A nonoscillation theorem for second-order nonlinear differential equations with decaying coefficients

The purpose of this paper is to give sufficient conditions for all nontrivial solutions of the nonlinear differential equation x " + a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfies the sign condition xg(x) > 0 if x not equal 0, but is not assumed to be monotone increasing. This differential equation includes the generalized Emden-Fowler equation as a special case. Our main result extends some nonoscillation theorems for the generalized Emden-Fowler equation. Proof is given by means of some Liapunov functions and phase-plane analysis.