Uniqueness, continuation, and nonoscillation for a second order nonlinear differential equation

John Willard Heidel

Vol. 32 (1970), No. 3, 715–721

DOI: 10.2140/pjm.1970.32.715

Abstract

This paper considers the differential equation (1) y′′ + f(t)y^γ = 0 where f(t) is nonnegative and continuous on [0,∞) and γ is the quotient of odd, positive integers. For this equation we discuss uniqueness of the zero solution, continuation of solutions to [0,∞), and nonoscillation of solutions. Using a relation between uniqueness and continuation on the one hand and nonoscillation on the other, we can show that the condition f′(t) ≦ 0 in Atkinson’s nonoscillation theorem (Pacific J. Math. 5 (1955), 643-647), and in a corresponding theorem for 0 < γ < 1, cannot be removed entirely.

Mathematical Subject Classification

Primary: 34.42

Milestones

Received: 18 March 1969

Published: 1 March 1970