Browder and Gohde fixed point theorem for G-nonexpansive mappings

In this paper, we prove the analog to Browder and Gohde fixed point theorem for G-nonexpansive mappings in complete hyperbolic metric spaces uniformly convex. In the linear case, this result is refined. Indeed, we prove that if X is a Banach space uniformly convex in every direction endowed with a graph G, then every G-nonexpansive mapping T : A → A, where A is a nonempty weakly compact convex subset of X, has a fixed point provided that there exists u0 ∈ A such that T (u0) and u0 are G-connected.