New Results for Oscillation of Solutions of Odd-Order Neutral Differential Equations

Abstract

Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system’s past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufficient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results.

1. Introduction

Consider the odd-order neutral delay differential equation (NDDE)

$${\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)=0,t\ge t_0,$$

(1)

where $n>1$ is odd, $\alpha$ is a quotient of odd positive integers, and

$${\rm Y}\left(t\right):=x\left(t\right)+p\left(t\right)x\left(\zeta \left(t\right)\right).$$

Throughout this paper, we assume the following:

(I₁)

$r,h$ and $\zeta$ are continuously differentiable real-valued functions on $\left[t_0,\infty \right),$ and $p,$ q and f are continuous real-valued functions on $\left[t_0,\infty \right)$.

(I₂)

$r\left(t\right)>0,$$r^{\prime }\left(t\right)\ge 0,$$0<p\left(t\right)\le p_0<\infty ,$$q\ge 0$ does not vanish identically, and $\eta \left(t_0\right)=\infty ,$ where

$$\eta \left(t\right):={\int }_t^{\infty }{r}^{-1/\alpha }\left(s\right)\mathrm{d}s;$$

(I₃)

$h\left(t\right)<t,$$\zeta \left(t\right)<t,$ and ${lim}_{t\to \infty }h\left(t\right)={lim}_{t\to \infty }\zeta \left(t\right)=\infty .$

(I₄)

$f\left(x\right)\ge k{x}^{\alpha }$ for all $x\ne 0$, where k is a positive constant (note that $x^{\alpha }=-{\left|x\right|}^{\alpha }$ for $x<0$).

By a solution of (1), we mean a continuous real-valued function $x\left(t\right)$ for $t\ge t_x\ge t_0$, which has the property: ${\rm Y}$ is continuously differentiable n times for $t\ge t_x$, $r{\left({{\rm Y}}^{\left(n-1\right)}\right)}^{\alpha }$ is continuously differentiable for $t\ge t_x$, and x satisfies (1) on $[t_x,\infty )$. We consider only the nontrivial solutions of (1) is present on some half-line $[t_x,\infty )$ and satisfying the condition $sup\{\left|x\left(t\right)\right|:t\le t<\infty \}>0$for any $t\ge t_x$.

On many occasions, symmetries have appeared in mathematical formulations that have become essential for solving problems or delving further into research. High quality studies that use nontrivial mathematics and their symmetries applied to relevant problems from all areas were presented. In fact, in recent years, many monographs and a lot of research papers have been devoted to the behavior of solutions of delay differential equations. This is due to its relevance for different life science applications and its effectiveness in finding solutions of real world problems such as natural sciences, technology, population dynamics, medicine dynamics, social sciences and genetic engineering. For some of these applications, we refer to [1,2,3]. A study of the behavior of solutions to higher order differential equations yield much fewer results than for the least order equations although they are of the utmost importance in a lot of applications, especially neutral delay differential equations. In the literature, there are many papers and books which study the oscillatory and asymptotic behavior of solutions of neutral delay differential equations by using different technique in order to establish some sufficient conditions which ensure oscillatory behavior of the solutions of (1), see [4,5,6].

The authors in [1,3,7] have studied the oscillatory behavior of the higher-order differential equation

$${\left({\left(x^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{{}^{\prime }}+q\left(t\right)x^{\beta }\left(h\left(t\right)\right)=0.$$

And the author of [8] extended the results to the following equation

$${\left(r\left(t\right){\left(x^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{{}^{\prime }}+q\left(t\right)x^{\alpha }\left(h\left(t\right)\right)=0.$$

Agarwal, Li and Rath [9,10,11,12] investigated the oscillatory behavior of quasi-linear neutral differential equation

$${\left(r\left(t\right){\left({\left(x\left(t\right)+p\left(t\right)x\left(\zeta \left(t\right)\right)\right)}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)x^{\alpha }\left(h\left(t\right)\right)=0,\mathrm{for}t\ge t_0,$$

under the condition

$$0\le p\left(t\right)<1.$$

The latter differential equation was studied by Xing et al. in [13] under the condition

$$0\le p\left(t\right)<\infty .$$

The aim of this paper is to study the oscillatory behavior of the solutions of odd-order NDDE (1). By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of (1) is either oscillatory or tends to zero.

2. Auxiliary Results

In order to prove our main results, we will employ the following lemmas.

Lemma 1

([14] Lemma (2.3)). Let $G\left(v\right)=Cv-D{v}^{\left(\alpha +1\right)/\alpha }$ where $C,D>0$. Then G attains its maximum value on $\mathbb{R}$ at $v^{*}={\left(\alpha C/\left(\alpha +1\right)D\right)}^{\alpha }$ and

$$\underset{v\in \mathbb{R}}{max}G\left(v\right)=G\left(v^{*}\right)=\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{C^{\alpha +1}}{D^{\alpha }}.$$

(2)

Lemma 2

([15]). Assume that $c_1,c_2\in [0,\infty )$ and $\gamma >0$. Then

$${\left({}_{i=1}^2{c}_i\right)}^{\gamma }\le {\mu }_{i=1}^2{c}_i^{\gamma },$$

(3)

where

$$\mu \left(\gamma \right):=\left\{\begin{array}{cc}1\hfill & if\gamma \le 1,\\ \hfill 2^{\gamma -1}& if\gamma >1.\end{array}\right.$$

Lemma 3.

Let $f\in C^n\left(\left[t_0,\infty \right),\left(0,\infty \right)\right).$ Assume that $f^{\left(n\right)}\left(t\right)$ is of fixed sign and not identically zero on $\left[t_0,\infty \right)$ and that there exists a $t_1\ge t_0$ such that $f^{\left(n-1\right)}\left(t\right)f^{\left(n\right)}\left(t\right)\le 0$ for all $t\ge t_1$. If ${lim}_{t\to \infty }f\left(t\right)\ne 0,$ then for every $\mu \in \left(0,1\right)$ there exists $t_{\mu }\ge t_1$ such that

$$f\left(t\right)\ge \frac{\mu }{\left(n-1\right)!}{t}^{n-1}\left|f^{\left(n-1\right)}\left(t\right)\right|\mathrm{for}t\ge t_{\mu }.$$

Lemma 4.

Let the function x be a positive solution to (1) on the interval $\left[t_0,\infty \right)$. Then there exists $t_1\ge t_0$ such that, for $t\ge t_1,$ ${\rm Y}\left(t\right)>0,$ ${\left(r{\left({\left({\rm Y}\right)}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }\le 0$ and there occur two cases for the derivatives of the function Υ:

$$\begin{array}{cc}\left(\mathbf{I}\right)& {{\rm Y}}^{\prime }\left(t\right)>0,{{\rm Y}}^{\prime \prime }\left(t\right)>0,{{\rm Y}}^{\left(n-1\right)}\left(t\right)>0,{{\rm Y}}^{\left(n\right)}\left(t\right)\le 0;\\ \left(\mathbf{II}\right)& {{\rm Y}}^{\prime }\left(t\right)<0,{{\rm Y}}^{\prime \prime }\left(t\right)>0,{{\rm Y}}^{\left(n-1\right)}\left(t\right)>0,{{\rm Y}}^{\left(n\right)}\left(t\right)\le 0.\end{array}$$

Proof. 

By the definition of a positive solution to (1) there exists a $t_1\in [t_0,\infty )$such that $x\left(t\right)>0,$ $x\left(h\left(t\right)\right)>0$ and $x\left(\zeta \left(t\right)\right)>0,$ for $t\ge t_1.$ By the definition of ${\rm Y}$, it is easy to see that ${\rm Y}\left(t\right)>0.$ Furthermore, from (1), we have ${\left(r{\left({\left({\rm Y}\right)}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }\le 0.$ The rest of the proof is similar to proof of ([3] Lemma 2). Thus, the proof completed. ☐

Lemma 5.

Let x be a positive solution of (1), ${\rm Y}$satisfy $\left(II\right)$ and put

$$\tilde{\eta }\left(t\right)=\frac{1}{r^{\frac{1}{\alpha }}\left(t\right)}{\left({\int }_t^{\infty }q\left(s\right)\mathrm{d}s\right)}^{\frac{1}{\alpha }}.$$

If

$${\int }_{t_0}^{\infty }\tilde{\eta }\left(s\right)s^{n-2}\mathrm{d}s=\infty ,$$

(4)

then ${lim}_{t\to \infty }x\left(t\right)={lim}_{t\to \infty }{\rm Y}\left(t\right)=0.$

Proof. 

Let x be a positive solution of (1). Using (I${}_4$) in (1), we have

$${\left(r{\left({\left({\rm Y}\right)}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }\left(t\right)+kq\left(t\right)x^{\alpha }\left(h\left(t\right)\right)\le 0.$$

(5)

From $\left(\mathbf{II}\right)$, we note that ${lim}_{t\to \infty }{\rm Y}\left(t\right)=c\ge 0,$ due to ${\rm Y}\left(t\right)>0$ and ${{\rm Y}}^{\prime }\left(t\right)<0.$ Assume that $c>0$. Then for any $ϵ>0$, we have $ϵ+c$ $>{\rm Y}\left(t\right)>c,$ eventually. By definition of ${\rm Y}\left(t\right),$ we have

$$x\left(t\right)={\rm Y}\left(t\right)-p\left(t\right)x\left(\zeta \left(t\right)\right)\ge {\rm Y}\left(t\right)-p\left(t\right){\rm Y}\left(\zeta \left(t\right)\right),$$

thus,

$$x\left(t\right)\ge c-p_0\left(ϵ+c\right)=\frac{c-p_0\left(ϵ+c\right)}{ϵ+c}\left(ϵ+c\right).$$

This implies that

$$x\left(t\right)\ge \varrho {\rm Y}\left(t\right),$$

(6)

where $\varrho =\frac{c-p_0\left(ϵ+c\right)}{ϵ+c}>0.$ Using (6) in (5), we obtain

$${\left(r{\left({\left({\rm Y}\right)}^{\left(n-1\right)}\right)}^{\alpha }\right)}^{\prime }\left(t\right)+k{\varrho }^{\alpha }q\left(t\right){{\rm Y}}^{\alpha }\left(h\left(t\right)\right)\le 0.$$

Integrating the above inequality from t to ∞, we obtain

$$r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\ge k{\varrho }^{\alpha }{\int }_t^{\infty }q\left(s\right){{\rm Y}}^{\alpha }\left(h\left(s\right)\right)\mathrm{d}s.$$

By ${lim}_{t\to \infty }{\rm Y}\left(t\right)>c$, it follows that

$${{\rm Y}}^{\left(n-1\right)}\left(t\right)\ge \varrho c{k}^{\frac{1}{\alpha }}\tilde{\eta }\left(t\right).$$

(7)

Integrating (7) twice from t to ∞, we have

$${{\rm Y}}^{\left(n-3\right)}\left(t\right)\ge \varrho c{k}^{\frac{1}{\alpha }}{\int }_t^{\infty }{\int }_u^{\infty }\tilde{\eta }\left(s\right)\mathrm{d}s\mathrm{d}u=\varrho c{k}^{\frac{1}{\alpha }}{\int }_t^{\infty }\tilde{\eta }\left(s\right)\left(s-t\right)\mathrm{d}s.$$

Repeating this procedure, we arrive at

$$-{{\rm Y}}^{\prime }\left(t\right)\ge \frac{\varrho c{k}^{\frac{1}{\alpha }}}{\left(n-3\right)!}{\int }_t^{\infty }\tilde{\eta }\left(s\right){\left(s-t\right)}^{n-3}\mathrm{d}s.$$

Now, integrating from $t_1$ to ∞, we see that

$${\rm Y}\left(t_1\right)\ge \frac{\varrho c{k}^{\frac{1}{\alpha }}}{\left(n-2\right)!}{\int }_{t_1}^{\infty }\tilde{\eta }\left(s\right){\left(s-t_1\right)}^{n-2}\mathrm{d}s\ge \frac{\varrho c{k}^{\frac{1}{\alpha }}}{2^{n-2}\left(n-2\right)!}{\int }_{2t_1}^{\infty }\tilde{\eta }\left(s\right)s^{n-2}\mathrm{d}s,$$

which contradicts (4), and so we have verified that ${lim}_{t\to \infty }{\rm Y}\left(t\right)=0.$ ☐

3. Main Results

In the following lemma, we will use the notation $\tilde{q}\left(t\right):=min\left\{q\left(t\right),q\left(h\left(t\right)\right)\right\},$ ${\tilde{q}}_2\left(t\right):=min\left\{q\left(h^{-1}\left(t\right)\right),q\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right\}$ and

$${\zeta }^{\prime }\ge {\zeta }_0>0;$$

(8)

$${\left(h^{-1}\left(t\right)\right)}^{\prime }\ge h_0>0.$$

(9)

Lemma 6.

Let x be a positive solution of the equation in (1). If (8) and the equality $h\circ \zeta =\zeta \circ h$ hold, then the following inequality is valid

$${\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }+\frac{p_0^{\alpha }}{{\zeta }_0}r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{k}{\mu }\tilde{q}\left(t\right){{\rm Y}}^{\alpha }\left(h\left(t\right)\right)\le 0.$$

(10)

Moreover, if (8) and (9) hold, then

$$\begin{array}{ccc}& & \frac{{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{h_0}+\frac{p_0^{\alpha }{\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }}{h_0{\zeta }_0}\hfill \\ & & +\frac{k}{\mu }{\tilde{q}}_2\left(t\right){{\rm Y}}^{\alpha }\left(t\right)\le 0.\hfill \end{array}$$

(11)

Proof. 

Let x be a positive solution of (1). Then, there exists $t_1\ge t_0$ such that $x\left(t\right)>0,$ $x\left(h\left(t\right)\right)>0$ and $x\left(\zeta \left(t\right)\right)>0$ for $t\ge t_1$. By the equality ${\rm Y}\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\zeta \left(t\right)\right)$ together with Lemma 2, we obtain the inequality

$${{\rm Y}}^{\alpha }\left(t\right)\le \mu \left(x^{\alpha }\left(t\right)+p_0^{\alpha }{x}^{\alpha }\left(\zeta \left(t\right)\right)\right).$$

(12)

From (5) and the properties $h\circ \zeta =\zeta \circ h$ and ${\zeta }^{\prime }\ge {\zeta }_0$, we obtain

$$\begin{array}{ccc}\hfill 0& \ge & \frac{p_0^{\alpha }}{{\zeta }^{\prime }\left(t\right)}{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+p_0^{\alpha }kq\left(\zeta \left(t\right)\right)x^{\alpha }\left(h\left(\zeta \left(t\right)\right)\right)\hfill \\ & \ge & \frac{p_0^{\alpha }}{{\zeta }_0}{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+p_0^{\alpha }kq\left(\zeta \left(t\right)\right)x^{\alpha }\left(\zeta \left(h\left(t\right)\right)\right).\hfill \end{array}$$

(13)

Using the latter inequalities and taking those in (5) and (13) into account as well, we obtain

$$\begin{array}{ccc}\hfill 0& \ge & {\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\alpha }}{{\zeta }_0}{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +kq\left(t\right)x^{\alpha }\left(h\left(t\right)\right)+p_0^{\alpha }kq\left(\zeta \left(t\right)\right)x^{\alpha }\left(\zeta \left(h\left(t\right)\right)\right)\hfill \\ & \ge & {\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\alpha }}{{\zeta }_0}{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +k\tilde{q}\left(t\right)\left(x^{\alpha }\left(h\left(t\right)\right)+p_0^{\alpha }{x}^{\alpha }\left(\zeta \left(h\left(t\right)\right)\right)\right),\hfill \end{array}$$

which with (12) gives

$$\begin{array}{ccc}\hfill 0& \ge & {\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\alpha }}{{\zeta }_0}{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{k}{\mu }\tilde{q}\left(t\right){{\rm Y}}^{\alpha }\left(h\left(t\right)\right)\hfill \\ & \ge & {\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }+\frac{p_0^{\alpha }}{{\zeta }_0}\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)\right)}^{\prime }+\frac{k}{\mu }\tilde{q}\left(t\right){{\rm Y}}^{\alpha }\left(h\left(t\right)\right).\hfill \end{array}$$

This proves the inequality in (10). In order to show inequality (11) we proceed as follows. From (8) and (9), we obtain

$$\begin{array}{ccc}\hfill 0& \ge & \frac{1}{{\left(h^{-1}\left(t\right)\right)}^{\prime }}{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+kq\left(h^{-1}\left(t\right)\right)x^{\alpha }\left(t\right)\hfill \\ & \ge & \frac{1}{h_0}{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+kq\left(h^{-1}\left(t\right)\right)x^{\alpha }\left(t\right).\hfill \end{array}$$

(14)

Moreover,

$$\begin{array}{ccc}\hfill 0& \ge & \frac{p_0^{\alpha }}{{\left(h^{-1}\left(\zeta \left(t\right)\right)\right)}^{\prime }}{\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +p_0^{\alpha }kq\left(h^{-1}\left(\zeta \left(t\right)\right)\right)x^{\alpha }\left(\zeta \left(t\right)\right)\hfill \\ & \ge & \frac{p_0^{\alpha }}{h_0{\zeta }_0}{\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }+p_0^{\alpha }kq\left(h^{-1}\left(\zeta \left(t\right)\right)\right)x^{\alpha }\left(\zeta \left(t\right)\right).\hfill \end{array}$$

(15)

Combining (14) with (15) and taking into account (12), we have

$$\begin{array}{ccc}\hfill 0& \ge & \frac{1}{h_0}{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{k}{\mu }{\tilde{q}}_2\left(t\right){\left(x\left(t\right)+p_{0}x\left(\zeta \left(t\right)\right)\right)}^{\alpha }\hfill \\ & & +\frac{p_0^{\alpha }}{h_0{\zeta }_0}{\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime },\hfill \end{array}$$

that is

$$\begin{array}{ccc}\hfill 0& \ge & \frac{1}{h_0}{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }+\frac{p_0^{\alpha }}{{\zeta }_0}r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\frac{k}{\mu }{\tilde{q}}_2\left(t\right){{\rm Y}}^{\alpha }\left(t\right).\hfill \end{array}$$

This proves (11) and completes the proof of Lemma 6. ☐

Theorem 1.

Suppouse that $h\left(t\right)\le \zeta \left(t\right)$, $h\circ \zeta =\zeta \circ h$, $h^{\prime }\left(t\right)>0$ and (8) hold. Morever, assume that (4) is satisfied and that there exists a function $\delta \in C^1\left(\left[t_0,\infty \right),\left(0,\infty \right)\right)$ with the property that for all sufficiently large $t_1\ge t_0,$ there exists $t_2\ge t_1$ such that

$$\underset{t\to \infty }{lim\; sup}{\int }_{t_1}^t\left[k\delta \left(s\right)\frac{\tilde{q}\left(s\right)}{\mu }-\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\mu }^{\alpha }{\left(\alpha +1\right)}^{\alpha +1}}\left(1+\frac{p_0^{\alpha }}{{\zeta }_0}\right)\frac{r\left(s\right){\left({\delta }^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\delta \left(s\right)h^{n-2}\left(s\right)h^{\prime }\left(s\right)\right)}^{\alpha }}\right]\mathrm{d}s=\infty .$$

(16)

Then, a solution $x\left(t\right)$ to (1) either oscillates or else tends to zero when $t\to \infty$.

Proof. 

Let x be a positive solution of (1). Then, there exist $t_1\ge t_0$ such that $x\left(t\right)>0,$ $x\left(h\left(t\right)\right)>0$ and $x\left(\zeta \left(t\right)\right)>0$ for $t\ge t_1$. Define the positive function $\omega$ by

$$\omega \left(t\right)=\delta \left(t\right)\frac{r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}.$$

(17)

Hence, by differentiating (17), we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& =& {\delta }^{\prime }\left(t\right)\frac{r{\left({\left({\rm Y}\right)}^{\left(n-1\right)}\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\delta \left(t\right)\frac{{\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }{{\rm Y}}^{\alpha -1}\left(h\left(t\right)\right){{\rm Y}}^{\prime }\left(h\left(t\right)\right)h^{\prime }\left(t\right)}{{{\rm Y}}^{2\alpha }\left(h\left(t\right)\right)}.\hfill \end{array}$$

(18)

Since ${{\rm Y}}^{{}^{\prime }}>0,$ ${{\rm Y}}^{{}^{\prime \prime }}>0,$we see that ${lim}_{t\to \infty }{{\rm Y}}^{{}^{\prime }}\left(t\right)\ne 0,$ using Lemma 3 with $f={{\rm Y}}^{{}^{\prime }}$, we obtain

$${{\rm Y}}^{{}^{\prime }}\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(t\right),$$

for every $\mu \in (0,1).$ Thus, by ${{\rm Y}}^{\left(n\right)}\left(t\right)\le 0,$ we obtain

$${{\rm Y}}^{{}^{\prime }}\left(h\left(t\right)\right)\ge \frac{\mu }{\left(n-2\right)!}{\left(h\left(t\right)\right)}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(h\left(t\right)\right)\ge \frac{\mu }{\left(n-2\right)!}{\left(h\left(t\right)\right)}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(t\right).$$

(19)

Substituting (17) and (19) into (18) implies

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & {\delta }^{\prime }\left(t\right)\frac{r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\delta \left(t\right)\frac{{\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}\hfill \\ & & -{\left(\frac{{{\rm Y}}^{\left(n-1\right)}\left(t\right)}{{\rm Y}\left(h\left(t\right)\right)}\right)}^{\alpha +1}\frac{\alpha \delta \left(t\right)r\left(t\right)\mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!}\hfill \\ & \le & \delta \left(t\right)\frac{{\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(t\right)\mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!}{\left(\frac{\omega \left(t\right)}{\delta \left(t\right)r\left(t\right)}\right)}^{\frac{\alpha +1}{\alpha }},\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & \delta \left(t\right)\frac{{\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)\hfill \\ & & -\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(t\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right).\hfill \end{array}$$

(20)

Now, define another positive function v by

$$v\left(t\right)=\delta \left(t\right)\frac{r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}.$$

(21)

By differentiating (21), we obtain

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& =& {\delta }^{\prime }\left(t\right)\frac{r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\frac{\delta \left(t\right){\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }{{\rm Y}}^{\alpha -1}\left(h\left(t\right)\right){{\rm Y}}^{\prime }\left(h\left(t\right)\right)h^{\prime }\left(t\right)}{{{\rm Y}}^{2\alpha }\left(h\left(t\right)\right)}.\hfill \end{array}$$

(22)

From (19), $h\left(t\right) \le \zeta \left(t\right)$ and ${{\rm Y}}^{\left(n\right)}\left(t\right) \le 0$, we have

$${{\rm Y}}^{{}^{\prime }}\left(h\left(t\right)\right)\ge \frac{\mu }{\left(n-2\right)!}{\left(h\left(t\right)\right)}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(h\left(t\right)\right) \ge \frac{\mu }{\left(n-2\right)!}{\left(h\left(t\right)\right)}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right).$$

(23)

Substituting (23) and (21) into (22), implies

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& \le & {\delta }^{\prime }\left(t\right)\frac{r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\delta \left(t\right)\frac{{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}\hfill \\ & & -{\left(\frac{{{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)}{{\rm Y}\left(h\left(t\right)\right)}\right)}^{\alpha +1}\frac{\alpha \delta \left(t\right)r\left(\zeta \left(t\right)\right)\mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!}\hfill \\ & \le & \delta \left(t\right)\frac{{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(\zeta \left(t\right)\right)\mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!}{\left(\frac{v\left(t\right)}{\delta \left(t\right)r\left(\zeta \left(t\right)\right)}\right)}^{\frac{\alpha +1}{\alpha }}.\hfill \end{array}$$

By $r^{{}^{\prime }}\left(t\right)>0,$ we obtain

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& \le & \delta \left(t\right)\frac{{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)\hfill \\ & & -\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(t\right)}{v}^{\left(\alpha +1\right)/\alpha }\left(t\right).\hfill \end{array}$$

(24)

Now, using inequalities (20) and (24), we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)+\frac{p_0^{\alpha }}{{\zeta }_0}{v}^{\prime }\left(t\right)& \le & \delta \left(t\right)\frac{{\left(r\left(t\right){\left({{\rm Y}}^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+\frac{{\delta }_0^{\alpha }}{{\zeta }_0}{\left(r\left(\zeta \left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(\zeta \left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(h\left(t\right)\right)}\hfill \\ & & +\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)-\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(t\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\hfill \\ & & +\frac{{\delta }_0^{\alpha }}{{\zeta }_0}\left(\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)-\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(t\right)}{v}^{\left(\alpha +1\right)/\alpha }\left(t\right)\right).\hfill \end{array}$$

(25)

By inserting the inequality in (10) in (26), we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)+\frac{p_0^{\alpha }}{{\zeta }_0}{v}^{\prime }\left(t\right)& \le & -\delta \left(t\right)\left(\frac{k\tilde{q}\left(t\right)}{\mu }\right)\hfill \\ & & +\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)-\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(t\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\hfill \\ & & +\frac{{\delta }_0^{\alpha }}{{\zeta }_0}\left(\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)-\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(\zeta \left(t\right)\right)}{v}^{\left(\alpha +1\right)/\alpha }\left(t\right)\right).\hfill \end{array}$$

(26)

By applying the Lemma 1 with

$$C=\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\mathrm{and}D=\frac{\alpha \mu h^{n-2}\left(t\right)h^{\prime }\left(t\right)}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(t\right)},$$

we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)+\frac{p_0^{\alpha }}{{\zeta }_0}{v}^{\prime }\left(t\right)& \le & -\delta \left(t\right)\left(\frac{k\tilde{q}\left(t\right)}{\mu }\right)+\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\mu }^{\alpha }{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\delta }^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\left(\delta \left(t\right)h^{n-2}\left(t\right)h^{\prime }\left(t\right)\right)}^{\alpha }}\hfill \\ & & +\frac{p_0^{\alpha }{\left(\left(n-2\right)!\right)}^{\alpha }}{{\zeta }_0{\mu }^{\alpha }{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left({\delta }^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\left(\delta \left(t\right)h^{n-2}\left(t\right)h^{\prime }\left(t\right)\right)}^{\alpha }}.\hfill \end{array}$$

Integrating last the inequality from $t_2$ to $t,$ we obtain

$${\int }_{t_2}^t\left[k\delta \left(s\right)\frac{\tilde{q}\left(s\right)}{\mu }-\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\mu }^{\alpha }{\left(\alpha +1\right)}^{\alpha +1}}\left(1+\frac{p_0^{\alpha }}{{\zeta }_0}\right)\frac{r\left(s\right){\left({\delta }^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\delta \left(s\right)h^{n-2}\left(s\right)h^{\prime }\left(s\right)\right)}^{\alpha }}\right]\mathrm{d}s\le \omega \left(t_2\right)+\frac{p_0^{\alpha }}{{\zeta }_0}v\left(t_2\right).$$

The proof is complete. ☐

Theorem 2.

Suppose that the functions h and ζ satisfy (8), (9) and $h\left(t\right)\le \zeta \left(t\right)$ for $t_0$. In addition, suppose that (4) is satisfied. If there exists a function $\delta \in C^1\left(\left[t_0,\infty \right),\left(0,\infty \right)\right)$ with the property that for all sufficiently large $t_1\ge t_0,$ there exists $t_2\ge t_1$ such that

$$\underset{t\to \infty }{lim\; sup}{\int }_{t_2}^t\left[k\delta \left(s\right)\frac{{\tilde{q}}_2\left(s\right)}{\mu }-\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\mu }^{\alpha }{h}_0{\left(\alpha +1\right)}^{\alpha +1}}\left(1+\frac{p_0^{\alpha }}{{\zeta }_0}\right)\frac{r\left(h^{-1}\left(s\right)\right){\left({\delta }^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\delta \left(s\right)s^{n-2}\right)}^{\alpha }}\right]\mathrm{d}s=\infty$$

(27)

is valid. Then a solution $x\left(t\right)$ of Equation (1) oscillates or tends to zero when $t\to \infty$.

Proof. 

Let x be a positive solution of (1). Then, there exist $t_1\ge t_0$ such that $x\left(t\right)>0,$ $x\left(h\left(t\right)\right)>0$ and $x\left(\zeta \left(t\right)\right)>0$ for $t\ge t_1$. Define the positive function $\omega$ by

$$\omega \left(t\right)=\delta \left(t\right)\frac{r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(t\right)}.$$

(28)

Hence, by differentiating (28), we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& =& {\delta }^{\prime }\left(t\right)\frac{r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(t\right)}+\delta \left(t\right)\frac{{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }{{\rm Y}}^{\alpha -1}\left(t\right){{\rm Y}}^{\prime }\left(t\right)}{{{\rm Y}}^{2\alpha }\left(t\right)}.\hfill \end{array}$$

(29)

Since ${{\rm Y}}^{{}^{\prime }}>0,$ ${{\rm Y}}^{{}^{\prime \prime }}>0,$we see that ${lim}_{t\to \infty }{{\rm Y}}^{{}^{\prime }}\ne 0,$ using Lemma 3 with $f={{\rm Y}}^{{}^{\prime }}$, we obtain

$${{\rm Y}}^{{}^{\prime }}\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(t\right),$$

(30)

for every $\mu \in (0,1).$ Thus, by $h^{-1}\left(t\right)>t$ and ${{\rm Y}}^{\left(n\right)}\left(t\right)\le 0,$ we obtain

$${{\rm Y}}^{{}^{\prime }}\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(t\right)\ge {{\rm Y}}^{{}^{\prime }}\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right).$$

(31)

Substituting (28) and (31) into (29) implies

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & {\delta }^{\prime }\left(t\right)\frac{r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(t\right)}+\delta \left(t\right)\frac{{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}\hfill \\ & & -{\left(\frac{{{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)}{{\rm Y}\left(t\right)}\right)}^{\alpha +1}\frac{\alpha \delta \left(t\right)r\left(h^{-1}\left(t\right)\right)\mu t^{n-2}}{\left(n-2\right)!}\hfill \\ & \le & \delta \left(t\right)\frac{{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(h^{-1}\left(t\right)\right)\mu t^{n-2}}{\left(n-2\right)!}{\left(\frac{\omega \left(t\right)}{\delta \left(t\right)r\left(h^{-1}\left(t\right)\right)}\right)}^{\frac{\alpha +1}{\alpha }},\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & \delta \left(t\right)\frac{{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)\hfill \\ & & -\frac{\alpha \mu t^{n-2}}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(t\right)\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right).\hfill \end{array}$$

(32)

Now, define another positive function v by

$$v\left(t\right)=\delta \left(t\right)\frac{r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(t\right)}.$$

(33)

By differentiating (33), we obtain

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& =& {\delta }^{\prime }\left(t\right)\frac{r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(t\right)}\hfill \\ & & +\frac{\delta \left(t\right){\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }{{\rm Y}}^{\alpha -1}\left(t\right){{\rm Y}}^{\prime }\left(t\right)}{{{\rm Y}}^{2\alpha }\left(t\right)}.\hfill \end{array}$$

(34)

From (30), $h^{-1}\left(\zeta \left(t\right)\right)\ge t$ and ${{\rm Y}}^{\left(n\right)}\left(t\right)\le 0$, we have

$${{\rm Y}}^{{}^{\prime }}\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right).$$

(35)

A similar method has been used in the work [16]. Substituting (35) and (33) into (34), implies

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& \le & {\delta }^{\prime }\left(t\right)\frac{r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }}{{{\rm Y}}^{\alpha }\left(t\right)}\hfill \\ & & +\frac{\delta \left(t\right){\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}\hfill \\ & & -{\left(\frac{{{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)}{{\rm Y}\left(t\right)}\right)}^{\alpha +1}\frac{\alpha \delta \left(t\right)r\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\mu t^{n-2}}{\left(n-2\right)!}\hfill \\ & \le & \frac{\delta \left(t\right){\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)\hfill \\ & & -\frac{\alpha \delta \left(t\right)r\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\mu t^{n-2}}{\left(n-2\right)!}{\left(\frac{v\left(t\right)}{\delta \left(t\right)r\left(h^{-1}\left(\zeta \left(t\right)\right)\right)}\right)}^{\frac{\alpha +1}{\alpha }}.\hfill \end{array}$$

By $r^{{}^{\prime }}\left(t\right)>0,$ we obtain

$$\begin{array}{ccc}\hfill v^{\prime }\left(t\right)& \le & \frac{\delta \left(t\right){\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }}{{{\rm Y}}^{\alpha }\left(t\right)}+\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)\hfill \\ & & -\frac{\alpha \mu t^{n-2}}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(t\right)\right)}{v}^{\left(\alpha +1\right)/\alpha }\left(t\right).\hfill \end{array}$$

(36)

Now, using inequalities (20) and (24), we obtain

$$\begin{array}{ccc}& & \frac{1}{h_0}{\omega }^{\prime }\left(t\right)+\frac{p_0^{\alpha }}{h_0{\zeta }_0}{v}^{\prime }\left(t\right)\hfill \\ & \le & \frac{\delta \left(t\right)}{h_0{{\rm Y}}^{\alpha }\left(t\right)}{\left(r\left(h^{-1}\left(t\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(t\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\alpha }}{h_0{\zeta }_0}{\left(r\left(h^{-1}\left(\zeta \left(t\right)\right)\right){\left({{\rm Y}}^{\left(n-1\right)}\left(h^{-1}\left(\zeta \left(t\right)\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +\frac{{\delta }^{\prime }\left(t\right)}{h_0\delta \left(t\right)}\omega \left(t\right)-\frac{\alpha \mu t^{n-2}}{h_0\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(t\right)\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\hfill \\ & & +\frac{p_0^{\alpha }}{h_0{\zeta }_0}\left(\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}v\left(t\right)-\frac{\alpha \mu t^{n-2}}{\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(t\right)\right)}{v}^{\left(\alpha +1\right)/\alpha }\left(t\right)\right).\hfill \end{array}$$

By (4), we obtain

$$\begin{array}{ccc}& & \frac{1}{h_0}{\omega }^{\prime }\left(t\right)+\frac{p_0^{\alpha }}{h_0{\zeta }_0}{v}^{\prime }\left(t\right)\hfill \\ & \le & -\delta \left(t\right)\left(\frac{k{\tilde{q}}_2\left(t\right)}{\mu }\right)\hfill \\ & & +\frac{{\delta }^{\prime }\left(t\right)}{h_0\delta \left(t\right)}\omega \left(t\right)-\frac{\alpha \mu t^{n-2}}{h_0\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(t\right)\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(t\right)\hfill \\ & & +\frac{p_0^{\alpha }}{{\zeta }_0}\left(\frac{{\delta }^{\prime }\left(t\right)}{h_0\delta \left(t\right)}v\left(t\right)-\frac{\alpha \mu t^{n-2}}{h_0\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(\zeta \left(t\right)\right)\right)}{v}^{\left(\alpha +1\right)/\alpha }\left(t\right)\right).\hfill \end{array}$$

By using Lemma 1 with

$$C=\frac{{\delta }^{\prime }\left(t\right)}{h_0\delta \left(t\right)}\mathrm{and}D=\frac{\alpha \mu t^{n-2}}{h_0\left(n-2\right)!{\delta }^{1/\alpha }\left(t\right)r^{1/\alpha }\left(h^{-1}\left(t\right)\right)},$$

we obtain

$$\begin{array}{ccc}\hfill \frac{1}{h_0}{\omega }^{\prime }\left(t\right)+\frac{p_0^{\alpha }}{h_0{\zeta }_0}{v}^{\prime }\left(t\right)& \le & -\delta \left(t\right)\left(\frac{k{\tilde{q}}_2\left(t\right)}{\mu }\right)\hfill \\ & & +\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\mu }^{\alpha }{h}_0{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(h^{-1}\left(t\right)\right){\left({\delta }^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\left(\delta \left(t\right)t^{n-2}\right)}^{\alpha }}\hfill \\ & & +\frac{p_0^{\alpha }{\left(\left(n-2\right)!\right)}^{\alpha }}{{\zeta }_0{h}_0{\mu }^{\alpha }{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(h^{-1}\left(t\right)\right){\left({\delta }^{\prime }\left(t\right)\right)}^{\alpha +1}}{{\left(\delta \left(t\right)t^{n-2}\right)}^{\alpha }}.\hfill \end{array}$$

Integrating both sides of the latter inequality from $t_2$ to $t,$ we obtain

$$\begin{array}{ccc}& & {\int }_{t_2}^t\left[\frac{k\delta \left(s\right)}{\mu }{\tilde{q}}_2\left(s\right)-\frac{{\left(\left(n-2\right)!\right)}^{\alpha }}{{\mu }^{\alpha }{h}_0{\left(\alpha +1\right)}^{\alpha +1}}\left(1+\frac{p_0^{\alpha }}{{\zeta }_0}\right)\frac{r\left(h^{-1}\left(s\right)\right){\left({\delta }^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\delta \left(s\right)s^{n-2}\right)}^{\alpha }}\right]\mathrm{d}s\hfill \\ & \le & \frac{1}{h_0}\omega \left(t_2\right)+\frac{p_0^{\alpha }}{h_0{\zeta }_0}v\left(t_2\right).\hfill \end{array}$$

The proof is complete. ☐

Example 1.

Consider the odd order neutral delay differential equation

$${{\rm Y}}^{\left(n\right)}\left(t\right)+\frac{q_0}{t^n}x\left(\frac{t}{e^2}\right)=0,t\ge 1,q_0>0,n\ge 3,$$

(37)

where ${\rm Y}\left(t\right)=x\left(t\right)+\frac{17}{18}x\left(\frac{t}{e}\right),$ and

$$k=\mu =\alpha =r\left(t\right)=1,{\tilde{q}}_2\left(s\right)=\frac{q_0}{{\left(e^{2}t\right)}^n},h\left(t\right)=\frac{t}{e^2},\zeta \left(t\right)=\frac{t}{e},andset\delta \left(t\right)=t^{n-1}.$$

Using Example 1 in [17], we find that every solution of (37) oscillates or tends to zero if

$$q_0>9\left(n-1\right)!e^{2n-3},$$

and using Example 2.11 in [13], we find that every solution of (37) oscillates or tends to zero if

$$q_0>\left(n-1\right)!\left(e^{2n-3}+\frac{17}{18}{e}^{2n-2}\right).$$

From condition (27) in Theorem 2, we see that every solution of (37) oscillates or tends to zero if

$$\underset{t\to \infty }{lim\; sup}\left[\frac{q_0}{e^{2n}}-\frac{\left(n-2\right)!\left(n-1\right)}{4e^2}\left(1+\frac{17}{18}e\right)\right]lnt=\infty ,$$

thus,

$$q_0>\frac{e}{4}\left(n-1\right)!\left(e^{2n-3}+\frac{17}{18}{e}^{2n-2}\right).$$

Hence, we can see that our results are better than ([17] Example 1) and ([13] Example 2.11).

4. Conclusions

In this work, we established the oscillation criteria for a class of odd-order delay differential equations. By using Riccati transformation, we presented some sufficient conditions which ensure that every solution of (1) is either oscillatory or tends to zero. The approach used does not need to be restricted by the condition $0<p\left(t\right)<1$, unlike many previous work.

For interested researchers, results presented in this paper may be extended to more general equations than (1). Another interesting problem for further research is to obtain new criteria for oscillatory solutions for (1) without requiring (8).