# Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense

- Xiaolong Qin
^{1}, - SunYoung Cho
^{2}and - JongKyu Kim
^{3}Email author

**2010**:186874

**DOI: **10.1155/2010/186874

© Xiaolong Qin et al. 2010

**Received: **15 October 2009

**Accepted: **23 February 2010

**Published: **30 March 2010

## Abstract

A new nonlinear mapping is introduced. The convergence of Ishikawa iterative processes for the class of asymptotically pseudocontractive mappings in the intermediate sense is studied. Weak convergence theorems are established. A strong convergence theorem is also established without any compact assumption by considering the so-called hybrid projection methods.

## 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 . The symbols and are denoted by strong convergence and weak convergence, respectively. denotes the weak -limit set of . Let be a nonempty closed and convex subset of and a mapping. In this paper, we denote the fixed point set of by .

*nonexpansive*if

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

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex and 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. [2]. It is known [3] that if is a nonempty close convex 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.

*strictly pseudocontractive*if there exists a constant such that

The class of strict pseudocontractions was introduced by Browder and Petryshyn [4] in a real Hilbert space. Marino and Xu [5] proved that the fixed point set of strict pseudocontractions is closed convex, and they also obtained a weak convergence theorem for strictly pseudocontractive mappings by Mann iterative process; see [5] 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 [6] in 1996 (see also [7]). Kim and Xu [8] proved that the fixed point set of asymptotically strict pseudocontractions is closed convex. They also obtained that the class of asymptotically strict pseudocontractions is demiclosed at the origin; see [8, 9] for more details.

*asymptotically strict pseudocontraction in the intermediate sense*if

They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings. Moreover, a strong convergence theorem was also established in a real Hilbert space by considering the so-called hybrid projection methods; see [10] for more details.

*asymptotically pseudocontractive*if there exists a sequence with as such that

The class of asymptotically pseudocontractive mapping was introduced by Schu [11] (see also [12]). In [13], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [13] for more details. In 1991, Schu [11] established the following classical results.

Theorem JS.

then converges strongly to some fixed point of .

Recently, Zhou [14] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point. Moreover, the fixed point set is closed and convex.

In this paper, we introduce and consider the following mapping.

Definition 1.1.

*asymptotically pseudocontractive mapping in the intermediate sense*if

We remark that if for each , then the class of asymptotically pseudocontractive mappings in the intermediate sense is reduced to the class of asymptotically pseudocontractive mappings.

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

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

Lemma 1.2 (see [15]).

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

Lemma 1.3.

From now on, we always use to denotes .

Lemma 1.4.

Let be a nonempty close convex subset of a real Hilbert space and a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences and as defined in (1.15). Then is a closed convex subset of .

Proof.

Letting in (1.23), we obtain that . Since is uniformly -Lipschitz, we see that This completes the proof of the convexity of . From the continuity of , we can also obtain the closedness of . The proof is completed.

Lemma 1.5.

Let be a nonempty close convex subset of a real Hilbert space and a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that is nonempty. Then is demiclosed at zero.

Proof.

Letting in (1.31), we see that . Since is uniformly -Lipschitz, we can obtain that This completes the proof.

## 2. Main Results

Theorem 2.1.

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

(a) , , where for each

(b) for some and some

then the sequence generated by (*) converges weakly to fixed point of .

Proof.

Since is bounded, we see that there exists a subsequence such that . From Lemma 1.5, we see that .

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

Next, we modify Ishikawa iterative processes to obtain a strong convergence theorem without any compact assumption.

Theorem 2.2.

where for each . Assume that the control sequences and are chosen such that for some and some Then the sequence generated in (**) converges strongly to a fixed point of .

Proof.

The proof is divided into seven steps.

Step 1.

Show that is closed and convex for each

It is easy to see that is convex for each . Hence, we obtain that is closed and convex for each This completes Step 1.

Step 2.

Show that for each .

where for each . This implies that for each . That is, for each

which implies that . That is, . This proves that for each . Hence, for each . This completes Step 2.

Step 3.

Show that exists.

This shows that the sequence is nondecreasing. Note that is bounded. It follows that exists. This completes Step 3.

Step 4.

Show that as

Hence, we have as This completes Step 4.

Step 5.

Show that as

This completes Step 5.

Step 6.

Show that as

From Step 5, we can conclude the desired conclusion. This completes Step 6.

Step 7.

Show that , where as

From Lemma of Yanes and Xu [16], we can obtain Step 7. This completes the proof.

Remark 2.3.

The results of Theorem 2.2 are more general which includes the corresponding results of Kim and Xu [17], Marino and Xu [5], Qin et al. [18], Sahu et al. [10], Zhou [14, 19] as special cases.

## Declarations

### Acknowledgment

This work was supported by the Kyungnam University Research Fund 2009.

## 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 - 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 - Liu QH:
**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 - 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 - Qin X, Cho YJ, Kang SM, Shang M:
**A hybrid iterative scheme for asymptotically -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):1902–1911. 10.1016/j.na.2008.02.090MathSciNetView 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 - Kim JK, Nam YM:
**Modified Ishikawa iterative sequences with errors for asymptotically set-valued pseudocontractive mappings in Banach spaces.***Bulletin of the Korean Mathematical Society*2006,**43**(4):847–860.MathSciNetView 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 - 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 - Yanes CM, Xu H-K:
**Strong convergence of the CQ method for fixed point iteration processes.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(5):1140–1152. 10.1016/j.na.2005.05.059MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y, Shang M:
**Strong convergence theorems for asymptotically nonexpansive mappings by hybrid methods.***Kyungpook Mathematical Journal*2008,**48**(1):133–142.MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**343**(1):546–556. 10.1016/j.jmaa.2008.01.045MathSciNetView 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.