ON THE CAUCHY PROBLEM FOR K-TH ORDER DISCONTINUOUS ORDINARY DIFFERENTIAL EQUATIONS

Given a nonempty set Y subset of R-n and a function f : [a, b] x (R-n)(k) x Y -> R, we are interested in the problem of finding u is an element of W-k,W-p([a,b],R-n) such that f (t, u(t), u'(t), ... , u((k)) (t)) = 0 for a.e. t is an element of [a, b], and u((i)) (t(0)) = u(0)((i)) for all i = 0, ... ,k - 1, where t(0) is an element of[a, b] and (u(0)((0)), u(0)((0)), ... , u(0)((k-1))) is an element of(R-n)(k) are given points. We prove an existence result where, for any fixed (t, y) is an element of[a, b] x Y, the function f (t, center dot, y) can be discontinuous even at all points xi is an element of (R-n)(k). The function f (t, xi, center dot) is only assumed to be continuous and locally nonconstant.