#### 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 [1–9]).

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.