Periodic solutions of damped hyperbolic equations at resonance: a translation along trajectories approach

Abstract

We develop an abstract averaging method for a periodic problem $$\left\{ \begin{array}{ll} \ddot u(t) + \beta \dot u(t) + A u + F(t,u(t)) = 0 , \ \ \ t\in [0,T] \\ u(0)=u(T), \dot u(0) = \dot u(T) , \end{array}\right. $$ where $A:D(A)\to X$ is a self-adjoint operator on a Hilbert space $X$ with compact resolvent and $\alpha>0$ such that $A+\alpha$ is strongly $m$-accretive, $\beta>0$ and $F:\mathbb R\times X^{1/2}\to X$, with $X^{1/2}$ being the fractional space of $A+\alpha$, is compact and $T$-periodic in the first variable. The so-called resonant case is considered, i.e., when ${\mathrm{Ker}}\, A\neq \{0\}$ and $F$ is bounded. By use of the topological degree applied to the translation along trajectories operator, effective conditions guaranteeing the existence of $T$-periodic solutions are given. The general result is applied to a damped hyperbolic partial differential equation and the Landesman-Lazer type criterion is derived.