Resonance phenomena in the second differential order equations with nonlinear impulsive effects

In this paper, we discuss the resonance phenomena in linear and weakly nonlinear impulsive differential equations. Using a topological approach, we obtain the Landesman–Lazer type necessary and sufficient conditions for the existence of periodic solutions and extend the Fredholm alternative for linear equations with impulses. Moreover, we show the coexistence of periodic and unbounded solutions of weakly nonlinear impulsive differential equations through the use of the successor map and a Lyapunov function.