Periodic solutions of a second order differential equation with discontinuities in the spatial variable

We assume f to be measurable, but make no continuity requirements on f . We use Filippov’s definition of a solution (see [4]). Many results for boundary value problems (BVP’s) with discontinuities only in the time variable were proved by using Carathéodory’s definition of a solution. Filippov’s definition of a solution is more general than that of Carathéodory and it includes it as a special case. A standard approach to boundary value problems with discontinuities in the spatial variable is to solve the problem on each side of the discontinuity separately and then try to match the resulting solutions. A totally different approach is used in [9]. Using Filippov’s theory, the BVP’s are reformulated as differential inclusions and then the existence principles proved in [7] are applied to obtain existence results for the periodic problem and for the Dirichlet problem. The results are further used to establish the existence of