Let (M,d) be a complete metric
space and S(M) the set of all nonempty bounded closed subsets of M. A set-valued
mapping f : M → S(M) will be called (uniformly) locally contractive if there exist 𝜖
and λ(𝜖 > 0,0 < λ < 1) such that D(f(x),f(y)) ≦ λd(x,y) whenever d(x,y) < 𝜖 and
where D(f(x),f(y)) is the distance between f(x) and f(y) in the Hausdorff metric
induced by d on S(M). It is shown in the first theorem that if M is “well-chained,”
then f has a fixed point is, that is, a point x ∈ M such that x ∈ f(x). This fact, in
turn, yields a fixed-point theorem for locally nonexpansive set-valued mappings on a
compact star-shaped subset of a Banach space. Both theorems are extensions of
earlier results.
