A solution y(t) of
 (1) 
is said to be oscillatory if for every T > 0 there exists t_{0} > T such that y(t_{0}) = 0. Let
ℱ be the class of solutions of (1) which are indefinitely continuable to the right,
i.e. y ∈ℱ implies y(t) exists as a solution to (1) on some interval of the
form [T_{y},∞). Equation (1) is said to be oscillatory if each solution from ℱ
is oscillatory. If no solution in ℱ is oscillatory, equation (1) is said to be
nonoscillatory.
THEOREM 1. Let f(t,x) be continuous and satisfy b(t)Ψ(x) ≧ f(t,x) ≧ a(t)Φ(αj)
for 0 ≦ t < ∞, −∞ < x < ∞, where
 a(t) ≧ 0,b(t) ≧ 0 are both locally integrable,
 Φ(x),Ψ(x) are nondecreasing and satisfy xΦ(x) > 0, xΨ(x) > 0 for x≠0
and, for some α ≧ 0,∫
_{α}^{∞}[Φ(u)]^{−1} du < ∞, ∫
_{−α}^{−∞}[Ψ(u)]^{−1} du < ∞.
Then equation (1) is oscillatory if and only if ∫
^{∞} ta(t)dt = ∫
^{∞} tb(t)dt =
∞.
Conditions on f(t,x) are also given (Theorem 2) which are sufficient for equation
(1) to be nonoscillatory.
