Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics

Volume 2013, Article ID 705814, 4 pages

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

Strong Convergence for Hybrid -Iteration Scheme

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., [510] 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.

References

  1. 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 · View at MathSciNet
  2. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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 · View at MathSciNet
  4. D. R. Sahu and A. Petruşel, “Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6012–6023, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. C. E. Chidume, “Approximation of fixed points of strongly pseudocontractive mappings,” Proceedings of the American Mathematical Society, vol. 120, no. 2, pp. 545–551, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. E. Chidume, “Iterative approximation of fixed points of Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2245–2251, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. E. Chidume and C. Moore, “Fixed point iteration for pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 127, no. 4, pp. 1163–1170, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831–840, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Schu, “Approximating fixed points of Lipschitzian pseudocontractive mappings,” Houston Journal of Mathematics, vol. 19, no. 1, pp. 107–115, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Zhou and Y. Jia, “Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption,” Proceedings of the American Mathematical Society, vol. 125, no. 6, pp. 1705–1709, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. S. Chang, “Some problems and results in the study of nonlinear analysis,” Nonlinear Analysis, vol. 30, no. 7, pp. 4197–4208, 1997. View at Google Scholar
  13. K. Delmling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 283–288, 1974. View at Google Scholar
  14. 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 · View at MathSciNet