- 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]).

## 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:

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

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## 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.