Boundedness of Solutions of Nonlinear Differential Equations

In this paper we are concerned with the boundedness of solutions of some nonlinear differential equations, which are neither dissipative nor conservative, having the formx″+(x, t) x′+g(x, t)=0, wherefandgare odd polynomials inxwith coefficients which are even and periodic int. Using the KAM theory for reversible systems, we prove that all the solutions are bounded whenever a sharp condition on the degrees offandgis satisfied. We also obtain a boundedness result whenf(x, t)≡0 (i.e., the equation is conservative), under some smoothness assumptions ongwhich improve the previously known ones. In this case, no symmetry conditions ongare needed.