Vol. 65, No. 2, 1976

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Fixed points of locally contractive and nonexpansive set-valued mappings

Peter K. F. Kuhfittig

Vol. 65 (1976), No. 2, 399–403

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.

Mathematical Subject Classification 2000
Primary: 54H25
Received: 12 August 1975
Published: 1 August 1976
Peter K. F. Kuhfittig