Oscillation criteria for third-order neutral dynamic equations with continuously distributed delay

Abstract

It is the purpose of this paper to give oscillation criteria for the third-order neutral dynamic equations with continuously distributed delay,

$${[r(t){\left({[x(t)+{\int }_a^{b}p(t,\eta )x[\tau (t,\eta )]\mathrm{\Delta }\eta ]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^{\gamma }]}^{\mathrm{\Delta }}+{\int }_c^{d}q(t,\xi )f\left(x[\varphi (t,\xi )]\right)\mathrm{\Delta }\xi =0,$$

on a time scale $\mathbb{T}$, where γ is the quotient of odd positive integers. By using a generalized Riccati transformation and an integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

1 Introduction

We are concerned with the oscillatory behavior of third-order neutral dynamic equations with continuously distributed delay,

$${[r(t){\left({[x(t)+{\int }_a^{b}p(t,\eta )x[\tau (t,\eta )]\mathrm{\Delta }\eta ]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^{\gamma }]}^{\mathrm{\Delta }}+{\int }_c^{d}q(t,\xi )f\left(x[\varphi (t,\xi )]\right)\mathrm{\Delta }\xi =0,$$

(1)

on an arbitrary time scale $\mathbb{T}$, where γ is a quotient of odd positive integers. Throughout this paper, we will assume the following hypotheses:

(H1) r and q are positive rd-continuous functions on $\mathbb{T}$ and

$${\int }_{t_0}^{\mathrm{\infty }}{\left(\frac{1}{r(t)}\right)}^{\frac{1}{\gamma }}\mathrm{\Delta }t=\mathrm{\infty };$$

(2)

(H2) $p(t,\eta )\in C_{rd}([t_0,\mathrm{\infty })\times [a,b],\mathbb{R})$, $0\le p(t)\equiv {\int }_a^{b}p(t,\eta )\mathrm{\Delta }\eta \le P<1$;

(H3) $\tau (t,\eta )\in C_{rd}([t_0,\mathrm{\infty })\times [a,b],\mathbb{T})$ is not a decreasing function for η and such that

$$\tau (t,\eta )\le t\text{and}\underset{t\to \mathrm{\infty }}{lim}\underset{\eta \in [a,b]}{min}\tau (t,\eta )=\mathrm{\infty };$$

(H4) $\varphi (t,\xi )\in C_{rd}([t_0,\mathrm{\infty })\times [c,d],\mathbb{T})$ is not decreasing function for ξ and such that

$$\varphi (t,\xi )\le t\text{and}\underset{t\to \mathrm{\infty }}{lim}\underset{\xi \in [c,d]}{min}\varphi (t,\xi )=\mathrm{\infty };$$

(H5) the function $f\in C_{rd}(\mathbb{T},\mathbb{R})$ is assumed to satisfy $uf(u)>0$ and there exists a positive rd-continuous function $\delta (t)$ on $\mathbb{T}$ such that $\frac{f(u)}{u^{\gamma }}\ge \delta$, for $u\ne 0$.

Define the function by

$$z(t)=x(t)+{\int }_a^{b}p(t,\eta )x[\tau (t,\eta )]\mathrm{\Delta }\eta .$$

(3)

Furthermore, (1) is like the following:

$${[r(t){\left({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^{\gamma }]}^{\mathrm{\Delta }}+{\int }_c^{d}q(t,\xi )f\left(x[\varphi (t,\xi )]\right)\mathrm{\Delta }\xi =0.$$

(4)

A solution $x(t)$ of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory.

Much recent attention has been given to dynamic equations on time scales, or measure chains, and we refer the reader to the landmark paper of Hilger [1] for a comprehensive treatment of the subject. Since then, several authors have expounded various aspects of this new theory; see the survey paper by Agarwal et al. [2]. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus. In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and non-oscillation of solutions of various equations on time scales; we refer the reader to the papers [4–19]. Candan [20] considered oscillation of second-order neutral dynamic equations with distributed deviating arguments of the form

$${(r(t){\left({(y(t)+p(t)y(\tau (t)))}^{\mathrm{\Delta }}\right)}^{\gamma })}^{\mathrm{\Delta }}+{\int }_c^{d}f(t,y(\theta (t,\xi )))\mathrm{\Delta }\xi =0,$$

where $\gamma >0$ is a ratio of odd positive integers with $r(t)$ and $p(t)$ real-valued rd-continuous positive functions defined on $\mathbb{T}$. He established some new oscillation criteria and gave sufficient conditions to ensure that all solutions of nonlinear neutral dynamic equation are oscillatory on a time scale $\mathbb{T}$.

To the best of our knowledge, there is very little known about the oscillatory behavior of third-order dynamic equations. Erbe et al. [21] are concerned with the oscillatory behavior of solutions of the third-order linear dynamic equation

$$x^{\mathrm{\Delta }\mathrm{\Delta }\mathrm{\Delta }}(t)+p(t)x(t)=0,$$

on an arbitrary time scale $\mathbb{T}$, where $p(t)$ is a positive real-valued rd-continuous function defined on $\mathbb{T}$. Li et al. [22] considered third-order nonlinear delay dynamic equation

$$x^{{\mathrm{\Delta }}^3}+p(t)x^{\gamma }(\tau (t))=0,$$

on a time scale $\mathbb{T}$, where $\gamma >0$ is quotient of odd positive integers.

Erbe et al. [23, 24] established some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation

$${(c(t){(a(t)x^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }})}^{\mathrm{\Delta }}+q(t)f(x(t))=0,$$

and the third-order dynamic equation

$${(c(t){\left({(a(t)x^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}\right)}^{\gamma })}^{\mathrm{\Delta }}+f(t,x(t))=0$$

oscillate or converge to zero. Li et al. [25] considered the third-order delay dynamic equations

$${(a(t){\left({[r(t)x^{\mathrm{\Delta }}(t)]}^{\mathrm{\Delta }}\right)}^{\gamma })}^{\mathrm{\Delta }}+f(t,x(\tau (t)))=0,$$

on a time scale $\mathbb{T}$, where $\gamma >0$ is quotient of odd positive integers, a and r are positive rd-continuous functions on $\mathbb{T}$, and the so-called delay function $\tau :\mathbb{T}\to \mathbb{T}$ satisfies $\tau (t)\le t$, and $\tau (t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$, $f(x)\in C_{rd}(\mathbb{T}\times \mathbb{R},\mathbb{R})$ is assumed to satisfy $uf(t,u)>0$, for $u\ne 0$, and there exists a function p on $\mathbb{T}$ such that $\frac{f(t,u)}{u^{\gamma }}\ge p(t)>0$, for $u\ne 0$.

Saker [26] considered the third-order nonlinear functional dynamic equations

$${(p(t){\left({[r(t)x^{\mathrm{\Delta }}(t)]}^{\mathrm{\Delta }}\right)}^{\gamma })}^{\mathrm{\Delta }}+q(t)f\left(x(\tau (t))\right)=0,$$

on a time scale $\mathbb{T}$, where $\gamma >0$ is quotient of odd positive integers. Recently Han et al. [27] and Grace et al. [28] considered the third-order neutral delay dynamic equation

$${(r(t){(x(t)-a(t)x(\tau (t)))}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\mathrm{\Delta }}+p(t)x^{\gamma }(\delta (t))=0,$$

on a time scale $\mathbb{T}$.

In this paper, we consider third-order neutral dynamic equation with continuously distributed delay on time scales which is not in literature. We obtain some conclusions which contribute to oscillation theory of third-order neutral dynamic equations.

2 Several lemmas

Before stating our main results, we begin with the following lemmas which play an important role in the proof of the main results. Throughout this paper, we let

$$d_{+}(t):=max\{0,d(t)\},d_{-}(t):=max\{0,-d(t)\},$$

and

$$\begin{array}{c}\beta (t):=b(t),0<\gamma \le 1,\beta (t):=b^{\gamma }(t),\gamma >1,\hfill \\ b(t)=\frac{t}{\sigma (t)},R(t,t_{\ast }):={\int }_{t_{\ast }}^t{\left(\frac{1}{r(s)}\right)}^{\frac{1}{\gamma }}\mathrm{\Delta }s,\hfill \end{array}$$

where we have sufficiently large $t_{\ast }\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$.

In order to prove our main results, we will use the formula

$${(z^{\gamma }(t))}^{\mathrm{\Delta }}=\gamma {\int }_0^1{[h{z}^{\sigma }+(1-h)z]}^{\gamma -1}{z}^{\mathrm{\Delta }}(t)dh,$$

where $z(t)$ is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rule (see Bohner and Peterson [3]).

Lemma 2.1 Let $x(t)$ be a positive solution of (1), $z(t)$ is defined as in (3). Then $z(t)$ has only one of the following two properties:

• (I) $z(t)>0$, $z^{\mathrm{\Delta }}(t)>0$, $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)>0$,

• (II) $z(t)>0$, $z^{\mathrm{\Delta }}(t)<0$, $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)>0$,

with $t\ge t_1$, $t_1$ sufficiently large.

Proof Let $x(t)$ be a positive solution of (1) on $[t_0,\mathrm{\infty })$, so that $z(t)>x(t)>0$, and

$${[r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }]}^{\mathrm{\Delta }}=-{\int }_c^{d}q(t,\xi )f\left(x[\varphi (t,\xi )]\right)\mathrm{\Delta }\xi <0.$$

Then $r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }$ is a decreasing function and therefore eventually of one sign, so $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)$ is either eventually positive or eventually negative on $t\ge t_1\ge t_0$. We assert that $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)>0$ on $t\ge t_1\ge t_0$. Otherwise, assume that $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)<0$, then there exists a constant $M>0$, such that

$$r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }\le -M<0.$$

By integrating the last inequality from $t_1$ to t, we obtain

$$z^{\mathrm{\Delta }}(t)\le z^{\mathrm{\Delta }}(t_1)-M^{\frac{1}{\gamma }}{\int }_{t_1}^t{\left(\frac{1}{r(s)}\right)}^{\frac{1}{\gamma }}\mathrm{\Delta }s.$$

Let $t\to \mathrm{\infty }$. Then from (H1), we have ${(z(t))}^{\mathrm{\Delta }}\to -\mathrm{\infty }$, and therefore eventually $z^{\mathrm{\Delta }}(t)<0$.

Since $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)<0$ and $z^{\mathrm{\Delta }}(t)<0$, we have $z(t)<0$, which contradicts our assumption $z(t)>0$. Therefore, $z(t)$ has only one of the two properties (I) and (II).

This completes the proof. □

Lemma 2.2 Let $x(t)$ be an eventually positive solution of (1), correspondingly $z(t)$ has the property (II). Assume that (2) and

$${\int }_{t_0}^{\mathrm{\infty }}{\int }_v^{\mathrm{\infty }}{[\frac{1}{r(u)}{\int }_u^{\mathrm{\infty }}{q}_1(s)\mathrm{\Delta }s]}^{\frac{1}{\gamma }}\mathrm{\Delta }u\mathrm{\Delta }v=\mathrm{\infty }$$

(5)

hold. Then ${lim}_{t\to \mathrm{\infty }}x(t)=0$.

Proof Let $x(t)$ be an eventually positive solution of (1). Since $z(t)$ has the property (II), then there exists finite ${lim}_{t\to \mathrm{\infty }}z(t)=I$. We assert that $I=0$. Assume that $I>0$, then we have $I+ϵ>z(t)>I$ for all $ϵ>0$. Choosing $ϵ<\frac{I(1-P)}{P}$ and using (3) and (H2), we obtain

$$\begin{array}{rcl}x(t)& =& z(t)-{\int }_a^{b}p(t,\eta )\left[x(\tau (t,\eta ))\right]\mathrm{\Delta }\eta \\ >& I-{\int }_a^{b}p(t,\eta )\left[x(\tau (t,\eta ))\right]\mathrm{\Delta }\eta \\ \ge & I-p(t)\left[z(\tau (t,a))\right]\\ \ge & I-P(I+ϵ)>Kz(t),\end{array}$$

(6)

where $K=\frac{I-P(1+ϵ)}{I+ϵ}>0$. Using (H5) and (6), we find from (1) that

$$\begin{array}{rcl}{[r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }]}^{\mathrm{\Delta }}& =& -{\int }_c^{d}q(t,\xi )f\left(x[\varphi (t,\xi )]\right)\mathrm{\Delta }\xi \\ \le & -{\int }_c^{d}q(t,\xi ){\left(x[\varphi (t,\xi )]\right)}^{\gamma }\delta \mathrm{\Delta }\xi \\ \le & -K^{\gamma }\delta {\int }_c^{d}q(t,\xi ){\left(z[\varphi (t,\xi )]\right)}^{\gamma }\mathrm{\Delta }\xi .\end{array}$$

Note that $z(t)$ has property (II) and (H4), and we have

$${[r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }]}^{\mathrm{\Delta }}\le -K^{\gamma }\cdot \delta \cdot {\left(z[\varphi (t,d)]\right)}^{\gamma }{\int }_c^{d}q(t,\xi )\mathrm{\Delta }\xi =-q_1(t){\left(z({\varphi }_1(t))\right)}^{\gamma },$$

(7)

where $q_1(t)=K^{\gamma }\delta {\int }_c^{d}q(t,\xi )\mathrm{\Delta }\xi$, ${\varphi }_1(t)=\varphi (t,d)$. Integrating inequality (7) from t to ∞, we obtain

$$r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }\ge {\int }_t^{\mathrm{\infty }}{q}_1(s){\left(z({\varphi }_1(s))\right)}^{\gamma }\mathrm{\Delta }s.$$

Using ${(z({\varphi }_1(s)))}^{\gamma }\ge I^{\gamma }$, we obtain

$$z^{\mathrm{\Delta }\mathrm{\Delta }}(t)\ge \frac{I}{r^{\frac{1}{\gamma }}}{[{\int }_t^{\mathrm{\infty }}{q}_1(s)]}^{\frac{1}{\gamma }}\mathrm{\Delta }(s).$$

(8)

Integrating inequality (8) from t to ∞, we have

$$-z^{\mathrm{\Delta }}(t)\ge I{\int }_t^{\mathrm{\infty }}{[\frac{1}{r(u)}{\int }_u^{\mathrm{\infty }}{q}_1(s)\mathrm{\Delta }(s)]}^{\frac{1}{\gamma }}\mathrm{\Delta }u.$$

Integrating the last inequality from $t_1$ to ∞, we obtain

$$z(t_1)\ge I{\int }_{t_1}^{\mathrm{\infty }}{\int }_v^{\mathrm{\infty }}{[\frac{1}{r(u)}{\int }_u^{\mathrm{\infty }}{q}_1(s)\mathrm{\Delta }(s)]}^{\frac{1}{\gamma }}\mathrm{\Delta }u\mathrm{\Delta }v.$$

Because (7) and the last inequality contradict (5), we have $I=0$. Since $0\le x(t)\le z(t)$, ${lim}_{t\to \mathrm{\infty }}x(t)=0$. This completes the proof. □

Lemma 2.3 Assume that $x(t)$ is a positive solution of (1), $z(t)$ is defined as in (3) such that $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)>0$, $z^{\mathrm{\Delta }}(t)>0$, on ${[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}$, $t_{\ast }\ge 0$. Then

$$z^{\mathrm{\Delta }}(t)\ge R(t,t_{\ast })r^{\frac{1}{\gamma }}(t)z^{\mathrm{\Delta }\mathrm{\Delta }}(t).$$

(9)

Proof Since $r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }$ is strictly decreasing on ${[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}$, we get for $t\in {[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}$

$$\begin{array}{rcl}{z}^{\mathrm{\Delta }}(t)& >& z^{\mathrm{\Delta }}(t)-z^{\mathrm{\Delta }}(t_{\ast })\\ =& {\int }_{t_{\ast }}^t\frac{{(r(s){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma })}^{\frac{1}{\gamma }}}{r^{\frac{1}{\gamma }}(s)}\mathrm{\Delta }s\\ \ge & {(r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma })}^{\frac{1}{\gamma }}{\int }_{t_{\ast }}^t{\left(\frac{1}{r(s)}\right)}^{\frac{1}{\gamma }}\mathrm{\Delta }s.\end{array}$$

Using the definition of $R(t,t_{\ast })$, we obtain

$$z^{\mathrm{\Delta }}(t)>R(t,t_{\ast })r^{\frac{1}{\gamma }}(t)z^{\mathrm{\Delta }\mathrm{\Delta }}(t)\text{on }{[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}.$$

 □

Lemma 2.4 Assume that $x(t)$ is a positive solution of (1), correspondingly $z(t)$ has the property (I). Such that $z^{\mathrm{\Delta }}(t)>0$, $z^{\mathrm{\Delta }\mathrm{\Delta }}(t)>0$, on ${[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}$, $t_{\ast }\ge t_0$. Furthermore,

$${\int }_{t_2}^t{q}_2(s){\varphi }_2^{\gamma }(s)\mathrm{\Delta }s=\mathrm{\infty }.$$

(10)

Then there exists a $T\in {[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}$, sufficiently large, so that

$$z(t)>t{z}^{\mathrm{\Delta }}(t),$$

$z(t)/t$ is strictly decreasing, $t\in {[T,\mathrm{\infty })}_{\mathbb{T}}$.

Proof Let $U(t)=z(t)-t{z}^{\mathrm{\Delta }}(t)$. Hence $U^{\mathrm{\Delta }}(t)=-\sigma (t)z^{\mathrm{\Delta }\mathrm{\Delta }}(t)<0$. We claim there exists a $t_1\in {[t_{\ast },\mathrm{\infty })}_{\mathbb{T}}$ such that $U(t)>0$, $z(\varphi (t,\xi ))>0$ on ${[t_1,\mathrm{\infty })}_{\mathbb{T}}$. Assume not. Then $U(t)<0$ on ${[t_1,\mathrm{\infty })}_{\mathbb{T}}$. Therefore,

$${\left(\frac{z(t)}{t}\right)}^{\mathrm{\Delta }}=\frac{t{z}^{\mathrm{\Delta }}(t)-z(t)}{t\sigma (t)}=-\frac{U(t)}{t\sigma (t)}>0,t\in {[t_1,\mathrm{\infty })}_{\mathbb{T}},$$

which implies that $z(t)/t$ is strictly increasing on ${[t_1,\mathrm{\infty })}_{\mathbb{T}}$. Pick $t_2\in {[t_1,\mathrm{\infty })}_{\mathbb{T}}$ so that $\varphi (t,\xi )\ge \varphi (t_1,\xi )$, for $t\ge t_2$. Then

$$\frac{z(\varphi (t,\xi ))}{\varphi (t,\xi )}\ge \frac{z(\varphi (t_1,\xi ))}{\varphi (t_1,\xi )}=d>0,$$

so that $z(\varphi (t,\xi ))>d\varphi (t,\xi )$, for $t\ge t_2$. By (1), (3), and (H2), we obtain

$$\begin{array}{rcl}x(t)& =& z(t)-{\int }_a^{b}p(t,\eta )x[\tau (t,\eta )]\mathrm{\Delta }\eta \\ \ge & z(t)-{\int }_a^{b}p(t,\eta )z[\tau (t,\eta )]\mathrm{\Delta }\eta \\ \ge & z(t)-z[\tau (t,b)]{\int }_a^{b}p(t,\eta )\mathrm{\Delta }\eta \\ \ge & (1-{\int }_a^{b}p(t,\eta )\mathrm{\Delta }\eta )z(t)\\ \ge & (1-P)z(t).\end{array}$$

(11)

Using (11), (H4), and (H5), we have

$$\begin{array}{rcl}{[r(t){\left({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^{\gamma }]}^{\mathrm{\Delta }}& =& -{\int }_c^{d}q(t,\xi )f\left(x[\varphi (t,\xi )]\right)\mathrm{\Delta }\xi \\ \le & -\delta {(1-P)}^{\gamma }{\int }_c^{d}q(t,\xi )z^{\gamma }(\varphi (t,\xi ))\mathrm{\Delta }\xi \\ \le & -\delta {(1-P)}^{\gamma }{z}^{\gamma }(\varphi (t,c)){\int }_c^{d}q(t,\xi )\mathrm{\Delta }\xi \\ \le & -q_2(t)z^{\gamma }({\varphi }_2(t)),\end{array}$$

(12)

where $q_2(t)=\delta {(1-P)}^{\gamma }{\int }_c^{d}q(t,\xi )\mathrm{\Delta }\xi$, ${\varphi }_2(t)=\varphi (t,c)$.

Now by integrating both sides of last equation from $t_2$ to t, we have

$$r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }-r(t_2){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t_2))}^{\gamma }+{\int }_{t_2}^t{q}_2(s)z^{\gamma }({\varphi }_2(s))\mathrm{\Delta }s\le 0.$$

This implies that

$$r(t_2){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t_2))}^{\gamma }\ge {\int }_{t_2}^t{q}_2(s){\left(z({\varphi }_2(s))\right)}^{\gamma }\mathrm{\Delta }s\ge d^{\gamma }{\int }_{t_2}^t{q}_2(s){\varphi }_2^{\gamma }(s)\mathrm{\Delta }s,$$

which contradicts (10). So $U(t)>0$ on $t\in {[t_1,\mathrm{\infty })}_{\mathbb{T}}$ and consequently,

$${\left(\frac{z(t)}{t}\right)}^{\mathrm{\Delta }}=\frac{t{z}^{\mathrm{\Delta }}(t)-z(t)}{t\sigma (t)}=-\frac{U(t)}{t\sigma (t)}<0,t\in {[t_1,\mathrm{\infty })}_{\mathbb{T}},$$

and we find that $z(t)/t$ is strictly decreasing on $t\in {[t_1,\mathrm{\infty })}_{\mathbb{T}}$. The proof is now complete. □

3 Main results

In this section we give some new oscillation criteria for (1).

Theorem 3.1 Assume that (2), (5), and (10) hold. Furthermore, assume that there exists a positive function $\rho \in C_{rd}^1({[t_0,\mathrm{\infty })}_{\mathbb{T}},\mathbb{R})$, for all sufficiently large $T_1\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$, there is a $T>T_1$ such that

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t[{\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }-\frac{{({({\rho }^{\mathrm{\Delta }}(s))}_{+})}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,t_{\ast }))}^{\gamma }}]\mathrm{\Delta }s=\mathrm{\infty }.$$

(13)

Then every solution of (1) is either oscillatory or tends to zero.

Proof Assume (1) has a non-oscillatory solution $x(t)$ on ${[t_0,\mathrm{\infty })}_{\mathbb{T}}$. We may assume without loss of generality that $x(t)>0$, $t\ge t_1$; $x(\tau (t,\eta ))>0$, $(t,\eta )\in [t_1,\mathrm{\infty })\times [a,b]$ and $x(\varphi (t,\xi ))>0$, $(t,\xi )\in [t_1,\mathrm{\infty })\times [c,d]$ for all $t_1\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$. $z(t)$ is defined as in (3). We suppose that $z(t)>0$. We shall consider only this case, since the proof when $z(t)$ is eventually negative is similar. Therefore Lemma 2.1 and Lemma 2.2, we get

$${[r(t){\left({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^{\gamma }]}^{\mathrm{\Delta }}<0,z^{\mathrm{\Delta }\mathrm{\Delta }}(t)>0,t\in {[t_1,\mathrm{\infty })}_{\mathbb{T}},$$

and either $z^{\mathrm{\Delta }}(t)>0$ for $t\ge t_2\ge t_1$ or ${lim}_{t\to \mathrm{\infty }}x(t)=0$. Let $z^{\mathrm{\Delta }}(t)>0$ on ${[t_2,\mathrm{\infty })}_{\mathbb{T}}$.

By (11) and (12), we have

$${[r(t){\left({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^{\gamma }]}^{\mathrm{\Delta }}\le -q_2(t)z^{\gamma }({\varphi }_2(t)),$$

where $q_2(t)=\delta {(1-P)}^{\gamma }{\int }_c^{d}q(t,\xi )\mathrm{\Delta }\xi$, ${\varphi }_2(t)=\varphi (t,c)$.

Define the function $w(t)$ by the Riccati substitution

$$w(t)=\rho (t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }}{z^{\gamma }(t)}.$$

(14)

Then

$$\begin{array}{rcl}{w}^{\mathrm{\Delta }}(t)& =& {\rho }^{\mathrm{\Delta }}(t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }}{z^{\gamma }(t)}+{\rho }^{\sigma }(t){\left[\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }}{z^{\gamma }(t)}\right]}^{\mathrm{\Delta }}\\ =& {\rho }^{\mathrm{\Delta }}(t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }}{z^{\gamma }(t)}+{\rho }^{\sigma }(t)\frac{{[r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }]}^{\mathrm{\Delta }}}{z^{\gamma \sigma }(t)}\\ -{\rho }^{\sigma }(t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }{(z^{\gamma }(t))}^{\mathrm{\Delta }}}{z^{\gamma }(t)z^{\gamma \sigma }(t)}.\end{array}$$

From (1), the definition of $w(t)$ and using the fact $z(t)/t$ is strictly decreasing for $t\in {[t_3,\mathrm{\infty })}_{\mathbb{T}}$, $t_3\ge t_2$, it follows that

$$\begin{array}{r}{w}^{\mathrm{\Delta }}(t)\le \frac{{\rho }^{\mathrm{\Delta }}(t)}{\rho (t)}w(t)-{\rho }^{\sigma }(t)q_2(t)\frac{z^{\gamma }({\varphi }_2(t))}{z^{\gamma \sigma }(t)}-{\rho }^{\sigma }(t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }{(z^{\gamma }(t))}^{\mathrm{\Delta }}}{z^{\gamma }(t)z^{\gamma \sigma }(t)},\\ w^{\mathrm{\Delta }}(t)\le \frac{{\rho }^{\mathrm{\Delta }}(t)}{\rho (t)}w(t)-{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }-{\rho }^{\sigma }(t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }{(z^{\gamma }(t))}^{\mathrm{\Delta }}}{z^{\gamma }(t)z^{\gamma \sigma }(t)}.\end{array}$$

(15)

Now we consider the following two cases: $0<\gamma \le 1$ and $\gamma >1$. In the first case $0<\gamma \le 1$. Using the Keller chain rule (see [3]), we have

$${(z^{\gamma }(t))}^{\mathrm{\Delta }}=\gamma {\int }_0^1{[h{z}^{\sigma }+(1-h)z]}^{\gamma -1}{z}^{\mathrm{\Delta }}(t)dh\ge \gamma {(z^{\sigma }(t))}^{\gamma -1}{z}^{\mathrm{\Delta }}(t),$$

(16)

in view of (16), Lemma 2.2, Lemma 2.3, and (9), we have

$$\begin{array}{rcl}{w}^{\mathrm{\Delta }}(t)& \le & -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)-\gamma {\rho }^{\sigma }(t)\frac{r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }{z}^{\mathrm{\Delta }}(t)z(t)}{z^{\gamma +1}(t)z^{\sigma }(t)}\\ \le & -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)\\ -\gamma {\rho }^{\sigma }(t)R(t,t_{\ast })\frac{r^{\frac{\gamma +1}{\gamma }}(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma +1}z(t)}{z^{\gamma +1}(t)z(\sigma (t))}\\ \le & -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)-\gamma {\rho }^{\sigma }(t)R(t,t_{\ast })\frac{t}{\sigma (t)}\frac{w^{\frac{\gamma +1}{\gamma }}(t)}{{\rho }^{\frac{\gamma +1}{\gamma }}(t)}.\end{array}$$

(17)

In the second case $\gamma >1$. Applying the Keller chain rule, we have

$${(z^{\gamma }(t))}^{\mathrm{\Delta }}=\gamma {\int }_0^1{[h{z}^{\sigma }+(1-h)z]}^{\gamma -1}{z}^{\mathrm{\Delta }}(t)dh\ge \gamma {(z(t))}^{\gamma -1}{z}^{\mathrm{\Delta }}(t),$$

(18)

in the view of (18), Lemma 2.2, Lemma 2.3, and (9), we have

$$\begin{array}{r}{w}^{\mathrm{\Delta }}(t)\le -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)\\ \phantom{w^{\mathrm{\Delta }}(t)\le }-\gamma {\rho }^{\sigma }(t)\frac{r(t){({[z(t)]}^{\mathrm{\Delta }\mathrm{\Delta }})}^{\gamma }{z}^{\mathrm{\Delta }}(t)z^{\gamma }(t)}{z^{\gamma +1}(t)z^{\gamma \sigma }(t)},\\ w^{\mathrm{\Delta }}(t)\le -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)\\ \phantom{w^{\mathrm{\Delta }}(t)\le }-\gamma {\rho }^{\sigma }(t){\left(\frac{t}{\sigma (t)}\right)}^{\gamma }R(t,t_{\ast })\frac{w^{\frac{\gamma +1}{\gamma }}(t)}{{\rho }^{\frac{\gamma +1}{\gamma }}(t)}.\end{array}$$

(19)

By (17), (19), and the definition of $b(t)$ and $\beta (t)$, we have, for $\gamma >0$,

$$w^{\mathrm{\Delta }}(t)\le -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)-\gamma {\rho }^{\sigma }(t)\beta (t)R(t,t_{\ast })\frac{w^{\lambda }(t)}{{\rho }^{\lambda }(t)},$$

(20)

where $\lambda :=\frac{\gamma +1}{\gamma }$. Define $A\ge 0$ and $B\ge 0$ by

$$\begin{array}{c}{A}^{\lambda }:=\gamma {\rho }^{\sigma }(t)\beta (t)R(t,t_{\ast })\frac{w^{\lambda }(t)}{{\rho }^{\lambda }(t)},\hfill \\ B^{\lambda -1}:=\frac{{\rho }^{\mathrm{\Delta }}(t)}{\lambda {(\gamma {\rho }^{\sigma }(t)\beta (t)R(t,t_{\ast }))}^{\frac{1}{\lambda }}}.\hfill \end{array}$$

Then using the inequality [15]

$$\lambda A{B}^{\lambda -1}-A^{\lambda }\le (\lambda -1)B^{\lambda },$$

(21)

which yields

$$\frac{{({\rho }^{\mathrm{\Delta }}(t))}_{+}}{\rho (t)}w(t)-\gamma {\rho }^{\sigma }(t)\beta (t)R(t,t_{\ast })\frac{w^{\lambda }(t)}{{\rho }^{\lambda }(t)}\le \frac{{({({\rho }^{\mathrm{\Delta }}(t))}_{+})}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (t){\rho }^{\sigma }(t)R(t,t_{\ast }))}^{\gamma }}.$$

From this last inequality and (20), we find

$$w^{\mathrm{\Delta }}(t)\le -{\rho }^{\sigma }(t)q_2(t){\left(\frac{{\varphi }_2(t)}{\sigma (t)}\right)}^{\gamma }+\frac{{({({\rho }^{\mathrm{\Delta }}(t))}_{+})}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (t){\rho }^{\sigma }(t)R(t,t_{\ast }))}^{\gamma }}.$$

Integrating both sides from T to t, we get

$${\int }_T^t[{\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }-\frac{{({({\rho }^{\mathrm{\Delta }}(s))}_{+})}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,t_{\ast }))}^{\gamma }}]\mathrm{\Delta }s\le w(T)-w(t)\le w(T),$$

which contradicts assumption (13). This completes the proof of Theorem 3.1. □

Remark 3.1 From Theorem 3.1, we can obtain different conditions for oscillation of (1) with different choices of $\rho (t)$.

Remark 3.2 The conclusion of Theorem 3.1 remains intact if assumption (13) is replaced by the two conditions

$$\begin{array}{c}\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t{\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s=\mathrm{\infty },\hfill \\ \underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t\frac{{({({\rho }^{\mathrm{\Delta }}(s))}_{+})}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)\psi (s,t_{\ast }))}^{\gamma }}\mathrm{\Delta }s<\mathrm{\infty }.\hfill \end{array}$$

For example, let $\rho (t)=t$. Now Theorem 3.1 yields the following results.

Corollary 3.1 Assume that (H1)-(H5), (5), and (10) hold. If

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t[\sigma (s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }-\frac{1}{{(\gamma +1)}^{\gamma +1}{(\beta (s)\sigma (s)R(s,t_{\ast }))}^{\gamma }}]\mathrm{\Delta }s=\mathrm{\infty }$$

(22)

holds, then every solution (1) is either oscillatory or ${lim}_{t\to \mathrm{\infty }}x(t)=0$.

For example, let $\rho (t)=1$. Now Theorem 3.1 yields the following results.

Corollary 3.2 Assume that (H1)-(H5), (5), and (10) hold. If

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t{q}_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s=\mathrm{\infty },$$

(23)

then every solution (1) is either oscillatory or ${lim}_{t\to \mathrm{\infty }}x(t)=0$.

Theorem 3.2 Assume that (2), (5), and (10) hold. Furthermore, suppose that there exist functions $H,h\in C_{rd}(\mathbb{D},\mathbb{R})$, where $\mathbb{D}\equiv (t,s):t\ge s\ge t_0$ such that

$$\begin{array}{c}H(t,t)=0,t\ge 0,\hfill \\ H(t,s)>0,t>s\ge t_0,\hfill \end{array}$$

and H has a nonpositive continuous Δ-partial derivative $H^{\mathrm{\Delta }s}(t,s)$ with respect to the second variable and satisfies

$$H^{\mathrm{\Delta }s}(\sigma (t),s)+H(\sigma (t),\sigma (s))\frac{{\rho }^{\mathrm{\Delta }}(s)}{\rho (s)}=-\frac{h(t,s)}{\rho (s)}H{(\sigma (t),\sigma (s))}^{\frac{\gamma }{\gamma +1}},$$

(24)

and for all sufficiently large $T_1\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$, there is a $T>T_1$ such that

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{H(\sigma (t),T)}{\int }_T^{\sigma (t)}K(t,s)=\mathrm{\infty },$$

(25)

where ρ is a positive Δ-differentiable function and

$$K(t,s)=H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }-\frac{{(h_{-}(t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,T_1))}^{\gamma }}\mathrm{\Delta }s=\mathrm{\infty }.$$

Then every solution of (1) is either oscillatory or tends to zero.

Proof Suppose that $x(t)$ is a non-oscillatory solution of (1) and $z(t)$ is defined as in (3). Without loss of generality, we may assume that there is a $t_1\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$ sufficiently large so that the conclusions of Lemma 2.1 hold and (24) holds for $t_2>t_1$. If case (1) of Lemma 2.1 holds then proceeding as in the proof of Theorem 3.1, we see that (20) holds for $t>t_2$. Multiplying both sides of (20) by $H(\sigma (t),\sigma (s))$ and integrating from T to $\sigma (t)$, we get

$$\begin{array}{r}{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s\\ \le -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))w^{\mathrm{\Delta }}(s)\mathrm{\Delta }s+{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\frac{{\rho }^{\mathrm{\Delta }}(s)}{\rho (s)}w(s)\mathrm{\Delta }s\\ -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(s)\beta (s)R(s,T_1)\frac{w^{\lambda }(s)}{{\rho }^{\lambda }(s)}\mathrm{\Delta }s(\lambda =\frac{\gamma +1}{\gamma }).\end{array}$$

(26)

Integrating by parts and using $H(t,t)=0$, we obtain

$${\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))w^{\mathrm{\Delta }}(s)\mathrm{\Delta }s=-H(\sigma (t),T)w(T)-{\int }_T^{\sigma (t)}{H}^{\mathrm{\Delta }s}(\sigma (t),s)w(s)\mathrm{\Delta }s.$$

It then follows from (26) that

$$\begin{array}{r}{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s\\ \le H(\sigma (t),T)w(T)+{\int }_T^{\sigma (t)}{H}^{\mathrm{\Delta }s}(\sigma (t),s)w(s)\mathrm{\Delta }s\\ +{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\frac{{\rho }^{\mathrm{\Delta }}(s)}{\rho (s)}w(s)\mathrm{\Delta }s\\ -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(s)\beta (s)R(s,T_1)\frac{w^{\lambda }(s)}{{\rho }^{\lambda }(s)}\mathrm{\Delta }s,\\ {\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s\\ \le H(\sigma (t),T)w(T)\\ +[{\int }_T^{\sigma (t)}{H}^{\mathrm{\Delta }s}(\sigma (t),s)+H(\sigma (t),\sigma (s))\frac{{\rho }^{\mathrm{\Delta }}(s)}{\rho (s)}]w(s)\mathrm{\Delta }s\\ -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(s)\beta (s)R(s,T_1)\frac{w^{\lambda }(s)}{{\rho }^{\lambda }(s)}\mathrm{\Delta }s.\end{array}$$

It then follows from (24) that

$$\begin{array}{r}{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s\\ \le H(\sigma (t),T)w(T)\\ +{\int }_T^{\sigma (t)}[-\frac{h(t,s)}{\rho (s)}H{(\sigma (t),\sigma (s))}^{\frac{\gamma }{\gamma +1}}]w(s)\mathrm{\Delta }s\\ -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(s)\beta (s)R(s,T_1)\frac{w^{\lambda }(s)}{{\rho }^{\lambda }(s)}\mathrm{\Delta }s\\ \le H(\sigma (t),T)w(T)+{\int }_T^{\sigma (t)}\left[\frac{h(t,s)}{\rho (s)}H{(\sigma (t),\sigma (s))}^{\frac{\gamma }{\gamma +1}}\right]w(s)\mathrm{\Delta }s\\ -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(s)\beta (s)R(s,T_1)\frac{w^{\lambda }(s)}{{\rho }^{\lambda }(s)}\mathrm{\Delta }s.\end{array}$$

Therefore, as in Theorem 3.1, by letting

$$\begin{array}{c}{A}^{\lambda }:=H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(t)\beta (t)R(t,T_1)\frac{w^{\lambda }(t)}{{\rho }^{\lambda }(t)},\hfill \\ B^{\lambda -1}:=\frac{h_{-}(t,s)}{\lambda {(\gamma {\rho }^{\sigma }(t)\beta (t)R(t,T_1))}^{\frac{1}{\lambda }}}.\hfill \end{array}$$

Then using the inequality [15]

$$\lambda A{B}^{\lambda -1}-A^{\lambda }\le (\lambda -1)B^{\lambda }.$$

We have

$$\begin{array}{r}{\int }_T^{\sigma (t)}\left[\frac{h_{-}(t,s)}{\rho (s)}H{(\sigma (t),\sigma (s))}^{\frac{\gamma }{\gamma +1}}\right]w(s)\mathrm{\Delta }s\\ -{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s))\gamma {\rho }^{\sigma }(s)\beta (s)R(s,T_1)\frac{w^{\lambda }(s)}{{\rho }^{\lambda }(s)}\mathrm{\Delta }s\\ ={\int }_T^{\sigma (t)}\frac{{(h_{-}(t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(t,T_1))}^{\gamma }}\mathrm{\Delta }s,\\ {\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s\\ \le H(\sigma (t),T)w(T)+{\int }_T^{\sigma (t)}\frac{{(h_{-}(t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(t,T_1))}^{\gamma }}\mathrm{\Delta }s.\end{array}$$

Then for $T>T_1$ we have

$$\begin{array}{r}{\int }_T^{\sigma (t)}[H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }-\frac{{(h_{-}(t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,T_1))}^{\gamma }}]\mathrm{\Delta }s\\ \le H(\sigma (t),T)w(T),\end{array}$$

and this implies that

$$\begin{array}{r}\frac{1}{H(\sigma (t),T)}{\int }_T^{\sigma (t)}[H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\\ -\frac{{(h_{-}(t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,T_1))}^{\gamma }}]\mathrm{\Delta }s<w(T),\end{array}$$

for all large T, which contradicts (25). This completes the proof of Theorem 3.2. □

Remark 3.3 The conclusion of Theorem 3.2 remains intact if assumption (25) is replaced by the two conditions

$$\begin{array}{c}\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{H(\sigma (t),T)}{\int }_T^{\sigma (t)}H(\sigma (t),\sigma (s)){\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }\mathrm{\Delta }s=\mathrm{\infty },\hfill \\ \underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{1}{H(\sigma (t),T)}{\int }_T^{\sigma (t)}\frac{{(h_{-}(t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,T_1))}^{\gamma }}\mathrm{\Delta }s<\mathrm{\infty }.\hfill \end{array}$$

Remark 3.4 Define w as (14), we also get

$$w^{\mathrm{\Delta }}(t)=r^{\sigma }(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma \sigma }{\left[\frac{\rho (t)}{z^{\gamma }(t)}\right]}^{\mathrm{\Delta }}+\frac{\rho (t)}{z^{\gamma }}{[r(t){(z^{\mathrm{\Delta }\mathrm{\Delta }}(t))}^{\gamma }]}^{\mathrm{\Delta }},$$

similar to the proofs of Theorem 3.1, we can obtain different results. We leave the details to the reader.

Example 3.1 Consider the following third-order neutral dynamic equation $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$:

$${(x(t)+{\int }_a^b{e}^{-t}x(t-\eta )\mathrm{\Delta }\eta )}^{\mathrm{\Delta }\mathrm{\Delta }\mathrm{\Delta }}+{\int }_c^d\frac{\beta \cdot t}{(t^2-t\xi ){(t^2-t\xi )}^{\sigma }}x(t-\xi )\mathrm{\Delta }\xi =0,$$

(27)

where $\gamma =1$, $r(t)=1$, $\tau (t,\eta )=t-\eta$, $\varphi (t,\xi )=t-\xi$, $\delta =1$, $q_2(t)=\frac{\beta }{t{\varphi }_2(t)}$, $p(t,\eta )=e^{-t}$, $q(t,\xi )=\beta \cdot t/(t^2-t\xi ){(t^2-t\xi )}^{\sigma }$.

It is clear that condition (2), (5), and (10) hold. Therefore, by Theorem 3.1, picking $\rho (t)=t$, we have

$$\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t[{\rho }^{\sigma }(s)q_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^{\gamma }-\frac{{({({\rho }^{\mathrm{\Delta }}(s))}_{+})}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}{(\beta (s){\rho }^{\sigma }(s)R(s,t_{\ast }))}^{\gamma }}]\mathrm{\Delta }s\\ =\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t[\frac{\beta }{s}-\frac{1}{{(\gamma +1)}^{(\gamma +1)}s(s-t_{\ast })}]\mathrm{\Delta }s=\mathrm{\infty }.\end{array}$$

Hence, by Theorem 3.1 every solution of (27) is oscillatory or tends to zero if $\beta >0$.

Example 3.2 Consider the following third-order neutral dynamic equation $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$:

$${\left[\frac{1}{t}{\left({[x(t)+{\int }_a^b\frac{1}{2}x\left[\tau \left(\frac{t}{2}\right)\right]\mathrm{\Delta }\eta ]}^{\mathrm{\Delta }\mathrm{\Delta }}\right)}^3\right]}^{\mathrm{\Delta }}+{\int }_c^{d}q(t,\xi )f\left(x\left[\varphi \left(\frac{t}{2}\right)\right]\right)\mathrm{\Delta }\xi =0,$$

(28)

where $\gamma =3$, $r(t)=\frac{1}{t}$, $\tau (t,\eta )=\frac{t}{2}$, $\varphi (t,\xi )=\frac{t}{2}$, $\delta =1$, $q_2(t)=\frac{\beta }{t}\frac{{\sigma }^3(s)}{{\varphi }_2^3(t)}$, $p(t,\eta )=\frac{1}{2}$.

It is clear that condition (2), (5), and (10) hold. Therefore, by Theorem 3.1, picking $\rho (t)=1$, we have

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t{q}_2(s){\left(\frac{{\varphi }_2(s)}{\sigma (s)}\right)}^3\mathrm{\Delta }s=\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t\frac{\beta }{s}\mathrm{\Delta }s=\mathrm{\infty }.$$

Hence, by Theorem 3.1 every solution of (28) is oscillatory or tends to zero if $\beta >0$.