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

DOI: 10.2140/pjm.1974.53.131

Abstract

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) Tαi≠λx 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

Milestones

Received: 18 June 1973

Revised: 12 September 1973

Published: 1 July 1974