Vol. 43, No. 3, 1972

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Fixed point theorems for set-valued mappings of contractive type

Nadim A. Assad and William A. Kirk

Vol. 43 (1972), No. 3, 553–562

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).

Mathematical Subject Classification 2000
Primary: 54H25
Received: 24 August 1971
Published: 1 December 1972
Nadim A. Assad
William A. Kirk