A fixed point theorem for positivek-set contractions

Abstract

The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressivek-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to thek-set contractions (k< 1). It should be remarked that J. D. Hamilton has extended the same result toA-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whetherk-set contractions areA-proper or not.