# Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

- Jing Zhao
^{1, 2}Email author and - Songnian He
^{1, 2}

**2010**:281070

https://doi.org/10.1155/2010/281070

© Jing Zhao and Songnian He. 2010

**Received: **23 June 2010

**Accepted: **19 October 2010

**Published: **24 November 2010

## Abstract

We study the convergence of Ishikawa iteration process for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others.

## 1. Introduction and Preliminaries

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] 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 contains properly the class of asymptotically nonexpansive mappings.

The class of asymptotically -strict pseudocontractions was introduced by Qihou [4] in 1996 (see also [5]). Kim and Xu [6] studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .

They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see [7] for more details.

In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense.

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

Lemma 1.2 (see [10]).

Let be a bounded sequence in a reflexive Banach space . If , then .

Lemma 1.3 (see [11]).

Let be a nonempty closed convex subset of a real Hilbert space . Given and , then if and only if , for all

Lemma 1.4 (see [11]).

For a real Hilbert space , the following identities hold:

Lemma 1.5 (see [7]).

Lemma 1.6.

Proof.

Lemma 1.7 (see [7]).

Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as , then as .

Lemma 1.8 (see [7, Proposition ]).

Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 1.9 (see [7]).

Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then is closed and convex.

## 2. Main Results

Theorem 2.1.

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

Then the sequence given by (2.1) converges weakly to an element of .

Proof.

By the boundedness of , there exist a subsequence of such that . Observe that is uniformly continuous and as , for any we have as . From Lemma 1.8, we see that .

which is a contradiction. We see and hence is a singleton. Thus, converges weakly to by Lemma 1.2.

Corollary 2.2.

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

Then the sequence given by (2.15) converges weakly to an element of .

Next, we modify Ishikawa iterative process to get a strong convergence theorem.

Theorem 2.3.

where , , and for each . Assume that the control sequences and are chosen such that for some and . Then the sequence given by (2.16) converges strongly to an element of .

Proof.

We break the proof into six steps.

Step 1 ( is closed and convex for each ).

it is easy to see that is convex for each . Hence, is closed and convex for each .

where , , and for each . Hence for each .

This implies that . That is, . By the principle of mathematical induction, we get and hence for all . This means that the iteration algorithm (2.16) is well defined.

Step 3 ( exists and is bounded).

This shows that the sequence is bounded. Therefore, the limit of exists and is bounded.

It follows from Step 4, (2.30) and Lemma 1.7 that as .

Since is reflexive and is bounded, we get that is nonempty. First, we show that is a singleton. Assume that is subsequence of such that . Observe that is uniformly continuous and as , for any we have as . From Lemma 1.8, we see that .

Hence by the uniqueness of the nearest point projection of onto . Since is an arbitrary weakly convergent subsequence, it follows that and hence . It is easy to see as (2.34) that . Since has the Kadec-Klee property, we obtain that , that is, as . This completes the proof.

## Declarations

### Acknowledgments

This research is supported by Fundamental Research Funds for the Central Universities (ZXH2009D021) and supported by the Science Research Foundation Program in Civil Aviation University of China (no. 09CAUC-S05) as well.

## 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 - Bruck RE, 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 - 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 - Tian YX, Chang S-S, Huang J, Wang X, Kim JK:
**Implicit iteration process for common fixed points of strictly asymptotically pseudocontractive mappings in Banach spaces.***Fixed Point Theory and Applications*2008,**2008:**-12.Google 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 - Osilike MO, Aniagbosor SC:
**Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings.***Mathematical and Computer Modelling*2000,**32**(10):1181–1191. 10.1016/S0895-7177(00)00199-0MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu H-K:
**The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1992,**114**(2):399–404. 10.1090/S0002-9939-1992-1068133-2MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D, Sahu DR:
**Iterative construction of fixed points of nearly asymptotically nonexpansive mappings.***Journal of Nonlinear and Convex Analysis*2007,**8**(1):61–79.MathSciNetMATHGoogle 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

## 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.