# Convergence Theorems of Three-Step Iterative Scheme for a Finite Family of Uniformly Quasi-Lipschitzian Mappings in Convex Metric Spaces

- Tian You-xian
^{1}Email author and - Yang Chun-de
^{1}

**2009**:891965

**DOI: **10.1155/2009/891965

© T. You-xian and Y. Chun-de. 2009

**Received: **9 December 2008

**Accepted: **25 March 2009

**Published: **31 March 2009

## Abstract

We consider a new Noor-type iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite family of uniformly quasi-Lipschitzian mappings. The results presented in this paper extend, improve and unify some main results in previous work.

## 1. Introduction and Preliminaries

Takahashi [1] introduced a notion of convex metric spaces and studied the fixed point theory for nonexpansive mappings in such setting. For the convex metric spaces, Kirk [2] and Goebel and Kirk [3] used the term "hyperbolic type space" when they studied the iteration processes for nonexpansive mappings in the abstract framework. For the Banach space, Petryshyn and Williamson [4] proved a sufficient and necessary condition for Picard iterative sequences and Mann iterative sequence to converge to fixed points for quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [5] extended the results of [4] and gave the sufficient and necessary condition for Ishikawa iterative sequence to converge to fixed points for quasi-nonexpansive mappings. Liu [6–8] proved some sufficient and necessary conditions for Ishikawa iterative sequence and Ishikawa iterative sequence with errors to converge to fixed point for asymptotically quasi-nonexpansive mappings in Banach space and uniform convex Banach space. Tian [9] gave some sufficient and necessary conditions for an Ishikawa iteration sequence for an asymptotically quasi-nonexpansive mapping to converge to a fixed point in convex metric spaces. Very recently, Wang and Liu [10] gave some iteration sequence with errors to approximate a fixed point of two uniformly quasi-Lipschitzian mappings in convex metric spaces. The purpose of this paper is to give some sufficient and necessary conditions for a new Noor-type iterative sequence with errors to approximate a common fixed point for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. The results presented in this paper generalize, improve, and unify some main results of [1–14].

First of all, let us list some definitions and notations.

Remark 1.1.

If is nonempty, then it follows from the above definitions that an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, and an asymptotically quasi-nonexpansive mapping must be a uniformly quasi-Lipschitzian with . However, the inverse is not true in general.

Definition 1.2 (see [ 9]).

Definition 1.3.

In order to prove our main results, the following lemmas will be needed.

Lemma 1.4.

Proof.

This completes the Proof.

Lemma 1.5 (see [7]).

Then

(1) exists;

(2)In addition, if , then .

Lemma 1.6.

Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant Let be the iterative sequence with errors defined by (1.10) and , be three bounded sequences in . Let , , , be sequences in [0,1] satisfying the following conditions:

(i) ;

(ii) ;

(iii)

*Then the following conclusions hold*:

*for all*and ,

*where*,

*for all*

*and*

*there exists a constant*

*such that*

*for all*
.

- (1)It follows from (1.7),(1.10), and Lemma 1.4 that(1.22)

- (2)Since for all , it follows from (1.26) that, for and ,(1.28)

This completes the proof.

## 2. Main Results

Theorem 2.1.

Let be a complete convex metric space and be a nonempty closed convex subset of Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant . Let be the iterative sequence with errors defined by (1.10) and , , be three bounded sequence in and , , , , , , , and be nine sequences in [ 0,1] satisfying the following conditions:

(i) , ,

(ii) ,

(iii) .

Then the sequence converges to a common fixed point if and only if , where

Proof.

By the arbitrariness of , we know that for all , that is, . This completes the Proof of Theorem 2.1.

Taking in Theorem 2.1, then we have the following theorem.

Theorem 2.2.

where

Taking in Theorem 2.1, then we have the following theorem.

Theorem 2.3.

where

Remark 2.4.

Theorems 2.1–2.3 generalize, improve, and unify some corresponding results in [1–14].

Similarly, we can obtain the following results.

Theorem 2.5.

where

Proof.

where is the sequence appeared in (1.5). Hence the conclusion of Theorem 2.5 can be obtained from Theorem 2.1 immediately. This completes the Proof.

Theorem 2.6.

where

## Declarations

### Acknowledgment

The authors would like to express their thanks to the referees for their helpful comments and suggestions.

## Authors’ Affiliations

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