Convexity properties of some integral operators

Abstract

In [I] D. Blackwell extended a theorem due to A. Liapunov [2]. Blackwell proved the ranges of certain vector measures, which took values in R”, were compact and convex. This result has been the basis of many important theorems in the theory of optimal control (see, e.g., [3] or [4]). A different application of Blackwell’s work can be found in [5]. There it is shown that the existence of fixed points for a class of nonlinear integral equations can be deduced from Schauder’s fixed point theorem (see, e.g., [6]) without a priori convexity assumptions. When an attempt was made to extend the work in [5] to more general classes of equations, particularly equations in a Banach space, it was discovered that the Liapunov theorem held only in a restricted sense (see e.g., [7]). Hence the purpose of this paper is to prove a simple approximate version of the Liapunov theorem (Property 1 and Theorem 1) and to use this version to extend the result in [5]. A secondary title for this paper might be “A Fixed Point Theorem for Some Integral Operators in the Absence of Convexity Assumptions.” The paper is divided into two parts. Part I is the statement of basic notation, definitions and results. Part II consists of the proofs of the results stated in Part I. A general reference for vector measures will be [8] and a general reference for function analytic concepts will be [9]. The only topology considered for Banach spaces will be the normed topology.