Harmonic perturbations with delay of periodic separated variables differential equations

Abstract

We study the structure of the set of harmonic solutions to perturbed, nonautonomous, $T$-periodic, separated variables ODEs on manifolds. The perturbing term, supposed to be $T$-periodic in time, is allowed to contain a finite delay.Our main result extends those of [M. Furi and M. Spadini, Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud. 9 (2009), No. 2, 263-276] and [M. Spadini, Harmonic solutions to perturbations of periodic separated variables ODEs on manifolds, Electron. J. Differential Equations, 2003 (2003), No. 88, 1-11] but it cannot be simply deduced from them: It emerges from of a combination of the techniques exposed in those two papers.