Oscillation of higher order functional differential equations with an advanced argument

Abstract

This paper is concerned with the oscillatory behavior of solutions of the $n$th-order nonlinear differential equations with an advanced argument $x^{\left(n\right)}\left(t\right)+\delta q\left(t\right)x^{\lambda }\left(g\left(t\right)\right)=0,t\ge t_0>0,$ where $n\ge 2$ is even or odd and $\delta =\pm 1$. Here $q\left(t\right)>0$, $g\left(t\right)\ge t$, and $\lambda$ is the ratio of odd positive integers. The results are illustrated with an example.

Introduction

The study of oscillation and asymptotic behavior of solutions of differential equations has long attracted researchers since the time of Sturm’s work in 1836. Latest estimates to date are that there has been close to 15,000 papers written on the subject. In 1955, the classic paper by Atkinson introduced this topic for nonlinear equations and generated a surge of interest that continues today. One desirable, but not always achievable, goal is to find sufficient conditions for the oscillation of solutions that works for second order equations that is easily generalizable to equations of higher orders as well.

In this paper we examine the oscillatory and asymptotic behavior of all solutions of the nonlinear higher-order differential equation with an advanced argument

$$x^{\left(n\right)}\left(t\right)+\delta q\left(t\right)x^{\lambda }\left(g\left(t\right)\right)=0,t\ge t_0>0,$$

where $n\ge 2$ is even or odd and $\delta =\pm 1$. Here $\lambda >0$ is the ratio of odd positive integers, $q$, $g\in C\left([t_0,\infty ),\mathbb{R}\right)$, $q\left(t\right)\ge 0$ with $q\left(t\right)$ not identically zero for large $t$, $g\left(t\right)\ge t$, and $g^{\prime }\left(t\right)\ge 0$ for $t\ge t_0$. In view of the fact that we may have $\delta =1$ or $\delta =-1$, the results here include both the higher order Emden–Fowler and Thomas–Fermi type equations. Our aim is to find a single condition that works in all four of the cases of $n$ even or odd with $\delta =1$ or $\delta =-1$.

By a solution of (1.1) we mean a function $x:[t_x,\infty )\to \mathbb{R}$, for some $t_x\ge t_0$, such that $x\in C^n\left([t_x,\infty ),\mathbb{R}\right)$ and which satisfies Eq. (1.1) on $[t_x,\infty )$. Without further mention, we will assume throughout that every solution $x\left(t\right)$ of (1.1) under consideration here is continuable to the right and nontrivial, i.e., $x\left(t\right)$ is defined on some ray $[t_x,\infty )$, for some $t_x\ge t_0$, and $sup\left\{\left|x\left(t\right)\right|:t\ge T\right\}>0$ for every $T\ge t_x$. Moreover, we tacitly assume that (1.1) possesses such solutions. Such a solution is said to be oscillatory if it has arbitrarily large zeros on $[t_x,\infty )$; otherwise it is said to be nonoscillatory.

Various forms of higher order equations on similar problems have recently appeared in [1], [2], [3], [4], [5], [6], [7], [8], [9]. For example, Agarwal et al. [3] study the oscillatory behavior of equations of the form

$${\left({\left|x^{(n-1)}\left(t\right)\right|}^{\alpha -1}{x}^{(n-1)}\left(t\right)\right)}^{\prime }+F(t,x\left[g\left(t\right)\right])=0,$$

where $n$ is even and $\alpha >0$ is a constant. In [3, Theorems 2.2 and 2.3], they obtain a result like our Theorem 2.1(i) below. However, due to their conditions on the function $F$, if $\alpha =1$ in Eq. (1.1), their equation becomes linear. In [1], a slightly less general equation is considered with $\delta =1$ and the authors obtain some oscillation results for delay equations by comparing their equation to first order equations whose oscillatory character is known. Their results for advanced equations require special behavior of the nonlinear term. In [2], the authors obtain some oscillation results for various combinations of $\delta$ and $n$ by comparison to first order equations. Their results for advanced equations require that the equation also involves delay terms. Bazighifan and Ramos [4] study even order delay equations with $\delta =1$. Koplatadze [6] studied advanced equations with $\delta =1$. Li and Thandapani [7] considered equations with $n$ odd and $\delta =1$, and in [8] they examined linear neutral equations again with $n$ odd and $\delta =1$. Sun [9] considered forced equations and the results did not apply to unforced equations.