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

Abstract

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

(x2a)

for all , .

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

(1.1)

(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]).

Let be a metric space, and . A mapping is said to be convex structure on , if for any and , the following inequality holds:

(1.2)

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]).

Let be a metric space, , and real sequences in with . A mapping is said to be convex structure on , if for any and , the following inequality holds:

(1.3)

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.

Let be a generalized convex metric space with a convex structure , and a nonempty closed convex subset of . Let be a pair of asymptotically nonexpansive mappings, and six sequences in with for any given , define a sequence as follows:

(1.4)

where are two sequences in satisfying the following condition. If for any nonnegative integers , , then

(x2ax2a)

where ,

(1.5)

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]).

Let be a nonempty closed convex subset of complete convex metric space , and uniformly quasi-Lipschitzian mappings with and , and (). Suppose that is the Ishikawa type iteration process with errors defined by (1.4), satisfy (**), and are six sequences in satisfying

(1.6)

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

Remark 1.7.

Let . A mapping is called uniformly quasi-Lipshitzian if there exists such that

(1.7)

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.

Theorem 2.1.

Let be a nonempty closed convex subset of complete generalized convex metric space , and a pair of asymptotically nonexpansive mappings with , and . Suppose as in (1.4), satisfy (**), and are six sequences in satisfying

(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.

Let , for all ,

(2.2)

implies

(2.3)

which yield (using the fact that and )

(2.4)

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

(2.5)

Since , we obtain . This completes the proof.

Remark 2.2.

• (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.

Let be a nonempty closed convex subset of Banach space , a pair of asymptotically nonexpansive mappings, that is,

(2.6)

with , and . For any given , is an Ishikawa type iteration process with errors defined by

(2.7)

where are two bounded sequences and are six sequences in satisfying

(2.8)

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

Proof.

From the proof of Theorem 2.1, we have

(2.9)

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.

Let be a nonempty closed convex subset of Banach space , a pair of asymptotically nonexpansive mappings, that is,

(2.10)

with , and . For any given , an Ishikawa type iteration process defined by

(2.11)

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.