Properties of Kneser solutions for third-order differential equations

Abstract

We study asymptotic behavior of nonoscillatory solutions for the third-order differential equations of the form

$${\left[r\left(t\right){\left(y^{\prime }\left(t\right)\right)}^{\gamma }\right]}^{\prime \prime }+p\left(t\right)y\left(t\right)=0.$$

For property A of studied equations we extend the information about the asymptotic properties of positive decreasing solutions.

Introduction

This paper is concerned with the asymptotic properties of the solutions for the third-order differential equations

$${\left[r\left(t\right){\left(y^{\prime }\left(t\right)\right)}^{\gamma }\right]}^{\prime \prime }+p\left(t\right)y\left(t\right)=0.$$

As usual, it is assumed that

• γ is a quotient of two positive odd numbers;

• r, p ∈ C([t₀, ∞)) are positive.

Moreover, we assume that (E) is in a canonical form, that is,

$${\int }_{t_0}^{\infty }\frac{1}{r^{1/\gamma }\left(s\right)}\mathrm{d}s=\infty .$$

By a proper solution of Eq. (E) we mean a function y(t) such that y(t) ∈ C¹([t₀, ∞)), moreover r(t)(y′(t))^γ ∈ C²([t₀, ∞)) and y(t) satisfies Eq. (E) on [t₀, ∞). We consider only those solutions y(t) of (E) which satisfy $\sup \{\left|y\right(t\left)\right|:t\ge T\}>0$ for all T ≥ t₀.

We assume that (E) possesses such a solution. A solution of (E) is called oscillatory if it has arbitrarily large zeros on (t₀, ∞) and otherwise it is called nonoscillatory.

Oscillation and asymptotic properties of the third order equations have been subject on the intensive research, see papers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. The criteria obtained essentially make use of the known Kiguradze lemma, that provides the structure of solution space for nonoscillatory (let us say positive) solutions. In particular case, for ordinary linear differential equation

$$y^{\prime \prime \prime }\left(t\right)+p\left(t\right)y\left(t\right)=0,$$

the set $\mathcal{N}$ of all positive solutions has the following decomposition

$$\mathcal{N}={\mathcal{N}}_0\cup {\mathcal{N}}_2,$$

where

$$y\left(t\right)\in {\mathcal{N}}_0\iff y\left(t\right)>0,y^{\prime }\left(t\right)<0,y^{\prime \prime }\left(t\right)>0y^{\prime \prime \prime }\left(t\right)<0$$

while

$$y\left(t\right)\in {\mathcal{N}}_2\iff y\left(t\right)>0,y^{\prime }\left(t\right)>0,y^{\prime \prime }\left(t\right)>0y^{\prime \prime \prime }\left(t\right)<0.$$

It is known that the class ${\mathcal{N}}_0\ne \varnothing$ for (E₁). Thus the effort of mathematicians has been aimed at obtaining criteria under which

$$\mathcal{N}={\mathcal{N}}_0.$$

In the literature such situation is denoted as property A and the solutions from class ${\mathcal{N}}_0$ are said to be Kneser solutions.

There are numerous results for property A of (E₁). We recall the known result of Kiguradze and Chanturija [11].

Theorem A

Assume that

$$\underset{t\to \infty }{lim\; inf}{t}^2{\int }_t^{\infty }p\left(s\right)\mathrm{d}s>\frac{2}{3\sqrt{3}}.$$

Then (E₁) has property A.

The rareness of this results consists in its application to the Euler equation that shows that the constant $2/3\sqrt{3}$ is the best one for the third order equation. In what follows, we extend and complete the above mentioned result and we derive asymptotic properties of quasi-derivatives for nonoscillatory solutions.