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Abstract

For the second-order equation y" +qy = 0, Wintner [6] proved that a sufficient condition for oscillation was that (1) tr1 I qix)it -x)dx-> oo as t-> oo.Independently Leighton [2] proved that a sufficient condition for oscillation of iry')'+qy = 0 was that q he positive for sufficiently large x and that /OO /» CO rix)~xdx = oo and I qix)dx = oo. SubsequentlyLeighton [3] proved that conditions (2) were sufficient without the restriction q be positive for sufficiently large x.In this paper we prove analogous theorems for the linear equation of order 2n 7r("))<") + (-l)n~1qy = 0. There will be nosign restrictions on the function q. For n = l, the earlier results of Wintner and Leighton will be contained in our theorems. The results of this paper parallel earlier results of the author for a fourth-order equation with middle term [l].Throughout r and q denote continuous, real-valued functions on a ray [a, w) and r is assumed to be positive-valued.If the real-valued function y has n continuous derivatives and ryM has n continuous derivatives, then we define Liy) by: