On the second order periodic problem at resonance with impulses

Abstract

In this paper we consider the periodic problem for the second order equation at resonance with impulses in the derivative. The impulses are considered at fixed times and depend on the actual value of the solution in a nonlinear way. We formulate rather general sufficient condition in terms of the asymptotic properties of both nonlinear restoring force and nonlinear impulses which generalizes classical Landesman–Lazer condition. Moreover, our condition implies existence results for some open problems with vanishing and oscillating nonlinearities.