Periodic solutions existence for a second order differential equation with a discontinuous nonlinearity

An existence theorem is obtained for a periodic solution of an ordinary second order differential equation with discontinuous nonlinearity in the resonance case. The solutions are considered in the sense of differential inclusion. It is assumed that the non-linearity is of Borel (mod 0) and is bounded. At infinity, it satisfies the one-dimensional analogue of the Landesman — Laser condition for resonant elliptic boundary value problems. The operator statement of the problem under consideration leads to the problem of the existence of fixed points in a multi-valued compact map. To describe the convexity of Nemytskiy operator generated by nonlinearity, the results of M.A. Krasnoselsky and A.V. Pokrovsky are used. The presence of a fixed point is established using the multi-valued version of the Leray — Schauder method.