A surjection theorem and a fixed point theorem for a class of positive operators

This paper is concerned with alpha-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay - Ax <= M(y - x) for theta <= x <= y <= nu(0), where theta denotes the zero element and nu(0) is a constant. Moreover, we prove a fixed point theorem for alpha (> 1)-convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations. (c) 2007 Elsevier Inc. All rights reserved.