On the Asymptotic Behavior of Noncanonical Third-Order Emden–Fowler Delay Differential Equations with a Superlinear Neutral Term

Abstract

The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that every solution of the equation considered either oscillates or converges to zero asymptotically. The results are illustrated with two examples.

1. Introduction

This paper deals with the asymptotic behavior of solutions to the noncanonical third-order Emden–Fowler delay differential equation with a superlinear neutral term

$${\left(r\left(t\right){\left[{\left(x\left(t\right)+p\left(t\right)x^{\beta }\left(\tau \left(t\right)\right)\right)}^{″}\right]}^{\alpha }\right)}^{\prime }+q\left(t\right)x^{\delta }\left(\sigma \left(t\right)\right)=0$$

(1)

for $t\ge t_0>0$. Throughout the paper, we assume that

($C_1$)

$\alpha$, $\beta$, and $\delta$ are the quotients of odd positive integers with $\beta \ge 1$;

($C_2$)

$r,p,q:[t_0,\infty )\to \mathbb{R}$ are continuous functions, $r\left(t\right)>0$, $p\left(t\right)\ge 1$, $p\left(t\right)\not\equiv 1$ for large t, $q\left(t\right)\ge 0$, $q\left(t\right)$ is not identical to zero for large t, and (1) is in noncanonical form, i.e.,

$$I_1(t,t_0):={\int }_{t_0}^t\frac{1}{r^{1/\alpha }\left(s\right)}ds<\infty \mathrm{as}t\to \infty ;$$

(2)

($C_3$)

$\tau ,\sigma :[t_0,\infty )\to \mathbb{R}$ are continuous functions, $\tau \left(t\right)\le t$, $\sigma \left(t\right)<\tau \left(t\right)$, $\tau$ is strictly increasing, and ${lim}_{t\to \infty }\tau \left(t\right)={lim}_{t\to \infty }\sigma \left(t\right)=\infty$;

($C_4$)

$h\left(t\right):={\tau }^{-1}\left(\sigma \left(t\right)\right)$, where ${\tau }^{-1}$ is the inverse function of $\tau$.

To simplify the notation, we define

$$z\left(t\right):=x\left(t\right)+p\left(t\right)x^{\beta }\left(\tau \left(t\right)\right).$$

By a solution of (1), we mean a function $x\in \mathrm{C}([t_x,\infty ),\mathbb{R})$ for some $t_x\ge t_0$ such that $z\in {\mathrm{C}}^2([t_x,\infty ),\mathbb{R})$, $r{\left(z^{″}\right)}^{\alpha }\in {\mathrm{C}}^1([t_x,\infty ),\mathbb{R})$, and x satisfies (1) on $[t_x,\infty )$. We consider only those proper solutions of (1) that exist on some half-line $[t_x,\infty )$ and satisfy

$$sup\left\{\left|x\left(t\right)\right|:T_1\le t<\infty \right\}>0\mathrm{for}\mathrm{all}{T}_1\ge t_x;$$

moreover, we tacitly suppose that (1) possesses such proper solutions. A solution $x\left(t\right)$ of (1) is termed oscillatory if it has arbitrarily large zeros on $[t_x,\infty )$; otherwise, it is called nonoscillatory.

The analysis of qualitative properties of neutral differential equations is not only of theoretical interest but also has significant practical importance. This is due to the fact that such equations arise in numerous applied problems in natural sciences, engineering, and control. For instance, (1) and its particular cases and modifications have applications in mathematical, theoretical, and chemical physics; see, e.g., [1,2] for more details on applications of Emden–Fowler differential equations and [3,4] for particular applications of differential equations with a nonlinear neutral term.

Oscillatory and asymptotic properties of solutions to various classes of second-order and third-order neutral differential equations have attracted the long-term interest of researchers; see, e.g., [1,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and the references cited therein. A commonly used assumption is that the neutral term is linear (i.e., $\beta =1$); see, e.g., [1,13,15,16] for more details. Several researchers were concerned with equations with a nonlinear neutral term (i.e., $\beta \ne 1$); see, e.g., [9,10] for equations with a sublinear neutral term (i.e., $\beta <1$) and [11,18] for equations with a superlinear neutral term (i.e., $\beta >1$). Using comparison principles and integral criteria, the authors in [11,18] presented several sufficient conditions which ensure that all nonoscillatory solutions to (1) converge to zero at infinity in the canonical cases where

$$\underset{t\to \infty }{lim}{I}_1(t,t_0)=\infty$$

and

$$r\left(t\right)=1,\alpha =1,$$

respectively.

Our objective in this paper is to analyze the asymptotic behavior of (1) via the Riccati technique as well as integral criteria. It is our hope that the present paper will contribute significantly to the study of asymptotic nature of third-order differential equations with a superlinear neutral term.

Since we are concerned with the oscillation and asymptotic behavior of solutions, all functional inequalities are assumed to hold eventually. Without loss of generality, we deal only with eventually positive solutions of (1) due to the fact that, under our hypotheses on $\alpha$, $\beta$, and $\delta$, if x is a solution, so is $-x$.

2. Main Results

The following lemmas are essential to establish our theorems. To present results in a compact form, we adopt the following notation:

$$\pi \left(t\right):={\int }_t^{\infty }\frac{1}{r^{1/\alpha }\left(s\right)}ds,I_2(t,t_{*}):={\int }_{t_{*}}^t\frac{1}{\pi \left(s\right)}ds\mathrm{for}t\ge t_{*}\ge t_0;$$

$$Q\left(t\right):=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\delta /\beta -\alpha =0,\hfill \\ d_1,\hfill & \mathrm{if}\delta /\beta -\alpha >0,\hfill \\ d_2{t}^{\delta /\beta -\alpha },\hfill & \mathrm{if}\delta /\beta -\alpha <0,\hfill \end{array}\right.$$

for all constants $d_1>0$ and $d_2>0$;

$$E_1\left(t\right):=\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left[1-\frac{k^{1/\beta -1}}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right]\ge 0,$$

(3)

$$E_2\left(t\right):=\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left[1-{\left(\frac{I_2({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right),t_{*})}{I_2({\tau }^{-1}\left(t\right),t_{*})}\right)}^{1/\beta }\frac{l^{1/\beta -1}}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right]\ge 0,$$

(4)

$$\begin{array}{c}\hfill E_3\left(t\right):=\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left[1-{\left(\frac{{\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)}{{\tau }^{-1}\left(t\right)}\right)}^{1/\left(\beta \theta \right)}\frac{m^{1/\beta -1}}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right]\ge 0\end{array}$$

(5)

for all constants $k>0$, $l>0$, $m>0$, for some constant $0<\theta <1$, for all sufficiently large $t_{*}\ge t_0$, and for all sufficiently large t;

$$\mathsf{\Omega }\left(t\right):=g\left(t\right)q\left(t\right)E_2^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right),g_{+}^{\prime }\left(t\right):=max\left\{0,g^{\prime }\left(t\right)\right\}$$

for some function $g\in {\mathrm{C}}^1([t_0,\infty ),(0,\infty ))$.

Lemma 1

(see [19]). If X and Y are non-negative and $\lambda >1$, then

$$X^{\lambda }+(\lambda -1)Y^{\lambda }-\lambda X{Y}^{\lambda -1}\ge 0,$$

where the equality is valid if and only if $X=Y$.

Lemma 2.

Let conditions ($C_1$)−($C_3$) be satisfied and suppose that x is an eventually positive solution of (1). Then, there exists a $t_1\ge t_0$ such that, for $t\ge t_1$, the corresponding function z satisfies one of the following three possibilities:

(I)

$z\left(t\right)>0$, $z^{\prime }\left(t\right)<0$, $z^{″}\left(t\right)>0$, and ${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le 0$;

(II)

$z\left(t\right)>0$, $z^{\prime }\left(t\right)>0$, $z^{″}\left(t\right)>0$, and ${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le 0$;

(III)

$z\left(t\right)>0$, $z^{\prime }\left(t\right)>0$, $z^{″}\left(t\right)<0$, and ${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le 0$.

Proof. 

Thanks to condition (2), the proof is straightforward, and thus is omitted. □

Lemma 3.

Let conditions ($C_1$)−($C_4$) be satisfied and suppose that x is an eventually positive solution of (1) with z satisfying case(I) of Lemma 2. If, for all constants $k>0$,

$${\int }_{t_0}^{\infty }q\left(s\right)E_1^{\delta /\beta }\left(\sigma \left(s\right)\right)ds=\infty ,$$

(6)

or

$${\int }_{t_0}^{\infty }{\int }_v^{\infty }\frac{1}{r^{1/\alpha }\left(u\right)}{\left({\int }_u^{\infty }q\left(s\right)E_1^{\delta /\beta }\left(\sigma \left(s\right)\right)ds\right)}^{1/\alpha }dudv=\infty ,$$

(7)

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

Proof. 

Assume that $x\left(t\right)$ is an eventually positive solution of (1) such that $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for $t\ge t_1\ge t_0$. By virtue of the definition of z,

$$x^{\beta }\left(\tau \left(t\right)\right)=\frac{1}{p\left(t\right)}(z\left(t\right)-x\left(t\right))\le \frac{z\left(t\right)}{p\left(t\right)},$$

which implies that

$$x\left({\tau }^{-1}\left(t\right)\right)\le \frac{z^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}.$$

Using this inequality in the definition of z, we obtain

$$\begin{array}{cc}\hfill x^{\beta }\left(t\right)& =\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left[z\left({\tau }^{-1}\left(t\right)\right)-x\left({\tau }^{-1}\left(t\right)\right)\right]\hfill \\ \hfill & \ge \frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left[z\left({\tau }^{-1}\left(t\right)\right)-\frac{z^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right].\hfill \end{array}$$

(8)

Since $\tau \left(t\right)\le t$ and $\tau$ is strictly increasing, ${\tau }^{-1}$ is also strictly increasing and $t\le {\tau }^{-1}\left(t\right)$. Hence,

$${\tau }^{-1}\left(t\right)\le {\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right).$$

(9)

It follows from (9) and the monotonicity of z that

$$z\left({\tau }^{-1}\left(t\right)\right)\ge z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right).$$

Combining the latter inequality and (8), we obtain

$$\begin{array}{c}\hfill x^{\beta }\left(t\right)\ge \frac{z\left({\tau }^{-1}\left(t\right)\right)}{p\left({\tau }^{-1}\left(t\right)\right)}\left[1-\frac{z^{1/\beta -1}\left({\tau }^{-1}\left(t\right)\right)}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right].\end{array}$$

(10)

Taking into account that $z>0$ and $z^{\prime }<0$, there exists a constant $\mu \ge 0$ such that

$$\underset{t\to \infty }{lim}z\left(t\right)=\mu <\infty .$$

If $\mu :=k>0$, then there exists a $t_2\ge t_1$ such that, for $t\ge t_2$,

$$z\left(t\right)\ge \mu ,$$

which yields

$$z^{1/\beta -1}\left(t\right)\le {\mu }^{1/\beta -1}.$$

From this inequality and (10), we deduce that

$$x^{\beta }\left(t\right)\ge E_1\left(t\right)z\left({\tau }^{-1}\left(t\right)\right).$$

Using the latter inequality in (1), we have, for $t\ge t_3\ge t_2$,

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)E_1^{\delta /\beta }\left(\sigma \left(t\right)\right)z^{\delta /\beta }\left(h\left(t\right)\right)\le -{\mu }^{\delta /\beta }q\left(t\right)E_1^{\delta /\beta }\left(\sigma \left(t\right)\right).$$

(11)

Integrating (11) from $t_3$ to ∞, we conclude that

$${\int }_{t_3}^{\infty }q\left(s\right)E_1^{\delta /\beta }\left(\sigma \left(s\right)\right)ds\le \frac{r\left(t_3\right){\left(z^{″}\left(t_3\right)\right)}^{\alpha }}{{\mu }^{\delta /\beta }}<\infty ,$$

which contradicts (6), and so $\mu =0$. Therefore, ${lim}_{t\to \infty }z\left(t\right)=0$. On the basis of $0<x\left(t\right)\le z\left(t\right)$, ${lim}_{t\to \infty }x\left(t\right)=0$.

Let us now consider the case when condition (6) is not satisfied. Integrating (11) from t to ∞ consecutively two times, we arrive at

$${\int }_t^{\infty }\frac{1}{r^{1/\alpha }\left(u\right)}{\left({\int }_u^{\infty }q\left(s\right)E_1^{\delta /\beta }\left(\sigma \left(s\right)\right)ds\right)}^{1/\alpha }du\le -\frac{z^{\prime }\left(t\right)}{{\mu }^{\delta /\left(\alpha \beta \right)}}.$$

One more integration of the latter inequality from $t_3$ to ∞ yields

$${\int }_{t_3}^{\infty }{\int }_v^{\infty }\frac{1}{r^{1/\alpha }\left(u\right)}{\left({\int }_u^{\infty }q\left(s\right)E_1^{\delta /\beta }\left(\sigma \left(s\right)\right)ds\right)}^{1/\alpha }dudv\le \frac{z\left(t_3\right)}{{\mu }^{\delta /\left(\alpha \beta \right)}},$$

which contradicts (7), and thus $\mu =0$. This completes the proof of Lemma 3. □

Lemma 4.

Let conditions ($C_1$)−($C_4$) be satisfied and suppose that x is an eventually positive solution of (1) with z satisfying case (II) of Lemma 2. Then, there exists a $t_{*}\ge t_0$ and three constants $l>0$, $d_1>0$, and $d_2>0$ such that, for large t,

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)E_2^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right)z^{\alpha }\left(h\left(t\right)\right)\le 0.$$

(12)

Proof. 

Assume that $x\left(t\right)$ is an eventually positive solution of (1) satisfying $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for $t\ge t_1\ge t_0$. Proceeding as in the proof of Lemma 3, we conclude that (8) and (9) hold. Since $r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }$ is nonincreasing on $[t_1,\infty )$, we have

$$z^{\prime }\left(t\right)=z^{\prime }\left(t_1\right)+{\int }_{t_1}^t\frac{{\left(r\left(s\right){\left(z^{″}\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{r^{1/\alpha }\left(s\right)}ds\ge {\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{1/\alpha }{I}_1(t,t_1).$$

(13)

It follows from $I_1(t,t_1)+\pi \left(t\right)=\pi \left(t_1\right)>0$ and ${lim}_{t\to \infty }\pi \left(t\right)=0$ that, for large t,

$$I_1(t,t_1)\ge \pi \left(t\right).$$

By virtue of this inequality and (13), we deduce that, for $t\ge t_2\ge t_1$,

$$z^{\prime }\left(t\right)\ge {\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{1/\alpha }\pi \left(t\right).$$

(14)

From (14), one can easily obtain ${\left(z^{\prime }\left(t\right)\pi \left(t\right)\right)}^{\prime }\le 0$ for $t\ge t_2$, which implies that

$$\begin{array}{c}\hfill z\left(t\right)=z\left(t_2\right)+{\int }_{t_2}^t\frac{z^{\prime }\left(s\right)\pi \left(s\right)}{\pi \left(s\right)}ds\ge z^{\prime }\left(t\right)\pi \left(t\right)I_2(t,t_2).\end{array}$$

(15)

Hence, for $t\ge t_3>t_2$,

$${\left(\frac{z\left(t\right)}{I_2(t,t_2)}\right)}^{\prime }=\frac{I_2(t,t_2)z^{\prime }\left(t\right)-z\left(t\right)/\pi \left(t\right)}{{\left(I_2(t,t_2)\right)}^2}\le 0.$$

Combining the latter inequality and (9), we arrive at

$$z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)\le \frac{I_2({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right),t_2)}{I_2({\tau }^{-1}\left(t\right),t_2)}z\left({\tau }^{-1}\left(t\right)\right).$$

Using the latter inequality in (8) yields that for $t\ge t_3$,

$$x^{\beta }\left(t\right)\ge \frac{z\left({\tau }^{-1}\left(t\right)\right)}{p\left({\tau }^{-1}\left(t\right)\right)}\left[1-{\left(\frac{I_2({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right),t_2)}{I_2({\tau }^{-1}\left(t\right),t_2)}\right)}^{1/\beta }\frac{z^{1/\beta -1}\left({\tau }^{-1}\left(t\right)\right)}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right].$$

(16)

Since $z\left(t\right)$ is positive and strictly increasing for $t\ge t_1$, there exist a $t_4\ge t_3$ and a constant $c>0$ such that for $t\ge t_4$,

$$z\left(t\right)\ge c.$$

(17)

Using now (17) in (16), we conclude that for $t\ge t_5\ge t_4$,

$$x^{\beta }\left(\sigma \left(t\right)\right)\ge E_2\left(\sigma \left(t\right)\right)z\left({\tau }^{-1}\left(\sigma \left(t\right)\right)\right),$$

where $t_2:=t_{*}$ and $c:=l$. Substituting this inequality into (1), we deduce that, for $t\ge t_5$,

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)E_2^{\delta /\beta }\left(\sigma \left(t\right)\right)z^{\delta /\beta -\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(h\left(t\right)\right).$$

(18)

Thanks to the fact that $r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }$ is positive and nonincreasing on $[t_1,\infty )$, there exist a $t_6\ge t_5$ and a constant $c_1>0$ such that, for $t\ge t_6$,

$$z^{″}\left(t\right)\le \frac{c_1}{r^{1/\alpha }\left(t\right)}.$$

Integrating the latter inequality from $t_6$ to t consecutively two times, we conclude that, for some constant $c_2>0$ and for large t,

$$z\left(t\right)\le c_{2}t.$$

(19)

Combining (17) and (19), there exist two constants $d_1>0$ and $d_2>0$ such that, for large t, inequality (18) takes the form

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)E_2^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right)z^{\alpha }\left(h\left(t\right)\right),$$

(20)

i.e., inequality (12) holds. This completes the proof of Lemma 4. □

Lemma 5.

Let conditions ($C_1$)−($C_4$) be satisfied and suppose that x is an eventually positive solution of (1) with z satisfying case (III) of Lemma 2. Then, there exist three constants $m>0$, $d_1>0$, and $d_2>0$ (the same as those in Lemma 4) such that, for every constant $0<\theta <1$ and for large t,

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)E_3^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right)z^{\alpha }\left(h\left(t\right)\right)\le 0.$$

(21)

Proof. 

Assume that $x\left(t\right)$ is an eventually positive solution of (1) satisfying $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for $t\ge t_1\ge t_0$. Proceeding as in the proof of Lemma 3, we deduce that (8) and (9) hold. Since $z\left(t\right)>0$, $z^{\prime }\left(t\right)>0$, and $z^{\prime }\left(t\right)<0$ on $[t_1,\infty )$, for every constant $0<\theta <1$, there exists a $t_{\theta }\ge t_1$ such that, for $t\ge t_{\theta }$,

$$z\left(t\right)\ge \theta t{z}^{\prime }\left(t\right).$$

(22)

It follows from (22) that $z\left(t\right)/t^{1/\theta }$ is nonincreasing for $t\ge t_{\theta }$. Hence, by virtue of (9),

$$z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)\le \frac{{\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}^{1/\theta }}{{\left({\tau }^{-1}\left(t\right)\right)}^{1/\theta }}z\left({\tau }^{-1}\left(t\right)\right).$$

(23)

Using (23) in (8), we obtain

$$\begin{array}{c}\hfill x^{\beta }\left(t\right)\ge \frac{z\left({\tau }^{-1}\left(t\right)\right)}{p\left({\tau }^{-1}\left(t\right)\right)}\left[1-{\left(\frac{{\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)}{{\tau }^{-1}\left(t\right)}\right)}^{1/\left(\beta \theta \right)}\frac{z^{1/\beta -1}\left({\tau }^{-1}\left(t\right)\right)}{p^{1/\beta }\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right].\end{array}$$

(24)

Taking into account that $z\left(t\right)$ is positive and strictly increasing for $t\ge t_1$, there exist a $t_2\ge t_{\theta }$ and a constant $b>0$ such that, for $t\ge t_2$,

$$z\left(t\right)\ge b.$$

(25)

Substituting (25) into (24) implies that for large t,

$$x^{\beta }\left(\sigma \left(t\right)\right)\ge E_3\left(\sigma \left(t\right)\right)z\left({\tau }^{-1}\left(\sigma \left(t\right)\right)\right),$$

where $b:=m$. Using now the latter inequality in (1), we conclude that for large t,

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)E_3^{\delta /\beta }\left(\sigma \left(t\right)\right)z^{\delta /\beta -\alpha }\left(h\left(t\right)\right)z^{\alpha }\left(h\left(t\right)\right).$$

(26)

Since $z^{\prime }\left(t\right)$ is positive and strictly decreasing on $[t_1,\infty )$, there exist a $t_3\ge t_2$ and a constant $b_1>0$ such that, for $t\ge t_3$,

$$z^{\prime }\left(t\right)\le b_1.$$

Integration of the latter inequality from $t_3$ to t yields that, for some constant $b_2>0$ and for large t,

$$z\left(t\right)\le b_{2}t.$$

(27)

In view of (25) and (27), there exist two constants $d_1>0$ and $d_2>0$ such that for large t, inequality (26) takes the form

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)E_3^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right)z^{\alpha }\left(h\left(t\right)\right),$$

(28)

i.e., inequality (21) holds. This completes the proof of Lemma 5. □

Theorem 1.

Let conditions ($C_1$)−($C_4$) be satisfied and σ be nondecreasing, and suppose that either (6) or (7) holds for all constants $k>0$. If there exist a function $g\in {\mathrm{C}}^1([t_0,\infty ),(0,\infty ))$ and a constant $0<\theta <1$ such that for all constants $d_1>0$, $d_2>0$, $l>0$, $m>0$, for all sufficiently large $t_{*}\ge t_0$, and for some $T\ge t_{*}$,

$$\underset{t\to \infty }{lim\; sup}{\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{g_{+}^{\prime }\left(s\right)}{{\pi }^{\alpha }\left(s\right)}\right]ds=\infty$$

(29)

and

$$\underset{t\to \infty }{lim\; inf}{\int }_{h\left(t\right)}^t\frac{1}{r^{1/\alpha }\left(u\right)}{\left({\int }_T^{u}q\left(s\right)E_3^{\delta /\beta }\left(\sigma \left(s\right)\right)Q\left(h\left(s\right)\right)h^{\alpha }\left(s\right)ds\right)}^{1/\alpha }du>\frac{1}{\mathrm{e}},$$

(30)

then every solution x of (1) either oscillates or satisfies ${lim}_{t\to \infty }x\left(t\right)=0$.

Proof. 

Assume that (1) has a nonoscillatory solution x, which is eventually positive. Then, using Lemma 2, $z\left(t\right)$ satisfies either case (I) or case (II) or case (III) for $t\ge t_1$. If case (I) holds, as in Lemma 3, then $x\left(t\right)\to 0$ as $t\to \infty$.

Next, we consider case (II). Then, (14), (15), and (20) hold for $t\ge T$. For $t\ge T$, define

$$y\left(t\right):=g\left(t\right)\frac{r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha }}.$$

(31)

Clearly, $y\left(t\right)>0$ and, by virtue of (20) and (31), we obtain

$$\begin{array}{c}\hfill y^{\prime }\left(t\right)\le g_{+}^{\prime }\left(t\right)\frac{r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha }}-\mathsf{\Omega }\left(t\right)\frac{z^{\alpha }\left(h\left(t\right)\right)}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha }}-\alpha g\left(t\right)r\left(t\right)\frac{{\left(z^{″}\left(t\right)\right)}^{\alpha +1}}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha +1}}.\end{array}$$

(32)

It follows from (14) and (32) that

$$y^{\prime }\left(t\right)\le \frac{g_{+}^{\prime }\left(t\right)}{{\pi }^{\alpha }\left(t\right)}-\mathsf{\Omega }\left(t\right)\frac{z^{\alpha }\left(h\left(t\right)\right)}{z^{\alpha }\left(t\right)}\frac{z^{\alpha }\left(t\right)}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha }}.$$

(33)

Using $h\left(t\right)<t$ and the fact that $z\left(t\right)/I_2(t,t_{*})$ is nonincreasing for large t, we arrive at

$$\frac{z\left(h\right(t\left)\right)}{z\left(t\right)}\ge \frac{I_2(h\left(t\right),t_{*})}{I_2(t,t_{*})}.$$

(34)

Applications of inequalities (15), (33), and (34) yield

$$y^{\prime }\left(t\right)\le \frac{g_{+}^{\prime }\left(t\right)}{{\pi }^{\alpha }\left(t\right)}-\mathsf{\Omega }\left(t\right)I_2^{\alpha }(h\left(t\right),t_{*}){\pi }^{\alpha }\left(t\right).$$

Integrating the latter inequality from T to t, we conclude that

$${\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{g_{+}^{\prime }\left(s\right)}{{\pi }^{\alpha }\left(s\right)}\right]ds\le y\left(T\right),$$

which contradicts (29).

Finally, we consider case (III). Proceeding as in the proof of Lemma 5, we deduce that (22) and (28) hold for every constant $0<\theta <1$ and for large t. Using (22) in (28), we have

$${\left(r\left(t\right){\left(z^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -{\theta }^{\alpha }q\left(t\right)E_3^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right)h^{\alpha }\left(t\right){\left(z^{\prime }\left(h\left(t\right)\right)\right)}^{\alpha }.$$

(35)

Letting $w:=z^{\prime }>0$, then $w^{\prime }=z^{″}<0$ and inequality (35) reduces to

$${\left(r\left(t\right){\left(w^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -{\theta }^{\alpha }q\left(t\right)E_3^{\delta /\beta }\left(\sigma \left(t\right)\right)Q\left(h\left(t\right)\right)h^{\alpha }\left(t\right)w^{\alpha }\left(h\left(t\right)\right).$$

(36)

The integration of inequality (36) from T to t implies that for every constant $0<\theta <1$, w is a positive solution of a first-order linear delay differential inequality

$$w^{\prime }\left(t\right)+\frac{\theta }{r^{1/\alpha }\left(t\right)}{\left({\int }_T^{t}q\left(s\right)E_3^{\delta /\beta }\left(\sigma \left(s\right)\right)Q\left(h\left(s\right)\right)h^{\alpha }\left(s\right)ds\right)}^{1/\alpha }w\left(h\left(t\right)\right)\le 0.$$

(37)

However, in view of (30), there exists a constant ${\theta }_1\in (0,1)$ such that

$$\underset{t\to \infty }{lim\; inf}{\int }_{h\left(t\right)}^t\frac{{\theta }_1}{r^{1/\alpha }\left(u\right)}{\left({\int }_T^{u}q\left(s\right)E_3^{\delta /\beta }\left(\sigma \left(s\right)\right)Q\left(h\left(s\right)\right)h^{\alpha }\left(s\right)ds\right)}^{1/\alpha }du>\frac{1}{\mathrm{e}}.$$

(38)

An application of the result due to ([5], Lemma 2.2.9) yields that inequality (37) cannot have positive solutions if (38) holds. This contradiction completes the proof. □

Theorem 2.

Let conditions ($C_1$)−($C_4$) be satisfied and σ be nondecreasing, and suppose that either (6) or (7) holds for all constants $k>0$. If there exist a function $g\in {\mathrm{C}}^1([t_0,\infty ),(0,\infty ))$ and a constant $0<\theta <1$ such that for all constants $d_1>0$, $d_2>0$, $l>0$, $m>0$, for all sufficiently large $t_{*}\ge t_0$, and for some $T\ge t_{*}$,

$$\underset{t\to \infty }{limsup}{\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left(g_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{g^{\alpha }\left(s\right)}\right]ds=\infty$$

(39)

and condition (30) holds, then conclusion of Theorem 1 remains intact.

Proof. 

Assume that x is an eventually positive solution of (1). Then, by Lemma 2, $z\left(t\right)$ satisfies either case (I) or case (II) or case (III) for $t\ge t_1$. The proofs of cases (I) and (III) are the same as those of Theorem 1. Next, we consider case (II). Defining again y by (31) and proceeding as in the proof of Theorem 1, we arrive at (32). On the basis of (31), inequality (32) becomes

$$y^{\prime }\left(t\right)\le \frac{g_{+}^{\prime }\left(t\right)}{g\left(t\right)}y\left(t\right)-\mathsf{\Omega }\left(t\right)\frac{z^{\alpha }\left(h\left(t\right)\right)}{z^{\alpha }\left(t\right)}\frac{z^{\alpha }\left(t\right)}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha }}-\frac{\alpha y^{(\alpha +1)/\alpha }\left(t\right)}{{\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha }}.$$

(40)

Since z satisfies case (II), inequalities (15) and (34) hold. Using (15) and (34) in (40), we conclude that for $t\ge T$,

$$y^{\prime }\left(t\right)\le \frac{g_{+}^{\prime }\left(t\right)}{g\left(t\right)}y\left(t\right)-\mathsf{\Omega }\left(t\right)I_2^{\alpha }(h\left(t\right),t_{*}){\pi }^{\alpha }\left(t\right)-\frac{\alpha y^{(\alpha +1)/\alpha }\left(t\right)}{{\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha }}.$$

(41)

An application of Lemma 1 with

$$X:=\frac{{\alpha }^{1/\lambda }y\left(t\right)}{{\left[{\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha }\right]}^{1/\lambda }},\lambda :=\frac{\alpha +1}{\alpha },\mathrm{and}Y:={\left[\frac{\alpha }{\alpha +1}\frac{{\left[{\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha }\right]}^{1/\lambda }}{{\alpha }^{1/\lambda }}\frac{g_{+}^{\prime }\left(t\right)}{g\left(t\right)}\right]}^{\alpha }$$

implies that

$$\frac{g_{+}^{\prime }\left(t\right)}{g\left(t\right)}y\left(t\right)-\frac{\alpha y^{(\alpha +1)/\alpha }\left(t\right)}{{\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha }}\le \frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left(g_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{g^{\alpha }\left(t\right)}.$$

Substituting this inequality into (41), we obtain

$$y^{\prime }\left(t\right)\le -\mathsf{\Omega }\left(t\right)I_2^{\alpha }(h\left(t\right),t_{*}){\pi }^{\alpha }\left(t\right)+\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(t\right){\left(g_{+}^{\prime }\left(t\right)\right)}^{\alpha +1}}{g^{\alpha }\left(t\right)}.$$

Integrating the latter inequality from T to t yields

$${\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{r\left(s\right){\left(g_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{g^{\alpha }\left(s\right)}\right]ds\le y\left(T\right),$$

which contradicts (39). The proof is complete. □

The next result relates to the asymptotic behavior of (1) in the case when $\alpha \ge 1$.

Theorem 3.

Let conditions ($C_1$)−($C_4$) and $\alpha \ge 1$ be satisfied and σ be nondecreasing, and suppose that either (6) or (7) holds for all constants $k>0$. If there exist a function $g\in {\mathrm{C}}^1([t_0,\infty ),(0,\infty ))$ and a constant $0<\theta <1$ such that for all constants $d_1>0$, $d_2>0$, $l>0$, $m>0$, for all sufficiently large $t_{*}\ge t_0$, and for some $T\ge t_{*}$,

$$\underset{t\to \infty }{limsup}{\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{r^{1/\alpha }\left(s\right)}{4\alpha {\pi }^{\alpha -1}\left(s\right)}\frac{{\left(g_{+}^{\prime }\left(s\right)\right)}^2}{g\left(s\right)}\right]ds=\infty$$

(42)

and condition (30) holds, then conclusion of Theorem 1 remains intact.

Proof. 

Let x be an eventually positive solution of (1). Then, by virtue of Lemma 2, $z\left(t\right)$ satisfies either case (I) or case (II) or case (III) for $t\ge t_1$. The proofs of cases (I) and (III) are the same as those of Theorem 1. Next, we analyze case (II). Defining again y by (31) and proceeding as in the proof of Theorem 2, we deduce that inequality (41) holds for $t\ge T$, which can be written in the form

$$y^{\prime }\left(t\right)\le \frac{g_{+}^{\prime }\left(t\right)}{g\left(t\right)}y\left(t\right)-\mathsf{\Omega }\left(t\right)I_2^{\alpha }(h\left(t\right),t_{*}){\pi }^{\alpha }\left(t\right)-\frac{\alpha y^2\left(t\right)y^{1/\alpha -1}\left(t\right)}{{\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha }}.$$

(43)

Since z satisfies case (II), inequality (14) holds. It follows from (14) and (31) that

$$\begin{array}{cc}\hfill y^{1/\alpha -1}\left(t\right)& ={\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(\frac{{\left(z^{″}\left(t\right)\right)}^{\alpha }}{{\left(z^{\prime }\left(t\right)\right)}^{\alpha }}\right)}^{1/\alpha -1}\hfill \\ \hfill & ={\left(g\left(t\right)r\left(t\right)\right)}^{1/\alpha -1}{\left(\frac{z^{\prime }\left(t\right)}{z^{″}\left(t\right)}\right)}^{\alpha -1}\hfill \\ \hfill & \ge g^{1/\alpha -1}\left(t\right){\pi }^{\alpha -1}\left(t\right).\hfill \end{array}$$

(44)

Using (44) in (43), we arrive at

$$y^{\prime }\left(t\right)\le \frac{g_{+}^{\prime }\left(t\right)}{g\left(t\right)}y\left(t\right)-\mathsf{\Omega }\left(t\right)I_2^{\alpha }(h\left(t\right),t_{*}){\pi }^{\alpha }\left(t\right)-\frac{\alpha {\pi }^{\alpha -1}\left(t\right)}{g\left(t\right)r^{1/\alpha }\left(t\right)}{y}^2\left(t\right).$$

(45)

Completing the square with respect to y, it follows now from (45) that

$$y^{\prime }\left(t\right)\le -\mathsf{\Omega }\left(t\right)I_2^{\alpha }(h\left(t\right),t_{*}){\pi }^{\alpha }\left(t\right)+\frac{r^{1/\alpha }\left(t\right)}{4\alpha {\pi }^{\alpha -1}\left(t\right)}\frac{{\left(g_{+}^{\prime }\left(t\right)\right)}^2}{g\left(t\right)}.$$

Integrating the latter inequality from T to t, we conclude that

$${\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{r^{1/\alpha }\left(s\right)}{4\alpha {\pi }^{\alpha -1}\left(s\right)}\frac{{\left(g_{+}^{\prime }\left(s\right)\right)}^2}{g\left(s\right)}\right]ds\le y\left(T\right),$$

which contradicts (42), and the proof is complete. □

We conclude this section with two examples and three remarks illustrating theoretical results. Our first example deals with an equation with a superlinear neutral term in the case where ${lim}_{t\to \infty }p\left(t\right)=\infty$, and the second example is concerned with an equation with a linear neutral term in the case when p is constant.

Example 1.

Consider a noncanonical third-order Emden–Fowler delay differential equation with a superlinear neutral term

$${\left(t^2{\left(x\left(t\right)+t{x}^3(t/2)\right)}^{″}\right)}^{\prime }+t{x}^3(t/4)=0,t\ge 1.$$

(46)

Here, $r\left(t\right)=t^2$, $p\left(t\right)=t$, $q\left(t\right)=t$, $\tau \left(t\right)=t/2$, $\sigma \left(t\right)=t/4$, $\alpha =1$, $\beta =3$, and $\delta =3$. It is not difficult to see that conditions($C_1$)−($C_4$)hold,

$$I_1(t,t_0)=I_1(t,1)=(t-1)/t,\pi \left(t\right)=1/t,I_2(t,t_{*})=(t^2-t_{*}^2)/2,$$

$$I_2({\tau }^{-1}\left(t\right),t_{*})=I_2(2t,t_{*})=(4t^2-t_{*}^2)/2,I_2({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right),t_{*})=I_2(4t,t_{*})=(16t^2-t_{*}^2)/2,$$

$$I_2(h\left(t\right),t_{*})=I_2(t/2,t_{*})=(t^2-4t_{*}^2)/8,$$

$$E_1\left(t\right)=\frac{1}{2t}\left[1-\frac{1}{k^{2/3}{\left(4t\right)}^{1/3}}\right],E_2\left(t\right)=\frac{1}{2t}\left[1-{\left(\frac{16t^2-t_{*}^2}{4t^2-t_{*}^2}\right)}^{1/3}\frac{1}{l^{2/3}{\left(4t\right)}^{1/3}}\right],$$

$$E_3\left(t\right)=\frac{1}{2t}\left[1-\frac{2}{m^{2/3}{\left(4t\right)}^{1/3}}\right],\mathrm{where}\mathrm{we}\mathrm{let}\theta =1/3.$$

Since $(16t^2-t_{*}^2)/(4t^2-t_{*}^2)$ is strictly decreasing, $E_2\left(t\right)$ can be written as

$$E_2\left(t\right)\ge \frac{1}{2t}\left[1-\frac{5^{1/3}}{l^{2/3}{\left(4t\right)}^{1/3}}\right]\mathrm{for}\mathrm{large}t.$$

It follows from

$${\int }_{t_0}^{\infty }q\left(s\right)E_1^{\delta /\beta }\left(\sigma \left(s\right)\right)ds=2{\int }_1^{\infty }\left[1-\frac{1}{k^{2/3}{s}^{1/3}}\right]ds=\infty$$

that condition (6) holds.

Letting $g\left(t\right)=1$, we deduce that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\int }_T^t\left[\mathsf{\Omega }\left(s\right)I_2^{\alpha }(h\left(s\right),t_{*}){\pi }^{\alpha }\left(s\right)-\frac{g_{+}^{\prime }\left(s\right)}{{\pi }^{\alpha }\left(s\right)}\right]ds\ge \frac{1}{4}{\int }_T^{\infty }\frac{s^2-4t_{*}^2}{s}\left[1-\frac{5^{1/3}}{l^{2/3}{s}^{1/3}}\right]ds=\infty ,\end{array}$$

i.e., conditions (29), (39), and (42) are satisfied (note that conditions (29), (39), and (42) have the same form in the case when $g\left(t\right)=1$).

Finally, it follows from

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{lim\; inf}{\int }_{h\left(t\right)}^t\frac{1}{r^{1/\alpha }\left(u\right)}{\left({\int }_T^{u}q\left(s\right)E_3^{\delta /\beta }\left(\sigma \left(s\right)\right)Q\left(h\left(s\right)\right)h^{\alpha }\left(s\right)ds\right)}^{1/\alpha }du\hfill \\ \hfill & =\underset{t\to \infty }{lim\; inf}{\int }_{t/2}^t\frac{1}{u^2}\left({\int }_T^{u}s\left[1-\frac{2}{m^{2/3}{s}^{1/3}}\right]ds\right)du\hfill \\ \hfill & =\underset{t\to \infty }{lim\; inf}{\int }_{t/2}^t\frac{1}{u^2}\left(\frac{u^2}{2}-\frac{6u^{5/3}}{5m^{2/3}}-\frac{T^2}{2}+\frac{6T^{5/3}}{5m^{2/3}}\right)du\hfill \\ \hfill & =\underset{t\to \infty }{lim\; inf}{t}^{2/3}\left(\frac{t^{1/3}}{4}-\frac{9}{5m^{2/3}}\frac{2^{2/3}-1}{2^{2/3}}-\frac{T^2}{2t^{5/3}}+\frac{6T^{5/3}}{5m^{2/3}{t}^{5/3}}\right)=\infty \hfill \end{array}$$

that condition (30) holds. That is, all assumptions of Theorems 1–5 are fulfilled. Therefore, every solution x of (46) is either oscillatory or satisfies ${lim}_{t\to \infty }x\left(t\right)=0$.

Example 2.

Consider a noncanonical third-order Emden–Fowler delay differential equation with a linear neutral term

$${\left(t^{2/3}{\left[{\left(x\left(t\right)+16x(t/2)\right)}^{″}\right]}^{1/3}\right)}^{\prime }+t^{2/3}{x}^3(t/4)=0,t\ge 1.$$

(47)

Here, $r\left(t\right)=t^{2/3}$, $p\left(t\right)=16$, $q\left(t\right)=t^{2/3}$, $\tau \left(t\right)=t/2$, $\sigma \left(t\right)=t/4$, $\alpha =1/3$, $\beta =1$, and $\delta =3$. Then, conditions ($C_1$)−($C_4$) hold,

$$E_1\left(t\right)=\frac{1}{16}\left[1-\frac{1}{16}\right]=\frac{15}{256},E_2\left(t\right)=\frac{1}{16}\left[1-\frac{1}{16}\frac{16t^2-t_{*}^2}{4t^2-t_{*}^2}\right],$$

$$E_3\left(t\right)=\frac{1}{16}\left[1-\frac{1}{16}{\left(\frac{4t}{2t}\right)}^2\right]=\frac{3}{64},\mathrm{where}\mathrm{we}\mathrm{let}\theta =1/2.$$

Since $(16t^2-t_{*}^2)/(4t^2-t_{*}^2)$ is strictly decreasing, $E_2\left(t\right)$ can be written as

$$E_2\left(t\right)\ge \frac{1}{16}\left[1-\frac{5}{16}\right]=\frac{11}{256}\mathrm{for}\mathrm{large}t.$$

It is not difficult to verify that all assumptions of Theorems 1 and 2 are fulfilled. Hence, every solution x of (47) is either oscillatory or satisfies ${lim}_{t\to \infty }x\left(t\right)=0$.

Remark 1.

The results obtained in this paper can be applied to (1) in the noncanonical case (i.e., (2) holds), which provide an answer to the question posed in ([11], Remark 2.11).

Remark 2.

Conditions (3)–(5) mean that ${lim}_{t\to \infty }p\left(t\right)=\infty$ when assuming $\beta >1$. It would be of interest to suggest a different method to further investigate (1) with different assumptions on the neutral coefficient p.

Remark 3.

Due to the fact that the sign of the derivative $z^{\prime }$ is not fixed, our theorems ensure that every solution x of (1) is either oscillatory or converges to zero as $t\to \infty$. It is not easy to establish criteria which guarantee that every solution of (1) is just oscillatory and does not tend to zero asymptotically. Neither is it possible to use the method reported in this paper for proving that all solutions of (1) only converge to zero as $t\to \infty$. These interesting problems remain open at the moment.