Unbounded Solutions to Systems of Differential Equations at Resonance

Abstract

We deal with a weakly coupled system of ODEs of the type xj′′+nj2xj+hj(x1,…,xd)=pj(t),j=1,…,d,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x_j'' + n_j^2 \\,x_j + h_j(x_1,\\ldots ,x_d) = p_j(t), \\qquad j=1,\\ldots ,d, \\end{aligned}$$\\end{document}with h_j locally Lipschitz continuous and bounded, p_j continuous and 2pi -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.