Behavior of Solutions of Second-Order Differential Equations with Sublinear Damping

We investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation where uf(u) > 0 for u ≠ 0, a(t) ≥ 0, and α is a positive constant with 0 < α ≥ 1. The case α = 1 has been investigated by a number of other authors. Here, it is shown that the behavior of solutions in the case of sublinear damping (0 < α < 1) is completely different from that in the case of linear damping (α = 1). Sufficient conditions for all nonoscillatory solutions to converge to zero and sufficient conditions for the existence of a nonoscillatory solution that does not converge to zero are given. We also give sufficient conditions for all solutions to be nonoscillatory. Some open problems for future research are also indicated. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 186–200, April–June, 2005.