Journal of Applied Mathematics

Volume 2012 (2012), Article ID 902601, 8 pages

doi:10.1155/2012/902601

Research Article

On Convergence Results for Lipschitz Pseudocontractive Mappings

Shin Min Kang¹ and Arif Rafiq²

¹Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

²Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan

Received 4 June 2012; Revised 3 September 2012; Accepted 3 September 2012

Academic Editor: Alicia Cordero

Copyright © 2012 Shin Min Kang and Arif Rafiq. 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 Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.

1. Introduction and Preliminaries

Let be a real Banach space and 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 mapping by .

Let be a mapping with domain in .

Definition 1.1. is said to be Lipschitz if there exists a constant such that

Definition 1.2. is said to be nonexpansive if

Definition 1.3. is said to be pseudocontractive if for all , there exists such that

Remark 1.4. It is well known that every nonexpansive mapping is pseudocontractive. Indeed, if is nonexpansive mapping, then for all ,, there exists such that

Rhoades [1] showed that the class of pseudocontractive mappings properly contains the class of nonexpansive mappings.

The class of pseudocontractions is, perhaps, the most important generalization of the class of nonexpansive mappings because of its strong relationship with the class of accretive mappings. A mapping is accretive if and only if is pseudocontractive.

For a nonempty convex subset of a normed space and a mapping .

The Mann iteration scheme [2]: the sequence is defined by

where is a sequence in .

The Ishikawa iteration scheme [3]: the sequence is defined by

where and are sequences in .

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 (e.g., [3]). Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (e.g., [4–6] and the references cited therein).

In 1974, Ishikawa [3] introduced an iteration scheme which, in some sense, is more general than that of Mann and which converges, under this setting, to a fixed point of . He proved the following result.

Theorem 1.5. Let be a compact convex subset of a Hilbert space and let be a Lipschitz pseudocontractive mapping. For arbitrary , let be a sequence defined iteratively by

where and are sequences satisfying conditions

(i); (ii); (iii).

Then converges strongly to a fixed point of   .

In [4], Chidume extended the results of Schu [7] from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of strongly pseudocontractive mappings.

In this paper, we establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.

2. Main Results

We will need the following results.

Lemma 2.1 2.1 (see [8]). Let be the normalized duality mapping. Then, for any , one has

Lemma 2.2 2.2 (see [9]). If there exists a positive integer such that for all ,   (the set of all positive integers)

where ,   and , then

We now prove our main results.

Theorem 2.3. Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive mapping such that . Let and be sequences in satisfying the conditions:

(iv)(v); (vi).

For arbitrary , let be defined iteratively by

Then the following conditions are equivalent:

(a) converges strongly to the fixed point of .(b) and are bounded.

Proof. Because is a fixed point of , then the set of fixed points of is nonempty.

Suppose that , then since is Lipschitz, so

which implies that . Therefore and are bounded.

Set

Obviously .

It is clear that . Let . Next we will prove that .

Consider

So, from the above discussion, we can conclude that the sequence is bounded. Let .

Denote . Obviously .

Now from Lemma 2.1 we obtain for all

where

Using (2.4) we have

From the conditions and (2.10), we obtain

and since is Lipschitz,

thus, we have

The real function defined by is increasing and convex. For all and ,   we have

Consider

Substituting (2.15) in (2.8), we get

where . Now, with the help of ,   , (2.13), and Lemma 2.2, we obtain from (2.16) that

This completes the proof.

Remark 2.4. Our technique of proofs is of independent interest.

Corollary 2.5. Let be a nonempty closed convex subset of a real Hilbert space and let be a Lipschitz pseudocontractive mapping such that . Let and be sequences in satisfying the conditions (iv), (v), and (vi).

For arbitrary , let be the sequence defined iteratively by (2.4). Then the following conditions are equivalent:

(a) converges strongly to the fixed point   of .(b) and are bounded.

The proof of the following result runs on the lines of proof of the Theorem 2.3, so is omitted.

Theorem 2.6. Let be a nonempty closed convex subset of a real Banach space and let be two Lipschitz pseudocontractive mappings such that . Let and be sequences in satisfying the conditions (iv), (v), and (vi). For arbitrary , let be a sequence defined iteratively by

Then the following conditions are equivalent:

(a) converges strongly to the common fixed point of and .(b) and are bounded.

Corollary 2.7. Let be a nonempty closed convex subset of a real Hilbert space and let be two Lipschitz pseudocontractive mappings such that . Let and be sequences in satisfying conditions (iv), (v), and (vi).

For arbitrary , let be the sequence defined iteratively by (2.18). Then the following conditions are equivalent:

(a) converges strongly to the common fixed point of and .(b) and are bounded.

Remark 2.8. It is worth to mentioning that we have the following.

(1)The results of Chidume [4] and Zhou and Jia [10] depend on the geometry of the Banach space, whereas in our case we do not need such geometry.(2)We remove the boundedness assumption on introduced in [4, 10].

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions.

References

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