Attracting mappings in Banach and hyperbolic spaces

In this paper we study spaces of mappings A: K --> K satisfying Ax = x for all x is an element of 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 a re equipped with natural complete uniform structures. We study the converge nce of powers of (F)-attracting mappings as well as the convergence of infi nite products of uniformly (F)-attracting sequences and show that if there exists an (F)-attracting mapping, then a generic mapping is also (F)-attrac ting. We also consider a finite sequence of subsets F-i subset of K, i = 1, ..., n, with a nonempty intersection F and a certain regularity property an d show that if each mapping A(i) is (F-i)-attracting, i = 1,..., n, then th eir product and convex combinations are (F)-attracting, (C) 2001 Academic P ress.