JOURNAL OF DIFFERENTIAL EQUATIONS 37, 89-97 (1980) On Differential Relations with Lower Continuous Right-Hand Side. An Existence Theorem* A. BRESSAN Istituto di Matematica Applicata, Universitci, Via Belzoni 7, 35100 Padova, Italy Received August 16, 1979 1. INTRODUCTION In this paper we consider differential relations of the form x’ E F(t, x), (1.1) where F is a multifunction that is defined on a suitable subset I x B of R”+l and takes values in the family of nonempty, compact but not necessarily convex, subsets of KY. Assuming that F is continuous with respect to X, the existence of solutions for (1.1) has been proved recently by several different techniques [l-6]. In particular, in [l] a selection theorem is first established, which yields the existence of a solution of (1.1) as a direct consequence of Schauder’s fixed- point theorem. Our present aim is to prove an existence theorem for lower continuous differential relations. Lower continuous orientor fields often arise in control theory. Namely, if x’ = f(t, x, u), UE@, (1.2) describes a control system, we may ask about solutions t -+ x(t, u(t)) of (1.2) for which x’(t, U) belongs a.e. to the closureF*(t, r) of the set of extreme points of qt, x) = (f(t, x, 24): u E 4%). Clearly, it F is HausdortI continuous and compact valued, F* is lower continuou: . In the linear case, this subject had a deep investigation leading to the bang- bang principle. In the general case nothing can be said unless some solutions for x’ E F*(t, x) are provided. Now let CL&.) denote a measure, for instance, a probability distribution on the values u’(t, x) of derivatives U’ at (t, x). If pt.% depends on * Supported by the CNR, Comitato per la Matematica. 89 0022-0396/80/070089-09$02.00/0 Copyright 0 1980by Academic Press, Inc. All rights of reproduction in any form reserved.