ON THE EXISTENCE OF SOLUTIONS WITH PRESCRIBED ASYMPTOTIC BEHAVIOUR FOR PERTURBED NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER

A global existence result for solutions u(t) of the differential equation x"+f(t,x)=p(t), t => t(0) => 1, that can be written as u(t)=P(t)+o(1) for all large t, where P"(t)=p(t), is established by means of the Schauder-Tikhonov theorem. It generalizes the recent work of Lipovan [On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations, Glasgow Math. J. 45 (2003), 179–87] and allows for a unifying treatment of the existence problems concerning asymptotically linear and oscillatory solutions of second order nonlinear differential equations.