Research Article On the Weak Relatively Nonexpansive Mappings in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 189751 7 pages doi 2010 189751 Research Article On the Weak Relatively Nonexpansive Mappings in Banach Spaces Yongchun Xu1 and Yongfu Su2 1 Department of Mathematics Hebei North University Zhangjiakou 075000 China 2 Department of Mathematics Tianjin Polytechnic University Tianjin 300160 China Correspondence should be addressed to Yongfu Su suyongfu@ Received 23 March 2010 Accepted 20 May 2010 Academic Editor Billy Rhoades Copyright 2010 Y. Xu and Y. Su. 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. In recent years the definition of weak relatively nonexpansive mapping has been presented and studied by many authors. In this paper we give some results about weak relatively nonexpansive mappings and give two examples which are weak relatively nonexpansive mappings but not relatively nonexpansive mappings in Banach space l2 and Lp 0 1 1 p x . 1. Introduction Let E be a smooth Banach space and let C be a nonempty closed convex subset of E. We denote by Ộ the function defined by ệ x ỳ x 2 - 2 x Jy y 2 for x y e E. L1 Following Alber 1 the generalized projection nC from E onto C is defined by nC x argminộ y x Vx e E. yeC The generalized projection He from E onto C is well defined single value and satisfies l x - y 2 Hx y x y 2 for x y e E. If E is a Hilbert space then ộ y x y - x 2 and nC is the metric projection of E onto C. 2 Fixed Point Theory and Applications Let C be a closed convex subset of E and let T be a mapping from C into itself. We denote by F T the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T 2-4 if C contains a sequence xn which converges weakly to p such that limn X Txn-x 0. The set of asymptotic fixed point of T will be denoted by b F T
