A short survey on open problems in metric fixed point theory and some related results for nonexpansive mappings (Q2047298)

Let \(K\) be a nonempty closed bounded convex subset of a normed linear space \(X:=(X,{\|\cdot\|})\). We say that \(K\) has the fixed point property if every nonexpansive mapping \(T:K\to K\), that is, \(\|Tx-Ty\|\le\|x-y\|\) for all \(x,y\in K\), has a fixed point. We also say that \(X\) has the fixed point property (weak fixed point property, resp.) if every closed bounded convex subset (weakly compact convex subset, resp.) of \(X\) has the fixed point property. In this paper, the authors discuss various sufficient conditions for the fixed point property and the weak fixed point property of a Banach space.