Remarks on approximate fixed points (Q435882)

The purpose of this article is to extend the Klee and Phu results concerning the existence of \(\gamma\)-invariant points of \(r\)-continuous mappings. Recall that a mapping \(f\) from a topological space \(X\) to a metric space \((M,d)\) is called \(r\)-continuous (\(r>0\)) if for any \(x\in X\) there exists a neighborhood \(U_x\) of \(x\) such that \(diam(f(U_x))\leq r\). The author proves analogues of those results for compact hyperconvex metric spaces and for geodesically bounded complete \(\mathbb R\)-trees. In particular, he proves that if \((M,d)\) is a compact hyperconvex metric space and \(f:M\to M\) is almost \(r\)-continuous, then there exists \(\bar{x}_0\in M\) such that NEWLINE\[NEWLINE d(\bar{x}_0,f(\bar{x}_0))<r. NEWLINE\]NEWLINE The author proves also a similar type result for Busemann spaces which include e.g. strictly convex normed spaces and \(CAT(0)\) spaces.