On periodic solutions of nonlinear second order vector differential equations

Abstract

SynopsisThis paper considers existence of periodic solutions for vector Liénard differential equationsIn our main result we writewhere Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d2/dt2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equationthe assumptions are weakened.Our approach is based on Leray-Schauder's degree theory and a priori estimates.