Research Article | Open Access

Shin Min Kang, Arif Rafiq, Young Chel Kwun, "Strong Convergence for Hybrid -Iteration Scheme", *Journal of Applied Mathematics*, vol. 2013, Article ID 705814, 4 pages, 2013. https://doi.org/10.1155/2013/705814

# Strong Convergence for Hybrid -Iteration Scheme

**Academic Editor:**D. R. Sahu

#### 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.

#### References

- S. Ishikawa, “Fixed points by a new iteration method,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 147–150, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 | MathSciNet - 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 Site | Google Scholar | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 - K. Delmling, “Zeros of accretive operators,”
*Manuscripta Mathematica*, vol. 13, pp. 283–288, 1974. View at: Google Scholar - 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 | Zentralblatt MATH | MathSciNet

#### Copyright

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.