Existence, Uniqueness, and Comparison Results for a Differential Equation with Discontinuous Nonlinearities

In this paper we shall show that the initial value problem x9sq x.g t, x., x 0.sxo has on an interval Jsw0, Tx, T)0, a unique solution which is increasing and right-continuous with respect to xo and g, under hypotheses which allow measurable discontinuities for q and g ?, z.and left-hand side discontinuities for g t,?.. If g t,?. is also increasing, the solution is obtained by a method of successive approximations.