Fixed point theorems for set-valued mapping of contractive type

Abstract

In this paper a new fixed point theorem is proved for contraction mappings in a complete metric space by observing that if the space is metrically convex, then significant weakenings may be made concerning the domain and range of the mapping considered. While the main theorem is formulated for set-valued mappings, its point-to-point analogue is also a new result. This result, proved in § 1, is the following: Suppose M is a complete, metrically convex, metric space, K a nonempty closed subset of M, and tp a contraction mapping from K into the family of nonempty closed bounded subsets of M supplied with the Hausdorff metric. Then if ϕ maps the boundary of K into subsets of K, ϕ has a fixed point in K, i.e., there is a point x0∈ K such that x0∈ ϕ (x0). © 1972 Pacific Journal of Mathematics.