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 φ a
contraction mapping from K into the family 𝒯 (M) 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).
