On weak recurrent points of ultimately nonexpansive mappings (Q1101668)

Let K be a weakly compact subset of a Banach space X and let f:K\(\to K\). A point \(x\in K\) is said to be a recurrent (weak recurrent) point of f if \(f^{n_ i}(x)\to x\) \((f^{n_ i}(x)\to x)\) for some subsequence \(\{f^{n_ i}(x)\}\) of \(\{f^ n(x)\}\). We say that f is nonexpansive (ultimately nonexpansive) self-mapping of K if \(\| f(x)-f(y)\| \leq \| x-y\|\) (f is continuous and \(\limsup_{n\to \infty}\| f^ n(x)-f^ n(y)\| \leq \| x-y\|)\) for all x,y\(\in K.\) The main result of the paper asserts that if K has the fixed point property for nonexpansive maps and f is ultimately nonexpansive self- mapping of K then f has a weak recurrent point. Examples are given which show that better results cannot be obtained.