Unbounded Solutions to Systems of Differential Equations at Resonance

Abstract

AbstractWe deal with a weakly coupled system of ODEs of the type $$\begin{aligned} x_j'' + n_j^2 \,x_j + h_j(x_1,\ldots ,x_d) = p_j(t), \qquad j=1,\ldots ,d, \end{aligned}$$ x j ′ ′ + n j 2 x j + h j ( x 1 , … , x d ) = p j ( t ) , j = 1 , … , d , with $$h_j$$ h j locally Lipschitz continuous and bounded, $$p_j$$ p j continuous and $$2\pi $$ 2 π -periodic, $$n_j \in {\mathbb {N}}$$ n j ∈ 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 $$h_1,\ldots ,h_d$$ h 1 , … , h d are assumed.