On the weak-approximate fixed point property

Let X be a Banach space and C a bounded, closed, convex subset of X, C is said to have the weak-approximate fixed point property if for any norm-continuous mapping f : C -> C, there exists a sequence {x(n)} in C such that (x(n) - f (x(n)))(n) converges to 0 weakly. It is known that every infinite-dimensional Banach space with the Schur property does not have the weak-approximate fixed point property. In this article. we show that every Asplund space has the weak-approximate fixed point property. Applications to the asymptotic fixed point theory are given. (c) 2009 Elsevier Inc. All rights reserved.