A NONOSCILLATION THEOREM FOR SUBLINEAR EMDEN-FOWLER EQUATIONS

Consider the Emden-Fowler equation (E) : y″ + a(x)|y|γ-1y = 0, where γ > 0 and a(x) is a positive continuous function on (0, ∞). I. T. Kiguradze showed in 1962 that if x(γ+3)/2+δ a(x) is nonincreasing for any δ > 0, then equation (E) is nonoscillatory when γ > 1. We prove in this paper that the same theorem remains valid in the sublinear case, i.e., equation (E) when 0 < γ < 1.

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 ≠ 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.

Nonoscillation theorems for functional differential equations of arbitrary order

The authors give sufficient conditions for all oscillatory solutions of a sublinear forced higher order nonlinear functional differential equation to converge to zero. They then prove a nonoscillation theorem for such equations. A few intermediate results are also obtained.

A nonoscillation theorem for a second order sublinear retarded differential equation

Sufficient conditions are obtained for all solutions of a class of second order nonlinear functional differential equations to be nonoscillatory.