Periodic solutions of a Lienard equation with forcing term

The problem of the existence of a 2π-periodic solution of the forced Lienard equation x″(t)+d/dt [F(x(t))]+g(t,x(t)) = e(t) under various conditions on F(·) and g(·,·) has been extensively discussed. It is assumed that g(t,s)s does not change sign for |s| large, there exists d>0 such that (s.t.) g(t,s)s≥0 for almost all (a.a.) t∈R, for all |s|≥d. To guarantee the existence of a periodic solution additional conditions were imposed. Two main types, either a dissipative condition: lim F(s)sign s = +∞ (or -∞) for |s| approaching ∞, g(s) arbitrary, or a nonresonance condition which, in the case g is independent of t, is lim sup g(s)/s<1 for |S| approaching ∞, F arbitrary, were considered.