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

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

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Copyright

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.