International Journal of Mathematics and Mathematical Sciences

Volume 2011, Article ID 409898, 10 pages

http://dx.doi.org/10.1155/2011/409898

## Strong and Weak Convergence Theorems for an Infinite Family of Lipschitzian Pseudocontraction Mappings in Banach Spaces

^{1}Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China^{2}Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 16 December 2010; Accepted 9 February 2011

Academic Editor: Giuseppe Marino

Copyright © 2011 Shih-sen Chang et al. 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.

#### Abstract

The purpose of this paper is to study the strong and weak convergence theorems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces.

#### 1. Introduction and Preliminaries

Throughout this paper, we assume that is a real Banach space, is the dual space of , is a nonempty closed convex subset of , is the set of nonnegative real numbers, and is the normalized duality mapping defined by
Let be a mapping. We use to denote the set of fixed points of . We also use “” to stand for strong convergence and “” for weak convergence. For a given sequence , let denote the *weak **-limit set*, that is,

*Definition 1.1. *(1) A mapping is said to be *pseudocontraction* [1], if for any , there exists such that

It is well known that [1] the condition (1.3) is equivalent to the following:
for all and all .

(2) is said to be *strongly pseudocontractive*, if there exists such that
for each and for some .

(3) is said to be *strictly pseudocontractive in the terminology of Browder and Petryshyn [1]*, if there exists such that
for every and for some .

In this case, we say is a *-strictly pseudocontractive mapping*.

(4) is said to be -*Lipschitzian, if there exists ** such that *

*Remark 1.2. *It is easy to see that if is a -strictly pseudocontractive mapping, then it is a -Lipschitzian mapping.

In fact, it follows from (1.6) that for any ,
Simplifying it, we have
that is,

Lemma 1.3 (see [2, Theorem 13.1] or [3]). *Let be a real Banach space, be a nonempty closed convex subset of , and be a continuous strongly pseudocontractive mapping. Then has a unique fixed point in .*

*Remark 1.4. *Let be a real Banach space, be a nonempty closed convex subset of and be a Lipschitzian pseudocontraction mapping. For every given and , define a mapping by
It is easy to see that is a continuous strongly pseudocontraction mapping. By using Lemma 1.3, there exists a unique fixed point of such that

The concept of pseudocontractive mappings is closely related to accretive operators. It is known that is pseudocontractive if and only if is accretive, where is the identity mapping. The importance of accretive mappings is from their connection with theory of solutions for nonlinear evolution equations in Banach spaces. Many kinds of equations, for example, Heat, wave, or Schrödinger equations can be modeled in terms of an initial value problem:
where is a pseudocontractive mapping in an appropriate Banach space.

In order to approximate a fixed point of Lipschitzian pseudocontractive mapping, in 1974, Ishikawa introduced a new iteration (it is called *Ishikawa iteration*). Since then, a question of whether or not the Ishikawa iteration can be replaced by the simpler Mann iteration has remained open. Recently Chidume and Mutangadura [4] solved this problem by constructing an example of a Lipschitzian pseudocontractive mapping with a unique fixed point for which every Mann-type iteration fails to converge.

Inspired by the implicit iteration introduced by Xu and Ori [5] for a finite family of nonexpansive mappings in a Hilbert space, Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9] proposed and studied convergence theorems for an implicit iteration process for a finite or infinite family of continuous pseudocontractive mappings.

The purpose of this paper is to study the strong and weak convergence problems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces. The results presented in this paper extend and improve some recent results of Xu and Ori [5], Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9].

For this purpose, we first recall some concepts and conclusions.

A Banach space is said to be uniformly convex, if for each , there exists a such that for any with , and , holds. The modulus of convexity of is defined by

Concerning the modulus of convexity of , Goebel and Kirk [10] proved the following result.

Lemma 1.5 (see [10, Lemma 10.1]). * Let be a uniformly convex Banach space with a modulus of convexity . Then is continuous, increasing, , for and
**
for all , and with . ** A Banach space is said to satisfy the Opial condition, if for any sequence with , then the following inequality holds: **
for any with .*

Lemma 1.6 (Zhou [8]). *Let be a real reflexive Banach space with Opial condition. Let be a nonempty closed convex subset of and be a continuous pseudocontractive mapping. Then is demiclosed at zero, that is, for any sequence , if and , then . *

Lemma 1.7 (Chang [11]). *Let be the normalized duality mapping, then for any ,
*

*Definition 1.8 (see [12]). *Let be a family of mappings with . We say satisfies the *AKTT-condition*, if for each bounded subset of the following holds:

*Lemma 1.9 (see [12]). Suppose that the family of mappings satisfies the AKTT-condition. Then for each , converges strongly to a point in . Moreover, let be the mapping defined by
Then, for each bounded subset , .*

*2. Main Results*

*Theorem 2.1. Let be a uniformly convex Banach space with a modulus of convexity , and be a nonempty closed convex subset of . Let be a family of -Lipschitzian and pseudocontractive mappings with and . Let be the sequence defined by
where is a sequence in . If the following conditions are satisfied: (i); (ii)there exists a compact subset such that ; (iii) satisfies the AKTT-condition, and , where is the mapping defined by (1.19). Then converges strongly to some point *

*Proof. * First, we note that, by Remark 1.4, the method is well defined. So, we can divide the proof in three steps. (I) For each the limit exists. In fact, since is pseudocontractive, for each , we have
Simplifying, we have that
Consequently, the limit exists, and so the sequence is bounded. (II) Now, we prove that . In fact, by virtue of (2.1) and (1.4), we have
Letting and , from (2.3), we know that , . It follows from (2.4) and Lemma 1.5 that
Simplifying, we have that
This implies that
Letting , if , the conclusion of Theorem 2.1 is proved. If , it follows from the property of modulus of convexity that . Therefore, from (2.1) and the condition (i), we have that
In view of (2.1) and (2.8), we have
(III) Now, we prove that converges strongly to some point in . In fact, it follows from (2.9) and condition (ii) that there exists a subsequence such that (as ), and (some point in ). Furthermore, by Lemma 1.9, we have . consequently, we have
This implies that , that is, . Since and the limit exists, we have .

This completes the proof of Theorem 2.1.

*Theorem 2.2. Let be a uniformly convex Banach space satisfying the Opial condition. Let be a nonempty closed convex subset of and be a family of -Lipschitzian pseudocontractive mappings with and . Let be the sequence defined by (2.1) and be a sequence in (0, 1). If the following conditions are satisfied: (i), (ii)for any bounded subset of Then the sequence converges weakly to some point .*

*Proof. * By the same method as given in the proof of Theorem 2.1, we can prove that the sequence is bounded and

Now, we prove that

Indeed, for each , we have
By (2.12) and condition (ii), we have
The conclusion of (2.13) is proved.

Finally, we prove that converges weakly to some point .

In fact, since is uniformly convex, and so it is reflexive. Again since is bounded, there exists a subsequence such that . Hence from (2.13), for any , we have
By virtue of Lemma 1.6, , for all . This implies that

Next, we prove that is a singleton. Let us suppose, to the contrary, that if there exists a subsequence such that and . By the same method as given above we can also prove that . Taking and in (2.12). We know that the following limits
exist. Since satisfies the Opial condition, we have
This is a contradiction, which shows that . Hence,
This implies that .

This completes the proof of Theorem 2.2.

*In the next lemma, we propose a sequence of mappings that satisfy condition (iii) in Theorem 2.1. Moreover, we apply this lemma to obtain a corollary of our main Theorem 2.1. *

* Let be a Banach space and be a nonempty closed convex subset of . From Definition 1.1(3), we know that if is a -strictly pseudocontractive mapping, then it is a -Lipschitzian pseudocontractive mapping. *

* On the other hand, by the same proof as given in [12] we can prove the following result. *

*Lemma 2.3 (see [12] or [9]). Let be a smooth Banach space, be a closed convex subset of . Let be a family of -strictly pseudocontractive mappings with and . For each define a mapping by:
where is sequence of nonnegative real numbers satisfying the following conditions: (i), for all ; (ii), for all ; (iii). Then, (1)each , is a -strictly pseudocontractive mapping; (2) satisfies the AKTT-condition; (3)if is the mapping defined by
Then and . *

*The following result can be obtained from Theorem 2.1 and Lemma 2.3 immediately. *

*Theorem 2.4. Let be a uniformly convex Banach space, be a nonempty closed convex subset of . Let be a family of -strictly pseudocontractive mappings with and . For each define a mapping by
where is a sequence of nonnegative real numbers satisfying the following conditions: (i), for all ; (ii), for all ; (iii). *

Let be the sequence defined by where is a sequence in . If the following conditions are satisfied: (i); (ii)there exists a compact subset such that . Then, converges strongly to some point .

*Proof. * Since is a family of -strictly pseudocontractive mappings with . Therefore, is a family of -strictly pseudocontractive mappings. By Remark 1.2, is a family of -Lipschitzian and strictly pseudocontractive mappings. Hence, by Lemma 2.3, defined by (2.21) is a family of -Lipschitzian, strictly pseudocontractive mappings with and it has also the following properties: (1) satisfies the AKTT-condition; (2)if is the mapping defined by (2.22), then and . Hence, by Definition 1.1, is also a family of -Lipschitzian and pseudocontractive mappings having the properties (1) and (2) and . Therefore, satisfies all the conditions in Theorem 2.1. By Theorem 2.1, the sequence converges strongly to some point .

This completes the proof of Theorem 2.4.

*Acknowledgment*

*Acknowledgment*

*This paper was supported by the Natural Science Foundation of Yibin University (no. 2009Z01).*

*References*

*References*

- F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 20, pp. 197–228, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Deimling,
*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. - S. Chang, Y. J. Cho, and H. Zhou,
*Iterative Methods for Nonlinear Operator Equations in Banach Spaces*, Nova Science, Huntington, NY, USA, 2002. - C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz pseudocontractions,”
*Proceedings of the American Mathematical Society*, vol. 129, no. 8, pp. 2359–2363, 2001. View at Publisher · View at Google Scholar - H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,”
*Numerical Functional Analysis and Optimization*, vol. 22, no. 5-6, pp. 767–773, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 294, no. 1, pp. 73–81, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Chen, Y. Song, and H. Zhou, “Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 314, no. 2, pp. 701–709, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zhou, “Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 10, pp. 2977–2983, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Boonchari and S. Saejung, “Construction of common fixed points of a countable family of $\lambda $-demicontractive mappings in arbitrary Banach spaces,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 173–178, 2010. View at Publisher · View at Google Scholar - K. Goebel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar - S.-S. Chang, “Some problems and results in the study of nonlinear analysis,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 30, no. 7, pp. 4197–4208, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH

*
*