- Research Article
- Open Access
Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions
© The Author(s). 2010
- Received: 8 October 2009
- Accepted: 22 January 2010
- Published: 28 January 2010
We introduce and study some new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. General results for asymptotically strict pseudocontractions are established. The main result extends the previous results.
- Hilbert Space
- Point Theorem
- Iterative Algorithm
- Nonnegative Integer
- Variable Coefficient
is called to be asymptotically nonexpansive  if there exists a sequence with and such that
In , Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.
Theorem 1 A.
Such algorithm in (1.4) is referred to be the (CQ) algorithm in , due to the fact that each iterate is obtained by projecting onto the intersection of the suitably constructed closed convex sets and It is known that the (CQ) algorithm in (1.4) is of independent interest, and the (CQ) algorithm has been extended to various mappings by many authors (cf., e.g., [3–11]).
Very recently, by extending the (CQ) algorithm, Takahashi et al.  studied a family of nonexpansive mappings and gave some good strong convergence theorems. Kim and Xu  extended the (CQ) algorithm to study asymptotically -strict pseudocontractions and established the following interesting result with the help of some boundedness conditions.
Theorem 1 B.
It is our purpose in this paper to try to obtain some new fixed point theorems for asymptotically strict pseudocontractions without the boundedness conditions as in Theorem B. Motivated by Nakajo and Takahashi , Takahashi et al. , and Kim and Xu , we introduce and study certain new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. Our results improve essentially the corresponding results of .
Throughout this paper,
We divide the proof into five steps.
By (2.18), (2.24), and (2.26), we know that (2.25) holds.
Theorem 2.4 improves [5, Theorem ] since the condition that is satisfied and the boundedness of is dropped off.
The authors are very grateful to the referee for his/her valuable suggestions and comments. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). This work is dedicated to W. Takahashi.
- Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972,35(1):171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar
- Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar
- Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar
- Ge CS, Liang J: Convergence theorems of new Ishikawa iterative procedures with errors for multi-valued -hemicontractive mappings. Communications in Mathematical Analysis 2009,7(1):12–20.MathSciNetMATHGoogle Scholar
- Kim T-H, Xu H-K: Convergence of the modified Mann's iteration method for asymptotically strict pseudocontractions. Nonlinear Analysis: Theory, Methods & Applications 2008,68(9):2828–2836. 10.1016/j.na.2007.02.029MathSciNetView ArticleMATHGoogle Scholar
- Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar
- Osilike MO, Udomene A, Igbokwe DI, Akuchu BG: Demiclosedness principle and convergence theorems for -strictly asymptotically pseudocontractive maps. Journal of Mathematical Analysis and Applications 2007,326(2):1334–1345. 10.1016/j.jmaa.2005.12.052MathSciNetView ArticleMATHGoogle Scholar
- Qin XL, Cho YJ, Kang SM, Shang MJ: A hybrid iterative scheme for asymptotically -strict pseudocontractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1902–1911. 10.1016/j.na.2008.02.090MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleMATHGoogle Scholar
- Zegeye H, Shahzad N: Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4496–4503. 10.1016/j.na.2007.11.005MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.