A Remark on Fixed Point Theorems for Lipschitzian Mappings

Abstract

The aim of this note is to give the following theorem: Let p > 1 and let E be a p-uniformly convex Banach space, C a nonempty bounded closed convex subset of E, and A = [an,k]n,k ≥ 1 a strongly ergodic matrix. If T: C → C is a mapping such that g = lim infn → ∞infm = 0,1,2… ∑∞k = 1an,k · ||Tk + m||p < 1 + c, where c > 0 is some constant, then T has a fixed point in C.