Oscillation of solutions to second-order nonlinear differential equations

Jack W. Macki and James Sai-Wing Wong

Vol. 24 (1968), No. 1, 111–117

DOI: 10.2140/pjm.1968.24.111

Abstract

A solution y(t) of

(1)

is said to be oscillatory if for every T > 0 there exists t₀ > T such that y(t₀) = 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

1. a(t) ≧ 0,b(t) ≧ 0 are both locally integrable,
2. 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.

Mathematical Subject Classification

Primary: 34.42

Milestones

Received: 28 December 1966

Published: 1 January 1968