Vol. 24, No. 1, 1968

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ISSN: 0030-8730
Oscillation of solutions to second-order nonlinear differential equations

Jack W. Macki and James Sai-Wing Wong

Vol. 24 (1968), No. 1, 111–117
Abstract

A solution y(t) of

y′′ +f (t,y) = 0
(1)

is said to be oscillatory if for every T > 0 there exists t0 > T such that y(t0) = 0. Let 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 [Ty,). Equation (1) is said to be oscillatory if each solution from is oscillatory. If no solution in 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 x0 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
Authors
Jack W. Macki
James Sai-Wing Wong