Journal of Applied Mathematics

Volume 2013 (2013), Article ID 705814, 4 pages

http://dx.doi.org/10.1155/2013/705814

Research Article

Strong Convergence for Hybrid -Iteration Scheme

Shin Min Kang,¹ Arif Rafiq,² and Young Chel Kwun³

¹Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

²School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore 54660, Pakistan

³Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 19 November 2012; Accepted 4 February 2013

Academic Editor: D. R. Sahu

Copyright © 2013 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 a strong convergence for the hybrid -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.

1. Introduction and Preliminaries

Let be a real Banach space and let be a nonempty convex subset of . Let denote the normalized duality mapping from to defined by

where denotes the dual space of and denotes the generalized duality pairing. We will denote the single-valued duality map by .

Let be a mapping.

Definition 1. The mapping is said to be Lipschitzian if there exists a constant such that

Definition 2. The mapping is said to be nonexpansive if

Definition 3. The mapping is said to be pseudocontractive if for all , there exists such that

Definition 4. The mapping is said to be strongly pseudocontractive if for all , there exists such that

Let be a nonempty convex subset of a normed space .

(a)The sequence defined by, for arbitrary ,

where and are sequences in , is known as the Ishikawa iteration process [1]. If for , then the Ishikawa iteration process becomes the Mann iteration process [2].(b)The sequence defined by, for arbitrary ,

where is a sequence in , is known as the -iteration process [3, 4].

In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration scheme (see, e.g., [1]). Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [5–10] and the references cited therein).

In 1974, Ishikawa [1] proved the following result.

Theorem 5. 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

(i), (ii)(iii)

Then the sequence converges strongly at a fixed point of .

In [6], Chidume extended the results of Schu [9] from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of (strongly) pseudocontractive mappings.

In [11], Zhou and Jia gave the more general answer of the question raised by Chidume [5] and proved the following.

If is a real Banach space with a uniformly convex dual , is a nonempty bounded closed convex subset of , and is a continuous strongly pseudocontractive mapping, then the Ishikawa iteration scheme converges strongly at the unique fixed point of .

In this paper, we establish the strong convergence for the hybrid -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces. We also improve the result of Zhou and Jia [11].

2. Main Results

We will need the following lemmas.

Lemma 6 (see [12]). Let be the normalized duality mapping. Then for any , one has

Lemma 7 (see [10]). Let be nonnegative sequence satisfying

where ,, and . Then

The following is our main result.

Theorem 8. Let be a nonempty closed convex subset of a real Banach space , let be nonexpansive, and let be Lipschitz strongly pseudocontractive mappings such that and

Let be a sequence in satisfying

(iv) (v) 

For arbitrary , let be a sequence iteratively defined by

Then the sequence converges strongly at the common fixed point of and .

Proof. For strongly pseudocontractive mappings, the existence of a fixed point follows from Delmling [13]. It is shown in [11] that the set of fixed points for strongly pseudocontractions is a singleton.

By (v), since , there exists such that for all ,

where . Consider

which implies that

where

and consequently from (16), we obtain

Substituting (18) in (15) and using (13), we get

So, from the above discussion, we can conclude that the sequence is bounded. Since is Lipschitzian, so is also bounded. Let . Also by (ii), we have

as , implying that is bounded, so let . Further,

which implies that is bounded. Therefore, is also bounded.

Set

Denote . Obviously, .

Now from (12) for all , we obtain

and by Lemma 6, we get

which implies that

because by (13), we have and . Hence, (23) gives us

For all , put

then according to Lemma 7, we obtain from (26) that

This completes the proof.

Corollary 9. Let be a nonempty closed convex subset of a real Hilbert space , let be nonexpansive, and let be Lipschitz strongly pseudocontractive mappings such that and the condition . Let be a sequence in satisfying the conditions (iv) and (v).

For arbitrary , let be a sequence iteratively defined by (12). Then the sequence converges strongly at the common fixed point of and .

Example 10. As a particular case, we may choose, for instance, .

Remark 11. (1) The condition is not new and it is due to Liu et al. [14].

(2) We prove our results for a hybrid iteration scheme, which is simple in comparison to the previously known iteration schemes.

Acknowledgment

This study was supported by research funds from Dong-A University.

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