Continuation methods in certain metric and geodesic spaces (Q300789)

In this paper the authors generalize the Granas continuation principle for contractions replacing the contractive assumption by the Browder condition. Using this generalization they prove a Leray-Schauder type principle for such generalized contractions in hyperbolic spaces. As an application of this Leray-Schauder principle they prove the following result for nonexpansive mappings in complete metric spaces.NEWLINENEWLINELet \(G\) be a bounded domain in a complete hyperbolic space \((X,d)\) and suppose that \(f: \bar{G} \to X\) is a nonexpansive mapping. Suppose also that there exists \(p \in G\) such that \(f\) satisfies the Leray-Schauder condition: \(x \notin (p,f(x))\) for all \(x \in \partial G\). Then \(\inf \{d(x, f(x)): x \in \bar{G}\}=0\).