Oscillations in superlinear differential equations of second order

Abstract

A new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t) + a(t)f[x(t)] = 0, where a ϵ C([t 0, ∞)), f ϵ C(R) with yf(y) > 0 for y ≠ 0 and ∝ ±1 ±∞ [ 1 f(y) ] dy < ∞, and f is continuously differentiable on R − {0} with f′( y) ⩾ 0 for all y ≠ 0. The coefficient a is not assumed to be eventually nonnegative and the oscillation cirterion obtained involves the average behavior of the integral of a. In the special case of the differential equation x″(t) + a(t) ¦x(t)¦ λ sgn x(t) = 0 (λ > 1) this criterion improves a recent oscillation result due to Wong [Oscillation theorems for second-order nonlinear differential equations, Proc. Amer. Math. Soc. 106 (1989), 1069–1077].