I really don't understand your question. You say "f is continuous and lipschitz in (y,y')". Without saying a specific set, that would be assumed to be on all x. In that case, there exist a unique solution, satisfying y(a)= y0, y'(a)= y1, what ever a is, for all x. I don't see how that has anything to do with the intervals [-3, 0] and [2, 3].