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Convergence Theorems of ThreeStep Iterative Scheme for a Finite Family of Uniformly QuasiLipschitzian Mappings in Convex Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 891965 (2009)
Abstract
We consider a new Noortype iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasiLipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite family of uniformly quasiLipschitzian 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 quasinonexpansive 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 quasinonexpansive 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 quasinonexpansive 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 quasinonexpansive 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 quasiLipschitzian mappings in convex metric spaces. The purpose of this paper is to give some sufficient and necessary conditions for a new Noortype iterative sequence with errors to approximate a common fixed point for a finite family of uniformly quasiLipschitzian 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.
Let be a given self mapping of a nonempty convex subset of an arbitrary real normed space.The sequence defined by
is called the Noor iterative procedure with errors [11], where and are appropriate sequences in [] with and ,, and are bounded sequences in . If then (1.1) reduces to the Ishikawa iterative procedure with errors [15] defined as follows:
If then (1.2) reduces to the following Mann type iterative procedure with errors [15]:
Let be a metric space. A mapping is said to be asymptotically nonexpansive, if there exists a sequence [1,], such that
Let be the set of fixed points of in and , a mapping is said to be asymptotically quasinonexpansive, if there exists with such that
Moreover, is said to be uniformly quasiLipschitzian, if there exists such that
Remark 1.1.
If is nonempty, then it follows from the above definitions that an asymptotically nonexpansive mapping must be asymptotically quasinonexpansive, and an asymptotically quasinonexpansive mapping must be a uniformly quasiLipschitzian with . However, the inverse is not true in general.
Definition 1.2 (see [ 9]).
Let be a metric space, and let [0,1],,, be real sequences in [] with . A mapping is said to be a convex structure on if, for any and ,
If is a metric space with a convex structure , then is called a convex metric space. Let be a convex metric space, a nonempty subset of is said to be convex if
Definition 1.3.
Let be a convex metric space with a convex structure and be a finite family of uniformly quasiLipschitzian mappings with . Let ,,, and be nine sequences in [with
For a given define a sequence as follows:
where , is a Lipschitz continuous mapping with a Lipschitz constant and , are any given three sequences in Then is called the Noortype iterative sequence with errors for a finite family of uniformly quasiLipschitzian mappings . If in (1.10), then the sequence defined by (1.10) can be written as follows:
If for all in (1.10), then for all and the sequence defined by (1.10) can be written as follows:
If and for all , then the sequence defined by (1.10) can be written as follows:
which is the Ishikawa type iterative sequence with errors considered in [9]. Further, if and for all , then for all and (1.10) reduces to the following Mann type iterative sequence with errors [9]:
In order to prove our main results, the following lemmas will be needed.
Lemma 1.4.
Let be a convex metric space, be a uniformly quasiLipschitzian mapping for such that . Then there exists a constant such that, for all
Proof.
In fact, for each , since is a uniformly quasiLipschitzian mapping, we have
where
This completes the Proof.
Lemma 1.5 (see [7]).
Let be three nonexpansive squences satisfying the following conditions:
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 quasiLipschitzian 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:
(1)for all and ,
where, for alland
(2)there exists a constantsuch that
for all.
Proof.

(1)
It follows from (1.7),(1.10), and Lemma 1.4 that
(1.22)
Substituting (1.23) into (1.22) and simplifying it, we have
Substituting (1.24) into (1.25) and simplifying it, we get
where

(2)
Since for all , it follows from (1.26) that, for and ,
(1.28)
where
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 quasiLipschitzian 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.
The necessity is obvious. Now prove the sufficiency. In fact, from Lemma 1.6, we have
where . By conditions (i) and (ii), we know that
It follows from Lemma 1.5 that exists. Since , we have
Next prove that is a Cauchy sequence in . In fact, for any given , there exists a positive integer such that
From (2.4), there exist and positive integer such that
Thus Lemma 1.6 implies that, for any positive integers with ,
This shows that is a Cauchy sequence in a nonempty closed convex subset of a complete convex metric space . Without loss of generality, we can assume that Next prove that . In fact, for any given , there exists a positive integer such that for all ,
Again from (2.7), there exist and positive integer such that
Thus, for any , from (2.7) and (2.8), we have
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.
Let be a complete convex metric space and be a nonempty closed convex subset of Let be a finite family of uniformly quasiLipschitzian mapping for such that . Let be the iterative sequence with errors defined by (1.11) and ,, be three bounded sequence in , and ,,,,,,,and be nine sequence in [ 0,1] satisfying the conditions (i)–(iii) of Theorem 2.1. Then the sequence converges to a common fixed point if and only if
where
Taking in Theorem 2.1, then we have the following theorem.
Theorem 2.3.
Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of uniformly quasiLipschitzian mapping for such that and be a contractive mapping with a contractive constant . Let be the iterative sequence with errors defined by (1.12) and , be two bounded sequences in and ,,,,, be nine sequences in [ ] satisfying the conditions (ii) and (iii) of Theorem 2.1 and for all . Then the sequence converges to a common fixed point if and only if
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.
Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of asymptotically quasinonexpansive 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 and ,,,,,, and be nine sequences in [] satisfying the conditions (i)–(iii) of Theorem 2.1. Then the sequence converges to a common fixed point if and only if
where
Proof.
From Remark 1.1, we know that each asymptotically quasinonexpansive mapping must be a uniformly quasiLipschitzian with
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.
Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of asymptotically quasinonexpansive mapping for, such that . Let be the iterative sequence with errors defined by (1.11) and ,, be three bounded sequence in and ,,,,,,, and be nine sequence in [] satisfying the conditions (i)–(iii) of Theorem 2.1. Then the sequence converges to a common fixed point if and only if
where
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The authors would like to express their thanks to the referees for their helpful comments and suggestions.
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Youxian, T., Chunde, Y. Convergence Theorems of ThreeStep Iterative Scheme for a Finite Family of Uniformly QuasiLipschitzian Mappings in Convex Metric Spaces. Fixed Point Theory Appl 2009, 891965 (2009). https://doi.org/10.1155/2009/891965
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DOI: https://doi.org/10.1155/2009/891965
Keywords
 Nonexpansive Mapping
 Common Fixed Point
 Nonempty Closed Convex Subset
 Convex Banach Space
 Iterative Sequence