Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces

Abstract

We establish strong convergence theorem for multi-step iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Our results extend and improve the recent ones announced by Plubtieng and Wangkeeree (2006), and many others.

1. Introduction

Let be a subset of real normal linear space . A mapping is said to be asymptotically nonexpansive on if there exists a sequence in with such that for each ,

(1.1)

If , then is known as a nonexpansive mapping. is called asymptotically nonexpansive in the intermediate sense [1] provided is uniformly continuous and

(1.2)

From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense.

Let be a nonempty subset of normed space , and Let be mappings. For a given and a fixed ( denotes the set of all positive integers), compute the iterative sequences defined by

(1.3)

where , are bounded sequences in and , , , are appropriate real sequences in such that for each .

The purpose of this paper is to establish a strong convergence theorem for common fixed points of the multistep iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in a uniformly convex Banach space. The results presented in this paper extend and improve the corresponding ones announced by Plubtieng and Wangkeeree [2], and many others.

2. Preliminaries

Definition 2.1 (see [1]).

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

(2.1)

Lemma 2.2 (see [3]).

Let , and be three nonnegative real sequences satisfying the following condition:

(2.2)

where and . Then

(1) exists;

(2)If , then .

Lemma 2.3 (see [4]).

Let be a uniformly convex Banach space and for all . Suppose that and are two sequences of such that

(2.3)

for some . Then

(2.4)

3. Main Results

Lemma 3.1.

Let be a uniformly convex Banach space, , are two sequences of , and be a real sequence. If there exists such that

(i) for all ;

(ii);

(iii);

(iv),

then .

Proof.

The proof is clear by Lemma 2.3.

Lemma 3.2.

Let be a uniformly convex Banach space, let be a nonempty closed bounded convex subset of , and let be asymptotically nonexpansive mappings in the intermediate sense such that . Put

(3.1)

so that . Let , , and be real sequences in satisfying the following condition:

(i) for all and ;

(ii) for all .

If is the iterative sequence defined by (1.3), then, for each , the limit exists.

Proof.

For each , we note that

(3.2)

where . Since

(3.3)

we see that

(3.4)

It follows from (3.2) that

(3.5)

where . Since

(3.6)

we see that

(3.7)

It follows from (3.5) that

(3.8)

where , and so

(3.9)

By continuing the above method, there are nonnegative real sequences such that

(3.10)

This together with Lemma 2.2 gives that exists. This completes the proof.

Lemma 3.3.

Let be a uniformly convex Banach space, let be a nonempty closed bounded convex subset of , and let be asymptotically nonexpansive mappings in the intermediate sense such that . Put

(3.11)

so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then we have the following:

(1);

(2).

Proof.

Taking each , it follows from Lemma 3.2 that exists. Let

(3.12)

for some . We note that

(3.13)

where is a nonnegative real sequence such that

(3.14)

It follows that

(3.15)

which implies that

(3.16)

Next, we observe that

(3.17)

Thus we have

(3.18)

Also,

(3.19)

gives that

(3.20)

Note that

(3.21)

This together with (3.18), (3.20), and Lemma 3.1, gives

(3.22)

This completes the proof of .

For each ,

(3.23)

Since

(3.24)

we obtain

(3.25)

It follows that

(3.26)

which implies that

(3.27)

On the other hand, we note that

(3.28)

where is a nonnegative real sequence such that

(3.29)

Thus we have

(3.30)

and hence

(3.31)

Next, we observe that

(3.32)

Thus we have

(3.33)

Also,

(3.34)

gives that

(3.35)

Note that

(3.36)

Therefore, it follows from (3.33), (3.35), and Lemma 3.1 that

(3.37)

This completes the proof.

Theorem 3.4.

Let be a uniformly convex Banach space and let be a nonempty closed bounded convex subset of . Let be asymptotically nonexpansive mappings in the intermediate sense such that and there exists one member in which is completely continuous. Put

(3.38)

so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then converges strongly to a common fixed point of the mappings .

Proof.

From Lemma 3.3, it follows that

(3.39)

which implies that

(3.40)

and so

(3.41)

It follows from (3.22), (3.37) that

(3.42)

Let for all . Then we have

(3.43)

Notice that . Thus and the above inequality becomes

(3.44)

and so

(3.45)

Since

(3.46)

we have

(3.47)

and so

(3.48)

Since is bounded and one of is completely continuous, we may assume that is completely continuous, without loss of generality. Then there exists a subsequence of such that as . Moreover, by (3.48), we have

(3.49)

which implies that as . By (3.48) again, we have

(3.50)

It follows that . Since exists, we have

(3.51)

that is,

(3.52)

Moreover, we observe that

(3.53)

for all and

(3.54)

Therefore,

(3.55)

for all . This completes the proof.

Remark 3.5.

Theorem 3.4 improves and extends the corresponding results of Plubtieng and Wangkeeree [2] in the following ways.

The iterative process defined by (1.3) in [2] is replaced by the new iterative process defined by (1.3) in this paper.

Theorem 3.4 generalizes Theorem of Plubtieng and Wangkeeree [2] from a asymptotically nonexpansive mappings in the intermediate sense to a finite family of asymptotically nonexpansive mappings in the intermediate sense.

Remark 3.6.

If and in Theorem 3.4, we obtain strong convergence theorem for Noor iteration scheme with error for asymptotically nonexpansive mapping in the intermediate sense in Banach space, we omit it here.