Vol. 53, No. 1, 1974

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A fixed point theorem for k-set-contractions defined in a cone

Juan A. Gatica and William A. Kirk

Vol. 53 (1974), No. 1, 131–136

Let X be a Banach space and H a solid closed cone in X with interior H0. Suppose B is a bounded open set in X containing the origin. For G = B H0, let HG denote the relative boundary of the closure G of G in H. In this paper mappings T : G H are considered where T is a k-set-contraction, k < 1. It is shown for such mappings that if (I tT)(G) is open, t [0,1], and if T satisfies (i) Tαiλx for all x HG and λ > 1, then T has a fixed point in G. In the special case when T is a contraction mapping, (I tT)(G) is always open and boundedness of B can be dispensed with.

Mathematical Subject Classification 2000
Primary: 47H10
Received: 18 June 1973
Revised: 12 September 1973
Published: 1 July 1974
Juan A. Gatica
William A. Kirk