Periodic solutions of a second order differential equation with discontinuities in the spatial variable (Q1918517)

The existence (in the sense of Filippov) of a solution of the periodic BVP \((*)\) \(x'= f(t, x, x')\), \(x(0)= x(1)\), \(x'(0)= x'(1)\), \(x\in \mathbb{R}\) is investigated. The function \(f\) is supposed to be only measurable and no continuity concerning \(f\) is required. The BVP \((*)\) is inserted into a more general class of problems \((**)\) \(x'\in F(t, x, x')\), \(x(0)= x(1)\), \(x'(0)= x'(1)\), \(x\in \mathbb{R}\) with a compact and convex set valued function \(F\) which is in a certain way related to \(f\). Using a Leray-Schauder-type principle, the existence of a solution of \((**)\) is proved and this result applied to \((*)\) gives conditions on \(f\) which guarantee the existence of a solution of \((*)\).