- Research Article
- Open Access
Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings
Journal of Inequalities and Applications volume 2006, Article number: 65983 (2006)
We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.
Bose SC: Weak convergence to the fixed point of an asymptotically nonexpansive map. Proceedings of the American Mathematical Society 1978,68(3):305–308. 10.1090/S0002-9939-1978-0493543-4
Chang S-S, Cho YJ, Zhou H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. Journal of the Korean Mathematical Society 2001,38(6):1245–1260.
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/S0002-9939-1972-0298500-3
Köthe G: Topological Vector Spaces. I, Die Grundlehren der mathematischen Wissenschaften. Volume 159. Springer, New York; 1969:xv+456.
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1991,43(1):153–159. 10.1017/S0004972700028884
Sun Z-H: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2003,286(1):351–358. 10.1016/S0022-247X(03)00537-7
Xu H-K, Ori RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization 2001,22(5–6):767–773. 10.1081/NFA-100105317
Zhang SS: On the iterative approximation problem of fixed points for asymptotically nonexpansive type mappings in Banach spaces. Applied Mathematics and Mechanics (English Edition) 2001,22(1):25–34.
Zhou Y, Chang S-S: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numerical Functional Analysis and Optimization 2002,23(7–8):911–921. 10.1081/NFA-120016276
About this article
Cite this article
Ceng, L.C., Wong, N.C. & Yao, J.C. Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings. J Inequal Appl 2006, 65983 (2006). https://doi.org/10.1155/JIA/2006/65983
- Banach Space
- Convex Subset
- Nonexpansive Mapping
- Strong Convergence
- Iteration Process