Abstract and Applied Analysis

Volume 2012, Article ID 804745, 7 pages

http://dx.doi.org/10.1155/2012/804745

## Strong Convergence of an Implicit -Iterative Process for Lipschitzian Hemicontractive Mappings

^{1}Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{2}School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg III, Lahore 54660, Pakistan

Received 22 October 2012; Accepted 15 November 2012

Academic Editor: Yongfu Su

Copyright © 2012 Shin Min Kang 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

We establish the strong convergence for the implicit -iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

#### 1. Introduction

Let be a Hilbert space and let be a mapping.

The mapping is called *Lipshitzian* if there exists such that

If , then is called *nonexpansive* and if , then is called *contractive*.

The mapping is said to be *pseudocontractive* ([1, 2]) if
and the mapping is said to be *strongly pseudocontractive* if there exists such that

Let and the mapping is called *hemicontractive* if and

It is easy to see the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings. For the importance of fixed points of pseudocontractions the reader may consult [1].

In 1974, Ishikawa [3] proved the following result.

Theorem 1.1. *Let be a compact convex subset of a Hilbert space and let be a Lipschitzian pseudocontractive mapping.**
For arbitrary , let be a sequence defined iteratively by**
where and are sequences satisfying the conditions:*(i) *,
*(ii) *,
*(iii) *. **
Then the sequence converges strongly to a fixed point of . *

Another iteration scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings.

In 2011, Sahu [4] and Sahu and Petruşel [5] introduced the -iterative process as follows.

Let be a nonempty convex subset of a normed space and let be a mapping. Then, for arbitrary , the -iterative process is defined by where is a real sequence in .

In this paper, we establish the strong convergence for the implicit -iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

#### 2. Main Results

We need the follwing lemma.

Lemma 2.1 (see [6]). *For all , and , the following well-known identity holds
*

Now we prove our main results.

Theorem 2.2. *Let be a compact convex subset of a real Hilbert space and let be a Lipschitzian hemicontractive mapping satisfying
**
Let be a sequence in satisfying*(iv)* ,*(v)* .**For arbitrary , let be a sequence defined iteratively by
**
Then the sequence converges strongly to the fixed point of .*

*Proof. *From Schauder’s fixed point theorem, is nonempty since is a convex compact set and is continuous, let . Using the fact that is hemicontractive we obtain

Now by (v), there exists such that for all ,
which implies that

With the help of (2.2), (2.3), and Lemma 2.1, we obtain the following estimates:

Substituting (2.7) in (2.4) we obtain

Also with the help of condition and (2.8), we have
which implies that
where
and consequently from (2.12), we obtain

Hence by (2.5), (2.10), (2.11), and (2.13), we have
which implies that
so that

Hence by conditions (iv) and (v), we get

It implies that

Consider
which implies that

The rest of the argument follows exactly as in the proof of Theorem of [3]. This completes the proof.

Theorem 2.3. *Let be a compact convex subset of a real Hilbert space and let be a Lipschitzian hemicontractive mapping satisfying the condition . Let be a sequence in satisfying the conditions (iv) and (v).**Assume that be the projection operator of onto . Let be a sequence defined iteratively by
**
Then the sequence converges strongly to a fixed point of . *

*Proof. *The operator is nonexpansive (see, e.g., [2]). is a Chebyshev subset of so that, is a single-valued mapping. Hence, we have the following estimate:

The set is compact and so the sequence is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof.

*Remark 2.4. *In main results, the condition is not new and it is due to Liu et al. [7].

#### Acknowledgment

The authors would like to thank the referees for thier useful comments and suggestions.

#### References

- F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in
*Nonlinear Functional Analysis*, American Mathematical Society, Providence, RI, USA, 1976. View at Google Scholar · View at Zentralblatt MATH - 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 - S. Ishikawa, “Fixed points by a new iteration method,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 147–150, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. R. Sahu, “Applications of the
*S*-iteration process to constrained minimization problems and split feasibility problems,”*Fixed Point Theory*, vol. 12, no. 1, pp. 187–204, 2011. View at Google Scholar - D. R. Sahu and A. Petruşel, “Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 74, no. 17, pp. 6012–6023, 2011. View at Publisher · View at Google Scholar - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Liu, C. Feng, J. S. Ume, and S. M. Kang, “Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings,”
*Taiwanese Journal of Mathematics*, vol. 11, no. 1, pp. 27–42, 2007. View at Google Scholar · View at Zentralblatt MATH