Open Access

Nonoscillation for higher-order nonlinear delay dynamic equations on time scales

Advances in Difference Equations20162016:58

https://doi.org/10.1186/s13662-016-0786-6

Received: 26 November 2015

Accepted: 21 February 2016

Published: 27 February 2016

Abstract

In this paper, we investigate the nonoscillation of the higher-order nonlinear delay dynamic equation
$$\begin{aligned} &\bigl(a_{n-1}(t) \bigl(a_{n-2}(t) \bigl(\cdots \bigl(a_{1}(t)x^{\Delta}(t)\bigr)^{\Delta}\cdots \bigr)^{\Delta}\bigr)^{\Delta}\bigr)^{\Delta} +u(t)g\bigl(x\bigl( \delta(t)\bigr)\bigr)=R(t) \\ &\quad\mbox{for } t\in [t_{0}, \infty)_{\mathbb{T}}, \end{aligned}$$
where \(\mathbb{T}\) is a scale with \(\sup\mathbb{T}=\infty\), \(t_{0}\in\mathbb{T}\), and \([t_{0},\infty)_{\mathbb{T}}= \{t\in\mathbb{T}:t\geq t_{0}\}\). We obtain some sufficient conditions for all solutions of this equation to be nonoscillatory.

Keywords

nonoscillation dynamic equation time scale

MSC

34K11 39A10 39A99

1 Introduction

A time scale \(\mathbb{ T}\) is an arbitrary nonempty closed subset of the real numbers. Thus, the set \(\mathbb{R}\) of all real numbers, the set \(\mathbb{N}\) of all natural numbers, and the set \(\mathbb{Z}\) of all integers are examples of time scales. On a time scale \(\mathbb{ T}\), the forward jump operator, the backward jump operator, and the graininess function are defined as
$$\sigma(t)=\inf\{s\in\mathbb{ T}:s>t\}, \qquad \rho(t)=\sup\{s\in \mathbb{ T}:s< t \}, \quad\mbox{and}\quad \mu(t)=\sigma(t)-t, $$
respectively.
In this paper, we investigate the nonoscillation of the higher-order nonlinear delay dynamic equation
$$\begin{aligned} &\bigl(a_{n-1}(t) \bigl(a_{n-2}(t) \bigl(\cdots \bigl(a_{1}(t)x^{\Delta}(t)\bigr)^{\Delta}\cdots \bigr)^{\Delta}\bigr)^{\Delta}\bigr)^{\Delta} +u(t)g\bigl(x\bigl( \delta(t)\bigr)\bigr)=R(t) \\ & \quad\mbox{for } t\in [t_{0}, \infty)_{\mathbb{T}}, \end{aligned}$$
(1.1)
where \(t_{0}\in{\mathbb { T}}\), the time scale interval \([t_{0},\infty)_{\mathbb{ T}}\equiv\{t\in\mathbb{ T}:t\geq t_{0}\}\), \(a_{i}\in C_{rd}([t_{0},\infty)_{\mathbb{ T}}, (0,\infty)) \) (\(1\leq i\leq n-1\)), \(u,R\in C_{rd}([t_{0},\infty)_{\mathbb{ T}}, {\mathbb{ R}})\), \(\delta\in C_{rd}([t_{0},\infty)_{\mathbb{T}},\mathbb{T})\) is surjective with \(\delta(t)\leq t\) and \(\delta(t)\rightarrow\infty \) as \(t\rightarrow\infty\), and \(g\in C([t_{0},\infty)_{\mathbb{ T}}\times{\mathbb{R}}, {\mathbb{ R}})\). Our goal is to obtain sufficient conditions for all solutions of (1.1) to be nonoscillatory.
We define
$$ R_{i}\bigl(t,x(t)\bigr)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} x(t) & \mbox{if } i=0,\\ a_{i}(t)R^{\triangle}_{i-1}(t,x(t)) & \mbox{if } 1\leq i\leq n-1. \end{array}\displaystyle \right . $$
(1.2)
Then (1.1) reduces to the equation
$$ R_{n-1}^{\triangle}\bigl(t,x(t)\bigr)+u(t)g\bigl(x\bigl(\delta(t) \bigr)\bigr)=R(t). $$
(1.3)
We can suppose the \(\sup{\mathbb{ T}}=\infty\) since we are interested in the oscillatory behavior of solutions near infinity. By a solution of (1.1) we mean a nontrivial real-valued function \(x\in C_{rd}([T_{x},\infty)_{\mathbb{ T}},{\mathbb{ R}})\), \(T_{x}\geq t_{0}\), such that \(R_{n-1}(t,x(t))\in C^{1}_{rd}([T_{x},\infty)_{\mathbb{ T}},{\mathbb{ R}})\) and satisfies (1.1) on \([T_{x},\infty)\). Since we are working on a time scale, the notion of oscillation takes the form of what is known as a generalized zero of a function. We say that \(x(t)\) has a generalized zero at a point T if \(x(T)x(\sigma(T))\leq0\). A function is said to be oscillatory if it has arbitrarily large generalized zeros and nonoscillatory otherwise.

In order to create a theory that can unify discrete and continuous analysis, the theory of time scale was initiated by Hilger’s landmark paper [1], which has received a lot of attention. There exist a variety of interesting time scales, and they give rise to many applications (see [2]). We refer the reader to [3, 4] for further results on time-scale calculus. In the thousands of papers in the literature, finding sufficient conditions for all solutions of an equation to be oscillatory have been a major focus of study (see [528]), but finding necessary and sufficient conditions for the existence of a nonoscillatory bounded solution of an equation are more rare (see [29]).

Zhu and Wang [21] studied the existence of nonoscillatory solutions to neutral dynamic equation
$$\bigl[x(t)+p(t)x\bigl(g (t)\bigr)\bigr]^{\Delta} +f\bigl(t,x\bigl(h(t) \bigr)\bigr)=0. $$
Karpuz and Öcalan [22] studied the asymptotic behavior of delay dynamic equations of the form
$$\bigl[x(t)+A(t)x\bigl(\alpha(t)\bigr)\bigr]^{\triangle} +B(t)F\bigl(x\bigl( \beta(t)\bigr)\bigr)-C(t)G\bigl(x\bigl(\gamma (t)\bigr)\bigr)=\varphi(t). $$
Wu et al. [25] investigated the oscillation of the higher-order dynamic equation
$$\bigl\{ r_{n}(t)\bigl[\bigl(r_{n-1}(t) \bigl(\cdots \bigl(r_{1}(t)x(t)^{\Delta}\bigr)^{\Delta}\cdots \bigr)^{\Delta}\bigr)^{\Delta}\bigr]^{\gamma}\bigr\} ^{\Delta} +F\bigl(t,x\bigl(\tau(t)\bigr)\bigr)=0. $$
Sun et al. [26] obtained some necessary and sufficient conditions for the existence of nonoscillatory solution for the higher-order equation
$$\bigl\{ a(t) \bigl[\bigl(x(t)-p(t)x\bigl(\tau(t)\bigr)\bigr)^{\Delta^{m}} \bigr]^{\alpha} \bigr\} ^{\Delta}+f\bigl(t,x\bigl(\delta(t)\bigr) \bigr)=0. $$

2 Auxiliary results

We state the following conditions, which are needed in the sequel.
(H1): 

There exist constants \(\alpha,\beta\geq0\) and \(\gamma\geq0\) such that \(|g(u)|\leq \alpha|u|^{\gamma}+\beta\).

(H2): 

\(\int_{t_{0}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})}\int_{t_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots\int_{t_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{t_{0}}^{s_{n-1}}|R(s_{n})|\Delta s_{n}<\infty\).

(H3): 

\(\int_{t_{0}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})}\int_{t_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots\int_{t_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{t_{0}}^{s_{n-1}}|u(s_{n})|\Delta s_{n}<\infty\).

We shall employ the following lemma.

Lemma 2.1

Let \(\mathbb{R}_{+}\equiv[0,\infty)\) and \(H=\{(t,s_{1},s_{2},\ldots,s_{n-1}):0\leq s_{n-1}\leq s_{n-2}\leq\cdots\leq s_{1}\leq t<\infty\}\). Suppose that \({r}\in C_{rd}([t_{0},\infty)_{\mathbb{T}},\mathbb{R}_{+})\), \(h\in C_{rd}(H,\mathbb{R}_{+})\), and that \(p\in C(\mathbb{R}_{+},\mathbb{R}_{+})\) is nondecreasing with \(p(r)>0\) for \(r>0\). If there exists a constant \(c>0 \) such that
$$ {r}(t)\leq c+ \int_{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})p\bigl({r}(s_{n})\bigr)\Delta s_{n}, $$
(2.1)
then
$${r}(t)\leq P^{-1} \biggl(P(c)+ \int_{t_{0}}^{t}\Delta s_{1} \int _{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n} \biggr), $$
where
$$P(w)= \int_{w_{0}}^{w}\frac{ds}{p(s)},\quad w_{0},w>0, $$
\(P^{-1}\) is the inverse function of P, and
$$ P(c)+ \int_{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}\in \operatorname{Dom}\bigl(P^{-1} \bigr). $$
(2.2)

Proof

Let \(z(t)\) denote the right side of inequality (2.1). Then \(z(t_{0})=c\), \({r}(t)\leq z(t)\), and
$$\begin{aligned} z^{\Delta} (t) =& \int_{t_{0}}^{t}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}}h(t,s_{2}, \ldots,s_{n})p\bigl({r}(s_{n})\bigr)\Delta s_{n} \\ \leq&p\bigl(z(t)\bigr) \int_{t_{0}}^{t}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}}h(t,s_{2}, \ldots,s_{n})\Delta s_{n}. \end{aligned}$$
Since \(z^{\Delta} (t)\geq0\) and p is nondecreasing, we obtain
$$ \frac{z^{\Delta} (t)}{p(z(t))}\leq \int_{t_{0}}^{t}\Delta s_{2} \int _{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}}h(t,s_{2},\ldots ,s_{n})\Delta s_{n}. $$
(2.3)
Noting that
$$P^{\Delta}\bigl(z(t)\bigr)=z^{\Delta}(t) \int_{0}^{1}\frac{dh}{p[hz(\sigma(t)) +(1-h)z(t)]}\leq\frac{z^{\Delta} (t)}{p(z(t))}, $$
we have
$$P\bigl(z(t)\bigr)\leq P\bigl(z(t_{0})\bigr)+ \int_{t_{0}}^{t}\Delta s_{1} \int _{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}. $$
Since \(P(w) \) is increasing, we have
$$z(t)\leq P^{-1} \biggl(P(c)+ \int_{t_{0}}^{t}\Delta s_{1} \int _{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n} \biggr). $$
The proof is complete. □
Notice that taking \(p(\nu)=\nu^{\xi}\) and \(\xi>1 \) in Lemma 2.1, we have
$$P\bigl(z(t)\bigr)-P\bigl(z(t_{0})\bigr)=\frac{1}{1-\xi} \bigl[z^{1-\xi}(t)-z^{1-\xi}(t_{0})\bigr]. $$
So
$$\frac{1}{1-\xi}z^{1-\xi}(t)\leq\frac{1}{1-\xi}z^{1-\xi}(t_{0})+ \int _{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}, $$
that is,
$$z^{1-\xi}(t)\geq z^{1-\xi}(t_{0})+(1-\xi) \int_{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}. $$
We have
$${r}(t)\leq \biggl[c^{1-\xi}-(\xi-1) \int_{t_{0}}^{t}\Delta s_{1} \int _{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n} \biggr]^{\frac{-1}{\xi-1}}, $$
provided that
$$ \int_{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}< \frac{c^{1-\xi}}{\xi-1}. $$
(2.4)

3 Main results

Now, we state and prove our main results.

Theorem 3.1

Assume that conditions (H1)-(H3) hold and for some \(k\geq0\),
$$ \int_{t_{0}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int _{t_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int _{t_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{t_{0}}^{s_{n-1}}\bigl|u(s_{n})\bigr| \delta^{k\gamma}(s_{n})\Delta s_{n}< \infty. $$
(3.1)
If \(x(t)\) is an oscillatory solution of (1.1) such that
$$ \bigl|x(t)\bigr|=O\bigl(t^{k}\bigr),\quad t\rightarrow\infty, $$
(3.2)
then \(x(t)\rightarrow0\) as \(t\rightarrow\infty\).

Proof

We will show \(\limsup_{t\longrightarrow\infty}x(t)=0 \) and \(\liminf_{t\longrightarrow\infty}x(t)=0\). Suppose that \(\limsup_{t\longrightarrow\infty}x(t) =L>0\). Then for any \(t_{1}\geq t_{0}\), there exists \(t_{2}\geq t_{1}\) such that \(x(t_{2})>\frac{L}{2}\). In view of conditions (H2), (H3), (3.1), and (3.2), there exist \(T_{0}\geq t_{0}\) and \(K>0\) such that \(|x(t)|\leq Kt^{k}\) (\(t\geq T_{0}\)) and
$$\begin{aligned} &\int_{T_{0}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{T_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{T_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \\ &\quad{}\times \int_{T_{0}}^{s_{n-1}}\bigl\{ \bigl|R(s_{n})\bigr|+\bigl|u(s_{n})\bigr| \bigl[\alpha K^{\gamma}\delta^{k\gamma}(s_{n})+\beta\bigr] \bigr\} \Delta s_{n}< \frac{L}{4}. \end{aligned}$$
(3.3)
Since \(x(t)\) is an oscillatory solution of (1.1), every \(R_{i}{(t,x(t))}\) is oscillatory for \(i=1,2,\ldots,n-1\). Choose \(T_{0}< T_{1}\leq T_{2}\leq\cdots\leq T_{n-1}\) such that
$$ R_{n-i}\bigl(T_{i},x(T_{i})\bigr)R_{n-i}\bigl( \sigma(T_{i}),x\bigl(\sigma(T_{i})\bigr)\bigr)\leq0,\quad i=1,2, \ldots,n-1, $$
(3.4)
and
$$ R_{n-i}\bigl(T_{i},x(T_{i})\bigr)\leq0,\quad i=1,2,\ldots,n-1. $$
(3.5)
Integrating (1.1) from \(T_{i}\) to t, \(i=1,2,\ldots,n-1\), successively \(n-1\) times with \(t>T_{n-1}\), we obtain
$$\begin{aligned} a_{1}x^{\Delta}(t) =&a_{1}(T_{n-1})x^{\Delta}(T_{n-1})+ \int _{T_{n-1}}^{t}\frac{R_{2}(T_{n-2},x(T_{n-2}))}{a_{2}(s_{n-2})}\Delta s_{n-2} \\ &{}+ \int_{T_{n-1}}^{t}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})} \int _{T_{n-2}}^{s_{n-2}}\frac {R_{3}(T_{n-3},x(T_{n-3}))}{a_{3}(s_{n-3})}\Delta s_{n-3} \\ &{}+\cdots \\ &{}+ \int_{T_{n-1}}^{t}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})} \int_{T_{n-2}}^{s_{n-2}}\frac{\Delta s_{n-3}}{a_{3}(s_{n-3})}\cdots \int_{T_{2}}^{s_{2}} \frac{R_{n-1}(T_{1},x(T_{1}))}{a_{n-1}(s_{1})}\Delta s_{1} \\ &{}+ \int_{T_{n-1}}^{t}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})} \int _{T_{n-2}}^{s_{n-2}}\frac{\Delta s_{n-3}}{a_{3}(s_{n-3})}\cdots \int _{T_{1}}^{s_{1}}\bigl[R(s)-u(s)g\bigl(x\bigl(\delta(s) \bigr)\bigr)\bigr]\Delta s \\ \leq& \int_{T_{n-1}}^{t}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})} \int_{T_{n-2}}^{s_{n-2}}\frac{\Delta s_{n-3}}{a_{3}(s_{n-3})}\cdots \int_{T_{1}}^{s_{1}}\bigl[R(s)-u(s)g\bigl(x\bigl(\delta(s) \bigr)\bigr)\bigr]\Delta s. \end{aligned}$$
(3.6)
Choose \(T_{n}>T_{n-1}\) so that
$$x(T_{n})x\bigl(\sigma(T_{n})\bigr)\leq0 \quad\mbox{and} \quad x(T_{n})\leq0. $$
Take \(T_{n+1}\geq T_{n}\) such that
$$x(T_{n+1})\geq\frac{L}{2} \quad\mbox{and}\quad x(t)>0 ,\quad t \in(T_{n},T_{n+1}). $$
Note that such \(T_{n+1}\) exists since \(\limsup_{t\longrightarrow\infty}x(t)>\frac{L}{2}\). Dividing (3.6) by \(a_{1}(t)\) and integrating once more from \(T_{n}\) to \(T_{n+1}\), we have
$$\begin{aligned} \frac{L}{2} \leq& x(T_{n+1})\leq \int_{T_{n}}^{T_{n+1}}\frac{\Delta s_{n-1}}{a_{1}(s_{n-1})} \int_{T_{n-1}}^{s_{n-1}}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})} \int_{T_{n-2}}^{s_{n-2}}\frac{\Delta s_{n-3}}{a_{3}(s_{n-3})} \\ &{}\cdots \int_{T_{1}}^{s_{1}}\bigl[R(s)-u(s)g\bigl(x\bigl(\delta(s) \bigr)\bigr)\bigr] \Delta s. \end{aligned}$$
(3.7)
It follows from (H1) that
$$\begin{aligned} \frac{L}{2} \leq& \int_{T_{n}}^{T_{n+1}}\frac{\Delta s_{n-1}}{a_{1}(s_{n-1})} \int_{T_{n-1}}^{s_{n-1}}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})}\cdots \int _{T_{1}}^{s_{1}}\bigl[\bigl|R(s)\bigr|+\bigl|u(s)\bigr|\bigl|g\bigl(x\bigl( \delta(s)\bigr)\bigr)\bigr|\bigr]\Delta s \\ \leq& \int_{T_{n}}^{T_{n+1}}\frac{\Delta s_{n-1}}{a_{1}(s_{n-1})} \int _{T_{n-1}}^{s_{n-1}}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})}\\ &{}\cdots \int _{T_{1}}^{s_{1}}\bigl\{ \bigl|R(s)\bigr|+\bigl|u(s)\bigr|\bigl[\alpha\bigl|x \bigl(\delta(s)\bigr)\bigr|^{\gamma}+\beta\bigr]\bigr\} \Delta s \\ \leq& \int_{T_{n}}^{T_{n+1}}\frac{\Delta s_{n-1}}{a_{1}(s_{n-1})} \int_{T_{n-1}}^{s_{n-1}}\frac{\Delta s_{n-2}}{a_{2}(s_{n-2})}\cdots \int_{T_{1}}^{s_{1}}\bigl\{ \bigl|R(s)\bigr|+\bigl|u(s)\bigr|\bigl[\alpha K^{\gamma}\delta^{k\gamma}(s)+\beta\bigr]\bigr\} \Delta s . \end{aligned}$$
In view of (3.3), we have a contradiction.

In a similar fashion, we can show that \(\liminf_{t\longrightarrow\infty }x(t)=0\). The proof is complete. □

Theorem 3.2

Assume that conditions (H1)-(H3) hold with \(\gamma\geq1\). Then every oscillatory solution of (1.1) is bounded.

Proof

Let \(x(t)\) be an oscillatory solution of (1.1), and \(d>0\) be a constant.

If \(\gamma>1\), then it follows from conditions (H2) and (H3) that there exists \(T^{*}\geq t_{0}\) such that
$$ \int_{T^{*}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int _{T^{*}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int _{T^{*}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T^{*}}^{s_{n-1}}\bigl[\bigl|R(s_{n})\bigr|+ \beta\bigl|u(s_{n})\bigr|\bigr]\Delta s_{n}< d $$
(3.8)
and
$$ \int_{T^{*}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{T^{*}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{T^{*}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T^{*}}^{s_{n-1}}\alpha\bigl|u(s_{n})\bigr|\Delta s_{n}< \frac{d^{1-\gamma}}{2(\gamma-1)} . $$
(3.9)
We will show that eventually for any interval on which \(x(t)\) is positive, we have that \(x(t)\) is bounded by a constant independent of \(x(t)\). Choose \(T^{*}< T_{1}\leq T_{2}\leq\cdots\leq T_{n-1}\leq T_{n}\) so that (3.4)-(3.5) are satisfied, \(\delta(t)>T_{n-1}\) for \(t\geq T_{n}\), and \(x(\delta(T_{n}))x(\delta(\sigma(T_{n})))\leq0\) with \(x(\delta(T_{n}))\leq0\). As in the proof of Theorem 3.1, using (3.8), we have
$$\begin{aligned} x\bigl(\delta(t)\bigr) \leq& \int_{T_{1}}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{T_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})} \\ &{}\cdots \int_{T_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T_{1}}^{s_{n-1}}\bigl\{ \bigl|R(s_{n})\bigr|+\bigl|u(s_{n})\bigr| \bigl[\alpha\bigl|x\bigl(\delta (s_{n})\bigr)\bigr|^{\gamma}+\beta\bigr] \bigr\} \Delta s_{n} \\ =& \int_{T_{1}}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int _{T_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int _{T_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T_{1}}^{s_{n-1}}\bigl[\bigl|R(s_{n})\bigr|+ \beta\bigl|u(s_{n})\bigr|\bigr]\Delta s_{n} \\ &{}+ \int_{T_{1}}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int _{T_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int _{T_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T_{1}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha\bigl|x\bigl( \delta(s_{n})\bigr)\bigr|^{\gamma}\Delta s_{n} \\ \leq& d+ \int_{T_{1}}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int _{T_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})} \\ &{}\cdots \int _{T_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T_{1}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha\bigl|x\bigl( \delta(s_{n})\bigr)\bigr|^{\gamma}\Delta s_{n}. \end{aligned}$$
(3.10)
We can apply Lemma 2.1 with \(c=d\), \(h(s_{1},s_{2},\ldots,s_{n})=\frac{\alpha |u(s_{n})|}{a_{1}(s_{1})a_{2}(s_{2})\cdots a_{n-1}(s_{n-1})}\), \(\xi=\gamma\), and \(p(s)=s^{\gamma}\). From condition (3.9) we have
$$\begin{aligned} &d^{1-\gamma}-(\gamma-1)\alpha \int_{T_{1}}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{T_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{T_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T_{1}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\Delta s_{n} \\ &\quad>d^{1-\gamma}-(\gamma-1)\frac{d^{1-\gamma}}{2(\gamma-1)} =\frac{d^{1-\gamma}}{2}>0. \end{aligned}$$
Thus, (2.4) holds. It follows from Lemma 2.1 that
$$\begin{aligned} &x\bigl(\delta(t)\bigr) \\ &\quad\leq \biggl[d^{1-\gamma}-(\gamma-1)\alpha \int_{T_{1}}^{\delta(t)}\frac {\Delta s_{1}}{a_{1}(s_{1})} \int_{T_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{T_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T_{1}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\Delta s_{n} \biggr]^{\frac{-1}{\gamma -1}} \\ &\quad\leq\frac{2^{\frac{1}{\gamma-1}}}{d}. \end{aligned}$$
So \(x(\delta(t))\) is bounded. A similar argument holds for intervals where \(x(t)\) is negative.
If \(\gamma=1\), then choose \(\hat{T}\geq t_{0}\) so that (3.8) holds with \(T^{*}\) replaced by and
$$\int_{\hat{T}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{\hat{T}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{\hat{T}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{\hat{T}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\Delta s_{n}< \frac{1}{\alpha +1}. $$
Choose \(T^{*}< T'_{1}\leq T'_{2}\leq\cdots\leq T'_{n-1}\leq T'_{n}\) so that \(R_{n-i}(T'_{i},x(T'_{i}))R_{n-i}(\sigma(T'_{i}),x(\sigma(T'_{i})))\leq 0\) with \(R_{n-i}(T'_{i},x(T'_{i}))\geq0\) for \(1\leq i\leq n\) and \(\delta(t)>T'_{n-1}\) for \(t\geq T'_{n}\). As in the proof of Theorem 3.1, using (3.8), we have
$$\begin{aligned} x\bigl(\delta(t)\bigr) \geq-d- \int_{T'_{1}}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{T'_{1}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{T'_{1}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{T'_{1}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha\bigl|x\bigl( \delta(s_{n})\bigr)\bigr|\Delta s_{n}. \end{aligned}$$
Combining (3.10) with this inequality, we obtain
$$\begin{aligned} \bigl|x\bigl(\delta(t)\bigr)\bigr| \leq& d+ \int_{L}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{L}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})} \\ &{}\cdots \int_{L}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{L}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha\bigl|x\bigl( \delta(s_{n})\bigr)\bigr|\Delta s_{n}, \end{aligned}$$
(3.11)
where \(L=\min\{T_{1},T'_{1}\}\). Denoting by \(z(t)\) the right side of inequality (3.11), we see that \(|x(\delta(t))|\leq z(t)\), \(z(\delta (t))\leq z(t)\), and
$$\begin{aligned} z(t) =&d+ \int_{L}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{L}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{L}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{L}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha\bigl|x\bigl( \delta(s_{n})\bigr)\bigr|\Delta s_{n} \\ \leq& d+ \int_{L}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{L}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{L}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{L}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha z(s_{n})\Delta s_{n} \\ \leq&d+z(t) \int_{L}^{\delta(t)}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{L}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{L}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{L}^{s_{n-1}}\bigl|u(s_{n})\bigr|\alpha\Delta s_{n} \\ \leq&d+\frac{\alpha}{\alpha+1}z(t), \end{aligned}$$
which implies \(x(\delta(t))\leq d(\alpha+1)\). The proof is complete. □

After seeing the proof of Theorem 3.2, the proof of the following Theorem 3.3 becomes obvious.

Theorem 3.3

Assume that conditions (H1)-(H3) hold with \(\gamma\geq1\). If (3.1) holds, then every oscillatory solution of (1.1) converges to zero as \(t\rightarrow\infty\).

In a similar fashion as before, we can show the following theorem.

Theorem 3.4

Assume that conditions (H1)-(H3) hold with \(0<\gamma< 1\). If (3.1) holds, then every oscillatory solution of (1.1) is bounded and converges to zero as \(t\rightarrow\infty\).

Proof

Notice that taking \(p(\nu)=\nu^{\xi}\) and \(0<\xi<1\) in Lemma 2.1, we have
$$P\bigl(z(t)\bigr)-P\bigl(z(t_{0})\bigr)=\frac{1}{1-\xi} \bigl[z^{1-\xi}(t)-z^{1-\xi}(t_{0})\bigr]. $$
So
$$\frac{1}{1-\xi}z^{1-\xi}(t)\leq\frac{1}{1-\xi}z^{1-\xi}(t_{0})+ \int _{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}, $$
that is,
$$z(t)\leq \biggl[z^{1-\xi}(t_{0})+(1-\xi) \int_{t_{0}}^{t}\Delta s_{1} \int _{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n} \biggr]^{\frac{1}{1-\xi}}. $$
We have
$${r}(t)\leq \biggl[z^{1-\xi}(t_{0})+(1-\xi) \int_{t_{0}}^{t}\Delta s_{1} \int_{t_{0}}^{s_{1}}\Delta s_{2} \int_{t_{0}}^{s_{2}}\cdots \int_{t_{0}}^{s_{n-1}} h(s_{1},s_{2}, \ldots,s_{n})\Delta s_{n}\biggr]^{\frac{1}{1-\xi}}. $$
Further, the proof is similar to that of Theorem 3.2, so we have
$$\begin{aligned} x\bigl(\delta(t)\bigr) \leq& \biggl[d^{1-\gamma}+(1-\gamma)\alpha \int_{t_{0}}^{{\delta(t)}}\frac {\Delta s_{1}}{a_{1}(s_{1})} \int_{t_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})} \\ &{}\cdots \int_{t_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{t_{0}}^{s_{n-1}}\bigl|u(s_{n})\bigr|\Delta s_{n}\biggr]^{\frac{1}{1-\gamma}}. \end{aligned}$$
So we can conclude that every oscillatory solution of (1.1) is bounded, and by Theorem 3.1 \(x(t)\) converges to zero as \(t\rightarrow\infty\). The proof is complete. □

Theorem 3.5

Assume that conditions (H1)-(H3) hold with \(g(0)=0\). If there exists \(N>0\) such that for all large T, either
$$ \liminf_{t\longrightarrow\infty} \int_{T}^{t}\bigl[R(s)-N\bigl|u(s)\bigr|\bigr]\Delta s>0 $$
(3.12)
or
$$ \limsup_{t\longrightarrow\infty} \int_{T}^{t}\bigl[R(s)+N\bigl|u(s)\bigr|\bigr]\Delta s< 0, $$
(3.13)
then all solutions of (1.1) are nonoscillatory.

Proof

For contradiction, let \(x(t)\) be an oscillatory solution of (1.1). By Theorem 3.3 and Theorem 3.4, \(x(t)\) converges to 0 as \(t\rightarrow\infty\). Hence, there exists \(T_{0}\geq t_{0}\) such that \(|g(x(\delta(t)))|\leq N\) for \(t\geq T_{0}\). From (1.3) we have
$$ R(t)-N\bigl|u(t)\bigr|\leq R^{ \Delta}_{n-1}\bigl(t,x(t)\bigr)\leq R(t)+N\bigl|u(t)\bigr|. $$
(3.14)
If (3.12) holds, then we choose \(T\geq T_{0}\) such that \(\delta(t)\geq T_{0}\) for \(t\geq T\),
$$ R_{n-2}\bigl(T,x(T)\bigr)R_{n-2}\bigl(\sigma(T),x\bigl( \sigma(T)\bigr)\bigr)\leq 0 ,\qquad R_{n-2}\bigl(T,x(T)\bigr) \geq0, $$
(3.15)
and integrating the left inequality in (3.14) from T to t, we obtain
$$R_{n-2}\bigl(T,x(T)\bigr)+ \int^{t}_{T}\bigl[R(s)-N\bigl|u(s)\bigr|\bigr]\Delta s\leq R_{n-2}\bigl(t,x(t)\bigr). $$
This is a contradiction since if \(x(t)\) is oscillatory, then \(R_{n-2}(t,x(t))\) is also oscillatory.

If (3.13) holds, then we choose T so that the second inequality in (3.15) is reversed. This completes the proof of the theorem. □

4 Example

In this section, we give an example to illustrate our main results.

Lemma 4.1

[23, 24]

Assume that \(s,t\in {\mathbb{ T}}\) and \(g\in C_{rd}({\mathbb{ T}}\times{\mathbb{ T}},{\mathbb{ R}})\). Then
$$\int_{s}^{t} \biggl[ \int_{\eta}^{t} g(\eta,\zeta)\Delta\zeta \biggr]\Delta \eta = \int_{s}^{t} \biggl[ \int_{s}^{\sigma(\zeta)}g(\eta ,\zeta)\Delta\eta \biggr]\Delta \zeta. $$

Example 4.1

Let \(\mathbb{ T}=\{q^{n}:n\in\mathbb{ Z}\}\cup\{0\}\) with \(q>1\). Consider the higher-order dynamic equation
$$ \bigl(t\bigl(t\bigl(\cdots\bigl(t^{2+\frac{1}{\gamma}}x^{\Delta} \bigr)^{\Delta}\cdots\bigr)^{\Delta }\bigr)^{\Delta} \bigr)^{\Delta}+\frac{1}{ t^{1+k\gamma+\frac{1}{\gamma}}} \biggl|x \biggl(\frac{t}{q} \biggr) \biggr|^{\gamma}\operatorname{sgn} x \biggl(\frac{t}{q} \biggr)= \frac{1}{t^{1+\frac{1}{\gamma}}}, $$
(4.1)
where \(t\in[q,\infty)_{\mathbb{ T}}\), \(\gamma >0\), \(k\geq0\), \(a_{1}(t)=t^{2+\frac{1}{\gamma}}\), \(a_{i}(t)=t\) (\(2\leq i\leq n-1\)), \(u(t)=\frac{1}{t^{1+k\gamma}+\frac{1}{\gamma}}\), \(\delta(t)=\frac{t}{q}\), \(R(t)=\frac{1}{t^{1+\frac{1}{\gamma}}}\), and \(g(u)=|u|^{\gamma}\operatorname{sgn}(u)\).
It is easy to verify that \(R(t)\) and \(u(t)\) satisfy the condition (3.12). We will use the following inequality: if \(s>t\geq q\), then
$$\begin{aligned} \int_{t}^{s}\frac{1}{\tau}\Delta\tau &= \int_{t}^{qt}\frac{1}{\tau}\Delta\tau+ \int_{qt}^{q^{2}t}\frac{1}{\tau }\Delta\tau+\cdots+ \int_{q^{n-1}t}^{q^{n}t=s}\frac{1}{\tau}\Delta\tau \\ &=\frac{qt-t}{t}+\frac{q^{2}t-qt}{qt}+\cdots+\frac {q^{n}t-q^{n-1}t}{q^{n-1}t} \\ &=n(q-1)\leq q^{n+1}\leq q^{n}t=s. \end{aligned}$$
Applying Lemma 4.1 and the last inequality, we have
$$\begin{aligned} & \int_{q}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{q}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{q}^{s_{n-1}}\bigl|u(s_{n})\bigr|\Delta s_{n} \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots \int_{q}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{q}^{s_{n-1}}\bigl|R(s_{n})\bigr|\Delta s_{n} \\ &\quad= \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots \int_{q}^{s_{n-2}}\frac{\Delta s_{n-1}}{s_{n-1}} \int_{q}^{s_{n-1}}\frac{1}{s_{n}^{1+\frac{1}{\gamma}}}\Delta s_{n} \\ &\quad= \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots{ \int_{q}^{s_{n-3}}\frac{\Delta s_{n-2}}{s_{n-2}} \int_{q}^{s_{n-2}}\Delta s_{n-1} \int_{q}^{s_{n-1}}\frac{1}{s_{n-1}}\frac{1}{s_{n}^{1+\frac{1}{\gamma }}} \Delta s_{n}} \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots{ \int_{q}^{s_{n-3}}\frac{\Delta s_{n-2}}{s_{n-2}}} \int_{q}^{s_{n-2}}\Delta s_{n-1} \int_{q}^{\sigma(s_{n-1})}\frac{1}{s_{n-1}}\frac{1}{s_{n}^{1+\frac {1}{\gamma}}} \Delta s_{n} \\ &\quad= \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots{ \int_{q}^{s_{n-3}}\frac{\Delta s_{n-2}}{s_{n-2}}} \int_{q}^{s_{n-2}}\frac{\Delta s_{n}}{s_{n}^{1+\frac{1}{\gamma}}} \int_{s_{n}}^{s_{n-2}}\frac{1}{s_{n-1}}\Delta s_{n-1} \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots \int_{q}^{s_{n-3}}\frac{\Delta s_{n-2}}{s_{n-2}} \int _{q}^{s_{n-2}}\frac{s_{n-2}}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n} \\ &\quad= \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots{ \int_{q}^{s_{n-4}}\frac{\Delta s_{n-3}}{s_{n-3}}} \int_{q}^{s_{n-3}}\Delta s_{n-2} \int_{q}^{s_{n-2}}\frac{1}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n} \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots{ \int_{q}^{s_{n-4}}\frac{\Delta s_{n-3}}{s_{n-3}}} \int_{q}^{s_{n-3}}\Delta s_{n-2} \int_{q}^{\sigma(s_{n-2})}\frac{1}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n} \\ &\quad= \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots{ \int_{q}^{s_{n-4}}\frac{\Delta s_{n-3}}{s_{n-3}}} \int_{q}^{s_{n-3}}\Delta s_{n} \int_{s_{n}}^{s_{n-3}}\frac{1}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n-2} \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma }}} \int_{q}^{s_{1}}\frac{\Delta s_{2}}{s_{2}}\cdots \int _{q}^{s_{n-3}}\frac{s_{n-3}}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n} \\ &\quad\cdots \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{2+\frac{1}{\gamma}}} \int_{q}^{s_{1}}\frac {s_{1}}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n}\leq \int_{q}^{\infty}\frac{\Delta s_{1}}{s_{1}^{1+\frac{1}{\gamma}}} \int_{q}^{\sigma (s_{1})}\frac{1}{s_{n}^{1+\frac{1}{\gamma}}} \Delta s_{n} \\ &\quad= \int_{q}^{\infty}\frac{\Delta s_{n}}{s_{n}^{1+\frac{1}{\gamma}}} \int_{s_{n}}^{\infty}\frac {1}{s_{1}^{1+\frac{1}{\gamma}}} \Delta s_{1} \\ &\quad\leq \int_{q}^{\infty}\frac{\Delta s_{n}}{s_{n}^{1+\frac{1}{\gamma}}} \int_{q}^{\infty}\frac {1}{s_{1}^{1+\frac{1}{\gamma}}} \Delta s_{1}= \biggl(\frac{q-1}{q^{\frac{1}{\gamma}}-1} \biggr)^{2}< \infty. \end{aligned}$$
Thus, conditions (H1)-(H3) and (3.1) hold. Then it follows from Theorem 3.5 that every solution \(x(t)\) of (4.1) is nonoscillatory.

Declarations

Acknowledgements

This project is supported by NNSF of China (11461003).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Guangxi University
(2)
College of Information and Statistics, Guangxi University of Finance and Economics
(3)
College of Electrical Engineering, Guangxi University

References

  1. Hilger, S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 18, 18-56 (1990) View ArticleMathSciNetMATHGoogle Scholar
  2. Kac, V, Chueng, P: Quantum Calculus. Universitext (2002) View ArticleMATHGoogle Scholar
  3. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001) View ArticleGoogle Scholar
  4. Bohner, M, Peterson, A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003) View ArticleMATHGoogle Scholar
  5. Bohner, M, Li, T: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math. 58, 1445-1452 (2015) View ArticleMathSciNetGoogle Scholar
  6. Zhang, C, Agarwal, RP, Bohner, M, Li, T: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58, 143-160 (2015) View ArticleMathSciNetMATHGoogle Scholar
  7. Zhang, C, Li, T: Some oscillation results for second-order nonlinear delay dynamic equations. Appl. Math. Lett. 26, 1114-1119 (2013) View ArticleMathSciNetMATHGoogle Scholar
  8. Agarwal, RP, Bohner, M, Saker, SH: Oscillation of second order delay dynamic equations. Can. Appl. Math. Q. 13, 1-17 (2005) MathSciNetMATHGoogle Scholar
  9. Bohner, M, Karpuz, B, Öcalan, Ö: Iterated oscillation criteria for delay dynamic equations of first order. Adv. Differ. Equ. 2008, 458687 (2008) View ArticleGoogle Scholar
  10. Erbe, L, Peterson, A, Saker, SH: Oscillation criteria for second-order nonlinear delay dynamic equations. J. Math. Anal. Appl. 333, 505-522 (2007) View ArticleMathSciNetMATHGoogle Scholar
  11. Han, Z, Shi, B, Sun, S: Oscillation criteria for second-order delay dynamic equations on time scales. Adv. Differ. Equ. 2007, 070730 (2007) View ArticleMathSciNetGoogle Scholar
  12. Han, Z, Sun, S, Shi, B: Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales. J. Math. Anal. Appl. 334, 847-858 (2007) View ArticleMathSciNetMATHGoogle Scholar
  13. Şahiner, Y: Oscillation of second-order delay differential equations on time scales. Nonlinear Anal. 63, e1073-e1080 (2005) View ArticleMATHGoogle Scholar
  14. Şahiner, Y: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales. Adv. Differ. Equ. 2006, 065626 (2006) Google Scholar
  15. Zhang, B, Zhu, S: Oscillation of second-order nonlinear delay dynamic equations on time scales. Comput. Math. Appl. 49, 599-609 (2005) View ArticleMathSciNetMATHGoogle Scholar
  16. Grace, SR, Agarwal, RP, Kaymakcalan, B, Sae-jie, W: On the oscillation of certain second order nonlinear dynamic equations. Math. Comput. Model. 50, 273-286 (2009) View ArticleMathSciNetMATHGoogle Scholar
  17. Erbe, L, Peterson, A, Saker, SH: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J. Comput. Appl. Math. 181, 92-102 (2005) View ArticleMathSciNetMATHGoogle Scholar
  18. Erbe, L, Peterson, A, Saker, SH: Hille and Nehari type criteria for third order dynamic equations. J. Math. Anal. Appl. 329, 112-131 (2007) View ArticleMathSciNetMATHGoogle Scholar
  19. Erbe, L, Peterson, A, Saker, SH: Oscillation and asymptotic behavior a third-order nonlinear dynamic equation. Can. Appl. Math. Q. 14, 129-147 (2006) MathSciNetMATHGoogle Scholar
  20. Karpuz, B, Öcalan, Ö, Rath, RN: Necessary and sufficient conditions on the oscillatory and asymptotic behaviour of solutions to neutral delay dynamic equation. Electron. J. Differ. Equ. 2009, 64 (2009) Google Scholar
  21. Zhu, Z, Wang, Q: Existence of nonoscillatory solutions to neutral dynamic equations on time scales. J. Math. Anal. Appl. 335, 751-762 (2007) View ArticleMathSciNetMATHGoogle Scholar
  22. Karpuz, B, Öcalan, Ö: Necessary and sufficient conditions on the asymptotic behaviour of solutions of forced neutral delay dynamic equations. Nonlinear Anal. 71, 3063-3071 (2009) View ArticleMathSciNetMATHGoogle Scholar
  23. Karpuz, B: Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations. Appl. Math. Comput. 215, 2174-2183 (2009) View ArticleMathSciNetMATHGoogle Scholar
  24. Karpuz, B: Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients. Electron. J. Qual. Theory Differ. Equ. 2009, 34 (2009) MathSciNetGoogle Scholar
  25. Wu, X, Sun, T, Xi, H, Cheng, C: Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales. Adv. Differ. Equ. 2013, 248 (2013) View ArticleGoogle Scholar
  26. Sun, T, Xi, H, Peng, X, Yu, W: Nonoscillatory solutions for higher-order neutral dynamic equations on time scales. Abstr. Appl. Anal. 2010, 428963 (2010) MathSciNetGoogle Scholar
  27. Zhang, Z, Dong, W, Li, Q, Liang, H: Positive solutions for higher order nonlinear neutral dynamic equations on time scales. Appl. Math. Model. 33, 2455-2463 (2009) View ArticleMathSciNetMATHGoogle Scholar
  28. Agarwal, RP, Bohner, M: Basic calculus on time scales and some of its applications. Results Math. 35, 3-22 (1999) View ArticleMathSciNetMATHGoogle Scholar
  29. Graef, JR, Hill, M: Nonoscillation of all solutions of a higher order nonlinear delay dynamic equation on time scales. J. Math. Anal. Appl. 423, 1693-1703 (2015) View ArticleMathSciNetMATHGoogle Scholar

Copyright

© Tao et al. 2016