Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 68616 10 pages doi 2007 68616 Research Article Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings C. E. Chidume and Bashir Ali Received 20 October 2006 Revised 30 January 2007 Accepted 31 January 2007 Recommended by Donal O Regan Let E be a real Banach space K a closed convex nonempty subset of E and T1 T2 . Tm K K asymptotically quasi-nonexpansive mappings with sequences resp. kin 1 satisfying kin 1 as n TO and ỵ TO 1 kin - 1 TO i 1 2 . m. Let an TO 1 be a sequence in e 1 - e e e 0 1 . Define a sequence xn by x1 e K xn 1 1 - an xn anT1 yn m-2 yn m-2 1 an xn anT2 yn m-3 . yn 1 an xn anTmxn n 1 m 2. Let Plm 1F Ti 0. Necessary and sufficient conditions for a strong convergence of the sequence xn to a common fixed point of the family Ti m i are proved. Under some appropriate conditions strong and weak convergence theorems are also proved. Copyright 2007 C. E. Chidume and B. Ali. 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. 1. Introduction Let K be a nonempty subset of a real normed space E. A self-mapping T K K is called nonexpansive if II Tx - Ty II x - y II for every x y e K and quasi-nonexpansive if F T x e K Tx x 0 and II Tx - pH l x - pH for every x e K and p e F T . The mapping T is called asymptotically nonexpansive if there exists a sequence kn c 1 to with kn 1 as n TO such that for every n e N 11 Tnx - Tny kn x - yll for every x y e K. If F T 0 and there exists a sequence kn c 1 to with kn 1 as n TO such that for 2 Journal of Inequalities and Applications n G N Tnx - p kn x- pH for every x G K and p G F T then T is called asymptotically quasi-nonexpansive mapping. Iterative methods for approximating fixed points of nonexpansive mappings and their .
