Theory and Modern Applications

# Oscillation criteria of third-order nonlinear dynamic equations with nonpositive neutral coefficients on time scales

## Abstract

In this paper, we establish oscillation criteria of third-order nonlinear dynamic equations with nonpositive neutral coefficients on time scales by a generalized Riccati transformation and employing functions in some function classes. Two examples are presented to show the significance of the results.

## Introduction

In this paper, we consider third-order nonlinear dynamic equations with nonpositive neutral coefficients of the form

$$\bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\varDelta }+f \bigl(t,x \bigl(h(t) \bigr) \bigr)=0,$$
(1)

where $$z(t)=x(t)-p(t)x(g(t))$$, on a time scale $$\mathbb{T}$$ satisfying $$\inf\mathbb{T}=t_{0}$$ and $$\sup\mathbb{T}=\infty$$. Throughout this paper we assume that:

1. (C1)

$$r_{1}, r_{2}\in C_{\mathrm{rd}}(\mathbb{T}, (0,\infty))$$ such that

\begin{aligned} \int^{\infty}_{t_{0}}\frac{1}{r_{1}^{1/\gamma_{1}}(t)}\varDelta t=\infty, \qquad \int^{\infty}_{t_{0}}\frac{1}{r_{2}^{1/\gamma_{2}}(t)}\varDelta t= \infty; \end{aligned}
2. (C2)

γ, $$\gamma_{1}$$, $$\gamma_{2}$$ are all quotients of odd positive integers, and $$\gamma=\gamma_{1}\cdot\gamma_{2}$$;

3. (C3)

$$p\in C_{\mathrm{rd}}(\mathbb{T}, [0,\infty))$$ and there exists a constant $$p_{0}$$ with $$0\leq p_{0}<1$$ such that

\begin{aligned} \lim_{t \rightarrow\infty}p(t)=p_{0}; \end{aligned}
4. (C4)

$$g\in C_{\mathrm{rd}}(\mathbb{T}, \mathbb{T})$$, $$g(t)\leq t$$, $$\lim_{t \rightarrow\infty}g(t)=\infty$$, and there exists a sequence $$\{c_{k}\}_{k\geq0}$$ such that $$\lim_{k \rightarrow\infty}c_{k}=\infty$$ and $$g(c_{k+1})=c_{k}$$;

5. (C5)

$$h\in C_{\mathrm{rd}}(\mathbb{T}, \mathbb{T})$$, and for any $$t\in \mathbb{T}$$,

\begin{aligned} h(t)\geq \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \sigma(t),& 0< \gamma< 1,\\ t, & \gamma\geq1; \end{array}\displaystyle \right . \end{aligned}
6. (C6)

$$f\in C(\mathbb{T}\times\mathbb{R}, \mathbb{R})$$ and there exists a function $$q\in C_{\mathrm{rd}}(\mathbb{T}, (0, \infty))$$ such that $$uf(t,u)\geq q(t)u^{\gamma+1}$$;

7. (C7)

When $$0<\gamma<1$$, it always satisfies

\begin{aligned} \int^{\infty}_{t_{0}}q(t)\varDelta t< \infty. \end{aligned}

### Definition 1.1

A solution x of (1) is said to have a generalized zero at $$t^{*}\in\mathbb{T}$$ if $$x(t^{*})x(\sigma(t^{*}))\leq0$$, and it is said to be nonoscillatory on $$\mathbb{T}$$ if there exists $$t_{0}\in\mathbb{T}$$ such that $$x(t)x(\sigma(t))>0$$ for all $$t>t_{0}$$. Otherwise, it is oscillatory. Equation (1) is said to be oscillatory if all solutions of (1) are oscillatory.

In 1988, the theory of time scales was introduced by Hilger in his Ph.D. thesis [1] to unify continuous and discrete analysis; see also [2]. Since then, the theory had received a lot of attention. The details of time scales can be found in [36] and are omitted here.

There has been many achievements of the study of oscillation of nonlinear dynamic equations on time scales in the last few years; see [716] and the references therein. Hassan [8], Erbe et al. [7], and Zhang and Wang [16] gave some oscillation criteria successively for the third-order nonlinear delay dynamic equation

\begin{aligned} \bigl(a(t) \bigl[ \bigl(r(t)x^{\varDelta }(t) \bigr)^{\varDelta } \bigr]^{\gamma } \bigr)^{\varDelta }+f \bigl(t,x \bigl(\tau (t) \bigr) \bigr)=0. \end{aligned}

Saker et al. [13] studied the oscillation of the second-order damped dynamic equation

\begin{aligned} \bigl(a(t)x^{\varDelta }(t) \bigr)^{\varDelta }+p(t)x^{\varDelta ^{\sigma }}(t)+q(t) \bigl(f\circ x^{\sigma} \bigr)=0. \end{aligned}

Qiu and Wang [10] considered second-order nonlinear dynamic equation

\begin{aligned} \bigl(p(t)\psi \bigl(x(t) \bigr)k\circ x^{\varDelta }(t) \bigr)^{\varDelta }+f \bigl(t, x \bigl(\sigma(t) \bigr) \bigr)=0. \end{aligned}

Employing a generalized Riccati transformation

\begin{aligned} u(t)=A(t)\frac{p(t)\psi(x(t))k\circ x^{\varDelta }(t)}{x(t)}+B(t), \end{aligned}

the authors established some Kamenev-type oscillation criteria. Şenel [14] investigated the oscillation of the second-order nonlinear dynamic equation of the form

$$\bigl(r(t) \bigl(x^{\varDelta }(t) \bigr)^{\gamma} \bigr)^{\varDelta }+p(t) \bigl(x^{\varDelta }(t) \bigr)^{\gamma}+f \bigl(t, x \bigl(g(t) \bigr) \bigr)=0.$$
(2)

Qiu and Wang [11] corrected some mistakes in [14] and established correct oscillation criteria for (2). Yu and Wang [15] considered the third-order nonlinear dynamic equation

$$\biggl(\frac{1}{a_{2}(t)} \biggl( \biggl(\frac{1}{a_{1}(t)} \bigl(x^{\varDelta }(t) \bigr)^{\alpha_{1}} \biggr)^{\varDelta } \biggr)^{\alpha_{2}} \biggr)^{\varDelta }+q(t)f \bigl(x(t) \bigr)=0$$
(3)

under the condition $$\alpha_{1}\alpha_{2}=1$$, and they established some sufficient conditions which guarantee that every solution x of (3) oscillates or converges to zero on a time scale $$\mathbb{T}$$. Li et al. [9] studied the second-order neutral delay differential equation

\begin{aligned} \bigl(r(t) \bigl(z'(t) \bigr)^{\alpha} \bigr)'+q(t)f \bigl(x \bigl(\sigma(t) \bigr) \bigr)=0,\quad t\geq t_{0}>0, \end{aligned}

where $$z(t)=x(t)-p(t)x(\tau(t))$$ and $$\alpha>0$$ is the ratio of two odd integers. Qiu [12] obtained some significant results for the existence of nonoscillatory solutions to the third-order nonlinear neutral dynamic equation of the form

\begin{aligned} \bigl(r_{1}(t) \bigl(r_{2}(t) \bigl(x(t)+p(t)x \bigl(g(t) \bigr) \bigr)^{\varDelta } \bigr)^{\varDelta } \bigr)^{\varDelta }+f \bigl(t,x \bigl(h(t) \bigr) \bigr)=0, \end{aligned}

where $$\lim_{t \rightarrow\infty}p(t)=p_{0}\in(-1,1)$$.

In this paper, motivated by [9, 10, 12, 14, 15], we will establish oscillation criteria of (1), which are more general than (3), by a generalized Riccati transformation, and give two examples to show the significance of the results.

For the sake of simplicity, we denote $$(a,b)\cap\mathbb{T}=(a,b)_{\mathbb{T}}$$ throughout the paper, where $$a, b\in\mathbb{R}$$, and $$[a,b]_{\mathbb{T}}$$, $$\left .[a,b) \right ._{\mathbb{T}}$$, $$\left .(a,b] \right ._{\mathbb{T}}$$ are similar notations.

## Preliminary results

To establish the oscillation criteria of (1), we give six lemmas in this section.

### Lemma 2.1

Suppose that $$x(t)$$ is an eventually positive solution of (1), and there exists a constant $$a\geq0$$ such that $$\lim_{t\rightarrow \infty}z(t)=a$$. Then we have

\begin{aligned} \lim_{t\rightarrow\infty}x(t)=\frac{a}{1-p_{0}}. \end{aligned}

### Proof

Suppose that $$x(t)$$ is an eventually positive solution of (1). In view of (C3) and (C5), there exist $$T\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ and $$p_{0}< p_{1}<1$$ such that $$x(t)>0$$, $$x(g(t))>0$$, and $$p(t)\leq p_{1}$$ for $$t\in [T,\infty)_{\mathbb{T}}$$. We claim that $$x(t)$$ is bounded on $$[T,\infty)_{\mathbb{T}}$$. Assume not; then there exists $$\{t_{n}\}\in[T,\infty)_{\mathbb{T}}$$ with $$t_{n}\rightarrow \infty$$ as $$n\rightarrow\infty$$ such that

\begin{aligned} x(t_{n})=\max_{t\in[T,t_{n}]_{\mathbb{T}}}x(t) \quad\mbox{and}\quad \lim _{n\rightarrow\infty}x(t_{n})=\infty. \end{aligned}

Noting that $$g(t)\leq t$$, we have

\begin{aligned} z(t_{n})=x(t_{n})-p(t_{n})x \bigl(g(t_{n}) \bigr)\geq (1-p_{1})x(t_{n}) \rightarrow\infty \end{aligned}

as $$n\rightarrow\infty$$, which contradicts the fact that $$\lim_{t\rightarrow\infty}z(t)=a$$. Therefore, $$x(t)$$ is bounded. Then assume that

\begin{aligned} \limsup_{t\rightarrow\infty}x(t)=\overline{x} \quad\mbox{and}\quad \liminf _{t\rightarrow\infty}x(t)=\underline{x}. \end{aligned}

Since $$0\leq p_{0}<1$$, we have

\begin{aligned} a\geq\overline{x}-p_{0}\overline{x} \quad\mbox{and}\quad a\leq \underline{x}-p_{0}\underline{x}, \end{aligned}

which implies that $$\overline{x}\leq\underline{x}$$. So $$\overline{x}=\underline{x}$$, and we see that $$\lim_{t\rightarrow \infty}x(t)$$ exists and $$\lim_{t\rightarrow \infty}x(t)=a/(1-p_{0})$$. The proof is complete. □

### Lemma 2.2

Assume that $$x(t)$$ is an eventually positive solution of (1), then there exists a sufficiently large $$T\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that, for $$t\in \left .[T,\infty) \right ._{\mathbb{T}}$$, we have

\begin{aligned} \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta }>0 \end{aligned}

and

\begin{aligned} z^{\varDelta }(t)>0 \quad\textit{or}\quad z^{\varDelta }(t)< 0. \end{aligned}

### Proof

Suppose that $$x(t)$$ is an eventually positive solution of (1). From (C3) and (C5), there exist $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ and $$p_{0}< p_{1}<1$$ such that $$x(t)>0$$, $$x(g(t))>0$$, $$x(h(t))>0$$, and $$p(t)\leq p_{1}$$ for $$t\in [t_{1},\infty)_{\mathbb{T}}$$. By (1) and (C6), it follows that, for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} \bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\varDelta } =-f \bigl(t,x \bigl(h(t) \bigr) \bigr)< 0. \end{aligned}
(4)

Hence, $$r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma _{2}} )^{\varDelta } )^{\gamma_{1}}$$ is strictly decreasing on $$\left .[t_{1},\infty) \right ._{\mathbb{T}}$$. We claim that

\begin{aligned} r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}}>0,\quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}. \end{aligned}
(5)

Assume not; then there exists $$t_{2}\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$ such that

\begin{aligned} r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}}< 0 \end{aligned}

for $$t\in \left .[t_{2},\infty) \right ._{\mathbb{T}}$$. So there exists a constant $$c<0$$ and we have $$t_{3}\in \left .[t_{2},\infty) \right ._{\mathbb{T}}$$ such that $$r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma _{2}} )^{\varDelta } )^{\gamma_{1}}\leq c$$ for $$t\in \left .[t_{3},\infty) \right ._{\mathbb{T}}$$, which means that

\begin{aligned} \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta }\leq \biggl(\frac{c}{r_{1}(t)} \biggr)^{1/\gamma_{1}},\quad t\in \left .[t_{3},\infty) \right ._{\mathbb{T}}. \end{aligned}
(6)

Substituting s for t, and integrating (6) from $$t_{3}$$ to $$t\in \left .[\sigma(t_{3}),\infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}}\leq r_{2}(t_{3}) \bigl(z^{\varDelta }(t_{3}) \bigr)^{\gamma_{2}}+c^{1/\gamma _{1}}\int^{t}_{t_{3}} \frac{\varDelta s}{r_{1}^{1/\gamma_{1}}(s)}. \end{aligned}

Letting $$t\rightarrow\infty$$, by (C1) we have $$r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}}\rightarrow -\infty$$. Then there exists $$t_{4}\in \left .[t_{3},\infty) \right ._{\mathbb{T}}$$ such that $$r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}}\leq r_{2}(t_{4}) (z^{\varDelta }(t_{4}) )^{\gamma_{2}}<0$$ for $$t\in \left .[t_{4},\infty) \right ._{\mathbb{T}}$$, which implies that

\begin{aligned} z^{\varDelta }(t)\leq r_{2}^{1/\gamma_{2}}(t_{4})z^{\varDelta }(t_{4}) \cdot \frac{1}{r_{2}^{1/\gamma_{2}}(t)}. \end{aligned}
(7)

Substituting s for t, and integrating (7) from $$t_{4}$$ to $$t\in \left .[\sigma(t_{4}),\infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} z(t)-z(t_{4})\leq r_{2}^{1/\gamma_{2}}(t_{4})z^{\varDelta }(t_{4}) \int_{t_{4}}^{t} \frac{\varDelta s}{r_{2}^{1/\gamma_{2}}(s)}. \end{aligned}

Letting $$t\rightarrow\infty$$, by (C1) we have $$z(t)\rightarrow -\infty$$. Then there exists $$t_{5}\in \left .[t_{4},\infty) \right ._{\mathbb{T}}$$ such that $$z(t)<0$$ or

\begin{aligned} x(t)< p(t)x \bigl(g(t) \bigr)\leq p_{1}x \bigl(g(t) \bigr), \quad t\in \left .[t_{5},\infty) \right ._{\mathbb{T}}. \end{aligned}

By (C4), we can choose some positive integer $$k_{0}$$ such that $$c_{k}\in \left .[t_{5},\infty) \right ._{\mathbb{T}}$$ for all $$k\geq k_{0}$$. Then for any $$k\geq k_{0}+1$$, we have

\begin{aligned} x(c_{k})&< p_{1}x \bigl(g(c_{k}) \bigr)=p_{1}x(c_{k-1})< p_{1}^{2}x \bigl(g(c_{k-1}) \bigr) =p_{1}^{2}x(c_{k-2})< \cdots \\ &< p_{1}^{k-k_{0}}x \bigl(g(c_{k_{0}+1}) \bigr)=p_{1}^{k-k_{0}}x(c_{k_{0}}). \end{aligned}

The inequality above implies that $$\lim_{k\rightarrow \infty}x(c_{k})=0$$. It follows that

\begin{aligned} \lim_{k\rightarrow\infty}z(c_{k})=0, \end{aligned}

and this contradicts $$\lim_{t\rightarrow\infty}z(t)=-\infty$$. So (5) holds, which implies that

\begin{aligned} \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta }>0,\quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}. \end{aligned}

Therefore, $$r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}}$$ is strictly increasing on $$\left .[t_{1},\infty) \right ._{\mathbb{T}}$$. It follows that $$r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}}$$ is eventually positive or $$r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}}<0$$ on $$\left .[t_{1},\infty) \right ._{\mathbb{T}}$$. Lemma 2.2 is proved. □

### Lemma 2.3

Assume that $$x(t)$$ is an eventually positive solution of (1), then $$z(t)$$ is eventually positive or $$\lim_{t\rightarrow \infty}x(t)=0$$.

### Proof

Suppose that $$x(t)$$ is an eventually positive solution of (1), by Lemma 2.2 there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$z^{\varDelta }(t)>0$$ or $$z^{\varDelta }(t)<0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$.

(i) $$z^{\varDelta }(t)>0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. Then it follows that $$z(t)$$ is eventually positive or eventually negative. If $$z(t)$$ is eventually positive, the lemma is proved. If $$z(t)$$ is eventually negative, we see that $$\lim_{t\rightarrow\infty}z(t)$$ exists. Assume that $$\lim_{t\rightarrow\infty}z(t)<0$$. Similarly as in the proof of Lemma 2.2, we will have the contradiction. Hence, $$\lim_{t\rightarrow\infty}z(t)=0$$. Then it follows that $$\lim_{t\rightarrow\infty}x(t)=0$$ by Lemma 2.1.

(ii) $$z^{\varDelta }(t)<0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. Similarly, we see that $$z(t)$$ is eventually positive or eventually negative. Assume that $$z(t)$$ is eventually negative, there exists a constant $$c<0$$ and we have $$t_{2}\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$ such that $$z(t)< c$$, $$t\in \left .[t_{2},\infty) \right ._{\mathbb{T}}$$. It will cause a similar contradiction as in the proof of Lemma 2.2. Hence, $$z(t)$$ is eventually positive and the lemma is proved.

The proof is complete. □

### Lemma 2.4

For $$0<\gamma<1$$, assume that $$x(t)$$ is an eventually positive solution of (1), and $$z(t)$$, $$z^{\varDelta }(t)$$ are both eventually positive. Then there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that

\begin{aligned} \biggl(\frac{z^{\varDelta }(t)}{z^{\sigma}(t)} \biggr)^{1-\gamma}\geq \alpha(t)= \biggl( \frac{\delta(t)}{r_{2}(t)} \biggr)^{(1-\gamma)/\gamma _{2}} \biggl(\int_{t}^{\infty}q(s) \varDelta s \biggr)^{(1-\gamma)/\gamma}, \quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}, \end{aligned}

where

\begin{aligned} \delta(t)=\int_{t_{1}}^{t}\frac{\varDelta s}{r_{1}^{1/\gamma_{1}}(s)}. \end{aligned}

### Proof

Suppose that $$x(t)$$ is an eventually positive solution of (1), and $$z(t)$$, $$z^{\varDelta }(t)$$ are both eventually positive, then there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$x(t)>0$$, $$x(g(t))>0$$, $$x(h(t))>0$$, $$z(t)>0$$, and $$z^{\varDelta }(t)>0$$ for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. By Lemma 2.2 we have

\begin{aligned} \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta }>0,\quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}. \end{aligned}

By $$z^{\varDelta }(t)>0$$ and $$z(t)=x(t)-p(t)x(g(t))\leq x(t)$$, it follows that, for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} &\bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\varDelta } \\ &\quad=-f \bigl(t,x \bigl(h(t) \bigr) \bigr) \leq-q(t)x^{\gamma} \bigl(h(t) \bigr)\leq-q(t)z^{\gamma} \bigl(h(t) \bigr)\leq -q(t)z^{\gamma} \bigl(\sigma(t) \bigr)< 0. \end{aligned}
(8)

Substituting s for t, and integrating (8) from $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$ to ∞, we obtain

\begin{aligned} r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}}\geq \int _{t}^{\infty}q(s)z^{\gamma} \bigl(\sigma(s) \bigr) \varDelta s\geq z^{\gamma} \bigl(\sigma(t) \bigr)\int_{t}^{\infty}q(s) \varDelta s. \end{aligned}

As $$r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma _{2}} )^{\varDelta } )^{\gamma_{1}}$$ is strictly decreasing on $$\left .[t_{1},\infty) \right ._{\mathbb{T}}$$, we have, for $$t\in \left .[\sigma(t_{1}),\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} =&r_{2}(t_{1}) \bigl(z^{\varDelta }(t_{1}) \bigr)^{\gamma_{2}} +\int _{t_{1}}^{t}\frac{r_{1}^{1/\gamma_{1}}(s) (r_{2}(s) (z^{\varDelta }(s) )^{\gamma_{2}} )^{\varDelta }}{r_{1}^{1/\gamma _{1}}(s)}\varDelta s \\ \geq& r_{1}^{1/\gamma_{1}}(t) \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta }\int _{t_{1}}^{t}\frac{1}{r_{1}^{1/\gamma _{1}}(s)}\varDelta s \\ \geq& \delta(t) \biggl(z^{\gamma} \bigl(\sigma(t) \bigr)\int _{t}^{\infty}q(s)\varDelta s \biggr)^{1/\gamma_{1}}= \delta(t)z^{\gamma_{2}} \bigl(\sigma(t) \bigr) \biggl(\int_{t}^{\infty}q(s) \varDelta s \biggr)^{1/\gamma_{1}}. \end{aligned}

Hence, when $$0<\gamma<1$$, we have

\begin{aligned} \frac{z^{\varDelta }(t)}{z^{\sigma}(t)}\geq \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{1/\gamma_{2}} \biggl( \int_{t}^{\infty}q(s)\varDelta s \biggr)^{1/\gamma}, \quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}, \end{aligned}

which implies that

\begin{aligned} \biggl(\frac{z^{\varDelta }(t)}{z^{\sigma}(t)} \biggr)^{1-\gamma}\geq \alpha(t), \quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}. \end{aligned}

Lemma 2.4 is proved. □

### Lemma 2.5

For $$\gamma\geq1$$, assume that $$x(t)$$ is an eventually positive solution of (1), and $$z^{\varDelta }(t)$$ is eventually negative. If it satisfies

$$\int_{t_{0}}^{\infty}q(t)\varDelta t=\infty,$$
(9)

then $$\lim_{t\rightarrow\infty}x(t)=0$$.

### Proof

Suppose that $$x(t)$$ is an eventually positive solution of (1) and $$z^{\varDelta }(t)$$ is eventually negative. By the proof of Lemma 2.3, we see that $$z(t)$$ is eventually positive. Then there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$x(t)>0$$, $$x(g(t))>0$$, $$x(h(t))>0$$, $$z(t)>0$$, and $$z^{\varDelta }(t)<0$$ for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. By Lemma 2.2 we have

\begin{aligned} \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta }>0,\quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}. \end{aligned}

By $$z^{\varDelta }(t)<0$$, we claim that there exists $$b\geq0$$ such that $$\lim_{t\rightarrow\infty}z(t)=b$$. Assume not; then there exists $$t_{2}\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$ such that $$z(t)<0$$ for $$t\in \left .[t_{2},\infty) \right ._{\mathbb{T}}$$. It will cause a similar contradiction as in the proof of Lemma 2.2. Then assuming $$b>0$$, by (8) and $$z(\sigma(t)), z(g(t))>b$$, we obtain

\begin{aligned} \bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\varDelta }\leq -q(t)z^{\gamma} \bigl( \sigma(t) \bigr)< -b^{\gamma}q(t). \end{aligned}
(10)

Letting $$v(t)=r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma _{2}} )^{\varDelta } )^{\gamma_{1}}$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, we have $$v(t)>0$$, and

\begin{aligned} v^{\varDelta }(t)< -b^{\gamma}q(t),\quad t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}. \end{aligned}
(11)

Substituting s for t, and integrating (11) from $$t_{1}$$ to $$t\in \left .[\sigma(t_{1}),\infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} v(t)< v(t_{1})-b^{\gamma}\int_{t_{1}}^{t}q(s) \varDelta s. \end{aligned}

By (9), there exists a sufficiently large $$t_{2}\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$ such that $$v(t)<0$$, $$t\in \left .[t_{2},\infty) \right ._{\mathbb{T}}$$, which contradicts $$v(t)>0$$. So $$b=0$$, and Lemma 2.5 is proved. □

### Lemma 2.6

Assume that $$x(t)$$ is an eventually positive solution of (1), and there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$x(t)>0$$, $$x(g(t))>0$$, $$x(h(t))>0$$, $$z(t)>0$$, and $$z^{\varDelta }(t)>0$$ for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. For $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, define

$$u(t)=A(t)\frac{r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}} )^{\varDelta } )^{\gamma_{1}}}{z^{\gamma}(t)}+B(t),$$
(12)

where $$A \in C^{1}_{\mathrm{rd}}(\mathbb{T}, (0, \infty))$$, $$B \in C^{1}_{\mathrm{rd}}(\mathbb{T},\mathbb{R})$$. Then $$u(t)$$ satisfies

\begin{aligned} u^{\varDelta }(t)+A(t)q(t)-B^{\varDelta }(t)- \varPhi _{0}(t)\leq0, \end{aligned}
(13)

where

\begin{aligned} \varPhi _{0}(t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} A^{\varDelta }(t) (\frac{u(t)-B(t)}{A(t)} )^{\sigma}-\gamma A(t)\alpha(t) (\frac{\delta(t)}{r_{2}(t)} )^{\gamma_{1}} [ (\frac{u(t)-B(t)}{A(t)} )^{\sigma} ]^{2},& 0< \gamma< 1,\\ A^{\varDelta }(t) (\frac{u(t)-B(t)}{A(t)} )^{\sigma}-\gamma A(t) (\frac{\delta(t)}{r_{2}(t)} )^{1/\gamma_{2}} [ (\frac{u(t)-B(t)}{A(t)} )^{\sigma} ]^{(1+\gamma)/\gamma},& \gamma\geq1. \end{array}\displaystyle \right . \end{aligned}

### Proof

Since $$x(t)$$ is an eventually positive solution of (1), and there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$x(t)>0$$, $$x(g(t))>0$$, $$x(h(t))>0$$, $$z(t)>0$$, and $$z^{\varDelta }(t)>0$$ for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, Lemmas 2.2 and 2.4 hold. Let $$u(t)$$ be defined by (12). Then, differentiating (12) and using (1), it follows that

\begin{aligned} u^{\varDelta }(t) =&\frac{A(t)}{z^{\gamma}(t)} \bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\varDelta } \\ &{} + \biggl(\frac{A(t)}{z^{\gamma}(t)} \biggr)^{\varDelta } \bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma _{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\sigma}+B^{\varDelta }(t) \\ =&-\frac{A(t)}{z^{\gamma}(t)}\cdot f \bigl(t,x \bigl(h(t) \bigr) \bigr)+B^{\varDelta }(t) \\ &{} +\frac{A^{\varDelta }(t)z^{\gamma}(t)-A(t)(z^{\gamma}(t))^{\varDelta }}{z^{\gamma}(t)z^{\gamma}(\sigma(t))} \bigl(r_{1}(t) \bigl( \bigl(r_{2}(t) \bigl(z^{\varDelta }(t) \bigr)^{\gamma_{2}} \bigr)^{\varDelta } \bigr)^{\gamma_{1}} \bigr)^{\sigma}. \end{aligned}

Using the fact that

\begin{aligned} f \bigl(t,x \bigl(h(t) \bigr) \bigr)\geq q(t)x^{\gamma} \bigl(h(t) \bigr) \geq q(t)z^{\gamma} \bigl(h(t) \bigr)\geq q(t)z^{\gamma}(t), \end{aligned}

we obtain

\begin{aligned} u^{\varDelta }(t) \leq& -A(t)q(t)+B^{\varDelta }(t)+A^{\varDelta }(t) \biggl(\frac{u(t)-B(t)}{A(t)} \biggr)^{\sigma} \\ &{} -A(t)\frac{(z^{\gamma}(t))^{\varDelta }}{z^{\gamma}(t)} \biggl(\frac {u(t)-B(t)}{A(t)} \biggr)^{\sigma}. \end{aligned}
(14)

When $$0<\gamma<1$$, using the Pötzsche chain rule (see [5]), we have

\begin{aligned} \bigl(z^{\gamma}(t) \bigr)^{\varDelta }=\gamma \int_{0}^{1} \bigl(z(t)+h\mu(t)z^{\varDelta }(t) \bigr)^{\gamma-1}\,dh \cdot z^{\varDelta }(t)\geq\gamma \bigl(z^{\sigma}(t) \bigr)^{\gamma-1}z^{\varDelta }(t), \end{aligned}

and it follows that

\begin{aligned} \frac{(z^{\gamma}(t))^{\varDelta }}{z^{\gamma}(t)}\geq\frac{\gamma (z^{\sigma}(t))^{\gamma-1}z^{\varDelta }(t)}{z^{\gamma}(t)}=\gamma \frac{z^{\varDelta }(t)}{z^{\sigma}(t)} \biggl( \frac{z^{\sigma }(t)}{z(t)} \biggr)^{\gamma}. \end{aligned}

By Lemmas 2.2 and 2.4, for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} \frac{z^{\varDelta }(t)}{z^{\sigma}(t)} =&\frac{1}{r_{2}^{\gamma _{1}}(t)}\frac{r_{2}^{\gamma_{1}}(t) (z^{\varDelta }(t) )^{\gamma }}{(z^{\sigma}(t))^{\gamma}} \biggl( \frac{z^{\varDelta }(t)}{z^{\sigma }(t)} \biggr)^{1-\gamma} \\ \geq&\alpha(t) \biggl(\frac{\delta(t) }{r_{2}(t)} \biggr)^{\gamma_{1}} \frac{r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}} )^{\varDelta } )^{\gamma _{1}}}{(z^{\gamma}(t))^{\sigma}} \\ \geq&\alpha(t) \biggl(\frac{\delta(t) }{r_{2}(t)} \biggr)^{\gamma_{1}} \frac{ (r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}} )^{\varDelta } )^{\gamma_{1}} )^{\sigma}}{(z^{\gamma}(t))^{\sigma}} \\ =&\alpha(t) \biggl(\frac{\delta(t) }{r_{2}(t)} \biggr)^{\gamma_{1}} \biggl( \frac{u(t)-B(t)}{A(t)} \biggr)^{\sigma} \end{aligned}

and

\begin{aligned} \frac{z^{\sigma}(t)}{z(t)}\geq1. \end{aligned}

So (14) becomes

\begin{aligned} u^{\varDelta }(t) \leq& -A(t)q(t)+B^{\varDelta }(t)+A^{\varDelta }(t) \biggl(\frac{u(t)-B(t)}{A(t)} \biggr)^{\sigma} \\ &{} -\gamma A(t)\alpha(t) \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{\gamma_{1}} \biggl[ \biggl(\frac{u(t)-B(t)}{A(t)} \biggr)^{\sigma} \biggr]^{2}. \end{aligned}
(15)

When $$\gamma\geq1$$, we have

\begin{aligned} \bigl(z^{\gamma}(t) \bigr)^{\varDelta }=\gamma \int_{0}^{1} \bigl(z(t)+h\mu(t)z^{\varDelta }(t) \bigr)^{\gamma-1}\,dh \cdot z^{\varDelta }(t)\geq\gamma z^{\gamma-1}(t)z^{\varDelta }(t), \end{aligned}

and it follows that

\begin{aligned} \frac{(z^{\gamma}(t))^{\varDelta }}{z^{\gamma}(t)}\geq\frac{\gamma z^{\gamma-1}(t)z^{\varDelta }(t)}{z^{\gamma}(t)}= \frac{\gamma z^{\varDelta }(t)}{z(t)}. \end{aligned}

By Lemmas 2.2 and 2.4, for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} \biggl(\frac{z^{\varDelta }(t)}{z(t)} \biggr)^{\gamma} =&\frac {1}{r_{2}^{\gamma_{1}}(t)} \frac{r_{2}^{\gamma_{1}}(t) (z^{\varDelta }(t) )^{\gamma}}{z^{\gamma}(t)} \\ \geq& \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{\gamma_{1}}\frac{r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}} )^{\varDelta } )^{\gamma_{1}}}{z^{\gamma}(t)} \\ \geq& \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{\gamma_{1}}\frac{ (r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma _{2}} )^{\varDelta } )^{\gamma_{1}} )^{\sigma}}{(z^{\gamma }(t))^{\sigma}} \\ =& \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{\gamma_{1}} \biggl(\frac {u(t)-B(t)}{A(t)} \biggr)^{\sigma}, \end{aligned}

which implies that

\begin{aligned} \frac{z^{\varDelta }(t)}{z(t)}\geq \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{1/\gamma_{2}} \biggl[ \biggl(\frac {u(t)-B(t)}{A(t)} \biggr)^{\sigma} \biggr]^{1/\gamma}. \end{aligned}

So (14) becomes

\begin{aligned} u^{\varDelta }(t) \leq& -A(t)q(t)+B^{\varDelta }(t)+A^{\varDelta }(t) \biggl(\frac{u(t)-B(t)}{A(t)} \biggr)^{\sigma} \\ & {} -\gamma A(t) \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{1/\gamma_{2}} \biggl[ \biggl(\frac{u(t)-B(t)}{A(t)} \biggr)^{\sigma} \biggr]^{(1+\gamma)/\gamma}. \end{aligned}
(16)

By (15) and (16), (13) holds. Lemma 2.6 is proved. □

## Main results

In this section, we establish oscillation criteria of (1) by a generalized Riccati transformation. Firstly, we give some definitions as follows.

Let $$D_{0}=\{s\in\mathbb{T}: s\geq0\}$$ and $$D=\{(t,s)\in\mathbb{T}^{2}: t\geq s\geq0\}$$. For any function $$f(t,s)$$: $$\mathbb{T}^{2}\rightarrow\mathbb{R}$$, denote by $$f^{\varDelta }_{2}$$ the partial derivative of f with respect to s. Define

\begin{aligned}& (\mathscr{A}, \mathscr{B})= \bigl\{ (A, B): A(s) \in C^{1}_{\mathrm{rd}} \bigl(D_{0}, (0, \infty) \bigr), B(s) \in C^{1}_{\mathrm{rd}}(D_{0}, \mathbb{R}), s\in D_{0} \bigr\} ; \\& \mathscr{H}= \bigl\{ H(t,s)\in C^{1} \bigl(D, [0, \infty )\bigr): H(t,t)=0, H(t,s)>0, H^{\varDelta }_{2}(t,s)\leq0, t>s\geq0 \bigr\} . \end{aligned}

These function classes will be used in the sequel. Now, we give our first theorem.

### Theorem 3.1

Assume that there exist $$(A,B)\in(\mathscr{A}, \mathscr{B})$$ and $$H\in\mathscr{H}$$ such that, for any $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} \limsup_{t\rightarrow\infty}\frac{1}{H(t,t_{1})}\int ^{t}_{t_{1}} \bigl[H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr) -H_{2}^{\varDelta }(t,s)B^{\sigma}(s)- \varPhi _{1}(s) \bigr]\varDelta s=\infty, \end{aligned}
(17)

where

\begin{aligned} \varPhi _{1}(s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} (\frac{r_{2}(s)}{\delta(s)} )^{\gamma_{1}}\frac{ (H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s) )^{2}}{4\gamma H(t,s)A(s)\alpha(s)},& 0< \gamma< 1,\\ (\frac{r_{2}(s)}{\delta(s)} )^{\gamma_{1}}\frac{1}{ (H(t,s)A(s) )^{\gamma}} (\frac{H_{2}^{\varDelta }(t,s)A^{\sigma}(s) +H(t,s)A^{\varDelta }(s)}{1+\gamma} )^{1+\gamma},& \gamma\geq 1. \end{array}\displaystyle \right . \end{aligned}

Then (1) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)$$ exists.

### Proof

Assume that (1) is not oscillatory. Without loss of generality, we may suppose that $$x(t)$$ is an eventually positive solution of (1). Then by Lemma 2.3, we have $$z(t)$$ is eventually positive or $$\lim_{t\rightarrow\infty}x(t)=0$$.

If $$\lim_{t\rightarrow\infty}x(t)=0$$, the theorem is proved. While $$z(t)$$ is eventually positive, it follows that there exists $$T\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$z(t)>0$$ for $$t\in \left .[T,\infty) \right ._{\mathbb{T}}$$. By Lemma 2.2, there exists $$t_{1}\in \left .[T,\infty) \right ._{\mathbb{T}}$$ such that either $$z^{\varDelta }(t)>0$$ or $$z^{\varDelta }(t)<0$$ holds for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. Assume that $$z^{\varDelta }(t)>0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. Let $$u(t)$$ be defined by (12). Then by Lemma 2.6, (13) holds.

Multiplying (13), where t is replaced by s, by H, and integrating it with respect to s from $$t_{1}$$ to t with $$t\in \left .[\sigma(t_{1}), \infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} &\int^{t}_{t_{1}}H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr)\varDelta s \\ &\quad\leq-\int^{t}_{t_{1}}H(t,s)u^{\varDelta }(s) \varDelta s+\int^{t}_{t_{1}}H(t,s)\varPhi _{0}(s) \varDelta s. \end{aligned}

Noting that $$H(t,t)=0$$, by the integration by parts formula we have

\begin{aligned} & \int^{t}_{t_{1}}H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr)\varDelta s \\ &\quad\leq H(t,t_{1})u(t_{1})+\int^{t}_{t_{1}} \bigl(H_{2}^{\varDelta }(t,s)u^{\sigma }(s)+H(t,s) \varPhi _{0}(s) \bigr) \varDelta s \\ &\quad=H(t,t_{1})u(t_{1})+\int^{t}_{t_{1}}H_{2}^{\varDelta }(t,s)B^{\sigma}(s) \varDelta s \\ &\qquad{} +\int^{t}_{t_{1}} \biggl(H_{2}^{\varDelta }(t,s)A^{\sigma}(s) \biggl(\frac {u(s)-B(s)}{A(s)} \biggr)^{\sigma}+H(t,s)\varPhi _{0}(s) \biggr) \varDelta s. \end{aligned}
(18)

When $$0<\gamma<1$$, we have

\begin{aligned} & H_{2}^{\varDelta }(t,s)A^{\sigma}(s) \biggl( \frac {u(s)-B(s)}{A(s)} \biggr)^{\sigma}+H(t,s)\varPhi _{0}(s) \\ &\quad= \bigl(H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s) \bigr) \biggl(\frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma} \\ &\qquad{} -\gamma H(t,s)A(s)\alpha(s) \biggl(\frac{\delta(s)}{r_{2}(s)} \biggr)^{\gamma _{1}} \biggl[ \biggl(\frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma} \biggr]^{2} \\ &\quad= \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}}\frac{ (H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s) )^{2}}{4\gamma H(t,s)A(s)\alpha(s)} \\ &\qquad{} -\gamma H(t,s)A(s)\alpha(s) \biggl(\frac{\delta(s)}{r_{2}(s)} \biggr)^{\gamma _{1}} \biggl[ \biggl(\frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma} \\ &\qquad{} - \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}} \frac{H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s)}{2\gamma H(t,s)A(s)\alpha(s)} \biggr]^{2} \\ &\quad\leq \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}}\frac{ (H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s) )^{2}}{4\gamma H(t,s)A(s)\alpha(s)}. \end{aligned}

When $$\gamma\geq1$$, we have

\begin{aligned} & H_{2}^{\varDelta }(t,s)A^{\sigma}(s) \biggl( \frac {u(s)-B(s)}{A(s)} \biggr)^{\sigma}+H(t,s)\varPhi _{0}(s) \\ &\quad= \bigl(H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s) \bigr) \biggl(\frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma} \\ &\qquad{} -\gamma H(t,s)A(s) \biggl(\frac{\delta(s)}{r_{2}(s)} \biggr)^{1/\gamma_{2}} \biggl[ \biggl(\frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma} \biggr]^{(1+\gamma)/\gamma}. \end{aligned}

Using the inequality

\begin{aligned} \lambda ab^{\lambda-1}-a^{\lambda}\leq(\lambda-1)b^{\lambda}, \end{aligned}

let $$\lambda=\frac{1+\gamma}{\gamma}$$, and

\begin{aligned}& a^{\lambda}=a^{(1+\gamma)/\gamma}=\gamma H(t,s)A(s) \biggl(\frac{\delta(s)}{r_{2}(s)} \biggr)^{1/\gamma_{2}} \biggl[ \biggl(\frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma} \biggr]^{(1+\gamma)/\gamma}, \\& b^{\lambda-1}=b^{1/\gamma}=\frac{\gamma}{1+\gamma} \biggl(\frac {r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}/(1+\gamma)} \frac{H_{2}^{\varDelta }(t,s)A^{\sigma}(s)+H(t,s)A^{\varDelta }(s)}{(\gamma H(t,s)A(s))^{\gamma/(1+\gamma)}}, \end{aligned}

then we have

\begin{aligned} &H_{2}^{\varDelta }(t,s)A^{\sigma}(s) \biggl( \frac{u(s)-B(s)}{A(s)} \biggr)^{\sigma}+H(t,s)\varPhi _{0}(s) \\ &\quad\leq \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}}\frac {1}{(H(t,s)A(s))^{\gamma}} \biggl( \frac{H_{2}^{\varDelta }(t,s)A^{\sigma }(s)+H(t,s)A^{\varDelta }(s)}{1+\gamma} \biggr)^{1+\gamma}. \end{aligned}

Therefore, for all $$\gamma>0$$, by (18) we have

\begin{aligned} &\int^{t}_{t_{1}}H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr)\varDelta s \\ &\quad\leq H(t,t_{1})u(t_{1})+\int^{t}_{t_{1}}H_{2}^{\varDelta }(t,s)B^{\sigma}(s) \varDelta s+\int^{t}_{t_{1}}\varPhi _{1}(s)\varDelta s, \end{aligned}

which implies that

\begin{aligned} \int^{t}_{t_{1}} \bigl[H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr)-H_{2}^{\varDelta }(t,s)B^{\sigma}(s)- \varPhi _{1}(s) \bigr]\varDelta s \leq H(t,t_{1})u(t_{1}). \end{aligned}

Hence,

\begin{aligned} \frac{1}{H(t,t_{1})}\int^{t}_{t_{1}} \bigl[H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr) -H_{2}^{\varDelta }(t,s)B^{\sigma}(s)- \varPhi _{1}(s) \bigr]\varDelta s \leq u(t_{1})< \infty, \end{aligned}

which contradicts (17). So $$z^{\varDelta }(t)<0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, and it is clear that $$\lim_{t\rightarrow\infty}z(t)$$ exists. By Lemma 2.1 we see that $$\lim_{t\rightarrow\infty}x(t)$$ exists. The proof is completed. □

When $$\gamma\geq1$$, if (9) holds, we have the following corollary on the basis of Lemma 2.5 and Theorem 3.1.

### Corollary 3.2

When $$\gamma\geq1$$, assume that (9) holds and there exist $$(A,B)\in(\mathscr{A}, \mathscr{B})$$ and $$H\in\mathscr{H}$$ such that, for any $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} \limsup_{t\rightarrow\infty}\frac{1}{H(t,t_{1})}\int ^{t}_{t_{1}} \bigl[H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr) -H_{2}^{\varDelta }(t,s)B^{\sigma}(s)- \varPhi _{1}(s) \bigr]\varDelta s=\infty, \end{aligned}
(19)

where

\begin{aligned} \varPhi _{1}(s)= \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}} \frac {1}{ (H(t,s)A(s) )^{\gamma}} \biggl(\frac{H_{2}^{\varDelta }(t,s)A^{\sigma}(s) +H(t,s)A^{\varDelta }(s)}{1+\gamma} \biggr)^{1+\gamma}. \end{aligned}

Then (1) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)=0$$.

### Remark 3.3

In Corollary 3.2, letting $$(A,B)=(1,0)$$, we can simplify (19) as

\begin{aligned} \limsup_{t\rightarrow\infty}\frac{1}{H(t,t_{1})}\int^{t}_{t_{1}} \biggl[H(t,s)q(s) - \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}}\frac{1}{H^{\gamma }(t,s)} \biggl( \frac{H_{2}^{\varDelta }(t,s)}{1+\gamma} \biggr)^{1+\gamma} \biggr]\varDelta s=\infty. \end{aligned}

When $$B=0$$, (12) is simplified as

$$u(t)=A(t)\frac{r_{1}(t) ( (r_{2}(t) (z^{\varDelta }(t) )^{\gamma_{2}} )^{\varDelta } )^{\gamma_{1}}}{z^{\gamma}(t)},\quad t\in \left .[t_{1}, \infty) \right ._{\mathbb{T}}.$$
(20)

Now we have the following theorem.

### Theorem 3.4

Assume that there exists $$A\in C^{1}_{\mathrm{rd}}(D_{0},(0, \infty))$$ such that, for any $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} \limsup_{t\rightarrow\infty}\int^{t}_{t_{1}} \bigl[A(s)q(s)-\varPhi _{2}(s) \bigr]\varDelta s=\infty, \end{aligned}
(21)

where

\begin{aligned} \varPhi _{2}(s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} (\frac{r_{2}(s)}{\delta(s)} )^{\gamma_{1}}\frac{ (A^{\varDelta }(s) )^{2}}{4\gamma A(s)\alpha(s)},& 0< \gamma< 1,\\ (\frac{r_{2}(s)}{\delta(s)} )^{\gamma_{1}}\frac{1}{A^{\gamma }(s)} (\frac{A^{\varDelta }(s)}{1+\gamma} )^{1+\gamma},& \gamma\geq1. \end{array}\displaystyle \right . \end{aligned}

Then (1) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)$$ exists.

### Proof

Assume that (1) is not oscillatory. Without loss of generality, we may suppose that $$x(t)$$ is an eventually positive solution of (1). Similarly as in the proof of Theorem 3.1, we have $$z(t)$$ is eventually positive or $$\lim_{t\rightarrow\infty}x(t)=0$$.

If $$\lim_{t\rightarrow\infty}x(t)=0$$, the theorem is proved. If $$z(t)$$ is eventually positive, there exists $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$ such that $$z(t)>0$$, and either $$z^{\varDelta }(t)>0$$ or $$z^{\varDelta }(t)<0$$ holds for $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$ by Lemma 2.2. Assume that $$z^{\varDelta }(t)>0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$. Let $$u(t)$$ be defined by (20). Then by Lemma 2.6, we have

\begin{aligned} u^{\varDelta }(t)+A(t)q(t)-\varPhi _{0}(t)\leq0, \end{aligned}

where $$\varPhi _{0}(t)$$ is simplified as

\begin{aligned} \varPhi _{0}(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} A^{\varDelta }(t) (\frac{u(t)}{A(t)} )^{\sigma}-\gamma A(t)\alpha(t) (\frac{\delta(t)}{r_{2}(t)} )^{\gamma_{1}} [ (\frac{u(t)}{A(t)} )^{\sigma} ]^{2},& 0< \gamma< 1,\\ A^{\varDelta }(t) (\frac{u(t)}{A(t)} )^{\sigma}-\gamma A(t) (\frac{\delta(t)}{r_{2}(t)} )^{1/\gamma_{2}} [ (\frac{u(t)}{A(t)} )^{\sigma} ]^{(1+\gamma)/\gamma},& \gamma\geq1. \end{array}\displaystyle \right . \end{aligned}

When $$0<\gamma<1$$, we have

\begin{aligned} u^{\varDelta }(t) \leq& -A(t)q(t)+A^{\varDelta }(t) \biggl(\frac{u(t)}{A(t)} \biggr)^{\sigma}-\gamma A(t)\alpha(t) \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{\gamma_{1}} \biggl[ \biggl(\frac{u(t)}{A(t)} \biggr)^{\sigma} \biggr]^{2} \\ =&-A(t)q(t)+ \biggl(\frac{r_{2}(t)}{\delta(t)} \biggr)^{\gamma_{1}}\frac { (A^{\varDelta }(t) )^{2}}{4\gamma A(t)\alpha(t)} \\ &{} -\gamma A(t)\alpha(t) \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{\gamma_{1}} \biggl[ \biggl(\frac{u(t)}{A(t)} \biggr)^{\sigma}- \biggl( \frac{r_{2}(t)}{\delta (t)} \biggr)^{\gamma_{1}} \frac{A^{\varDelta }(t)}{2\gamma A(t)\alpha(t)} \biggr]^{2} \\ \leq& -A(t)q(t)+ \biggl(\frac{r_{2}(t)}{\delta(t)} \biggr)^{\gamma_{1}} \frac { (A^{\varDelta }(t) )^{2}}{4\gamma A(t)\alpha(t)}. \end{aligned}

When $$\gamma\geq1$$, we have

\begin{aligned} u^{\varDelta }(t)\leq -A(t)q(t)+A^{\varDelta }(t) \biggl(\frac{u(t)}{A(t)} \biggr)^{\sigma}-\gamma A(t) \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{1/\gamma_{2}} \biggl[ \biggl(\frac{u(t)}{A(t)} \biggr)^{\sigma} \biggr]^{(1+\gamma)/\gamma}. \end{aligned}

Using the inequality

\begin{aligned} \lambda ab^{\lambda-1}-a^{\lambda}\leq(\lambda-1)b^{\lambda}, \end{aligned}

let $$\lambda=\frac{1+\gamma}{\gamma}$$, and

\begin{aligned}& a^{\lambda}=a^{(1+\gamma)/\gamma}=\gamma A(t) \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{1/\gamma_{2}} \biggl[ \biggl(\frac{u(t)}{A(t)} \biggr)^{\sigma} \biggr]^{(1+\gamma)/\gamma}, \\& b^{\lambda-1}=b^{1/\gamma}=\frac{\gamma}{1+\gamma} \biggl(\frac {r_{2}(t)}{\delta(t)} \biggr)^{\gamma_{1}/(1+\gamma)}\frac{A^{\varDelta }(t)}{(\gamma A(t))^{\gamma/(1+\gamma)}}, \end{aligned}

then we have

\begin{aligned} u^{\varDelta }(t)\leq -A(t)q(t)+ \biggl(\frac{r_{2}(t)}{\delta(t)} \biggr)^{\gamma_{1}}\frac {1}{A^{\gamma}(t)} \biggl(\frac{A^{\varDelta }(t)}{1+\gamma} \biggr)^{1+\gamma}. \end{aligned}

Therefore, for all $$\gamma>0$$, we always have

\begin{aligned} u^{\varDelta }(t)\leq-A(t)q(t)+\varPhi _{2}(t), \end{aligned}

which implies that

\begin{aligned} A(t)q(t)-\varPhi _{2}(t)\leq-u^{\varDelta }(t). \end{aligned}
(22)

Letting t be replaced by s, and integrating (22) with respect to s from $$t_{1}$$ to $$t\in \left .[\sigma(t_{1}),\infty) \right ._{\mathbb{T}}$$, we obtain

\begin{aligned} \int^{t}_{t_{1}} \bigl[A(s)q(s)- \varPhi _{2}(s) \bigr]\varDelta s\leq -\int^{t}_{t_{1}}u^{\varDelta }(s) \varDelta s= u(t_{1})-u(t)< u(t_{1})< \infty, \end{aligned}

which is a contradiction of (21). So $$z^{\varDelta }(t)<0$$, $$t\in \left .[t_{1},\infty) \right ._{\mathbb{T}}$$, and as before, $$\lim_{t\rightarrow\infty}z(t)$$ and $$\lim_{t\rightarrow\infty}x(t)$$ exist. The proof is completed. □

When $$\gamma\geq1$$, if (9) holds, from Lemma 2.5 and Theorem 3.4 we have the following result.

### Corollary 3.5

When $$\gamma\geq1$$, assume that (9) holds and there exists $$A \in C^{1}_{\mathrm{rd}}(D_{0}, (0, \infty))$$ such that, for any $$t_{1}\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$$,

\begin{aligned} \limsup_{t\rightarrow\infty}\int^{t}_{t_{1}} \biggl[A(s)q(s) - \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}} \frac{1}{A^{\gamma }(s)} \biggl(\frac{A^{\varDelta }(s)}{1+\gamma} \biggr)^{1+\gamma} \biggr]\varDelta s=\infty. \end{aligned}
(23)

Then (1) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)=0$$.

### Remark 3.6

It is not difficult to satisfy the conditions in Corollary 3.5. Indeed, letting $$A=1$$, by (9) we have (23). The condition (23) can be deleted in Corollary 3.5. Therefore, when $$\gamma\geq1$$, assume that (9) holds, then it follows that (1) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)=0$$.

### Remark 3.7

Take $$r_{1}(t)=1/a_{2}(t)$$, $$r_{2}(t)=1/a_{1}(t)$$, $$\gamma_{1}=\alpha_{2}$$, $$\gamma_{2}=\alpha_{1}$$, $$\gamma=1$$, $$p(t)=0$$, $$h(t)=t$$, and $$f(t,x)=q(t)f_{0}(x)$$, where $$f_{0}$$ is equivalent to f in Yu and Wang [15]. It is obvious that the conclusions in this paper extend the ones in [15]. Meanwhile, the proofs and results above may provide some enlightenment to the study of oscillation of higher-order nonlinear dynamic equations with nonpositive neutral coefficients on time scales.

## Examples

In this section, the application of our oscillation criteria will be shown in two examples. Now we give the first example to demonstrate Theorem 3.1 (or Corollary 3.2).

### Example 4.1

Let $$\mathbb{T}=\bigcup_{n=1}^{\infty}[2n-1,2n]$$. Consider the equation

\begin{aligned} \biggl(t \biggl( \biggl(\frac{1}{t} \biggl( \biggl(x(t)- \frac {t-1}{2t}x(t-2) \biggr)^{\varDelta } \biggr)^{1/3} \biggr)^{\varDelta } \biggr)^{5} \biggr)^{\varDelta } +\frac{2+\sin t}{t}x^{5/3} \bigl(h(t) \bigr)=0, \end{aligned}
(24)

where $$r_{1}(t)=t$$, $$r_{2}(t)=1/t$$, $$p(t)=(t-1)/2t$$, $$g(t)=t-2$$, $$\gamma_{1}=5$$, $$\gamma_{2}=1/3$$, $$\gamma=5/3$$, $$h(t)\geq t$$, and $$t_{0}=1$$. By (C3) we have $$p_{0}=1/2$$, and by (C6) we take $$q(t)=1/t$$. Since

\begin{aligned} \int^{\infty}_{t_{0}}\frac{1}{r_{1}^{1/\gamma_{1}}(t)}\varDelta t=\int ^{\infty}_{1}\frac{1}{t^{1/5}}\varDelta t=\infty, \qquad \int^{\infty}_{t_{0}}\frac{1}{r_{2}^{1/\gamma_{2}}(t)}\varDelta t=\int ^{\infty}_{1}t^{3}\varDelta t=\infty \end{aligned}

and

\begin{aligned} \int^{\infty}_{t_{0}}q(t)\varDelta t=\int ^{\infty}_{1}\frac{1}{t}\varDelta t=\infty, \end{aligned}

it is obvious that the coefficients of (24) satisfy (C1)-(C6) and (9). Letting $$H(t,s)=(t-s)^{2}$$ and $$(A,B)=(s,0)$$, we have

\begin{aligned} \delta(t)=\int_{t_{1}}^{t}\frac{\varDelta s}{r_{1}^{1/\gamma_{1}}(s)} =\int _{t_{1}}^{t}\frac{\varDelta s}{s^{1/5}}=O \bigl(t^{4/5} \bigr) \end{aligned}

and

\begin{aligned} \varPhi _{1}(s) = & \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}} \frac {1}{ (H(t,s)A(s) )^{\gamma}} \biggl(\frac{H_{2}^{\varDelta }(t,s)A^{\sigma}(s) +H(t,s)A^{\varDelta }(s)}{1+\gamma} \biggr)^{1+\gamma} \\ =& \biggl(\frac{s^{-1}}{O(s^{4/5})} \biggr)^{5}\frac{1}{ ((t-s)^{2}s )^{5/3}} \biggl( \frac{O(s)\cdot O(s)+(t-s)^{2}}{8/3} \biggr)^{8/3}=O \bigl(s^{-26/3} \bigr). \end{aligned}

Hence,

\begin{aligned} &\limsup_{t\rightarrow\infty}\frac{1}{H(t,t_{1})}\int^{t}_{t_{1}} \bigl[H(t,s) \bigl(A(s)q(s)-B^{\varDelta }(s) \bigr)-H_{2}^{\varDelta }(t,s)B^{\sigma }(s)- \varPhi _{1}(s) \bigr]\varDelta s \\ &\quad=\limsup_{t\rightarrow\infty}\frac{1}{(t-t_{1})^{2}}\int ^{t}_{t_{1}} \bigl[(t-s)^{2}-O \bigl(s^{-26/3} \bigr) \bigr]\varDelta s=\infty. \end{aligned}

That is, (19) holds. By Theorem 3.1 (or Corollary 3.2) we see that (24) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)=0$$.

The second example illustrates Theorem 3.4.

### Example 4.2

Let $$\mathbb{T}=\bigcup_{n=0}^{\infty}[3^{n},2\cdot3^{n}]$$. Consider the equation

\begin{aligned} \biggl(\frac{1}{t^{2}} \biggl( \biggl(\sqrt{t} \biggl( \biggl(x(t)-\frac {1}{t}x \biggl(\frac{t}{3} \biggr) \biggr)^{\varDelta } \biggr)^{5/3} \biggr)^{\varDelta } \biggr)^{1/5} \biggr)^{\varDelta } +\frac{2+t^{2}}{t^{2} (1+t^{2} )}x^{1/3} \bigl(h(t) \bigr)=0, \end{aligned}
(25)

where $$r_{1}(t)=1/t^{2}$$, $$r_{2}(t)=\sqrt{t}$$, $$p(t)=1/t$$, $$g(t)=t/3$$, $$\gamma_{1}=1/5$$, $$\gamma_{2}=5/3$$, $$\gamma=1/3$$, $$h(t)\geq\sigma(t)$$, and $$t_{0}=1$$. By (C3) we have $$p_{0}=0$$, and by (C6) we take $$q(t)=1/t^{2}$$. Since

\begin{aligned} \int^{\infty}_{t_{0}}\frac{1}{r_{1}^{1/\gamma_{1}}(t)}\varDelta t=\int ^{\infty}_{1}t^{10}\varDelta t=\infty,\qquad \int ^{\infty}_{t_{0}}\frac{1}{r_{2}^{1/\gamma_{2}}(t)}\varDelta t=\int ^{\infty}_{1}\frac{1}{t^{3/10}}\varDelta t=\infty \end{aligned}

and

\begin{aligned} \int^{\infty}_{t_{0}}q(t)\varDelta t=\int ^{\infty}_{1}\frac{1}{t^{2}}\varDelta t< \infty, \end{aligned}

it is obvious that the coefficients of (25) satisfy (C1)-(C7). Then, letting $$(A,B)=(s^{2},0)$$, we obtain

\begin{aligned}& \delta(t)=\int_{t_{1}}^{t}\frac{\varDelta s}{r_{1}^{1/\gamma_{1}}(s)} =\int _{t_{1}}^{t}s^{10}\varDelta s=O \bigl(t^{11} \bigr), \\& \begin{aligned}[b] \alpha(t)&= \biggl(\frac{\delta(t)}{r_{2}(t)} \biggr)^{(1-\gamma)/\gamma _{2}} \biggl(2\int_{t}^{\infty}q(s)\varDelta s \biggr)^{(1-\gamma)/\gamma} \\ &= \biggl(\frac{O(t^{11})}{\sqrt{t}} \biggr)^{2/5} \bigl(O \bigl(t^{-1} \bigr) \bigr)^{2}=O \bigl(t^{11/5} \bigr), \end{aligned} \end{aligned}

and

\begin{aligned} \varPhi _{2}(s) =& \biggl(\frac{r_{2}(s)}{\delta(s)} \biggr)^{\gamma_{1}} \frac { (A^{\varDelta }(s) )^{2}}{4\gamma A(s)\alpha(s)} \\ =& \biggl(\frac{\sqrt{s}}{O(s^{11})} \biggr)^{1/5}\frac{O(s^{2})}{4/3\cdot s^{2}\cdot O(s^{11/5})}=O \bigl(s^{-43/10} \bigr). \end{aligned}

Therefore,

\begin{aligned} \limsup_{t\rightarrow\infty}\int^{t}_{t_{1}} \bigl[A(s)q(s)-\varPhi _{2}(s) \bigr]\varDelta s=\limsup_{t\rightarrow\infty} \int^{t}_{t_{1}} \bigl[1-O \bigl(s^{-43/10} \bigr) \bigr]\varDelta s=\infty. \end{aligned}

That is, (21) holds. By Theorem 3.4 we see that (25) is oscillatory or $$\lim_{t\rightarrow\infty}x(t)$$ exists.

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## Acknowledgements

This project was supported by the NNSF of China (no. 11271379).

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Correspondence to Yang-Cong Qiu.

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Qiu, YC. Oscillation criteria of third-order nonlinear dynamic equations with nonpositive neutral coefficients on time scales. Adv Differ Equ 2015, 299 (2015). https://doi.org/10.1186/s13662-015-0636-y