Vol. 68, No. 2, 1977

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The essential uniqueness of bounded nonoscillatory solutions of certain even order differential equations

Garret J. Etgen and Willie Taylor

Vol. 68 (1977), No. 2, 339–346
Abstract

Let n be a positive integer, let p be a positive continuous function on [0,), and consider the 2n-th order linear differential equation

u(2n) − p(x)u = 0.
(1)

It is well known that this equation has a solution w = w(x) satisfying

(− 1)kw (k)(x) > 0, k = 0,1,⋅⋅⋅ ,2n− 1,
(2)

on [0,), and it is clear that w is positive and bounded. The purpose of this paper is to investigate the essential uniqueness of the solution w, where the statement “w is essentially unique” means that if y is any other solution of (1) which satisfies (2), then y = kw for some nonzero constant k.

Mathematical Subject Classification 2000
Primary: 34C10
Milestones
Received: 17 September 1976
Published: 1 February 1977
Authors
Garret J. Etgen
Willie Taylor