Best approximation, coincidence and fixed point theorems for set-valued maps in R-trees

Abstract

Suppose X is a closed, convex and geodesically bounded subset of a complete R-tree M, and suppose F : X {multimap} M is an almost lower semicontinuous set-valued map whose values are nonempty closed convex. Suppose also G : X {multimap} X is a continuous, onto quasiconvex set-valued map with compact, convex values. Then there exists x0 ∈ X such that d (G (x0), F (x0)) = under(inf, x ∈ X) d (x, F (x0)) . As applications, we give some coincidence and fixed point results for weakly inward set-valued maps. Our results generalize some well-known results in literature. © 2009.