Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces

Abstract

Let be a uniformly convex Banach space, and let be a pair of mean nonexpansive mappings. In this paper, it is proved that the sequence of Ishikawa iterations associated with and converges to the common fixed point of and .

1. Introduction and Preliminaries

Let be a Banach space and let , be mappings from to . The pair of mean nonexpansive mappings was introduced by Bose in [1]:

(1.1)

for all , , .

The Ishikawa iteration sequence of and was defined by

(1.2)

where , . The recursion formulas (1.2) were first introduced in 1994 by Rashwan and Saddeek [2] in the framework of Hilbert spaces.

In recent years, several authors (see [2–6]) have studied the convergence of iterations to a common fixed point for a pair of mappings. Rashwan has studied the convergence of Mann iterations to a common fixed point (see [5]) and proved that the Ishikawa iterations converge to a unique common fixed point in Hilbert spaces (see [2]). Recently, Ćirić has proved that if the sequence of Ishikawa iterations sequence associated with and converges to , then is the common fixed point of and (see [7]). In [4, 6], the authors studied the same problem. In [1], Bose defined the pair of mean nonexpansive mappings, and proved the existence of the fixed point in Banach spaces. In particular, he proved the following theorem.

Theorem 1.1 ([1]).

Let be a uniformly convex Banach space and a nonempty closed convex subset of , and are a pair of mean nonexpansive mappings, and . Then,

• (i) and have a common fixed point ;

• (ii) further, if , then

  • (a) is the unique common fixed point and unique as a fixed point of each and ,

  • (b) the sequence defined by , for any , converges strongly to .

It is our purpose in this paper to consider an iterative scheme, which converges to a common fixed point of the pair of mean nonexpansive mappings. Theorem 2.1 extends and improves the corresponding results in [1].

2. Main Results

Now we prove the following theorem which is the main result of this paper.

Theorem 2.1.

Let be a uniformly convex Banach space, and are a pair of mean nonexpansive with a nonempty common fixed points set; if , then the Ishikawa sequence converges to the common fixed point of and .

Proof.

First, we show that the sequence is bounded. For a common fixed point of and , we have

(2.1)

Let , by , it is easy to see that , thus and .

Similarly, we have ,

(2.2)

So

(2.3)

Hence, is bounded.

Second, we show that

(2.4)

We recall that Banach space is called uniformly convex if for every , where the modulus of convexity of is defined by

(2.5)

for every with . It is easy to see that Banach space is uniformly convex if and only if for any implies .

Assume that , then there exist a subsequence of and a real number , such that

(2.6)

On the other hand, for a common fixed point of and , we have

(2.7)

Thus,

(2.8)

Because

(2.9)

we know is bounded, then there exists , such that . Thus, .

Furthermore, we have

(2.10)

From

(2.11)

and the fact that is uniformly convex Banach space, there exists , such that

(2.12)

Thus,

(2.13)

Using (2.3), we obtain that

(2.14)

So

(2.15)

Let , then we have . It is a contradiction. Hence, .

Third, we show that

(2.16)

Since

(2.17)

we have

(2.18)

Let , then

(2.19)

So

(2.20)

Using (2.4), we get that

(2.21)

Forth, we show that if the Ishikawa sequence converges to some point , then is the common fixed point of and . By

(2.22)

we have . Since is a convergent sequence, we get . It is easy to see that and . On the other hand,

(2.23)

By (1.1), we obtain

(2.24)

Since

(2.25)

we get

(2.26)

So

(2.27)

Let , Since , we have

(2.28)

It is easy to see that

(2.29)

Note that , then we get

(2.30)

So .

Let , then . By (1.1), we have

(2.31)

Let , then we get

(2.32)

Since , it follows that

(2.33)

Similarly, we can prove that . So is the common fixed point of and .

Finally, we show that is a Cauchy sequence. For any ,

(2.34)

Since , thus we get Simplify, then we have

(2.35)

where , and By (2.16) and (2.30), we know that

(2.36)

So it is easy to see that . Thus, , that is is a Cauchy sequence. Hence, there exists , such that . We know that and is the common fixed point of and . This completes the proof of the theorem.