Abstract

Weak and strong convergence theorems are established for hybrid implicit iteration for a finite family of non-self-nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper extend and improve some recent results.

1. Introduction

The convergence problem of an implicit (or nonimplicit) iterative process to a common fixed point for a finite family of nonexpansive mappings (or asymptotically nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces has been considered by many authors (see [19]).

In 2001, Xu and Ori [1] introduced the following implicit iteration scheme for common fixed points of a finite family of nonexpansive mappings in Hilbert spaces: where , and they proved the weak convergence theorem.

In 2005, Zeng and Yao [2] introduced the following implicit iteration process with a perturbed mapping in Hilbert space .

For an arbitrary initial point , the sequence is generated as follows: where .

Using this iteration process, they proved the following weak and strong convergence theorems for a family of nonexpansive mappings in Hilbert spaces.

Theorem 1 (see [2]). Let be a real Hilbert space and let be a mapping such that, for some constants , is -Lipschitzian and -strongly monotone. Let be nonexpansive self-mappings of such that . Let and . Let and satisfying conditions and , , for some . Then the sequence defined by (2) converges weakly to a common fixed point of the mappings .

Theorem 2 (see [2]). Let be a real Hilbert space and let be a mapping such that, for some constants , is -Lipschitzian and -strongly monotone. Let be nonexpansive self-mappings of such that . Let and . Let and satisfying conditions and , , for some . Then the sequence defined by (2) converges strongly to a common fixed point of the mappings if and only if .

The purpose of this paper is to extend Theorems 1 and 2 from Hilbert spaces to uniformly convex Banach spaces and from self-mappings to non-self-mappings. Our results are more general and applicable than the results of Zeng and Yao [2] because the strong monotonicity condition imposed on by them is not required in this paper.

2. Preliminaries

Throughout this paper, we assume that is a real Banach space. is a mapping, where is the domain of . denotes the set of fixed points of a mapping .

Recall that is said to satisfy Opial’s condition [10], if, for each sequence in , the condition that the sequence weakly implies that for all with .

Definition 3. Let be a closed subset of and let , be two mappings.(1)is said to be demiclosed at the origin, if, for each sequence in , the condition weakly and strongly implies .(2) is said to be semicompact, if, for any bounded sequence in , such that , there exists a subsequence converging to some in .(3) is said to be nonexpansive, if for any .(4) is said to be -Lipschitzian if there exists constant such that for any .

Definition 4. A nonempty subset of is said to be a retract of , if there exists a continuous mapping such that , for any . And is called the retraction of onto .

Remark 5 (see [3]). It is known that every nonempty closed convex subset of a uniformly convex Banach space is a retract of and the retraction is a nonexpansive mapping.
Suppose that is a nonempty closed convex subset of , which is also a retract of . Let be any given point. Let be nonexpansive mappings with . Let be an -Lipschitzian mapping. Assume that is a sequence in (0,1) and , given . Then the sequence defined by is called hybrid implicit iteration for a finite family of nonexpansive mappings in Banach spaces, where and is a fixed constant.

The purpose of this paper is to study weak and strong convergence of hybrid implicit iteration defined by (4) to a common fixed point of in Banach spaces. The results we obtained in this paper extend and improve the corresponding results of Xu and Ori [1], Zeng and Yao [2], and others.

In order to prove our main results of this paper, we need the following lemmas.

Lemma 6 (see [4]). Let , , and be three nonnegative sequences satisfying If and , then exists.

Lemma 7 (see [5]). Let be a uniformly convex Banach space. Let , be two constants with . Suppose that is a sequence in [b, c] and , are two sequences in . Then the conditions imply that , where is some constant.

Lemma 8 (see [6]). Let be a nonempty closed convex subset of real Banach space and a nonexpansive mapping. If has a fixed point, then is demiclosed at zero, where is the identity mapping of .

3. Main Results

Theorem 9. Suppose that is a real uniformly convex Banach space satisfying Opial’s condition and is a nonempty closed convex subset of with a nonexpansive retraction . Let be nonexpansive mappings with and let be an -Lipschitzian mapping. Assume that is a sequence in and satisfying the following conditions: (i);(ii)there exist constants such that Then, the implicit iterative process defined by (4) converges weakly to a common fixed point of in .

Proof. Since , for each , we have Simplifying we have By condition (ii), ; hence from (9) we have By condition (i), we know that and as ; therefore there exists a positive integer such that , for all ; then we have It follows from (11) that Taking , , and and by using , it is easy to see that It follows from Lemma 6 that exists for each . Hence, there exists , such that We can assume that where is some number. Since is a convergent sequence, is a bounded sequence in . Since by condition (i) and (8) and (15), that Since is a uniformly convex Banach space, from (15)–(17) and Lemma 7 we know that By (18), we have that It follows from (18) and (19) that By (14), we know From (20), (22), and condition (i) we have Consequently, for any , from (21) and (23) we have This implies that the sequence Since, for each , is a subsequence of , therefore we have Since is uniformly convex, every bounded subset of is weakly compact. Since is a bounded sequence in , there exists a subsequence such that converges weakly to . From (26) we have By Lemma 8, we know that . By the arbitrariness of , we have that .
Suppose that there exists some subsequence such that weakly and . From Lemma 8, . By (12) we know that and exist. Since satisfies Opial’s condition, we have which is a contradiction. Hence . This implies that converges weakly to a common fixed point of in .

Theorem 10. Suppose that is a real uniformly convex Banach space and is a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be nonexpansive mappings with and let be an -Lipschitzian mapping. Assume that is a sequence in and satisfying the following conditions: (i);(ii)there exist constants such that Then, the implicit iterative process defined by (4) converges strongly to a common fixed point of if and only if (for all ).

Proof. From (12) and (14) in the proof of Theorem 9, we have where , , and . Hence, . Since , it follows from Lemma 6 that exists.
If converges strongly to a common fixed point of , then . Since we know that .
Conversely, suppose ; then . Moreover, we have ; thus for arbitrary , there exists a positive integer such that and for all . It follows from (30) that, for all and for all , we have Taking infimum over all , we obtain Thus, is a Cauchy sequence. Letting , then, from Lemma 8, we have . This completes the proof of the theorem.

Theorem 11. Suppose that is a real uniformly convex Banach space and is a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be nonexpansive mappings with and at least there exists a , , which is semicompact. Let be -Lipschitzian mapping. Assume that is a sequence in and satisfying the following conditions: (i);(ii)there exist constants such that Then, the implicit iterative process defined by (4) converges strongly to a common fixed point of in .

Proof. From the proof of Theorem 9, is bounded, and , for all . We especially have By the assumption of Theorem 11, we may assume that is semicompact, without loss of generality. Then, it follows from (35) that there exists a subsequence of such that converges strongly to . Thus from (26) we have This implies that . In addition, since exists, therefore ; that is, converges strongly to a fixed point of in . The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

The main idea of this paper was proposed by Qiaohong Jiang. All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

Acknowledgment

The research was supported by the Fujian Nature Science Foundation under Grant no. 2014J01008.