- Research Article
- Open Access

# Convergence of Three-Step Iterations Scheme for Nonself Asymptotically Nonexpansive Mappings

- Seyit Temir
^{1}Email author

**2010**:783178

https://doi.org/10.1155/2010/783178

© Seyit Temir. 2010

**Received:**15 February 2010**Accepted:**30 June 2010**Published:**13 July 2010

## Abstract

Weak and strong convergence theorems of three-step iterations are established for nonself asymptotically nonexpansive mappings in uniformly convex Banach space. The results obtained in this paper extend and improve the recent ones announced by Suantai (2005), Khan and Hussain (2008), Nilsrakoo and Saejung (2006), and many others.

## Keywords

- Banach Space
- Nonexpansive Mapping
- Strong Convergence
- Nonempty Closed Convex Subset
- Convex Banach Space

## 1. Introduction

Suppose that is a real uniformly convex Banach space, is a nonempty closed convex subset of . Let be a self-mapping of .

for all .

for all and .

The class of asymptotically nonexpansive maps which is an important generalization of the class nonexpansive maps was introduced by Goebel and Kirk [1]. They proved that every asymptotically nonexpansive self-mapping of a nonempty closed convex bounded subset of a uniformly convex Banach space has a fixed point.

*uniformly*

*-Lipschitzian*if there exists a constant such that , the following inequality holds:

for all .

where , , , , , , , and in satisfy certain conditions.

where , , , , , and in satisfy certain conditions.

Let
be a real normed space and
be a nonempty subset of
. A subset
of
is called a *retract* of
if there exists a continuous map
such that
for all
. Every closed convex subset of a uniformly convex Banach space is a rectract. A map
is called a retraction if
. In particular, a subset
is called a *nonexpansive retract* of
if there exists a *nonexpansive retraction*
such that
for all
.

where and for some .

*asymptotically nonexpansive*if there exists a sequence with such that

for all , and . From the above definition, it is obvious that nonself asymptotically nonexpansive mappings are uniformly -Lipschitzian.

Now, we give the following nonself-version of (1.4):

, where , , , , , , , and in satisfy certain conditions.

The aim of this paper is to prove the weak and strong convergence of the three-step iterative sequence for nonself asymptotically nonexpansive mappings in a real uniformly convex Banach space. The results presented in this paper improve and generalize some recent papers by Suantai [7], Khan and Hussain [10], Nilsrakoo and Saejung [6], and many others.

## 2. Preliminaries

*uniformly convex*if the modulus of convexity of is as follows:

for all (i.e., is a function

*Opial's condition*[13] if, for each sequence in , the condition weakly as and for all with implies that

Lemma 2.1 (see [12]).

Let be a uniformly convex Banach space, a nonempty closed convex subset of and a nonself asymptotically nonexpansive mapping with a sequence and , then is demiclosed at zero.

Lemma 2.2 (see [12]).

Let be a real uniformly convex Banach space, a nonempty closed subset of with as a sunny nonexpansive retraction and a mapping satisfying weakly inward condition, then .

Lemma 2.3 (see [14]).

Let , , and be sequences of nonnegative real sequences satisfying the following conditions: , , where and , then exists.

Lemma 2.4 (see [6]).

for all , and with .

Lemma 2.5 (See [7], Lemma ).

Let be a Banach space which satisfies Opial's condition and let be a sequence in . Let be such that and . If , are the subsequences of which converge weakly to , respectively, then .

## 3. Main Results

In this section, we prove theorems of weak and strong of the three-step iterative scheme given in (1.12) to a fixed point for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. In order to prove our main results the followings lemmas are needed.

Lemma 3.1.

If and are sequences in such that and is sequence of real numbers with for all and , then there exists a positive integer and such that for all .

Proof.

from which we have . Put , then we have for all .

Lemma 3.2.

Let be a real Banach space and a nonempty closed and convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12), then we have, for any , exists.

Proof.

Since and from Lemma 2.3, it follows that exits.

Lemma 3.3.

Let be a real uniformly convex Banach space and a nonempty closed and convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12), then one has the following conclusions.

If , then

If either or and , then

If the following conditions

,

either and or and are satisfied, then

Proof.

Let .

Therefore, the assumption implies that .

This completes the proof.

Next, we show that .

Lemma 3.4.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12) with the following restrictions:

and ,

,

then .

Proof.

In the next result, we prove our first strong convergence theorem as follows.

Theorem 3.5.

Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12) with the following restrictions:

and ,

.

If, in addition, is either completely continuous or demicompact, then converges strongly to a fixed point of .

Proof.

It folows that . Moreover, since exists, then , that is, converges strongly to a fixed point of .

We assume that is demicompact. Then, using the same ideas and argument, we also prove that converges strongly to a fixed point of .

Finally, we prove the weak convergence of the iterative scheme (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.

Theorem 3.6.

Let be a real uniformly convex Banach space satisfying Opial's condition and a nonempty closed convex subset of . Let be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set and a sequence of real numbers such that and . Let , , , , , and be real sequences in , such that and in for all . Let be a sequence in defined by (1.12) with the following restrictions:

and ,

,

then converges weakly to a fixed point of .

Proof.

Let . Then as in Lemma 3.2, exists. We prove that has a unique weak subsequential limit in . We assume that and are weak limits of the subsequences , , or , respectively. By Lemma 3.4, and is demiclosed by Lemma 2.1, and in the same way, . Therefore, we have . It follows from Lemma 2.5 that . Thus, converges weakly to an element of This completes the proof.

## Authors’ Affiliations

## References

- Goebel K, Kirk WA:
**A fixed point theorem for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1972,**35:**171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar - Schu J:
**Weak and strong convergence to fixed points of asymptotically nonexpansive mappings.***Bulletin of the Australian Mathematical Society*1991,**43**(1):153–159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar - Noor MA:
**New approximation schemes for general variational inequalities.***Journal of Mathematical Analysis and Applications*2000,**251**(1):217–229. 10.1006/jmaa.2000.7042MathSciNetView ArticleMATHGoogle Scholar - Glowinski R, Le Tallec P:
*Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics*.*Volume 9*. SIAM, Philadelphia, Pa, USA; 1989:x+295.View ArticleMATHGoogle Scholar - Xu B, Noor MA:
**Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*2002,**267**(2):444–453. 10.1006/jmaa.2001.7649MathSciNetView ArticleMATHGoogle Scholar - Nilsrakoo W, Saejung S:
**A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings.***Applied Mathematics and Computation*2006,**181**(2):1026–1034. 10.1016/j.amc.2006.01.063MathSciNetView ArticleMATHGoogle Scholar - Suantai S:
**Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2005,**311**(2):506–517. 10.1016/j.jmaa.2005.03.002MathSciNetView ArticleMATHGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Khan SH, Hussain N:
**Convergence theorems for nonself asymptotically nonexpansive mappings.***Computers & Mathematics with Applications*2008,**55**(11):2544–2553. 10.1016/j.camwa.2007.10.007MathSciNetView ArticleMATHGoogle Scholar - Wang L:
**Strong and weak convergence theorems for common fixed point of nonself asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2006,**323**(1):550–557. 10.1016/j.jmaa.2005.10.062MathSciNetView ArticleMATHGoogle Scholar - Chidume CE, Ofoedu EU, Zegeye H:
**Strong and weak convergence theorems for asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2003,**280**(2):364–374. 10.1016/S0022-247X(03)00061-1MathSciNetView ArticleMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar

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