all and ; (2)If and , then , for all ; (3)For if and only if .

Remark 1.2. If is a real Hilbert space , then and (the metric projection of onto ).

A mapping is said to be closed, if for any sequence with and , then .

Definition 1.3. Let be the nonexpansive retraction.

(1) is said to be quasi--nonexpansive nonself mapping, if and

(2) is said to be quasi--asymptotically nonexpansive nonself mapping, if and there exists a real sequence with such that

(3) is said to be total quasi--asymptotically nonexpansive nonself mapping, if and there exists nonnegative real sequence with as and a strictly increasing continuous function with such that for all

(4) A countable family of nonself mappings is said to be uniformly total quasi--asymptotically nonexpansive, if and there exists nonnegative real sequence with (as ) and a strictly increasing continuous function with such that for each and all

Remark 1.4. From the definitions, it is easy to know that

(1) If is a quasi--nonexpansive nonself mapping, then it must be a quasi--asymptotically nonexpansive nonself mapping with .(2) Taking , and , then (1.9) can be rewritten as

This implies that each quasi--asymptotically nonexpansive nonself mapping must be a total quasi--asymptotically nonexpansive nonself mapping, but the converse is not true.

A nonself mapping is said to be uniformly L-Lipschitz continuous, if there exists a constant such that

Lemma 1.5 (see [2]). Let be a smooth and uniformly convex Banach space and let , be two sequences of . If as and either or is bounded, then as ).

Lemma 1.6. Let be a smooth, strictly convex, and reflexive Banach space and be a nonempty closed and convex subset . Let be a closed and total quasi--asymptotically nonexpansive nonself mapping with nonnegative real sequence and a strictly increasing continuous function such that and . Then the fixed point set is a closed and convex subset of .

Proof. Let be a sequence in such that (as ). Since , by the closeness of , we have , that is, . This shows that is a closed set in .

Next, we prove that is convex. For any , putting , we prove that . Indeed, let be a sequence generated by

we have

Since

Substituting (1.16) into (1.15), and simplifying we have

By Lemma 1.5, we have . This implies that .

Since and is closed, we have . Since , , thus . this implies that is a convex set in .

Concerning the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi--nonexpansive and quasi--asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see e.g., [2–19]).

The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of of total quasi--asymptotically nonexpansive nonself mappings and to have the strong convergence under removing is a convex set of condition and a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results of Chang et al. [4–7], W. P. Guo and W. Guo [8], Hao et al. [9], Kamimura and Takahashi [10], Kiziltunc and Temir [11], Nilsrakoo and Saejung [2], Pathak et al. [12], Qin et al. [13], Su et al. [14], Thianwan [15], Wang et al. [16], Yıldırım and Özdemir [17], Yang and Xie [18], Zegeye et al. [19], Kanjanasamranwong et al. [20], Saewan and Kumam [21–24] and Wattanawitoon and Kumam [25].

2. Main Results

Theorem 2.1. Let be a real uniformly convex and uniformly smooth Banach space, and be a nonempty closed convex subset . Let be a family of closed and uniformly total quasi--asymptotically nonexpansive nonself mappings with nonnegative real sequence and a strictly increasing continuous function such that and , and for each , be uniformly -Lipschitz continuous. Let be a sequence in and be a sequence in satisfying the following conditions:

(a) ; (b).

Let be a sequence generated by

where , for all , . If is a nonempty-bounded subset in , then converges strongly to .

Proof. We divide the proof of Theorem 2.1 into five steps.

  and    are closed and convex subset in .

In fact, it follows from Lemma 1.6 that , is closed and convex subset of . Therefore is a closed and convex subset in .

Again by the assumption that is closed and convex. Suppose that is closed and convex for some . In view of the definition of we have that

This implies that is closed and convex. The conclusion is proved.

  Now we prove that  .

In fact, it is obvious that . Suppose that for some . Letting

it follows from (1.6) that for any we have

therefore we have

where . This shows that , and so . The conclusion is proved.

  Next we prove that    is a Cauchy sequence in .

In fact, since , from Lemma 1.1(2) we have

Again since for all , we have

It follows from Lemma 1.1(1) that for each and for each

Therefore is bounded. By virtue of (1.5), is also bounded.

Since and , we have for all . This implies that is nondecreasing. Hence the limit exists. By the construction of , for any positive integer , we have and . This shows that

It follows from Lemma 1.5 that . Hence is a Cauchy sequence in . Since is a nonempty closed subset of Banach space , it is complete, without loss of generality, we can assume that .

By the assumption, it is easy to see that

  Now we prove that  .

In fact, since and , it follows from (2.1) and (2.11) that

Since , by virtue of Lemma 1.5 for each ,we have

Since is bounded, is uniformly total quasi--asymptotically nonexpansive nonself mappings with nonnegative real sequence and a strictly increasing continuous function such that , and , for any given , we have

This implies that is uniformly bounded. Since

This implies that is also uniformly bounded.

Since , from (2.1), for each we have

Since is uniformly continuous on each bounded subset of , it follows from (2.13) and (2.16) that

Since is uniformly continuous on each bounded subset of , we have

By condition (b), we have that

Since is uniformly continuous, this shows that uniformly in .

Again by the assumptions that for each , is uniformly -Lipschitz continuous, thus we have

Since and , these together with (2.20) imply that and , that is,

In view continuity of , it yields that . Since . This shows that . By the arbitrariness of , we have .

  Finally we prove that  .

Let . Since and , we have , for all . This implies that

In view of the definition of , from (2.22) we have . Therefore . This completes the proof of Theorem 2.1.

Theorem 2.2. Let , , , be the same as in Theorem 2.1. Let be a family of closed and uniformly quasi--asymptotically nonexpansive nonself mappings with sequence , and for each , be uniformly -Lipschitz continuous. Let be a sequence generated by

where , . If is a nonempty bounded subset in , then converges strongly to .

Proof. By Remark 1.4 be a family of closed and uniformly quasi--asymptotically nonexpansive nonself mappings that it is a family of closed and uniformly total quasi--asymptotically nonexpansive nonself mappings with taking , and . Therefore all conditions in Theorem 2.1 are satisfied. By the similar methods as given in the proof of Theorem 2.1, we can prove that the sequence defined by (2.23) converges strongly to .

Theorem 2.3. Let , , , be the same as in Theorem 2.2. Let be a family of quasi--nonexpansive nonself mappings such that and for each , be uniformly -Lipschitz continuous. Let be a sequence generated by

Then converges strongly to .

Proof. By Remark 1.4 be a family of quasi--nonexpansive nonself mappings that it is a family of uniformly quasi--asymptotically nonexpansive nonself mappings with sequence . Hence Therefore all conditions in Theorem 2.2 are satisfied. By the similar methods, we can prove that the sequence defined by (2.24) converges strongly to .

3. Application and Example

In this section we utilize the results presented in Section 2 to prove a strong convergence theorem concerning maximal monotone operators in Hilbert spaces.

Let be a real Hilbert space and let be a maximal monotone operator from to . For each , we can define a single valued mapping by and such a mapping is called the resolvent of . It is easy to prove that is a nonexpansive mapping and for all . Therefore it is a uniformly -Lipschitz continuous and quasi--nonexpansive mapping. Hence for each and , we have

and . These show that all conditions in Theorem 2.3 are satisfied. Hence from Theorem 2.3 we have the following.

Theorem 3.1. Let be a real Hilbert space. Let be two maximal monotone operators from to such that . Let and be the resolvent of and , respectively, where . Let , be the same as in Theorem 2.3 and be the sequence defined by

where is the metric projection from onto the subset . Then the sequence defined by (3.2) converges strongly to .

Acknowledgments

The authors would like to express their thanks to the referees and the Editor for their helpful comments and suggestions. This Project was supported by the Scientific Research Fund of SiChuan Provincial Education Department (no. 11ZA222).

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