Effect of resonance on the existence of periodic solutions for strongly damped wave equation

Abstract

We are interested in the differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(t,u(t))$, where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We introduce new geometrical conditions for the nonlinearity $F$ and use topological degree methods to find $T$-periodic solutions for this equation as fixed points of Poincar\'e operator.