- Research Article
- Open Access

# Strong Convergence of an Implicit Iteration Process for a Finite Family of Uniformly -Lipschitzian Mappings in Banach Spaces

- Feng Gu
^{1}Email author

**2010**:801961

https://doi.org/10.1155/2010/801961

© Feng Gu. 2010

**Received:**19 September 2009**Accepted:**13 December 2009**Published:**5 January 2010

## Abstract

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly -Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by Chang (2001), Cho et al. (2005), Ofoedu (2006), Schu (1991) and Zeng (2003 and 2005), and many others.

## Keywords

- Banach Space
- Positive Integer
- Real Number
- Positive Constant
- Nonexpansive Mapping

## 1. Introduction and Preliminaries

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

where denotes the duality pairing between and . The single-valued normalized duality mapping is denoted by .

Definition 1.1.

Let be a mapping. Therefore, the following are given.

Remark 1.2.

( ) It is easy to see that if is an asymptotically nonexpansive mapping, then is a uniformly -Lipschitzian mapping, where . And every asymptotically nonexpansive mapping is asymptotically pseudocontractive, but the inverse is not true, in general.

( ) The concept of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [7], while the concept of asymptotically pseudocontractive mappings was introduced by Schu [4] who proved the following theorem.

Theorem 1.3 (see Schu [4]).

Let be a Hilbert space, be a nonempty bounded closed convex subset of , and let be a completely continuous, uniformly -Lipschitzian and asymptotically pseudocontractive mapping with a sequence satisfying the following conditions:

(i) as ,

(ii) , where .

Then converges strongly to a fixed point of in .

In [1], the first author extended Theorem to a real uniformly smooth Banach space and proved the following theorem.

Theorem 1.4 (see Chang [1]).

Let be a uniformly smooth Banach space, be a nonempty bounded closed convex subset of , and be an asymptotically pseudocontractive mapping with a sequence with , and let , where is the set of fixed points of in . Let be a sequence in satisfying the following conditions:

(i) ,

(ii) .

where is some fixed point of in , then as .

Very recently, in [3] Ofoedu proved the following theorem.

Theorem 1.5 (see Ofoedu [3]).

Let be a real Banach space, let be a nonempty closed convex subset of , and let be a uniformly -Lipschitzian asymptotically pseudocontractive mapping with a sequence , such that , where is the set of fixed points of in . Let be a sequence in satisfying the following conditions:

(i) ,

(ii) ,

(iii) .

then converges strongly to .

Remark 1.6.

It should be pointed out that although Theorem 1.5 extends Theorem 1.4 from a real uniformly smooth Banach space to an arbitrary real Banach space, it removes the boundedness condition imposed on .

In [8], Xu and Ori introduced the following implicit iteration process for a finite family of nonexpansive mappings (here ), with as a real sequence in (0, 1), and an initial point :

where (here the mod function takes values in ). Xu and Ori proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space.

Chidume and Shahzad [9] and Zhou and Chang [10] studied the weak and strong convergences of this implicit process to a common fixed point for a finite family of nonexpansive mappings, respectively.

Recently, Feng Gu [11] introduced a composite implicit iteration process with errors for a finite family of strictly pseudocontractive mappings as follows:

where , , , , , are four real sequences in satisfying and for all , and are two bounded sequences in , and is a given point in . Feng Gu proved the strong convergence of this process to a common fixed point for a finite family of strictly pseudocontractive mappings in a real Banach space.

Inspired and motivated by the abovesaid facts, we introduced a two-step implicit iteration process with errors for a finite family of -Lipschitzian mappings as follows:

where , , , and , are four real sequences in satisfying and for all , and are two bounded sequences in , and is a given point in .

Observe that if is a nonempty closed convex subset of and be uniformly -Lipschitzian mappings. If , where , then for given , and , the mapping defined by

is a contractive mapping. In fact, the following are observed

Since for all , hence is a contractive mapping. By Banach contractive mapping principle, there exists a unique fixed point such that

Therefore, if , then the iterative sequence (1.12) can be employed for the approximation of common fixed points for a finite family of uniformly -Lipschitzian mappings.

Especially, if and are two sequences in satisfying for all , is a bounded sequence in , and is a given point in , then the sequence defined by

is called the one-step implicit iterative sequence with errors for a finite family of operators .

The purpose of this paper is, by using a simple and quite different method, to study the convergence of implicit iterative sequence defined by (1.12) and (1.16) to a common fixed point for a finite family of -Lipschitzian mappings instead of the assumption that is a uniformly -Lipschitzian and asymptotically pseudocontractive mapping in a Banach space. Our results extend and improve some recent results in [1–6]. Even in the case of , for all or are also new.

For the main results, the following lemmas are given.

Lemma 1.7 (see Petryshyn [12]).

Lemma 1.8 (see Moore and Nnoli [13]).

where is some nonnegative integer and is a sequence of nonnegative number such that , then as .

Lemma 1.9.

where is a sequence in (0, 1) with . If , then exists.

## 2. Main Results

In this section, we shall prove our main theorems in this paper.

Theorem 2.1.

Let be a real Banach space, be a nonempty closed convex subset of , be uniformly -Lipschitzian mappings with , where is the set of fixed points of in , and let be a point in . Let be a sequence with . Let , , , and be four sequences in satisfying the following conditions: , , for all . Let and be two bounded sequences in , and let be the iterative sequence with errors defined by (1.12), then the following conditions are satisfied:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) ,

(vii) , for all , where .

for all and , then converges strongly to .

Proof.

- (i)
First, we prove that the sequence defined by (1.12) is bounded.

- (ii)
Now, we consider (2.9) and prove that .

that is, as . This completes the proof of Theorem 2.1.

Remark 2.2.

( ) Theorem 2.1 extends and improves the corresponding results in Chang [1], Cho et al. [2], Ofoedu [3], Schu [4], and Zeng [5, 6].

( ) The method given by the proof of Theorem 2.1 is quite different from the method given in Ofoedu [3].

( ) Theorem 2.1 extends and improves Theorem of Ofoedu [3]; it abolishes the assumption that is an asymptotically pseudocontractive mapping.

The following theorem can be obtained from Theorem 2.1 immediately.

Theorem 2.3.

Let be a real Banach space, let be a nonempty closed convex subset of , let be uniformly -Lipschitzian mappings with , where is the set of fixed points of in , and let be a point in . Let be a sequence with . Let and be two sequences in satisfying the following conditions: , for all . Let be a bounded sequence in , and let be the iterative sequence with errors defined by (1.16), then the following conditions are satisfied:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,for all , where .

for all and , then converges strongly to .

Proof.

Taking in Theorem 2.1, then the conclusion of Theorem 2.3 can be obtained from Theorem 2.1 immediately. This completes the proof of Theorem 2.3.

Theorem 2.4.

Let be a real Banach space, let be a nonempty closed convex subset of , let be a uniformly -Lipschitzian mappings with , where is the set of fixed points of in , and let be a point in . Let be a sequence with . Let and be two sequences in satisfying the following condition: , , and let be a bounded sequence in satisfying the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) .

(v) , .

for all and , then converges strongly to .

Proof.

Taking in Theorem 2.3, then the conclusion of Theorem 2.4 can be obtained from Theorem 2.3 immediately. This completes the proof of Theorem 2.4.

Remark 2.5.

In Theorem 2.4 without the assumption that is an asymptotically pseudocontractive mapping, Theorem 2.4 extends and improves Theorem of Ofoedu [3].

## Declarations

### Acknowledgments

The present studies were supported by the National Natural Science Foundation of China (10771141) and the Natural Science Foundation of Zhejiang Province (Y605191).

## Authors’ Affiliations

## References

- Chang SS: Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings.
*Proceedings of the American Mathematical Society*2001, 129(3):845–853. 10.1090/S0002-9939-00-05988-8MATHMathSciNetView ArticleGoogle Scholar - Cho YJ, Kang JI, Zhou H: Approximating common fixed points of asymptotically nonexpansive mappings.
*Bulletin of the Korean Mathematical Society*2005, 42(4):661–670.MATHMathSciNetView ArticleGoogle Scholar - Ofoedu EU: Strong convergence theorem for uniformly -Lipschitzian asymptotically pseudocontractive mapping in real Banach space.
*Journal of Mathematical Analysis and Applications*2006, 321(2):722–728. 10.1016/j.jmaa.2005.08.076MATHMathSciNetView ArticleGoogle Scholar - Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings.
*Journal of Mathematical Analysis and Applications*1991, 158(2):407–413. 10.1016/0022-247X(91)90245-UMATHMathSciNetView ArticleGoogle Scholar - Zeng LC: Iterative approximation of fixed points of asymptotically pseudo-contractive mappings in uniformly smooth Banach spaces.
*Chinese Annals of Mathematics. Series A*2005, 26(2):283–290.MATHMathSciNetGoogle Scholar - Zeng LC: On the approximation of fixed points for asymptotically nonexpansive mappings in Banach spaces.
*Acta Mathematica Scientia*2003, 23: 31–37.MATHGoogle Scholar - Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings.
*Proceedings of the American Mathematical Society*1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MATHMathSciNetView ArticleGoogle Scholar - Xu H-K, Ori RG: An implicit iteration process for nonexpansive mappings.
*Numerical Functional Analysis and Optimization*2001, 22(5–6):767–773. 10.1081/NFA-100105317MATHMathSciNetView ArticleGoogle Scholar - Chidume CE, Shahzad N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 62(6):1149–1156. 10.1016/j.na.2005.05.002MATHMathSciNetView ArticleGoogle Scholar - Zhou Y, Chang S-S: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces.
*Numerical Functional Analysis and Optimization*2002, 23(7–8):911–921. 10.1081/NFA-120016276MATHMathSciNetView ArticleGoogle Scholar - Gu F: The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings.
*Journal of Mathematical Analysis and Applications*2007, 329(2):766–776. 10.1016/j.jmaa.2006.07.005MATHMathSciNetView ArticleGoogle Scholar - Petryshyn WV: A characterization of strict convexity of Banach spaces and other uses of duality mappings.
*Journal of Functional Analysis*1970, 6: 282–291. 10.1016/0022-1236(70)90061-3MATHMathSciNetView ArticleGoogle Scholar - Moore C, Nnoli BVC: Iterative solution of nonlinear equations involving set-valued uniformly accretive operators.
*Computers & Mathematics with Applications*2001, 42(1–2):131–140. 10.1016/S0898-1221(01)00138-9MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.