Vol. 18, No. 2, 1966

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ISSN: 0030-8730
Oscillation criteria for third order nonlinear differential equations

Paul Elvis Waltman

Vol. 18 (1966), No. 2, 385–389
Abstract

Two oscillation theorems are provided for certain nonlinear nonautonomous third order differential equations. Both results involve integral conditions and are of the form that any solution which has one zero is oscillatory. Theorem. Let p(t), q(t) be continuous and let p(t) be nonpositive. Define Q(t) = 0tq(s)ds. If A + Bt t0tQ(s)ds < 0 for t sufficiently large, any A and B, if γ is positive and the quotient of two odd integers, then any continuable solution of y′′′ + p(t)y+ q(t)yγ = 0 which has a zero is oscillatory.

Theorem. Let p(t) and q(t) be continuous and nonnegative and let f(y)∕y α > 0 for some α. If αq(t) p(t) is positive and if t(αq(t) p)(t)dt = then any continuable solution of y′′′ + p(t)y+ q(t)f(y) = 0 which has a zero is oscillatory.

Mathematical Subject Classification
Primary: 34.42
Milestones
Received: 17 December 1964
Revised: 5 May 1965
Published: 1 August 1966
Authors
Paul Elvis Waltman