On the second order equations with nonlinear impulses. Fredholm alternative type results

Abstract

We consider the semilinear homogeneous Dirichlet boundary value problem for the second order equation on a finite interval with nonlinear impulses in the derivative at prescribed points. We introduce Landesman-Lazer type necessary and sufficient conditions for resonance problems and generalize the Fredholm alternative results for linear operators. An interaction between nonlinear restoring force and nonlinear impulses is presented.