On Discontinuous Di erential Equations

Consider the Cauchy problem for an ordinary diierential equation _ x = g(t; x); x(0) = x; t 2 0; T]: (1:1) When g is continuous, the local existence of solutions is provided by Peano's theorem. Several existence and uniqueness results are known also in the case of a discontinuous right hand side 7]. We recall here the classical theorem of Carath eodory 8]: Theorem A. Let g : 0; T] IR n 7 ! IR n be a bounded function. (i) If the map t 7 ! g(t; x) is measurable for each x and the map x 7 ! g(t; x) is continuous for each t, then the Cauchy problem (1.1) has at least one solution. (ii) If the map t 7 ! g(t; x) is measurable for each x and the map x 7 ! g(t; x) is Lipschitz continuous for each t, with a uniform Lipschitz constant, then the Cauchy problem (1.1) has a unique solution, depending Lipschitz continuously on the initial data x. By a solution of (1.1) we mean an absolutely continuous function x : 0; T] 7 ! IR n such that x(t) = x + Z t 0 g ? t; x(t) dt for all t 2 0; T]: (1:2) More recent results rely on the notions of directional continuity and of bounded directional variation of a vector eld. More precisely, given a closed convex cone ? IR m , we say that a (possibly discontinuous) map : IR m 7 ! IR n is directionally continuous if at each point p 2 IR m one has lim p 0 !p; p 0 ?p2? (p 0) = (p):