# Convergence Theorems for the Unique Common Fixed Point of a Pair of Asymptotically Nonexpansive Mappings in Generalized Convex Metric Space

- Chao Wang
^{1}Email author, - Jin Li
^{2}Email author and - Daoli Zhu
^{2}

**2010**:281890

**DOI: **10.1155/2010/281890

© ChaoWang et al. 2010

**Received: **21 September 2009

**Accepted: **13 December 2009

**Published: **5 January 2010

## Abstract

## 1. Introduction and Preliminaries

Let be a metric space, a pair of asymptotically nonexpansive mappings if there exists such that

Bose [1] first defined a pair of mean nonexpansive mappings in Banach space, that is,

(let in (*)), and then they proved several convergence theorems for commom fixed points of mean nonexpansive mappings. Gu and Li [2] also studied the same problem; they considered the Ishikawa iteration process to approximate the common fixed point of mean nonexpansive mappings in uniformly convex Banach space. Takahashi [3] first introduced a notion of convex metric space, which is more general space, and each linear normed space is a special example of the space. Late on, Ciric et al.[4] proved the convergence of an Ishikawa type iteration process to approximate the common fixed point of a pair of mappings (under condition (B), which is also a special example of (*)) in convex metric space. Very recently, Wang and Liu [5] give some sufficiency and necessary conditions for an Ishikawa type iteration process with errors to approximate a common fixed point of two mappings in generalized convex metric space.

Inspired and motivated by the above facts,we will consider the Ishikawa type iteration process with errors, which converges to the unique common fixed point of the pair of asymptotically nonexpansive mappings in generalized convex metric space. Our results extend and improve the corresponding results in [1–6].

First of all, we will need the following definitions and conclusions.

Definition 1.1 (see [3]).

If is a metric space with a convex structure , then is called a convex metric space. Moreover, a nonempty subset of is said to be convex if , for all .

Definition 1.2 (see [6]).

If is a metric space with a convex structure , then is called a generalized convex metric space. Moreover, a nonempty subset of is said to be convex if , for all .

Remark 1.3.

It is easy to see that every generalized convex metric space is a convex metric space (let ).

Definition 1.4.

then
is called the Ishikawa type iteration process with errors of a pair of asymptotically nonexpansive mappings *S* and *T*.

Remark 1.5.

Note that the iteration processes considered in [1, 2, 4, 6] can be obtained from the above process as special cases by suitably choosing the space, the mappings, and the parameters.

Theorem 1.6 (see [5]).

then converge to a fixed point of and if and only if where .

Remark 1.7.

## 2. Main Results

Now, we will prove the strong convergence of the iteration scheme (1.4) to the unique common fixed point of a pair of asymptotically nonexpansive mappings and in complete generalized convex metric spaces.

Theorem 2.1.

then converge to the unique common fixed point of and if and only if where .

Proof.

The necessity of conditions is obvious. Thus, we will only prove the sufficiency.

where . Similarly, we also have .

By Remark 1.7, we get that and are two uniformly quasi-Lipschitzian mappings (with ). Therefore, from Theorem 1.6, we know that converges to a common fixed point of and .

Finally, we prove the uniqueness. Let , , then, by (*), we have

Since , we obtain . This completes the proof.

- (i)
We consider a sufficient and necessary condition for the Ishikawa type iteration process with errors in complete generalized convex metric space; our mappings are the more general mappings (a pair of asymptotically nonexpansive mappings), so our result extend and generalize the corresponding results in [1–4, 6].

- (ii)
Since converges to the unique fixed point of and , we have improved Theorem 1.6 in [5].

Corollary 2.3.

Then, converges to the unique common fixed point of and if and only if , where .

Proof.

where . Hence, and are two uniformly quasi-Lipschitzian mappings in Banach space. Since Theorem 1.6 also holds in Banach spaces, we can prove that there exists a such that . The proof of uniqueness is the same to that of Theorem 2.1. Therefore, converges to the unique common fixed point of and .

Corollary 2.4.

where are two sequences in satisfying Then, converges to the unique common fixed point of and if and only if , where .

Proof.

Let and . The result can be deduced immediately from Corollary 2.3. This completes the proof.

## Declarations

### Acknowledgments

The authors would like to thank the referee and the editor for their careful reading of the manuscript and their many valuable comments and suggestions. The research was supported by the Natural Science Foundation of China (no. 70432001) and Shanghai Leading Academic Discipline Project (B210).

## Authors’ Affiliations

## References

- Bose SC:
**Common fixed points of mappings in a uniformly convex Banach space.***Journal of the London Mathematical Society*1978,**18**(1):151–156. 10.1112/jlms/s2-18.1.151MathSciNetView ArticleMATHGoogle Scholar - Gu Z, Li Y:
**Approximation methods for common fixed points of mean nonexpansive mapping in Banach spaces.***Fixed Point Theory and Applications*2008,**2008:**-7.Google Scholar - Takahashi W:
**A convexity in metric space and nonexpansive mappings. I.***Kodai Mathematical Seminar Reports*1970,**22:**142–149. 10.2996/kmj/1138846111MathSciNetView ArticleMATHGoogle Scholar - Ciric LB, Ume JS, Khan MS:
**On the convergence of the Ishikawa iterates to a common fixed point of two mappings.***Archivum Mathematicum*2003,**39**(2):123–127.MathSciNetMATHGoogle Scholar - Wang C, Liu LW:
**Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):2067–2071. 10.1016/j.na.2008.02.106MathSciNetView ArticleMATHGoogle Scholar - Tian Y-X:
**Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings.***Computers & Mathematics with Applications*2005,**49**(11–12):1905–1912. 10.1016/j.camwa.2004.05.017MathSciNetView ArticleMATHGoogle Scholar

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