Oscillatory behaviour of third order nonlinear differential equations with a nonlinear nonpositive neutral term

Abstract

The authors present some new oscillation criteria for third order nonlinear differential equations with a nonlinear nonpositive neutral term. The results obtained are new and improve known oscillation criteria appearing in the literature even for equations not having a neutral term. Using comparison methods and integral conditions, the authors obtain five new theorems on the oscillatory behaviour of the solutions. Suggestions for future research are included.

Keywords:

• Oscillation
• third order
• neutral differential equation
• nonlinear nonpositive neutral term

2000 Mathematics Subject Classifications:

• 34C10
• 34K11

1. Introduction

This paper is concerned with the oscillatory behaviour of solutions of the nonlinear third order differential equations with a nonlinear nonpositive neutral term (1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) where $y\left(t\right)=x\left(t\right)-p\left(t\right)x^{\alpha }\left(\sigma \right(t\left)\right).$ Here we are assuming that:

1. α, γ, and β are the ratios of odd positive integers with $\gamma \ge \beta$ and $0<\alpha \le 1$;

2. a, p, $q:[t_0,\mathrm{\infty })\to (0,\mathrm{\infty })$ are continuous functions with $0<p\left(t\right)\le p_0<1$;

3. τ, $\sigma :[t_0,\mathrm{\infty })\to \mathbb{R}$ are continuous functions with $\tau \left(t\right)\le t$, $\sigma \left(t\right)\le t$, ${\tau }^{\mathrm{\prime }}\left(t\right)>0$, ${\sigma }^{\mathrm{\prime }}\left(t\right)>0$, and $\underset{t\to \mathrm{\infty }}{lim}\tau \left(t\right)=\underset{t\to \mathrm{\infty }}{lim}\sigma \left(t\right)=\mathrm{\infty }$.

We let $A(v,u)={\int }_u^v\frac{1}{a^{1/\gamma }\left(s\right)}\mathrm{d}s,v\ge u\ge t_0,$ and assume that (2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) By a solution of Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ), we mean a function $x\in C\left(\right[T_x,\mathrm{\infty }),\mathbb{R})$ for some $T_x\ge t_0$ having the properties that $y\in C^2\left(\right[T_x,\mathrm{\infty }),\mathbb{R})$, $a(y^{\mathrm{\prime }\mathrm{\prime }}{)}^{\gamma }\in C^1([T_x,\mathrm{\infty }),\mathbb{R})$, and which satisfies (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) on $[T_x,\mathrm{\infty })$. We consider only those solutions of (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) that satisfy $sup\left\{\left|x\left(t\right)\right|:T\le t<\mathrm{\infty }\right\}>0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}T\ge T_x,$ and we assume that (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) possesses such solutions. Such a solution $x\left(t\right)$ of (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is said to be oscillatory if it has arbitrarily large zeros, i.e. for any $t_1\in [t_0,\mathrm{\infty })$ there exists $t_2\ge t_1$ such that $x\left(t_2\right)=0$; otherwise it is called nonoscillatory, i.e. it is eventually of one sign. Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is said to be oscillatory if all its solutions are oscillatory.

In recent years, the oscillation theory of functional differential equations has received much attention since it has a great number of applications in engineering and natural sciences. For some related contributions on the oscillatory behaviour of various classes of functional differential equations, we refer the reader to [1–16] and the references cited therein.

Neutral delay differential equations have applications to electric networks containing lossless transmission lines; such networks appear in high speed computers. Lossless transmission lines are used to interconnect switching circuits. These equations also occur in problems dealing with vibrating masses attached to elastic bar and as the Euler–Lagrange equations for variational problems with delays (see [17]).

The problem of the oscillation of solutions of differential equations has been widely studied by many authors using a wide variety of techniques ever since the pioneering work of Sturm [18] on second order linear differential equations. In the past 30 years, oscillation theory for second order neutral delay differential equations and third-order retarded delay differential equations has been well developed; see, for example, the monographs [19, 20] and papers [3–11] as well as the references contained therein. Compared to second order neutral delay differential equations, it seems that considerably less has been done on the oscillation and asymptotic behaviour of solutions of third order neutral differential equations [4, 10].

In this paper, we establish some new oscillation theorems for third order nonlinear differential equations with a nonlinear nonpositive neutral term of the type (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ). We accomplish this by a comparison to first order equations whose oscillatory behaviour are known, or via a comparison to second order inequalities with solutions having certain properties. The results obtained are new and improve many known oscillation criteria that have appeared in the literature even for the case of Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) with $p\left(t\right)=0$.

2. Main results

We begin with the following lemma that will play an important role in establishing our main results.

Lemma 2.1

Assume that $f\left(t\right)>0,$ $f^{\mathrm{\prime }}\left(t\right)>0$ and $f^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)<0$ for $t\ge t_1$ for some $t_1\ge t_0$. Then there exists $t_2\in [t_1,\mathrm{\infty })$ such that $\frac{f\left(t\right)}{f^{\mathrm{\prime }}\left(t\right)}\ge \frac{t}{2}\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$

Proof.

Since $f\left(t\right)$ is positive and $f^{\mathrm{\prime }}\left(t\right)$ is decreasing for $t\ge t_1$ for some $t_1\ge t_0$, we see that $f\left(t\right)=f\left(t_1\right)+{\int }_{t_1}^t{f}^{\mathrm{\prime }}\left(s\right)\mathrm{d}s\ge (t-t_1)f^{\mathrm{\prime }}\left(t\right).$ From this it follows that for all $t\ge t_2:=2t_1$, $f\left(t\right)\ge \frac{t}{2}{f}^{\mathrm{\prime }}\left(t\right),$ which completes the proof.

Our first result is the following.

Theorem 2.1

Let $h\left(t\right)={\sigma }^{-1}\left(\tau \right(t\left)\right)\le t,$ $h^{\mathrm{\prime }}\left(t\right)\ge 0$ for $t\ge t_0,$ and (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold. Assume that there exist continuous functions ρ, η, $\xi :[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<\eta \left(t\right)<\xi \left(t\right)<t$ and $h\left(t\right)<\rho \left(t\right)<t$ for $t\ge t_0$. If the first order equations (3) $\begin{array}{rl}& Y^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }{Y}^{\beta /\gamma }\left(\tau \right(t\left)\right)=0,\end{array}$(3) (4) $\begin{array}{rl}& W^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left[\left(\eta \right(t)-\tau (t\left)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta }{W}^{\beta /\gamma }\left(\xi \right(t\left)\right)=0,\end{array}$(4) and (5) $Z^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }{Z}^{\beta /\alpha \gamma }\left(\rho \right(t\left)\right)=0,$(5) are oscillatory, then Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ), say $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$. It follows from Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) that (6) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1,$(6) and hence $a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }$ is decreasing and eventually of one sign on $[t_1,\mathrm{\infty })$, so $y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)$ is eventually of one sign on $[t_1,\mathrm{\infty })$. That is, there exists $t_2\ge t_1$ such that $y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)>0$ or $y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)<0$ for $t\ge t_2$. We claim that $y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)>0$ for $t\ge t_2$. To prove this claim, suppose that $y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)<0$ for $t\ge t_2$. By (6(6) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1,$(6) ), $a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\le a(t_2\left)\right(y^{\mathrm{\prime }\mathrm{\prime }}\left(t_2\right){)}^{\gamma }=-C_1<0$ for $t\ge t_2$. An integration gives $y^{\prime }\left(t\right)\le y\left(t_2\right)-C_1^{1/\gamma }{\int }_{t_2}^t{a}^{-1/\gamma }\left(s\right)\mathrm{d}s.$ An application of condition (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) and a second integration show that $\underset{t\to \mathrm{\infty }}{lim}y\left(t\right)=-\mathrm{\infty }$.

Now, we consider the following two cases:

Case 1: If $x\left(t\right)$ is unbounded, then there exists an increasing sequence $\left\{T_k\right\}$ such that $T_1>t_2$, $\underset{k\to \mathrm{\infty }}{lim}{T}_k=\mathrm{\infty }$, $x\left(T_1\right)>1$, and $\underset{k\to \mathrm{\infty }}{lim}x\left(T_k\right)=\mathrm{\infty }$, where $x\left(T_k\right)=max\left\{x\right(s):t_0\le s\le T_k\}$. Since $\sigma \left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$, we have $\sigma \left(T_k\right)>t_0$ for sufficiently large k. From the fact that $\sigma \left(t\right)\le t$, we see that $x\left(\sigma \right(T_k\left)\right)\le max\left\{x\right(s):t_0\le s\le T_k\}=x\left(T_k\right).$ Therefore, for sufficiently large k, we obtain $\begin{array}{rl}y\left(T_k\right)& =x\left(T_k\right)-p\left(T_k\right)x^{\alpha }\left(\sigma \right(T_k\left)\right)\ge x\left(T_k\right)-p_0{x}^{\alpha }\left(T_k\right)\\ & \ge x\left(T_k\right)(1-p_0)>0\end{array}$ since $x\left(T_k\right)>1$ and $\alpha \le 1$. This contradicts the fact that $\underset{t\to \mathrm{\infty }}{lim}y\left(t\right)=-\mathrm{\infty }$.

Case 2: If $x\left(t\right)$ is bounded, then $y\left(t\right)$ is also bounded, which again contradicts the fact that $\underset{t\to \mathrm{\infty }}{lim}y\left(t\right)=-\mathrm{\infty }$. This proves the claim and so $y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)>0$ for $t\ge t_2$.

Next, we have two cases to consider: (I) $y\left(t\right)>0$ for $t\ge t_2$ or (II) $y\left(t\right)<0$ for $t\ge t_2$.

Case (I): Suppose that $y\left(t\right)>0$ for $t\ge t_2$. From the definition of y, we see that $x\left(t\right)\ge y\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2,$ and so (7) $x\left(\tau \right(t\left)\right)\ge y\left(\tau \right(t\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(7) where $\tau \left(t\right)\ge t_2$ for $t\ge t_3$ for some $t_3\ge t_2$. Using (7(7) $x\left(\tau \right(t\left)\right)\ge y\left(\tau \right(t\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(7) ) in (6(6) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_1,$(6) ) gives (8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) and so by the well known Kiguradze's Lemma (see [21]), we distinguish the following two cases: $\begin{array}{rl}& \left(\mathrm{a}\right)y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)>0\text{and}{y}^{\mathrm{\prime }}\left(t\right)>0\text{for}t\ge t_3,\text{or}\\ & \left(\mathrm{b}\right)y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)>0\text{and}{y}^{\mathrm{\prime }}\left(t\right)<0\text{for}t\ge t_3.\end{array}$ Suppose (a) holds. Then, $\begin{array}{rl}{y}^{\prime }\left(t\right)& =y^{\prime }\left(t_3\right)+{\int }_{t_3}^t{y}^{″}\left(s\right)\mathrm{d}s\\ & ={\int }_{t_3}^t\frac{\left[a\right(s\left)\right(y^{\mathrm{\prime }\mathrm{\prime }}\left(s\right){)}^{\gamma }{]}^{1/\gamma }}{a^{1/\gamma }\left(s\right)}\mathrm{d}s\\ & \ge \left[a\right(t\left)\right(y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right){)}^{\gamma }{]}^{1/\gamma }{\int }_{t_3}^t\frac{1}{a^{1/\gamma }\left(s\right)}\mathrm{d}s\\ & =A(t,t_3)\left[a\right(t\left)\right(y^{\mathrm{\prime }\mathrm{\prime }}\left(t\right){)}^{\gamma }{]}^{1/\gamma }.\end{array}$ Integrating this inequality from $t_3$ to t yields $y\left(t\right)\ge \left({\int }_{t_3}^{t}A(s,t_3)\mathrm{d}s\right){\left[a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right]}^{1/\gamma },$ and hence (9) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right){\left[a\left(\tau \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\tau \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_4,\end{array}$(9) where $\tau \left(t\right)\ge t_3$ for $t\ge t_4$. Using (9(9) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right){\left[a\left(\tau \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\tau \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_4,\end{array}$(9) ) in (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ) gives (10) $Y^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }{Y}^{\beta /\gamma }\left(\tau \right(t\left)\right)\le 0,$(10) where $Y\left(t\right)=a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }>0$. The function $Y\left(t\right)$ is clearly strictly decreasing on $[t_4,\mathrm{\infty })$ (see (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) )), so by Theorem 1 in [22], we conclude that there exists a positive solution $Y\left(t\right)$ of Equation (3(3) $\begin{array}{rl}& Y^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }{Y}^{\beta /\gamma }\left(\tau \right(t\left)\right)=0,\end{array}$(3) ). This contradicts the fact that Equation (3(3) $\begin{array}{rl}& Y^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }{Y}^{\beta /\gamma }\left(\tau \right(t\left)\right)=0,\end{array}$(3) ) is oscillatory.

Next, we consider (b). For $v\ge u\ge t_3$, we have (11) $y\left(u\right)-y\left(v\right)={\int }_u^v-y^{\mathrm{\prime }}\left(s\right)\mathrm{d}s.$(11) Setting $u=\tau \left(t\right)$ and $v=\eta \left(t\right)$ in (11(11) $y\left(u\right)-y\left(v\right)={\int }_u^v-y^{\mathrm{\prime }}\left(s\right)\mathrm{d}s.$(11) ), we obtain (12) $y\left(\tau \right(t\left)\right)\ge \left(\eta \left(t\right)-\tau \left(t\right)\right)\left(-y^{\mathrm{\prime }}\left(\eta \right(t)\right).$(12) Also, $\begin{array}{rl}-y^{\mathrm{\prime }}\left(u\right)\ge y^{\mathrm{\prime }}\left(v\right)-y^{\mathrm{\prime }}\left(u\right)& ={\int }_u^v{a}^{-1/\gamma }\left(s\right)\left(a^{1/\gamma }\left(s\right)y^{\mathrm{\prime }\mathrm{\prime }}\left(s\right)\right)\mathrm{d}s\\ & \ge A(v,u){\left[a\left(v\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(v){)}^{\gamma }\right]}^{1/\gamma },\end{array}$ and hence $-y^{\mathrm{\prime }}\left(u\right)\ge A(v,u){\left[a\left(v\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(v){)}^{\gamma }\right]}^{1/\gamma }.$ Letting $u=\eta \left(t\right)$ and $v=\xi \left(t\right)$ in the last inequality, we have (13) $-y^{\mathrm{\prime }}\left(\eta \right(t\left)\right)\ge A\left(\xi \right(t),\eta (t\left)\right){\left[a\left(\xi \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\xi \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(13) Combining (12(12) $y\left(\tau \right(t\left)\right)\ge \left(\eta \left(t\right)-\tau \left(t\right)\right)\left(-y^{\mathrm{\prime }}\left(\eta \right(t)\right).$(12) ) and (13(13) $-y^{\mathrm{\prime }}\left(\eta \right(t\left)\right)\ge A\left(\xi \right(t),\eta (t\left)\right){\left[a\left(\xi \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\xi \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(13) ) gives (14) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left(\eta \left(t\right)-\tau \left(t\right)\right)A\left(\xi \right(t),\eta (t\left)\right)\\ & \times {\left[a\left(\xi \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\xi \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.\end{array}$(14) Now using (14(14) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left(\eta \left(t\right)-\tau \left(t\right)\right)A\left(\xi \right(t),\eta (t\left)\right)\\ & \times {\left[a\left(\xi \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\xi \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.\end{array}$(14) ) in (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ) yields (15) $W^{\mathrm{\prime }}\left(t\right)\le -q\left(t\right){\left[\left(\eta \right(t)-\tau (t\left)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta }{W}^{\beta /\gamma }\left(\xi \right(t\left)\right),$(15) where $W\left(t\right)=a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }>0$. Proceeding as in the proof of case (a), we again get a contradiction.

Case (II). Suppose that $y\left(t\right)<0$ for $t\ge t_2$. Let $z\left(t\right)=-y\left(t\right)>0$ for $t\ge t_2$. Then, from Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ), we see that (16) ${\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}=q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)>0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(16) From the definition of $y\left(t\right)$, we have $z\left(t\right)=-y\left(t\right)=p\left(t\right)x^{\alpha }\left(\sigma \right(t\left)\right)-x\left(t\right)\le p\left(t\right)x^{\alpha }\left(\sigma \right(t\left)\right),$ and so $x\left(\sigma \right(t\left)\right)\ge z^{1/\alpha }\left(t\right)\mathrm{o}\mathrm{r}x\left(t\right)\ge z^{1/\alpha }\left({\sigma }^{-1}\right(t\left)\right),$ from which we have $x\left(\tau \right(t\left)\right)\ge z^{1/\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3$ for some $t_3\ge t_2$. Using the last inequality in (16(16) ${\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}=q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)>0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_2.$(16) ) gives (17) $\begin{array}{rl}& {\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right)z^{\beta /\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)=q\left(t\right)z^{\beta /\alpha }\left(h\right(t\left)\right)\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(17) Clearly, we have $z^{\mathrm{\prime }}\left(t\right)>0$ and $z^{\mathrm{\prime }\mathrm{\prime }}\left(t\right)<0$ for $t\ge t_3$ since $z\left(t\right)>0$. So in view of Lemma 2.1, we have $z\left(t\right)\ge \frac{1}{2}t{z}^{\mathrm{\prime }}\left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_4:=2t_3,$ or (18) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)z^{\mathrm{\prime }}\left(h\right(t\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_5$(18) for some $t_5\ge t_4$. Also, for $t_5\le u\le v$, we see that (19) $\begin{array}{rl}& z^{\mathrm{\prime }}\left(u\right)-z^{\mathrm{\prime }}\left(v\right)=-{\int }_u^v{a}^{-1/\gamma }\left(s\right)\left(a^{1/\gamma }\left(s\right)z^{\mathrm{\prime }\mathrm{\prime }}\left(s\right)\right)\mathrm{d}s\\ & \ge A(v,u){\left[a\left(v\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(v){)}^{\gamma }\right]}^{1/\gamma }.\end{array}$(19) With $u=h\left(t\right)$ and $v=\rho \left(t\right)$, (19(19) $\begin{array}{rl}& z^{\mathrm{\prime }}\left(u\right)-z^{\mathrm{\prime }}\left(v\right)=-{\int }_u^v{a}^{-1/\gamma }\left(s\right)\left(a^{1/\gamma }\left(s\right)z^{\mathrm{\prime }\mathrm{\prime }}\left(s\right)\right)\mathrm{d}s\\ & \ge A(v,u){\left[a\left(v\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(v){)}^{\gamma }\right]}^{1/\gamma }.\end{array}$(19) ) becomes (20) $z^{\mathrm{\prime }}\left(h\right(t\left)\right)\ge A\left(\rho \right(t),h(t\left)\right){\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(20) Combining (18(18) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)z^{\mathrm{\prime }}\left(h\right(t\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_5$(18) ) and (20(20) $z^{\mathrm{\prime }}\left(h\right(t\left)\right)\ge A\left(\rho \right(t),h(t\left)\right){\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(20) ) gives (21) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right){\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(21) Using (21(21) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right){\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(21) ) in (17(17) $\begin{array}{rl}& {\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right)z^{\beta /\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)=q\left(t\right)z^{\beta /\alpha }\left(h\right(t\left)\right)\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(17) ) gives (22) $Z^{\mathrm{\prime }}\left(t\right)\le -q\left(t\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }{Z}^{\beta /\alpha \gamma }\left(\rho \right(t\left)\right),$(22) where $Z\left(t\right)=a\left(t\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(t){)}^{\gamma }>0$. The rest of the proof is similar to that of case (a) in Case (I) and hence is omitted. This completes the proof of the theorem.

From [23, Theorem 1], it is well known that if F, $\tau \in C\left(\right[t_0,\mathrm{\infty }),\mathbb{R})$ with $F\left(t\right)\ge 0$, $\tau \left(t\right)\le t$, $\underset{t\to \mathrm{\infty }}{lim}\tau \left(t\right)=\mathrm{\infty }$, and $\underset{t\to \mathrm{\infty }}{lim inf}{\int }_{\tau \left(t\right)}^{t}F\left(s\right)\mathrm{d}s>\frac{1}{e},$ then the first-order delay differential equation $x^{\mathrm{\prime }}\left(t\right)+F\left(t\right)x\left(\tau \right(t\left)\right)=0$ is oscillatory. Thus, from Theorem 2.1, we have the following result.

Corollary 2.1

Let $h\left(t\right)={\sigma }^{-1}\left(\tau \right(t\left)\right)\le t,$ $h^{\mathrm{\prime }}\left(t\right)\ge 0$ for $t\ge t_0,$ (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold, $\alpha =1,$ and $\beta =\gamma$. Assume that there exist continuous functions ρ, η, $\xi :[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<\eta \left(t\right)<\xi \left(t\right)<t$ and $h\left(t\right)<\rho \left(t\right)<t$ for $t\ge t_0$. If $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim inf}{\int }_{\tau \left(t\right)}^{t}q\left(s\right){\left({\int }_{t_0}^{\tau \left(s\right)}A(u,t_0)\mathrm{d}u\right)}^{\beta }\mathrm{d}s>\frac{1}{e},\\ & \underset{t\to \mathrm{\infty }}{lim inf}{\int }_{\xi \left(t\right)}^{t}q\left(s\right){\left[\left(\eta \right(s)-\tau (s\left)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\mathrm{d}s>\frac{1}{e},\end{array}$ and $\underset{t\to \mathrm{\infty }}{lim inf}{\int }_{\rho \left(t\right)}^{t}q\left(s\right){\left[\frac{1}{2}h\left(s\right)A\left(\rho \right(s),h(s\left)\right)\right]}^{\beta }\mathrm{d}s>\frac{1}{e},$ then Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is oscillatory.

Corollary 2.2

Let $h\left(t\right)={\sigma }^{-1}\left(\tau \right(t\left)\right)\le t,$ $h^{\mathrm{\prime }}\left(t\right)\ge 0$ for $t\ge t_0,$ and (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold. Assume that there exist continuous functions ρ, η, $\xi :[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<\eta \left(t\right)<\xi \left(t\right)<t$ and $h\left(t\right)<\rho \left(t\right)<t$ for $t\ge t_0$. If $\begin{array}{rl}& {\int }_{t_0}^{\mathrm{\infty }}q\left(s\right){\left({\int }_{t_0}^{\tau \left(s\right)}A(u,t_0)\mathrm{d}u\right)}^{\beta }\mathrm{d}s=\mathrm{\infty }\mathrm{f}\mathrm{o}\mathrm{r}\beta <\gamma ,\\ & {\int }_{t_0}^{\mathrm{\infty }}q\left(s\right){\left[\left(\eta \right(s)-\tau (s\left)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\\ & \mathrm{d}s=\mathrm{\infty }\mathrm{f}\mathrm{o}\mathrm{r}\beta <\gamma ,\end{array}$ and ${\int }_{t_0}^{\mathrm{\infty }}q\left(s\right){\left[h\left(s\right)A\left(\rho \right(s),h(s\left)\right)\right]}^{\beta /\alpha }\mathrm{d}s=\mathrm{\infty }\mathrm{f}\mathrm{o}\mathrm{r}\beta <\alpha \gamma ,$ then Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is oscillatory.

The above corollary follows from (10(10) $Y^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }{Y}^{\beta /\gamma }\left(\tau \right(t\left)\right)\le 0,$(10) ), (15(15) $W^{\mathrm{\prime }}\left(t\right)\le -q\left(t\right){\left[\left(\eta \right(t)-\tau (t\left)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta }{W}^{\beta /\gamma }\left(\xi \right(t\left)\right),$(15) ), (22(22) $Z^{\mathrm{\prime }}\left(t\right)\le -q\left(t\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }{Z}^{\beta /\alpha \gamma }\left(\rho \right(t\left)\right),$(22) ) and the fact that $\tau \left(t\right)<t$, $\xi \left(t\right)<t$ and $\rho \left(t\right)<t$; we omit its proof.

Next, we take $\alpha =1$ and let $\begin{array}{rl}& Q\left(t\right)\le min\left\{{\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta },\right.\\ & \left.{\left[\left(\eta \right(t)-\tau (t\left)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta },{\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta }\right\}.\end{array}$ Then, Theorem 2.1 is equivalent to the following.

Theorem 2.2

Let $h\left(t\right)={\sigma }^{-1}\left(\tau \right(t\left)\right)\le t,$ $h^{\mathrm{\prime }}\left(t\right)\ge 0$ for $t\ge t_0,$ and (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold. Assume that there exist continuous functions ρ, η, $\xi :[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<\eta \left(t\right)<\xi \left(t\right)<t$ and $h\left(t\right)<\rho \left(t\right)<t$ for $t\ge t_0$. If the first order equation (23) $X^{\mathrm{\prime }}\left(t\right)+q\left(t\right)Q\left(t\right)X^{\beta /\gamma }\left(\tau \right(t\left)\right)=0$(23) is oscillatory, then Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is oscillatory.

For the non-neutral equation of type (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ), i.e. the equation (24) ${\left(a\left(t\right)\left(x^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(24) we have the following immediate result.

Theorem 2.3

Let (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold and assume that there exist continuous functions $\eta ,\xi :[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<\eta \left(t\right)<\xi \left(t\right)<t$ for $t\ge t_0$. If the first order Equations (3(3) $\begin{array}{rl}& Y^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }{Y}^{\beta /\gamma }\left(\tau \right(t\left)\right)=0,\end{array}$(3) ) and (4(4) $\begin{array}{rl}& W^{\mathrm{\prime }}\left(t\right)+q\left(t\right){\left[\left(\eta \right(t)-\tau (t\left)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta }{W}^{\beta /\gamma }\left(\xi \right(t\left)\right)=0,\end{array}$(4) ) are oscillatory, then Equation (24(24) ${\left(a\left(t\right)\left(x^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(24) ) is oscillatory.

Next, we present the following interesting result in which we assume that ξ is an increasing function.

Theorem 2.4

Let $h\left(t\right)={\sigma }^{-1}\left(\tau \right(t\left)\right)\le t,$ $h^{\mathrm{\prime }}\left(t\right)\ge 0$ for $t\ge t_0,$ and (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold. Assume that there exist continuous functions ρ, η, $\xi :[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<\eta \left(t\right)<\xi \left(t\right)<t,$ ${\xi }^{\mathrm{\prime }}\left(t\right)>0,$ and $h\left(t\right)<\rho \left(t\right)<t$ for $t\ge t_0$. If (25) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}\left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\gamma ,\end{array}\right.\end{array}$(25) (26) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}{\int }_{\xi \left(t\right)}^{t}q\left(s\right){\left[\left(\eta \right(s)-\tau (s\left)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\mathrm{d}s\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\gamma ,\end{array}\right.\end{array}$(26) and (27) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}\left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\alpha =1,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\alpha \gamma ,\end{array}\right.\end{array}$(27) then Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ), say $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$ and $x\left(\sigma \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$. As in the proof of Theorem 2.1, we again have two cases to consider: (I) $y\left(t\right)>0$ or (II) $y\left(t\right)<0$ for $t\ge t_2$. If Case (I) holds, then proceeding as in the proof of Theorem 2.1, we see that (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ) holds, and so we again have the two cases (a) or (b) to consider.

Suppose that (a) holds. Then, we again arrive at (9(9) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right){\left[a\left(\tau \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\tau \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_4,\end{array}$(9) ) for $t\ge t_4$. Integrating (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ) from t to $u\ge t$ and letting $u\to \mathrm{\infty }$ gives (28) $a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right)y^{\beta }(\tau \left(t\right)).$(28) Using (9(9) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right){\left[a\left(\tau \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\tau \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_4,\end{array}$(9) ) in (28(28) $a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right)y^{\beta }(\tau \left(t\right)).$(28) ), we obtain $\begin{array}{rl}a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }& \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }\\ & {\left[a\left(\tau \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\tau \left(t\right)){)}^{\gamma }\right]}^{\beta /\gamma }\\ & \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }\\ & {\left[a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right]}^{\beta /\gamma },\end{array}$ from which, we see that (29) $\begin{array}{rl}& {\left[a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right]}^{1-\beta /\gamma }\\ & \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }.\end{array}$(29) If $\beta =\gamma$, the contradiction to condition (25(25) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}\left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\gamma ,\end{array}\right.\end{array}$(25) ) is clear. If $\beta <\gamma$, then from (29(29) $\begin{array}{rl}& {\left[a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right]}^{1-\beta /\gamma }\\ & \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }.\end{array}$(29) ) and the fact that $a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }$ is decreasing, we see that, for $t\ge t_4$, $\begin{array}{rl}& \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_3}^{\tau \left(t\right)}A(s,t_3)\mathrm{d}s\right)}^{\beta }\\ & \le {\left[\left(a\right(t_4\left)\right(y^{\mathrm{\prime }\mathrm{\prime }}\left(t_4\right){)}^{\gamma }\right]}^{1-\beta /\gamma }<\mathrm{\infty },\end{array}$ which again contradicts condition (25(25) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}\left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left({\int }_{t_0}^{\tau \left(t\right)}A(s,t_0)\mathrm{d}s\right)}^{\beta }\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\gamma ,\end{array}\right.\end{array}$(25) ).

Next, we consider case (b). Then, (14(14) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left(\eta \left(t\right)-\tau \left(t\right)\right)A\left(\xi \right(t),\eta (t\left)\right)\\ & \times {\left[a\left(\xi \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\xi \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.\end{array}$(14) ) holds for $t\ge t_3$. Using (14(14) $\begin{array}{rl}y\left(\tau \right(t\left)\right)& \ge \left(\eta \left(t\right)-\tau \left(t\right)\right)A\left(\xi \right(t),\eta (t\left)\right)\\ & \times {\left[a\left(\xi \right(t\left)\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(\xi \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.\end{array}$(14) ) in (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ) gives (30) $\begin{array}{rl}{\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}& \le -q\left(t\right){\left[\left(\eta \left(t\right)-\tau \left(t\right)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta }\\ & {\left[a\left(\xi \right(t\left)\right){\left(y^{\mathrm{\prime }\mathrm{\prime }}\left(\xi \right(t\left)\right)\right)}^{\gamma }\right]}^{\beta /\gamma }.\end{array}$(30) Integrating (30(30) $\begin{array}{rl}{\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}& \le -q\left(t\right){\left[\left(\eta \left(t\right)-\tau \left(t\right)\right)A\left(\xi \right(t),\eta (t\left)\right)\right]}^{\beta }\\ & {\left[a\left(\xi \right(t\left)\right){\left(y^{\mathrm{\prime }\mathrm{\prime }}\left(\xi \right(t\left)\right)\right)}^{\gamma }\right]}^{\beta /\gamma }.\end{array}$(30) ) from $\xi \left(t\right)$ to t yields (31) $\begin{array}{rl}& {\left[a\left(\xi \right(t\left)\right){\left(y^{\mathrm{\prime }\mathrm{\prime }}\left(\xi \right(t\left)\right)\right)}^{\gamma }\right]}^{1-\beta /\gamma }\\ & \ge {\int }_{\xi \left(t\right)}^{t}q\left(s\right){\left[\left(\eta \left(s\right)-\tau \left(s\right)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\mathrm{d}s.\end{array}$(31) Taking the $limsup$ as $t\to \mathrm{\infty }$ in (31(31) $\begin{array}{rl}& {\left[a\left(\xi \right(t\left)\right){\left(y^{\mathrm{\prime }\mathrm{\prime }}\left(\xi \right(t\left)\right)\right)}^{\gamma }\right]}^{1-\beta /\gamma }\\ & \ge {\int }_{\xi \left(t\right)}^{t}q\left(s\right){\left[\left(\eta \left(s\right)-\tau \left(s\right)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\mathrm{d}s.\end{array}$(31) ), as in the case (a) we obtain a contradiction to (26(26) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}{\int }_{\xi \left(t\right)}^{t}q\left(s\right){\left[\left(\eta \right(s)-\tau (s\left)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\mathrm{d}s\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\gamma ,\end{array}\right.\end{array}$(26) ).

Next, assume that Case (II) holds. Then, as in the proof of Theorem 2.1, we see that (17(17) $\begin{array}{rl}& {\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right)z^{\beta /\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)=q\left(t\right)z^{\beta /\alpha }\left(h\right(t\left)\right)\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(17) ) and (21(21) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right){\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(21) ) hold for $t\ge t_4\ge t_3$. Integrating (17(17) $\begin{array}{rl}& {\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right)z^{\beta /\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)=q\left(t\right)z^{\beta /\alpha }\left(h\right(t\left)\right)\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(17) ) from t to $u\ge t$ and letting $u\to \mathrm{\infty }$ gives (32) $\begin{array}{rl}a\left(t\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(t){)}^{\gamma }& \ge {\int }_t^{\mathrm{\infty }}q\left(s\right)z^{\beta /\alpha }\left(h\right(s\left)\right)\mathrm{d}s\\ & \ge z^{\beta /\alpha }\left(h\right(t\left)\right){\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s.\end{array}$(32) Using (21(21) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right){\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1/\gamma }.$(21) ) in (32(32) $\begin{array}{rl}a\left(t\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(t){)}^{\gamma }& \ge {\int }_t^{\mathrm{\infty }}q\left(s\right)z^{\beta /\alpha }\left(h\right(s\left)\right)\mathrm{d}s\\ & \ge z^{\beta /\alpha }\left(h\right(t\left)\right){\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s.\end{array}$(32) ) gives (33) $\begin{array}{rl}a\left(t\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(t){)}^{\gamma }& \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }\\ & {\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{\beta /\alpha \gamma }.\end{array}$(33) Using the fact that $\rho \left(t\right)<t$ and $a\left(t\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(t){)}^{\gamma }$ is decreasing, we obtain from (33(33) $\begin{array}{rl}a\left(t\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(t){)}^{\gamma }& \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }\\ & {\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{\beta /\alpha \gamma }.\end{array}$(33) ) that (34) $\begin{array}{rl}& {\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1-\beta /\alpha \gamma }\\ & \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }.\end{array}$(34) Taking the $limsup$ as $t\to \mathrm{\infty }$ in (34(34) $\begin{array}{rl}& {\left[a\left(\rho \right(t\left)\right)(-z^{\mathrm{\prime }\mathrm{\prime }}(\rho \left(t\right)){)}^{\gamma }\right]}^{1-\beta /\alpha \gamma }\\ & \ge \left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }.\end{array}$(34) ), we obtain a contradiction to (27(27) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim sup}\left({\int }_t^{\mathrm{\infty }}q\left(s\right)\mathrm{d}s\right){\left[\frac{1}{2}h\left(t\right)A\left(\rho \right(t),h(t\left)\right)\right]}^{\beta /\alpha }\\ & \times \left\{\begin{array}{ll}>1,& \mathrm{i}\mathrm{f}\beta =\gamma ,\alpha =1,\\ =\mathrm{\infty },& \mathrm{i}\mathrm{f}\beta <\alpha \gamma ,\end{array}\right.\end{array}$(27) ) as before. This proves the theorem.

Theorem 2.5

Let $h\left(t\right)={\sigma }^{-1}\left(\tau \right(t\left)\right)\le t,$ $h^{\mathrm{\prime }}\left(t\right)\ge 0$ for $t\ge t_0,$ and (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold. Assume that there exists a continuous function ${\rho }^{\ast }:[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<{\rho }^{\ast }\left(t\right)<t$ for $t\ge t_0$. If the second order inequality (35) ${\left(a\left(t\right)\left(Y^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right){\left[\tau \left(t\right)-\tau \left(\tau \right(t\left)\right)\right]}^{\beta }{Y}^{\beta }\left(\tau \right(\tau \left(t\right))$(35) has no positive increasing solution; the second order inequality (36) ${\left(a\left(t\right)\left(W^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right){\left[{\rho }^{\ast }\left(t\right)-\tau \left(t\right)\right]}^{\beta }{W}^{\beta }\left({\rho }^{\ast }\right(t\left)\right)$(36) has no positive decreasing solution; and the second order inequality (37) ${\left(a\left(t\right)\left(Z^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right){\left[\frac{1}{2}h\left(t\right)\right]}^{\beta /\alpha }{Z}^{\beta /\alpha }\left(h\right(t\left)\right)$(37) has no positive decreasing solution, then Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) is oscillatory.

Proof.

Let $x\left(t\right)$ be a nonoscillatory solution of (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ), say $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$ and $x\left(\sigma \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$. As in the proof of Theorem 2.1, we again have two cases to consider: (I) $y\left(t\right)>0$ or (II) $y\left(t\right)<0$ for $t\ge t_2$. If Case (I) holds, proceeding as in the proof of Theorem 2.1, we see that (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ) holds. We again have the two cases (a) or (b) to consider.

Suppose that (a) holds. For $v\ge u\ge t_3$, we get (38) $y\left(v\right)=y\left(u\right)+{\int }_u^v{y}^{\mathrm{\prime }}\left(s\right)\mathrm{d}s\ge (v-u)y^{\mathrm{\prime }}\left(u\right).$(38) Setting $u=\tau \left(t\right)$ and v=t in (38(38) $y\left(v\right)=y\left(u\right)+{\int }_u^v{y}^{\mathrm{\prime }}\left(s\right)\mathrm{d}s\ge (v-u)y^{\mathrm{\prime }}\left(u\right).$(38) ), we obtain $y\left(t\right)\ge (t-\tau (t\left)\right)y^{\mathrm{\prime }}\left(\tau \right(t\left)\right),$ from which we see that (39) $y\left(\tau \right(t\left)\right)\ge \left[\tau \left(t\right)-\tau \left(\tau \right(t\left)\right)\right]y^{\mathrm{\prime }}\left(\tau \right(\tau \left(t\right)\left)\right).$(39) Letting $Y\left(t\right)=y^{\mathrm{\prime }}\left(t\right)$ and using (39(39) $y\left(\tau \right(t\left)\right)\ge \left[\tau \left(t\right)-\tau \left(\tau \right(t\left)\right)\right]y^{\mathrm{\prime }}\left(\tau \right(\tau \left(t\right)\left)\right).$(39) ) in (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ), we see that the inequality (40) ${\left(a\left(t\right)\left(Y^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right){\left[\tau \left(t\right)-\tau \left(\tau \right(t\left)\right)\right]}^{\beta }{Y}^{\beta }\left(\tau \right(\tau \left(t\right))$(40) has a positive increasing solution, which is a contradiction.

Next, we consider case (b). For $v\ge u\ge t_3$, (41) $y\left(u\right)=y\left(v\right)+{\int }_u^v-y^{\mathrm{\prime }}\left(s\right)\mathrm{d}s\ge (v-u)(-y^{\mathrm{\prime }}(v\left)\right).$(41) Putting $u=\tau \left(t\right)$ and $v={\rho }^{\ast }\left(t\right)$ in (41(41) $y\left(u\right)=y\left(v\right)+{\int }_u^v-y^{\mathrm{\prime }}\left(s\right)\mathrm{d}s\ge (v-u)(-y^{\mathrm{\prime }}(v\left)\right).$(41) ) gives (42) $y\left(\tau \right(t\left)\right)\ge \left({\rho }^{\ast }\left(t\right)-\tau \left(t\right)\right)\left(-y^{\mathrm{\prime }}\left({\rho }^{\ast }\right(t\left)\right)\right).$(42) Letting $W\left(t\right)=-y^{\mathrm{\prime }}\left(t\right)$ and using (42(42) $y\left(\tau \right(t\left)\right)\ge \left({\rho }^{\ast }\left(t\right)-\tau \left(t\right)\right)\left(-y^{\mathrm{\prime }}\left({\rho }^{\ast }\right(t\left)\right)\right).$(42) ) in (8(8) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right)y^{\beta }\left(\tau \right(t\left)\right)<0\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3,$(8) ), we see that the inequality (43) ${\left(a\left(t\right)\left(W^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right){\left[{\rho }^{\ast }\left(t\right)-\tau \left(t\right)\right]}^{\beta }{W}^{\beta }\left({\rho }^{\ast }\right(t\left)\right)$(43) has a positive decreasing solution, which is a contradiction.

Finally, assume that Case (II) holds. Then, as in the proof of Theorem 2.1, we see that (17(17) $\begin{array}{rl}& {\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right)z^{\beta /\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)=q\left(t\right)z^{\beta /\alpha }\left(h\right(t\left)\right)\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(17) ) and (18(18) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)z^{\mathrm{\prime }}\left(h\right(t\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_5$(18) ) hold. Using (18(18) $z\left(h\right(t\left)\right)\ge \frac{1}{2}h\left(t\right)z^{\mathrm{\prime }}\left(h\right(t\left)\right)\mathrm{f}\mathrm{o}\mathrm{r}t\ge t_5$(18) ) in (17(17) $\begin{array}{rl}& {\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right)z^{\beta /\alpha }\left({\sigma }^{-1}\right(\tau \left(t\right)\left)\right)=q\left(t\right)z^{\beta /\alpha }\left(h\right(t\left)\right)\\ & \mathrm{f}\mathrm{o}\mathrm{r}t\ge t_3.\end{array}$(17) ) gives (44) ${\left(a\left(t\right)\left(z^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right){\left[\frac{1}{2}h\left(t\right)\left(z^{\mathrm{\prime }}\left(h\right(t\left)\right)\right)\right]}^{\beta /\alpha }.$(44) Letting $Z\left(t\right)=z^{\mathrm{\prime }}\left(t\right)>0$ in the last inequality, we see that the inequality (45) ${\left(a\left(t\right)\left(Z^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right){\left[\frac{1}{2}h\left(t\right)\right]}^{\beta /\alpha }{Z}^{\beta /\alpha }\left(h\right(t\left)\right)$(45) has a positive decreasing solution, which is a contradiction. This completes the proof.

For Equation (24(24) ${\left(a\left(t\right)\left(x^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(24) ), we have the following result.

Theorem 2.6

Let (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) hold and assume that there exists continuous function ${\rho }^{\ast }:[t_0,\mathrm{\infty })\to \mathbb{R}$ such that $\tau \left(t\right)<{\rho }^{\ast }\left(t\right)<t$ for $t\ge t_0$. If the second order inequality (35(35) ${\left(a\left(t\right)\left(Y^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\le -q\left(t\right){\left[\tau \left(t\right)-\tau \left(\tau \right(t\left)\right)\right]}^{\beta }{Y}^{\beta }\left(\tau \right(\tau \left(t\right))$(35) ) has no positive increasing solution, and the second order inequality (36(36) ${\left(a\left(t\right)\left(W^{\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}\ge q\left(t\right){\left[{\rho }^{\ast }\left(t\right)-\tau \left(t\right)\right]}^{\beta }{W}^{\beta }\left({\rho }^{\ast }\right(t\left)\right)$(36) ) has no positive decreasing solution, then Equation (24(24) ${\left(a\left(t\right)\left(x^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(24) ) is oscillatory.

Example 2.1

Consider the differential equation with a nonpositive neutral term (46) ${\left[{\left({\left(x\left(t\right)-\frac{1}{7}{x}^{1/3}\left(t/2\right)\right)}^{\mathrm{\prime }\mathrm{\prime }}\right)}^5\right]}^{\mathrm{\prime }}+t^{2}x\left(t/6\right)=0,t\ge 1.$(46) Here we have $a\left(t\right)=1$, $p\left(t\right)=1/7$, $\sigma \left(t\right)=t/2$, $q\left(t\right)=t^2$, $\alpha =1/3$, $\gamma =5$, $\beta =1$, and $\tau \left(t\right)=t/6$. Then, ${\sigma }^{-1}\left(t\right)=2t$, $h\left(t\right)=t/3$, and (2(2) $A(t,t_0)\to \mathrm{\infty }\mathrm{a}\mathrm{s}t\to \mathrm{\infty }.$(2) ) holds. Letting, $\eta \left(t\right)=t/5$, $\xi \left(t\right)=t/4$, and $\rho \left(t\right)=t/2$, we see that $\begin{array}{rl}& {\int }_{t_0}^{\mathrm{\infty }}q\left(s\right){\left({\int }_{t_0}^{\tau \left(s\right)}A(u,t_0)\mathrm{d}u\right)}^{\beta }\mathrm{d}s\\ & ={\int }_1^{\mathrm{\infty }}{s}^2\left({\int }_1^{s/6}(u-1)\mathrm{d}u\right)\mathrm{d}s\\ & =\frac{1}{72}{\int }_1^{\mathrm{\infty }}{s}^2\left(s^2-12s+36\right)\mathrm{d}s=\mathrm{\infty },\\ {\int }_{t_0}^{\mathrm{\infty }}q\left(s\right)& {\left[\left(\eta \right(s)-\tau (s\left)\right)A\left(\xi \right(s),\eta (s\left)\right)\right]}^{\beta }\mathrm{d}s\\ & =\frac{1}{600}{\int }_1^{\mathrm{\infty }}{s}^4\mathrm{d}s=\mathrm{\infty },\end{array}$ and $\begin{array}{rl}& {\int }_{t_0}^{\mathrm{\infty }}q\left(s\right){\left[h\left(s\right)A\left(\rho \right(s),h(s\left)\right)\right]}^{\beta /\alpha }\mathrm{d}s\\ & =\frac{1}{{18}^3}{\int }_1^{\mathrm{\infty }}{s}^8\mathrm{d}s=\mathrm{\infty }.\end{array}$ Thus, all conditions of Corollary 2.2 are satisfied, so Equation (46(46) ${\left[{\left({\left(x\left(t\right)-\frac{1}{7}{x}^{1/3}\left(t/2\right)\right)}^{\mathrm{\prime }\mathrm{\prime }}\right)}^5\right]}^{\mathrm{\prime }}+t^{2}x\left(t/6\right)=0,t\ge 1.$(46) ) is oscillatory.

Remarks

1. The results of this paper are presented in a form that can be extended to higher order equations of the form $\begin{array}{rl}& {\left[{\left(a\left(t\right){\left(x\left(t\right)-p\left(t\right)x^{\alpha }\left(\sigma \right(t\left)\right)\right)}^{(n-1)}\right)}^{\gamma }\right]}^{\mathrm{\prime }}\\ & +q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,\end{array}$ where n is a positive integer, a, p, q, α, γ, β, σ and τ are defined as in this paper.

2. It would be of interest to study Equation (1(1) ${\left(a\left(t\right)\left(y^{\mathrm{\prime }\mathrm{\prime }}\right(t){)}^{\gamma }\right)}^{\mathrm{\prime }}+q\left(t\right)x^{\beta }\left(\tau \right(t\left)\right)=0,t\ge t_0>0,$(1) ) with $\beta >\gamma$.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

John R. Graef http://orcid.org/0000-0002-8149-4633

Ercan Tunç  http://orcid.org/0000-0001-8860-608X