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

We are interested in the differential equation ü(t)=−Au(t)−cAu̇(t)+λu(t)+F(t,u(t)), where c>0c>0 is a damping factor, AA is a sectorial operator and FF is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that λλ is an eigenvalue of AA and FF is a bounded map. We introduce new geometrical conditions for the nonlinearity FF and use topological degree methods to find TT-periodic solutions for this equation as fixed points of Poincaré operator.