Weighted Second-Order Differential Inequality on Set of Compactly Supported Functions and Its Applications

Aigerim Kalybay,Ryskul Oinarov,Yaudat Sultanaev

doi:10.3390/math9212830

Abstract

In the paper, we establish the oscillatory and spectral properties of a class of fourth-order differential operators in dependence on integral behavior of its coefficients at zero and infinity. In order to obtain these results, we investigate a certain weighted second-order differential inequality of independent interest.

Highlights

The aim of the paper is to establish inequality (3) in the case when the functions υ−p and r−1 are weakly singular at infinity and regular at zero, so that there exists the values f (0) = Dr1 f (0) = Dr1 f (∞) = 0, and in the symmetric case when the functions υ−p and r−1 are weakly singular at zero and regular at infinity, so that there exist the values Dr1 f (0) = f (∞) = Dr1 f (∞) = 0, on the basis of the obtained results in terms of the coefficients to derive necessary and sufficient conditions for strong non-oscillation and oscillation of Equation (4), and to find conditions for boundedness from below and discreteness of the spectrum of the operator L
We investigate inequality (3) under the following pairs of conditions [(iv)+, (iii)−] and [(iii)+, (iv)−], the obtained results that we apply to study the oscillatory and spectral properties of fourth-order differential operators
One of the most important problems in the theory of singular differential operators is to find conditions which guarantee that any self-adjoint extension L of the operator Lmin has a spectrum, which is discrete and bounded below; the so-called property BD [8]

Summary

Introduction

The aim of the paper is to establish inequality (3) in the case when the functions υ−p and r−1 are weakly singular at infinity and regular at zero, so that there exists the values f (0) = Dr1 f (0) = Dr1 f (∞) = 0, and in the symmetric case when the functions υ−p and r−1 are weakly singular at zero and regular at infinity, so that there exist the values Dr1 f (0) = f (∞) = Dr1 f (∞) = 0, on the basis of the obtained results in terms of the coefficients to derive necessary and sufficient conditions for strong non-oscillation and oscillation of Equation (4), and to find conditions for boundedness from below and discreteness of the spectrum of the operator L.

Preliminaries

Spectral Characteristics of Differential Operator L

Conclusions