An abstract fixed point theorem for nonexpansive mappings

Kirk, W. A.

doi:10.1090/S0002-9939-1981-0614894-6

An abstract fixed point theorem for nonexpansive mappings
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Proc. Amer. Math. Soc. 82 (1981), 640-642 Request permission

Abstract:

A class $\mathcal {S}$ of subsets of a bounded metric space is said to be normal if each member of $\mathcal {S}$ contains a nondiametral point. An induction proof is given for the following. Suppose $M$ is a nonempty bounded metric space which contains a class $\mathcal {S}$ of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in $M$. Then every nonexpansive self-mapping of $M$ has a fixed point.

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