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

This paper is concerned with α-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 θ x y v0, where θ denotes the zero element and v0 is a constant. Moreover, we prove a fixed point theorem for α (>1)-convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations. © 2007 Elsevier Inc. All rights reserved.