Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings

Abstract

We introduce a new three-step iterative scheme with errors. Several convergence theorems of this scheme are established for common fixed points of nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our theorems improve and generalize recent known results in the literature.

1. Introduction

Let be a nonempty closed convex subset of real normed linear space . Recall that a mapping is called asymptotically nonexpansive if there exists a sequence , with such that for all and . Moreover, it is uniformly -Lipschitzian if there exists a constant such that for all and each . Denote and define by the set of fixed points of . Suppose . A mapping is called asymptotically quasi-non-expansive if there exists a sequence , with such that for all , , and .

It is clear from the above definitions that an asymptotically nonexpansive mapping must be uniformly -Lipschitzian as well as asymptotically quasi-non-expansive, but the converse does not hold. Iterative technique for asymptotically nonexpansive self-mapping in Hilbert spaces and Banach spaces including Mann-type and Ishikawa-type iteration processes has been studied extensively by many authors; see, for example, [1–6].

Recently, Chidume etal. [7] have introduced the concept of nonself asymptotically nonexpansive mappings, which is the generalization of asymptotically nonexpansive mappings. Similarly, the concept of nonself asymptotically quasi-non-expansive mappings can also be defined as the generalization of asymptotically quasi-non-expansive mappings and nonself asymptotically nonexpansive mappings. These mappings are defined as follows.

Definition 1.1.

Let be a nonempty closed convex subset of real normed linear space , let be the nonexpansive retraction of onto , and let be a nonself mapping.

• (i) is said to be a nonself asymptotically nonexpansive mapping if there exists a sequence , with such that

  

  (1.1)

for all and .

• (ii) is said to be a nonself uniformly -Lipschitzian mapping if there exists a constant such that

  

  (1.2)

for all and .

• (iii) is said to be a nonself asymptotically quasi-non-expansive mapping if and there exists a sequence , with such that

  

  (1.3)

for all , , and .

By studying the following iteration process (Mann-type iteration):

(1.4)

where , Chidume etal. [7] obtained many convergence theorems for the fixed points of nonself asymptotically nonexpansive mapping . Later on, Wang [8] generalized the iteration process (1.4) as follows (Ishikawa-type iteration):

(1.5)

where are nonself asymptotically nonexpansive mappings and . Also, he got several convergence theorems of the iterative scheme (1.5) under proper conditions.

In 2000, Noor [9] first introduced a three-step iterative sequence and studied the approximate solutions of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. Glowinski and Tallec [10] showed that the three-step iterative schemes perform better than the Mann-type and Ishikawa-type iterative schemes. On the other hand, Xu and Noor [11] introduced and studied a three-step scheme to approximate fixed points of asymptotically nonexpansive mappings in Banach spaces. Cho etal. [12] and Plubtieng etal. [13] extended the work of Xu and Noor to the three-step iterative scheme with errors, and gave weak and strong convergence theorems for asymptotically nonexpansive mappings in Banach spaces.

Inspired and motivated by these facts, a new class of three-step iterative schemes with errors, for three nonself asymptotically quasi-non-expansive mappings, is introduced and studied in this paper. This scheme can be viewed as an extension for (1.4), (1.5), and others. This scheme is defined as follows.

Let be a nonempty convex subset of real normed linear space , let be the nonexpansive retraction of onto , and let be three nonself asymptotically quasi-non-expansive mappings. Compute the sequences , , and by

(1.6)

where , , , , , , , , and are real sequences in with , and , , and are bounded sequences in .

Remark 1.2.

• (i) If , , and , then scheme (1.6) reduces to the Mann-type iteration (1.4).

• (ii) If , , and , then scheme (1.6) reduces to the Ishikawa-type iteration (1.5).

• (iii) If , and are three self-asymptotically nonexpansive mappings, then scheme (1.6) reduces to the three-step iteration with errors defined by [12, 13], and others.

The purpose of this paper is to study the iterative sequences (1.6) to converge to a common fixed point of three nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our results extend and improve the corresponding results in [5, 7, 8, 11–13], and many others.

2. Preliminaries and Lemmas

In this section, we first recall some well-known definitions.

A real Banach space is said to be uniformly convex if the modulus of convexity of :

(2.1)

for all (i.e., is a function ).

A subset of is said to be a retract if there exists continuous mapping such that , for all , and every closed convex subset of a uniformly convex Banach space is a retract. A mapping is said to be a retraction if .

A mapping with is said to satisfy condition (A) (see [14]) if there exists a nondecreasing function with , for all , such that

(2.2)

for all , where .

We modify this condition for three mappings as follows. Three mappings , where is a subset of , are said to satisfy condition (B) if there exist a real number and a nondecreasing function with , for all , such that

(2.3)

for all , where . Note that condition (B) reduces to condition (A) when and .

A mapping is said to be semicompact if, for any sequence in such that , there exists subsequence of such that converges strongly to .

Next we state the following useful lemmas.

Lemma 2.1 (see [5]).

Let , , and be sequences of nonnegative real numbers satisfying the inequality

(2.4)

If and , then exists.

Lemma 2.2 (see [15]).

Let be a real uniformly convex Banach space and , for all positive integer . Suppose that and are two sequences of such that , , and hold, for some ; then .

3. Main Results

In this section, we will prove the strong convergence of the iteration scheme (1.6) to a common fixed point of nonself asymptotically quasi-non-expansive mappings , and . We first prove the following lemmas.

Lemma 3.1.

Let be a nonempty closed convex subset of a real normed linear space . Let be nonself asymptotically quasi-non-expansive mappings with sequences such that , for all . Suppose that is defined by (1.6) with , , and . If , then exists, for all .

Proof..

Let . Since , and are bounded sequences in , therefore there exists such that

(3.1)

Let and Then and . By (1.6), we have

(3.2)

(3.3)

and similarly, we also have

(3.4)

Substituting (3.4) into (3.3), we obtain

(3.5)

where . Since and , then . Substituting (3.5) into (3.2), we have

(3.6)

where and . Since , , and , then and . It follows from Lemma 2.1 that exists. This completes the proof.

Lemma 3.2.

Let be a nonempty closed convex subset of a real uniformly convex Banach space . Let be uniformly -Lipschitzian nonself asymptotically quasi-non-expansive mappings with sequences such that , for all . Suppose that is defined by (1.6) with , , and , where , and are three sequences in , for some . If , then

(3.7)

Proof..

For any , by Lemma 3.1, we see that exists. Assume , for some . For all , let and

Then, and . From (3.5), we have

(3.8)

Taking on both sides in (3.8), since and , we obtain

(3.9)

so that

(3.10)

Next consider

(3.11)

Since , we have

(3.12)

In addition,

(3.13)

This implies that

(3.14)

Further, observe that

(3.15)

By Lemma 2.2, (3.12), (3.14), and (3.15), we have

(3.16)

Next we will prove that . Since

(3.17)

and , we obtain

(3.18)

Thus, it follows from (3.10) and (3.18) that

(3.19)

On the other hand, from (3.4), we have

(3.20)

By boundedness of the sequence and by , we have

(3.21)

so that

(3.22)

Next consider

(3.23)

Thus, we have

(3.24)

This implies that

(3.25)

Note that

(3.26)

It follows from Lemma 2.2, (3.24), and (3.25) that

(3.27)

Similarly, by using the same argument as in the proof above, we obtain

(3.28)

Hence,

(3.29)

and this implies that

(3.30)

Since is uniformly -Lipschitzian mapping, then we have

(3.31)

(3.32)

It follows from (3.30), (3.31), and (3.32) that

(3.33)

Next consider

(3.34)

(3.35)

It follows from (3.30), (3.34), and (3.35) that

(3.36)

Finally, we consider

(3.37)

It follows from (3.29), (3.30), and (3.37) that

(3.38)

Therefore,

(3.39)

This completes the proof.

Now, we give our main theorems of this paper.

Theorem 3.3.

Let be a nonempty closed convex subset of a real uniformly convex Banach space . Let be uniformly -Lipschitzian and nonself asymptotically quasi-non-expansive mappings with sequences such that , for all satisfying condition (B). Suppose that is defined by (1.6) with , , and , where , and are three sequences in , for some . If , then converges strongly to a common fixed point of , and .

Proof..

It follows from Lemma 3.2 that . Since , and satisfy condition (B), we have .

From Lemma 3.1 and the proof of Qihou [5], we can obtain that is a Cauchy sequence in . Assume that . Since , by the continuity of , and , we have , that is, is a common fixed point of , and . This completes the proof.

Corollary 3.4.

Let be a nonempty closed convex subset of a real uniformly convex Banach space . Let be nonself asymptotically nonexpansive mappings with sequences such that , for all satisfying condition (B). Suppose that is defined by (1.6) with , , and , where , and are three sequences in for some . If , then converges strongly to a common fixed point of , and .

Proof..

Since every nonself asymptotically nonexpansive mapping is uniformly -Lipschitzian and nonself asymptotically quasi-non-expansive, the result can be deduced immediately from Theorem 3.3. This completes the proof.

Theorem 3.5.

Let be a nonempty closed convex subset of a real uniformly convex Banach space . Let be uniformly -Lipschitzian and nonself asymptotically quasi-non-expansive mappings with sequences such that for all . Suppose that is defined by (1.6) with , , and , where , and are three sequences in for some . If and one of , and is demicompact, then converges strongly to a common fixed point of , and .

Proof..

Without loss of generality, we may assume that is demicompact. Since , there exists a subsequence such that . Hence, from (3.39), we have

(3.40)

This implies that . By the arbitrariness of , from Lemma 3.1, and taking , similarly we can prove that

(3.41)

where is some nonnegative number. From , we know that , that is, . This completes the proof.

Corollary 3.6.

Let be a nonempty closed convex subset of a real uniformly convex Banach space . Let be nonself asymptotically nonexpansive mappings with sequences such that for all . Suppose that is defined by (1.6) with , , and , where , and are three sequences in for some . If and one of , and is demicompact, then converges strongly to a common fixed point of , and .