PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 104, Number 3, November 1988 UNIQUE SOLUTIONS FOR A CLASS OF DISCONTINUOUS DIFFERENTIAL EQUATIONS ALBERTO BRESSAN ABSTRACT. This paper is concerned with the Cauchy Problem x(t) = ¡(t,x(t)), i(i0) = io £ R", where the vector field / may be discontinuous with respect to both variables t, x. If the total variation of / along certain directions is locally finite, we prove the existence of a unique solution, depending continuously on the initial data. 1. Introduction. Let / be a vector field on R™. By definition, a Carathéodory solution of the Cauchy Problem (1.1) i(t) = f(t, x(t)), x(t0) = x0G R", is an absolutely continuous function t —► x(t) which takes the value xo at t = to and satisfies the differential equation in (1.1) at almost every t. If / is not continuous, Peano's theorem does not apply and (1.1) may not have any solution. Some authors have thus introduced new definitions of generalized or relaxed solutions for (1.1), for which a satisfactory existence theorem could then be proven [5, 6, 9]. An alternative approach to discontinuous O.D.E.'s, pursued in [4, 7, 8], relies on the study of certain conditions on / which are weaker than continuity, yet sufficient to guarantee the existence of Carathéodory solutions. This led to the investigation of directional continuity. For a fixed M > 0, consider the cone (1.2) TM = {(t,x)GRn+1;\\x\\<Mt}. We say that a map /: R"+1 -» R" is T^-continuous if, for every (t0,x0) G Rn+1 and £ > 0, there exists <5 > 0 such that to < t < to + 6, \\x -x0\\< M(t - to) => \\f(t, x) - f(t0, x0)\\ < £■ Assuming that ||/(i,z)|| < L < M for all t,x, solutions of O.D.E.'s with TM- continuous right-hand sides were obtained in [7] as limits of polygonal approxima- tions, in [2] through an application of Schauder's fixed point theorem, and in [3] by means of an upper semicontinuous, convex-valued regularizaron. These results acquire additional interest in connection with the theory of multivalued differen- tial equations. Indeed, the existence of directionally continuous selections for lower semicontinuous multifunctions now provides a very effective tool for the study of differential inclusions [2, 3]. The present paper is concerned with the problem of uniqueness and continuous dependence. In the classical theory, the uniqueness of solutions of (1.1) is proved assuming that / is locally Lipschitz continuous. Here we consider a much weaker Received by the editors November 2, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 34A10. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 772 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use