Convergence Theorems for the Unique Common Fixed Point of a Pair of Asymptotically Nonexpansive Mappings in Generalized Convex Metric Space
© ChaoWang et al. 2010
Received: 21 September 2009
Accepted: 13 December 2009
Published: 5 January 2010
Let be a generalized convex metric space, and let , be a pair of asymptotically nonexpansive mappings. In this paper, we will consider an Ishikawa type iteration process with errors to approximate the unique common fixed point of and .
1. Introduction and Preliminaries
Let be a metric space, a pair of asymptotically nonexpansive mappings if there exists such that
for all , .
Bose  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  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  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. 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  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 ).
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 ).
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 .
It is easy to see that every generalized convex metric space is a convex metric space (let ).
then is called the Ishikawa type iteration process with errors of a pair of asymptotically nonexpansive mappings S and T.
Theorem 1.6 (see ).
then converge to a fixed point of and if and only if where .
for all , .
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.
then converge to the unique common fixed point of and if and only if where .
The necessity of conditions is obvious. Thus, we will only prove the sufficiency.
Let , for all ,
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.
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].
Since converges to the unique fixed point of and , we have improved Theorem 1.6 in .
Then, converges to the unique common fixed point of and if and only if , where .
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 .
where are two sequences in satisfying Then, converges to the unique common fixed point of and if and only if , where .
Let and . The result can be deduced immediately from Corollary 2.3. This completes the proof.
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).
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