- Research Article
- Open Access

# A Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

- Xiaolong Qin
^{1}, - SunYoung Cho
^{2}and - ShinMin Kang
^{3}Email author

**2011**:859795

https://doi.org/10.1155/2011/859795

© Xiaolong Qin et al. 2011

**Received:**13 December 2010**Accepted:**1 February 2011**Published:**27 February 2011

## Abstract

The modified Ishikawa iterative process is investigated for the class of total asymptotically pseudocontractive mappings. A weak convergence theorem of fixed points is established in the framework of Hilbert spaces.

## Keywords

- Hilbert Space
- Banach Space
- Convex Subset
- Nonexpansive Mapping
- Real Hilbert Space

## 1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space, whose inner product and norm are denoted by and . and are denoted by strong convergence and weak convergence, respectively. Let be a nonempty closed convex subset of and a mapping. In this paper, we denote the fixed point set of by .

Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point.

where is a continuous and nondecreasing function such that is positive on , , and . We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex bounded subset of a real uniformly convex Banach space and is an asymptotically nonexpansive mapping on , then has a fixed point.

*asymptotically nonexpansive in the intermediate sense*if it is continuous and the following inequality holds:

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [4] (see also [5]). It is known [6] that if is a nonempty closed convex bounded subset of a uniformly convex Banach space and is asymptotically nonexpansive in the intermediate sense, then has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous; see [5, 7].

where is a continuous and strictly increasing function with and and are nonnegative real sequences such that and as . The class of mapping was introduced by Alber et al. [8]. From the definition, we see that the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings and the class of asymptotically nonexpansive mappings in the intermediate sense as special cases; see [9, 10] for more details.

The class of strict pseudocontractions was introduced by Browder and Petryshyn [11] in a real Hilbert space. In 2007, Marino and Xu [12] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings; see [12] for more details.

*asymptotically strict pseudocontraction*if there exist a constant and a sequence with as such that

The class of asymptotically strict pseudocontractions was introduced by Qihou [13] in 1996. Kim and Xu [14] proved that the class of asymptotically strict pseudocontractions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [14] for more details.

*asymptotically strict pseudocontraction in the intermediate sense*if there exist a constant and a sequence with as such that

The class of mappings was introduced by Sahu et al. [15]. They proved that the class of asymptotically strict pseudocontractions in the intermediate sense is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [15] for more details.

The class of asymptotically pseudocontractive mapping was introduced by Schu [16] (see also [17]). In [18], Rhoades gave an example to showed that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [18] for more details. Zhou [19] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point.

*asymptotically pseudocontractive mapping in the intermediate sense*if there exists a sequence with as such

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin et al. [20]. Weak convergence theorems of fixed points were established based on iterative methods; see [20] for more details.

In this paper, we introduce the following mapping.

Definition 1.1.

*total asymptotically pseudocontractive*if there exist sequences and with and as such that

where is a continuous and strictly increasing function with .

Remark 1.2.

Remark 1.3.

If , then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense.

where is a mapping, is an initial value, and and are real sequences in .

The purpose of this paper is to consider total asymptotically pseudocontractive mappings based on the modified Ishikawa iterative process. Weak convergence theorems are established in real Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma 1.4.

Lemma 1.5 (see [21]).

where is some nonnegative integer. If and , then exists.

## 2. Main Results

Now, we are ready to give our main results.

Theorem 2.1.

where and are sequences in . Assume that the following restrictions are satisfied:

(a) and ,

(b) for some and some .

Then, the sequence generated in (2.1) converges weakly to fixed point of .

Proof.

Letting in (2.24), we see that . Since is uniformly -Lipschitz, we can obtain that .

This derives a contradiction. It follows that . This completes the proof.

Remark 2.2.

Demiclosedness principle of the class of total asymptotically pseudocontractive mappings can be deduced from Theorem 2.1.

Remark 2.3.

Since the class of total asymptotically pseudocontractive mappings includes the class of strict pseudocontractions, the class of asymptotically strict pseudocontractions, the class of pseudocontractive mappings, the class of asymptotically pseudocontractive mappings and the class of asymptotically pseudocontractive mappings in the intermediate sense as special cases, Theorem 2.1 improves the corresponding results in Marino and Xu [12], Kim and Xu [14], Sahu et al. [15], Schu [16], Zhou [19], and Qin et al. [20].

Remark 2.4.

It is of interest to improve the main results of this paper to a Banach space.

## Declarations

### Acknowledgment

The authors thank the referees for useful comments and suggestions.

## Authors’ Affiliations

## References

- Alber YaI, Guerre-Delabriere S:
**On the projection methods for fixed point problems.***Analysis*2001,**21**(1):17–39.MathSciNetView ArticleMATHGoogle Scholar - Rhoades BE:
**Some theorems on weakly contractive maps.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(4):2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleMATHGoogle Scholar - 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 - Bruck R, Kuczumow T, Reich S:
**Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property.***Colloquium Mathematicum*1993,**65**(2):169–179.MathSciNetMATHGoogle Scholar - Kirk WA:
**Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type.***Israel Journal of Mathematics*1974,**17:**339–346. 10.1007/BF02757136MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Existence and convergence for fixed points of mappings of asymptotically nonexpansive type.***Nonlinear Analysis: Theory, Methods & Applications*1991,**16**(12):1139–1146. 10.1016/0362-546X(91)90201-BMathSciNetView ArticleMATHGoogle Scholar - Liu Z, Kim JK, Kim KH:
**Convergence theorems and stability problems of the modified Ishikawa iterative sequences for strictly successively hemicontractive mappings.***Bulletin of the Korean Mathematical Society*2002,**39**(3):455–469.MathSciNetView ArticleMATHGoogle Scholar - Alber YaI, Chidume CE, Zegeye H:
**Approximating fixed points of total asymptotically nonexpansive mappings.***Fixed Point Theory and Applications*2006,**2006:**-20.Google Scholar - Chidume CE, Ofoedu EU:
**A new iteration process for approximation of common fixed points for finite families of total asymptotically nonexpansive mappings.***International Journal of Mathematics and Mathematical Sciences*2009,**2009:**-17.Google Scholar - Chidume CE, Ofoedu EU:
**Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2007,**333**(1):128–141. 10.1016/j.jmaa.2006.09.023MathSciNetView ArticleMATHGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Qihou L:
**Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings.***Nonlinear Analysis: Theory, Methods & Applications*1996,**26**(11):1835–1842. 10.1016/0362-546X(94)00351-HMathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(9):2828–2836. 10.1016/j.na.2007.02.029MathSciNetView ArticleMATHGoogle Scholar - Sahu DR, Xu H-K, Yao J-C:
**Asymptotically strict pseudocontractive mappings in the intermediate sense.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(10):3502–3511. 10.1016/j.na.2008.07.007MathSciNetView ArticleMATHGoogle Scholar - Schu J:
**Iterative construction of fixed points of asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1991,**158**(2):407–413. 10.1016/0022-247X(91)90245-UMathSciNetView 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 - Rhoades BE:
**Comments on two fixed point iteration methods.***Journal of Mathematical Analysis and Applications*1976,**56**(3):741–750. 10.1016/0022-247X(76)90038-XMathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(9):3140–3145. 10.1016/j.na.2008.04.017MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho SY, Kim JK:
**Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense.***Fixed Point Theory and Applications*2010,**2010:**-14.Google 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

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.