Attracting Mappings in Banach and Hyperbolic Spaces

In 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 ofŽF.-attracting mappings as well as the convergence of infinite products of uniformly ŽF.-attracting sequences and show that if there exists an ŽF.-attracting mapping, then a generic mapping is also ŽF.-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 Aiis ŽFi.-attracting, i 1, . . . , n, then their product and convex combinations are ŽF.-attracting.