Vol. 63, No. 1, 1976

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Bounds and quantitative comparison theorems for nonoscillatory second order differential equations

Thomas Thornton Read

Vol. 63 (1976), No. 1, 231–245
Abstract

A lower bound is given for the positive increasing solution of y′′ + 2ry′−q2y = 0 on the interval [0,) and an upper bound is given for the positive decreasing solution of this equation. These are used to estimate z∕y and z0∕y0 where y and z (respectively y0 and z0) are positive nonprincipal (respectively principal) solutions of nonoscillatory equations y′′p1y = 0 and z′′p2z = 0 for which p2 p1. A special case of one result is that if p1 is bounded and if

      ∫
x       1∕2
lim inf( 0 (p2 − p1) dt∕x) > 0

as x →∞ then z∕y increases exponentially and z0∕y0 decreases exponentially.

Mathematical Subject Classification 2000
Primary: 34C10
Milestones
Received: 18 June 1975
Revised: 19 December 1975
Published: 1 March 1976
Authors
Thomas Thornton Read