Unbounded solutions to systems of differential equations at resonance

Abstract

We deal with a weakly coupled system of ODEs of the type $$x_j^{\prime \prime }+n_j^2{x}_j+h_j(x_1,\dots ,x_d)=p_j(t),j=1,\dots ,d,$$ with $h_j$ locally Lipschitz continuous and bounded, $p_j$ continuous and $2\pi$-periodic, $n_j\in \mathbb{N}$ (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms $h_1,\dots ,h_d$ are assumed.