Oscillation and spectral properties of some classes of higher order differential operators and weighted $n$th order differential inequalities

Abstract

In this paper, we obtain strong oscillation and non-oscillation conditions for a class of higher order differential equations in dependence on an integral behavior of its coefficients in a neighborhood of infinity. Moreover, we establish some spectral properties of the corresponding higher order differential operator. In order to prove these we establish a certain weighted differential inequality of independent interest.

Keywords

• higher order differential operator
• oscillation
• non-oscillation
• variational principle
• weighted inequality
• eigenvalues
• spectrum discreteness
• spectrum positive definiteness
• nuclear operator