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 form x" + f(x,t)x' + g(x,t) = 0, where f and g are odd polynomials in x with coefficients which are even and periodic in t. Using the KAM theory for reversible systems, we prove that all the solutions are bounded whenever a sharp condition on the degrees of f and g is satisfied. We also obtain a boundedness result when f(x,t) = 0 (i.e., the equation is conservative), under some smoothness assumptions on g which improve the previously known ones. In this case, no symmetry conditions on g are needed. (C) 1998 Academic Press.