On the Cauchy problem for k-th order discontinuous ordinary differential equations

Given a nonempty set Y ⊆ R n and a function f: [a, b] × (Rn)k × Y - > R, we are interested in the problem of finding u ε Wk,p([a,b],Rn) such that f(t,u(t),u'(t),...,u(k)(t)) = 0 for a.e. t ε [a,b], and u(i)(t0) = u0(i) for a11 i = 0 , . . . ,k - 1, where t0 ε [a, b] and (u0(0) ,u0(1) , . . . ,u0(k-1)) ε (Rn)k are given points. We prove an existence result where, for any fixed (t, y) ε [a, b] × Y, the function f(t, ·, y) can be discontinuous even at all points ξ ε (Rn)k. The function f(t, ξ, ·) is only assumed to be continuous and locally nonconstant.