Abstract and Applied Analysis

Volume 2012 (2012), Article ID 242354, 12 pages

http://dx.doi.org/10.1155/2012/242354

## Strong Convergence of Non-Implicit Iteration Process with Errors in Banach Spaces

School of Mathematics, Physics, and Information Science, Zhejiang Ocean University, Zhoushan 316004, China

Received 1 September 2012; Accepted 17 October 2012

Academic Editor: Xiaolong Qin

Copyright © 2012 Yan Hao 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

The purpose of this paper is to study the strong convergence of a non-implicit iteration process with errors for asymptotically *I*-nonexpansive mappings in the intermediate sense in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results recently announced.

#### 1. Introduction and Preliminaries

Let be a nonempty, closed, and convex subset of a real Banach space and let be a mapping. In this paper, we use to stand for the set of fixed points of , that is .

Recall that is said to be nonexpansive if

is said to be asymptotically nonexpansive if there exists a sequence with with such that

is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

Observe that if we define , then as and (1.3) reduces to

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive. And every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediated sense. In [1], Goebel and Kirk proved that, if is a nonempty closed convex bounded subset of a real uniformly convex Banach space , and is an asymptotically nonexpansive self-mapping on , then has a fixed point in . The class of mappings which are asymptotically nonexpansive in the intermediat sense was investigated by Bruck et al. [2] and Kirk [3]. Since then, many authors have investigated the fixed point problem of these mappings based on implicit iterative methods or non-implicit iterative methods; see, for example, [4–21].

Let be a mapping. Recall that is said to be asymptotically -nonexpansive if there exists a sequence with with such that Recently, weak and strong convergence theorems for fixed points of -nonexpansive mappings, and asymptotically -nonexpansive mappings have been established by many scholar, see, for example, [22–25].

In this paper, we consider a new mapping based on asymptotically nonexpansive mappings in the intermediate sense and asymptotically -nonexpansive mappings.

Let be two mappings. is said to be asymptotically -nonexpansive in the intermediate sense if it is continuous and the following inequality holds: Observe that if we define , , then as and (1.6) reduces to Note that if , where Id is the identity mapping, then (1.7) reduces to (1.4).

In this paper, we investigate asymptotically -nonexpansive mappings in the intermediate sense based on a non-implicit iterative algorithm. Strong convergence of the implicit iterative algorithm is obtained in the framework of Banach spaces.

In order to prove our main results, we need the following lemmas.

Lemma 1.1 (see [21]). * let be a uniformly convex Banach space. Let and be two constants with . Suppose that is a sequence in . Let and be two sequences in such that
**
hold for some , then . *

Lemma 1.2 (see [26]). * Let , , and be three nonnegative sequences satisfying the following condition:
**
where is some nonnegative integer, and . Then the limit exists.*

#### 2. Main Results

Lemma 2.1. * Let be a real Banach space and a nonempty closed and convex subset of . Let be a asymptotically -nonexpansive in the intermediate sense and a asymptotically nonexpansive in the intermediate sense. Assume that . Let and . Let , , , , , be six real number sequences in . Let be a sequence generated in the following iterative process:
**
where and be two bounded sequences in . Assume that the following restrictions are satisfied: *(a)*;
*(b)*; *(c)*. ** Then exists for all . *

* Proof. *Letting , we see that
Substituting (2.2) into (2.3),we obtain that
Let , , and
It follows from (2.4) that
In view of the restrictions (b) and (c), we see that . We can easily conclude the desired conclusion with the aid of Lemma 1.2. This completes the proof of Lemma 2.1.

Theorem 2.2. *Let be a real Banach space and a nonempty closed and convex subset of . Let be a asymptotically -nonexpansive in the intermediate sense and a asymptotically nonexpansive in the intermediate sense. Assume that . Let and . Let be six real number sequences in . Let be a sequence generated in the following iterative process:
**
where and be two bounded sequences in . Assume that the following restrictions are satisfied: *(a)*;
*(b)*; *(c)*. ** If both and are continuous, then the sequence strongly converges to a common fixed point of and if and only if
*

*Proof. *The necessity is obvious. Next, we prove the sufficiency part of the theorem. Note that continuity of and implies that the set and are closed. It follows from (2.6) that
This implies in turn that
Now applying Lemma 1.2 to (2.10), we obtain the existence of the limit . By condition (2.8), we have

Next we prove that the sequence is a Cauchy sequence in . For any positive integers , , from (2.9) it follows that
Since , and , for any given , there exists a positive integer such that
Therefore there exists such that , . Consequently, for any and for all , we have
This implies that is a Cauchy sequence in . Let . Since is closed, this implies that . This shows that strongly converges to a common fixed of and . This completes the proof of Theorem 2.2.

Lemma 2.3. *Let be a real Banach space and a nonempty closed and convex subset of . Let be a asymptotically -nonexpansive in the intermediate sense and a asymptotically nonexpansive in the intermediate sense. Assume that . Let and . Let , , , , , be six real number sequences in . Let be a sequence generated in the following iterative process:
**
where and be two bounded sequences in . Assume that the following restrictions are satisfied: *(a)*, ; *(b)*, ; *(c)*there exist constants such that , , ; *(d)*, . **Then
*

*Proof. *According to Lemma 2.1, for any , we have exists. Without loss of generality, we may assume that
where is some constant. It follows that
Notice that
It follows from the restriction (d) and (2.18) that
Notice that
In view of (2.19), (2.21) and (2.22), we obtain from Lemma 1.1 that
Notice that
It follows from (2.23) and the restriction (d) that
Notice that
It follows that
On the other hand, we have
Notice that
It follows that
Notice that
It follows from (2.27) that
In view of (2.28), (2.30), and (2.32), we obtain from Lemma 1.1 that
On the other hand, we have
In view of (2.23) and (2.33), we have . This completes the proof of Lemma 2.3.

Theorem 2.4. * Let be a real Banach space and a nonempty closed and convex subset of . Let be a asymptotically -nonexpansive in the intermediate sense and a asymptotically nonexpansive in the intermediate sense. Assume that . Let and . Let , , , , , be six real number sequences in . Assume that both and are Lipschitz continuous. Let are a sequence generated in the following iterative process:
**
where and be two bounded sequences in . Assume that the following restrictions are satisfied: *(a)*; *(b)*; *(c)*there exist constants such that ; *(d)*. ** If at least one of the mappings and is compact, then the sequence convergence strongly to a common fixed point of and . *

*Proof. *Without loss of generality, we may assume that is compact; this means that there exists a subsequence of such that converges strongly to , then (2.16) implies that converges strongly to . Since is continuous, then converges strongly to . On the other hand, according to (2.17) and the continuity of , we obtain that , converge strongly to , , respectively. Since , then
Observe that
Taking limit as in the above inequality, we find , which means . However, due to Lemma 2.1, the limit exists, therefore
which means that converges strongly to . This completes the proof of Theorem 2.4.

#### Acknowledgment

The work was supported by Natural Science Foundation of Zhejiang Province (Y6110270).

#### References

- K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mapping,”
*Proceedings of the American Mathematical Society*, vol. 35, pp. 171–174, 1972. View at Publisher · View at Google Scholar - B. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach space with the uniform Opial property,”
*Colloquium Mathematicum*, vol. 65, pp. 169–179, 1993. View at Google Scholar - W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type,”
*Israel Journal of Mathematics*, vol. 17, no. 4, pp. 339–346, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - X. Qin, S. S. Chang, and Y. J. Cho, “Iterative methods for generalized equilibrium problems and fixed point problems with applications,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 4, pp. 2963–2972, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Y. Cho and S. M. Kang, “Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process,”
*Applied Mathematics Letters*, vol. 24, no. 2, pp. 224–228, 2011. View at Publisher · View at Google Scholar · View at Scopus - S. Y. Cho and S. M. Kang, “Approximation of common solutions of variational inequalities via strict pseudocontractions,”
*Acta Mathematica Scientia*, vol. 32, no. 4, pp. 1607–1618, 2012. View at Publisher · View at Google Scholar - J. K. Kim, Y. M. Nam, and J. Y. Sim, “Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 71, no. 12, pp. e2839–e2848, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. M. Kang, S. Y. Cho, and Z. Liu, “Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 827082, 16 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - J. K. Kim, S. Y. Cho, and X. Qin, “Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 312602, 18 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Y. Cho, S. M. Kang, and X. Qin, “Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions,”
*Applied Mathematics Letters*, vol. 25, no. 5, pp. 854–857, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Su and S. Li, “Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 320, no. 2, pp. 882–891, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - X. Qin, S. Y. Cho, and S. M. Kang, “Iterative algorithms for variational inequality and equilibrium problems with applications,”
*Journal of Global Optimization*, vol. 48, no. 3, pp. 423–445, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Yang and W. Li, “Iterative solutions of a system of equilibrium problems in Hilbert spaces,”
*Advances in Fixed Point Theory*, vol. 1, no. 1, pp. 15–26, 2011. View at Google Scholar - X. Qin, S. Y. Cho, and S. M. Kang, “Strong convergence of shrinking projection methods for quasi-
*ϕ*-nonexpansive mappings and equilibrium problems,”*Journal of Computational and Applied Mathematics*, vol. 234, no. 3, pp. 750–760, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. Ye and J. Huang, “Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces,”
*Journal of Mathematical and Computational Science*, vol. 1, no. 1, pp. 1–18, 2011. View at Google Scholar - S. Husain and S. Gupta, “A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities,”
*Advances in Fixed Point Theory*, vol. 2, no. 1, pp. 18–28, 2012. View at Google Scholar - X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,”
*Journal of Computational and Applied Mathematics*, vol. 225, no. 1, pp. 20–30, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - X. Qin, S. Y. Cho, and S. M. Kang, “On hybrid projection methods for asymptotically quasi-
*ϕ*-nonexpansive mappings,”*Applied Mathematics and Computation*, vol. 215, no. 11, pp. 3874–3883, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - F. Gu, “Some convergence theorems of non-implicit iteration process with errors for a finite families of
*I*-asymptotically nonexpansive mappings,”*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 161–172, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Lv and C. Wu, “Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping,”
*Engineering Mathematics Letters*, vol. 1, no. 1, pp. 44–57, 2012. View at Google Scholar - J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 43, no. 1, pp. 153–159, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Mukhamedov and M. Saburov, “Strong convergence of an explicit iteration process for a totally asymptotically
*I*-nonexpansive mapping in Banach spaces,”*Applied Mathematics Letters*, vol. 23, no. 12, pp. 1473–1478, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Temir, “On the convergence theorems of implicit iteration process for a finite family of
*I*-asymptotically nonexpansive mappings,”*Journal of Computational and Applied Mathematics*, vol. 225, no. 2, pp. 398–405, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - F. Mukhamedov and M. Saburov, “Weak and strong convergence of an implicit iteration process for an asymptotically quasi-
*I*-nonexpansive mapping in Banach space,”*Fixed Point Theory and Applications*, vol. 2010, Article ID 719631, 13 pages, 2010. View at Google Scholar - N. Shahzad, “Generalized
*I*-nonexpansive maps and best approximations in Banach spaces,”*Demonstratio Mathematica*, vol. 37, pp. 597–600, 2004. View at Google Scholar · View at Zentralblatt MATH - K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,”
*Journal of Mathematical Analysis and Applications*, vol. 178, no. 2, pp. 301–308, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus