International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 11, Pages 685-688
doi:10.1155/S016117120100638X
On the fixed points of affine nonexpansive mappings
Department of Mathematics, Faculty of Sciences, Istanbul University, Vezneciler, Istanbul 34459, Turkey
Received 9 December 2000
Copyright © 2001 Hülya Duru. 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
Let K be a closed convex bounded subset of a Banach space X and let T:K→K be a continuous affine mapping. In this note, we show that (a) if T is nonexpansive then it has a fixed point, (b) if T has only one fixed point then the mapping A=(I+T)/2 is a focusing mapping; and (c) a continuous mapping S:K→K has a fixed point if and only if, for each x∈k, ‖(An∘S)(x)−(S∘An)(x)‖→0for some strictly nonexpansive affine mapping T.