Weighted differential inequality and oscillatory properties of fourth order differential equations

Aigerim Kalybay,Yaudat Sultanaev,Ryskul Oinarov

doi:10.1186/s13660-021-02731-7

Abstract

In this paper, we investigate the oscillatory properties of two fourth order differential equations in dependence on boundary behavior of its coefficients at infinity. These properties are established based on two-sided estimates of the least constant of a certain weighted differential inequality.

Highlights

We investigate the oscillatory properties of the following half-linear and linear fourth order differential equations: v(t)y (t) y (t) p–2 – u(t)y(t) y(t) p–2 = 0, t ∈ I, (1)
One of the directions of the qualitative theory of differential equations is the investigation of their oscillatory properties, which have important applications in physics, technology, medicine, biology, and in other scientific applications
[14, 15], and [16] have investigated the oscillatory properties of second order impulsive differential equations, the research of which has been significantly developed in recent decades

Summary

Introduction

We investigate the oscillatory properties of the following half-linear and linear fourth order differential equations: v(t)y (t) y (t) p–2 – u(t)y(t) y(t) p–2 = 0, t ∈ I,. C in inequality (3), and from the obtained estimates, in terms of the coefficients, we derive the oscillatory properties of equations (1) and (2). Let us consider the following second order differential inequality:. On the basis of Lemma 3.1 of the work [7], we have the following statement connecting the oscillatory properties of equation (2) to the least constant CT in inequality (3). Since from Lemma 1 it follows that the oscillatory properties of equations (1) and (2) depend on the least constant CT in (3), we first find two-sided estimates of CT of independent interest.

Assume that

It is easy to see that z