This paper concerns the dynamics of non-expansive maps on strictly convex finite dimensional normed spaces. By using results of Edelstein and Lyubich, we show that if X = (R, ‖ · ‖) is strictly convex and X has no 1-complemented Euclidean plane, then every bounded orbit of a non-expansive map f : X → X , converges to a periodic orbit. By putting extra assumptions on the derivatives of the norm,...