Abstract and Applied Analysis

Volume 2012 (2012), Article ID 264103, 11 pages

http://dx.doi.org/10.1155/2012/264103

## An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

^{1}School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China^{2}Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 26 April 2012; Accepted 11 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Youli Yu 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 First introduce a three-step iterative algorithm for approximating the fixed points of the hemicontractive mappings in Banach spaces. Consequently, we prove the strong convergence of the proposed algorithm under some assumptions. Since three-step iterations include Ishikawa iterations as special cases, our result continue to hold for these problems. Our main results can be viewed as an important refinement of the previously known results.

#### 1. Introduction

In recent years, several convergence results have been proved on iterative methods for approximating fixed points of pseudocontractive mappings (see, e.g., [1–10] and references therein). It is worth mentioning that such iterative type methods are known as Mann iterations and Ishikawa iterations. It is clear that a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iteration sequence failed to converge, but it does converge for the sequence obtained by the Ishikawa iterations, see [11]. In 2000, Noor [12] suggested and analyzed three-step iterative methods for finding the approximate solution of a continuous mapping in the Hilbert space using the technique of updating the solution. Three-step iterations are also known as Noor iteration. It is well known [13] that three-step iterative schemes include one-step (Mann) and two-step (Ishikawa) iterations as special cases. It raises an interesting question. Is there exist any Lipschitz pseudo-contractive mapping with a unique fixed point for which Ishikawa iteration sequence fail to convergence, but Does convergence for the sequence obtained from Noor iteration? This is an open and challenging problem. To the best of our knowledge, main result of Ishikawa [14], see Theorem IS has never been extended to more general Banach spaces. Motivated and inspired by the recent research activities in this filed, we suggest and analyze a three-step iterative scheme associated with hemi-contractive mappings in Banach spaces. We also prove the strong convergence of the sequence generated by the three-step iterations under mild conditions. Since three-step iterations include Ishikawa iterations as special cases, our results continue to hold for these problems. It is worth mentioning that our results may be considered as very significant, interesting and important extensions of the previously known results concerning pseudo-contractive mappings.

Let be a real Banach space and be its dual space. The normalized duality mapping from to is defined by where denotes the generalized duality pairing.

Let be a nonempty subset of , a mapping is called pseudo-contractive if there exists such that for all . Let . A mapping is called hemicontractive if and It is easy to see that the class of pseudo-contractive mappings with fixed points is a subclass of the class of hemicontractions. A mapping is called Lipschitzian if there exists a constant such that for each .

In 1974, Ishikawa [14] proved the following result for the pseudo-contractive mappings.

Theorem 1.1 (see [14]). *If is a compact convex subset of a Hilbert space , is a Lipschitzian pseudo-contractive mapping. For , define the sequence iteratively by
**
where , are sequences of positive numbers satisfying the conditions **
(i) ; **
(ii) ; **
(iii) . **Then the sequence defined by (1.4) converges strongly to a fixed point of . *

Since its publication in 1974, Theorem IS, as far as we know, has never been extended to more general Banach spaces.

In this paper, we suggest and analyze a three-step iteration below Algorithm 1.2 associated with hemi-contractive mappings having a strong convergence in the setting of Banach spaces under some appropriate conditions.

*Algorithm 1.2. *For arbitrary , let the sequence be generated by

It is clear that Algorithm 1.2 includes Mann (one-step) and Ishikawa (two-step) iterations as special cases. For some related works, please refere to [15–19].

#### 2. Preliminaries

Let be a Banach space, the modulus of convexity of is the function defined by
A Banach space is called uniformly convex if and only if for all . For , the (generalized) duality mapping is defined as . In particular, is the normalized duality mapping on . It is known that , . A Banach space is called *p*-uniformly convex if there exists a constant such that , . It is known (see e.g., [12]) that is

For proving our main results, we shall need the following lemmas.

Lemma 2.1 (see [20]). *Let be a given real number. Then the following statements about a Banach space are equivalent: **
(i) is p-uniformly convex; **
(ii) there is a constant such that for every , , the following inequality holds:
*

*Remark 2.2. *Replacing by , by in inequality (2.3) and using the Cauchy-Schwarz inequality, we can obtain

Lemma 2.3 (see [20]). *Let be a given real number. Let be a -uniformly convex Banach space. Then, there exists a constant such that
**
for all and , where . *

Lemma 2.4 (see [4]). *Let be two nonnegative sequences and for all integers (for some fixed ), . **
(i) if , then exists; **
(ii) if and has a sequence converging to zero, then . *

#### 3. Main Results

In the sequel, and will denote the constants appearing in inequalities (2.3) and (2.5), respectively. For the rest of this paper, we shall assume that be a real *p*-uniformly convex Banach space such that and . For spaces with , the following inequalities hold (see [12, pages 1131-1132]):
for and for all , where , and for , is the unique solution of the equation .

*Remark 3.1. *We observe that the function defined by is increasing on (), hence for , we have and . Therefore, the conditions and are satisfied.

Lemma 3.2. *Let be a real -uniformly convex Banach space, nonempty closed convex and bounded, and a hemi-contractive mapping with . Then, for each and for each integer , the following inequality holds:
*

*Proof. * Replacing by and by in inequality (2.3), we can get
Since so that
This completes the proof.

*Remark 3.3. *We note that the function defined by is strictly increasing on . Hence, it has at most one zero on , provided that . In this case, since , it follows that the zero .

Lemma 3.4. *Let be a real -uniformly convex Banach space such that and . Let be a nonempty closed convex and bounded subset of , let be a Lipschitz hemi-contractive mapping with Lipschitz constant and . Let , and be three real sequences in satisfying the following conditions:
**
for all integers , some and , where is the unique solution of the equation:
**
on .**For arbitrary , let the sequence be generated by
**
Then, . *

*Proof. * We shall use to denote the possible different constants appearing in the following reasoning.

Let . Using inequality (2.5), we have
From (3.2), we have
Moreover, we also have
At the same time, applying (2.4), we can obtain the following estimates:
Substitute (3.13) and (3.14) into (3.11) to get
this together with (3.9) implies that
Set , . Then,
From (3.12), (3.14), and (3.15), we have

Substitution of (3.18) and (3.19) into (3.10) yields
Substitution of this inequality into (3.8) now gives
that is,
Observe that and that by condition (3.5), since , we get . so that
Since is Lipschitzian, we have
By the assumption , hence
Since , it follows that . We can choose some such that . Then, condition (3.5) implies . Furthermore, inequality (3.25) now yields the following estimates:
Since , it follows from Lemma 2.3 that exists. Let . Inequality (3.26) also yields
Hence, . This completes the proof.

*Remark 3.5. *The interest and importance of Lemma 3.2 lie in the fact that strong convergence of the sequence is achieved under certain mild compactness assumptions either on or on its domain.

Now, we give a strong convergence theorem as follows.

Theorem 3.6. *Let be a real p-uniformly convex Banach space such that and . Let be a nonempty closed convex and bounded subset of , be a completely continuous Lipschitz hemi-contractive mapping with Lipschitz constant and . Let , , and be three real sequences in satisfying the following conditions:
**
for all integers , some and , where is the unique solution of the equation:
**
on .**For arbitrary , let the sequence be defined by (3.7). Then, converges strongly to a fixed point of . *

*Proof. *By Lemma 3.2 . Since is completely continuous, there exists a subsequence of such that . This implies, by Lemma 3.2 that
By the continuity of and Lemma 3.2, we obtain , that is, is a fixed point of . Replacing the by in inequality (3.26), we obtain that

From (3.30), we know that has a sequence converging to zero. We note that the condition . Hence from inequality (3.31) and Lemma 2.3, we can conclude that as , that is, converges to a fixed point of . This completes the proof.

From Theorem 3.6, we can obtain the following result.

Corollary 3.7. *Let be a real p-uniformly convex Banach space such that and . Let be a nonempty closed convex and bounded subset of , be a completely continuous Lipschitz hemi-contractive mapping with Lipschitz constant and . Let and be two real sequences in satisfying the following condition:
**
for all integers , some and , where is the unique solution of the equation:
**
on .**For arbitrary , let the sequence be defined by (1.4). Then, converges strongly to a fixed points of . *

#### Acknowledgment

This research was partially supported by the Youth Foundation of Taizhou University (2011QN11).

#### References

- H. Zhou, “Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 10, pp. 2977–2983, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y.-C. Liou, and R. Chen, “Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 67, no. 12, pp. 3311–3317, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Gu, “The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 2, pp. 766–776, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Hu and L. Liu, “A new iterative algorithm for common solutions of a finite family of accretive operators,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 6, pp. 2344–2351, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Liu and S. M. Kang, “Iterative solutions of nonlinear equations with $\phi $-strongly accretive operators in uniformly smooth Banach spaces,”
*Computers & Mathematics with Applications*, vol. 45, no. 4-5, pp. 623–634, 2003. View at Publisher · View at Google Scholar - Y. J. Cho, S. M. Kang, and X. Qin, “Some results on $k$-strictly pseudo-contractive mappings in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 5, pp. 1956–1964, 2009. View at Publisher · View at Google Scholar - Z. Liu and S. M. Kang, “Convergence and stability of perturbed three-step iterative algorithm for completely generalized nonlinear quasivariational inequalities,”
*Applied Mathematics and Computation*, vol. 149, no. 1, pp. 245–258, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,”
*Computers & Mathematics with Applications*, vol. 47, no. 4-5, pp. 707–717, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. C. Ceng, P. Cubiotti, and J. C. Yao, “An implicit iterative scheme for monotone variational inequalities and fixed point problems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 8, pp. 2445–2457, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Ciric, A. Rafiq, N. Cakić, and J. S. Ume, “Implicit Mann fixed point iterations for pseudo-contractive mappings,”
*Applied Mathematics Letters*, vol. 22, no. 4, pp. 581–584, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz pseudocontractions,”
*Proceedings of the American Mathematical Society*, vol. 129, no. 8, pp. 2359–2363, 2001. View at Publisher · View at Google Scholar - M. A. Noor, “New approximation schemes for general variational inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Aslam Noor, “Some developments in general variational inequalities,”
*Applied Mathematics and Computation*, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - Y. Yao and N. Shahzad, “Implicit and explicit methods for finding fixed points of strictly pseudo-contractive mappings in Banach spaces,”
*Journal of Nonlinear and Convex Analysis*, vol. 13, no. 1, pp. 183–194, 2012. View at Google Scholar - Y. Yao, Y. C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,”
*Journal of Global Optimization*. In press. - Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triplehierarchical constrained optimization problem,”
*Mathematical & Computer Modelling*, vol. 55, pp. 1506–1515, 2012. View at Google Scholar - Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,”
*European Journal of Operational Research*, vol. 212, no. 2, pp. 242–250, 2011. View at Publisher · View at Google Scholar - Y. Yao and H.-K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,”
*Optimization*, vol. 60, no. 6, pp. 645–658, 2011. View at Publisher · View at Google Scholar - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH