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. 1998 Academic Press

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