Existence of solutions for second-order differential equations with discontinuous right-hand side

Abstract

We consider an ordinary differential equation of second order with constant coefficients and a discontinuous right-hand side. First we use the point mapping method defining first return functions, then we use the phase-plane method. We establish both the existence and non-existence of periodic solutions (including stable ones) and oscillatory solutions depending on the coefficients of the equation. By the variational method, we prove the existence of nonzero semiregular solutions for a boundary-value problem.