Unbounded solutions to systems of differential equations at resonance

We deal with a weakly coupled system of ODEs of the type $x_j'' + n_j^2 \,x_j + h_j(x_1,\ldots,x_d) = p_j(t), \qquad j=1,\ldots,d,$ with $h_j$ locally Lipschitz continuous and bounded, $p_j$ continuous and $2π$-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 $h_1,\ldots,h_d$ are assumed.

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