Attracting Mappings in Banach and Hyperbolic Spaces

Abstract

AbstractIn this paper we study spaces of mappings A:K→K satisfying Ax=x for all x∈F, where K is a closed convex subset of a hyperbolic complete metric space and F is a closed convex subset of K. These spaces are equipped with natural complete uniform structures. We study the convergence of powers of (F)-attracting mappings as well as the convergence of infinite products of uniformly (F)-attracting sequences and show that if there exists an (F)-attracting mapping, then a generic mapping is also (F)-attracting. We also consider a finite sequence of subsets Fi⊂K, i=1,…,n, with a nonempty intersection F and a certain regularity property and show that if each mapping Ai is (Fi)-attracting, i=1,…,n, then their product and convex combinations are (F)-attracting