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The Kamenev type oscillation criteria of nonlinear differential equations under impulse effects

Advances in Difference Equations20162016:227

https://doi.org/10.1186/s13662-016-0911-6

  • Received: 11 April 2016
  • Accepted: 26 June 2016
  • Published:

Abstract

In this paper, a general class of second-order nonlinear damped differential equations under impulse effects is studied. By introducing an auxiliary function of two variables and using Riccati transformation, several Kamenev type interval oscillation criteria are established, which generalize and extend some known ones, such as those of Huang and Feng. Moreover, examples are given to illustrate the effectiveness and non-emptiness of our results.

Keywords

  • interval oscillation
  • Riccati transformation
  • impulse
  • damping

MSC

  • 34C15
  • 34A37
  • 65L03

1 Introduction and preliminaries

In 2003, Rogovchenko and Rogovchenko [1] first studied the oscillation of the nonlinear second-order differential equation with nonlinear damping of the form
$$ \bigl(r(t)k_{1}\bigl(x(t),x'(t)\bigr) \bigr)'+p(t)k_{2}\bigl(x(t),x'(t) \bigr)x'(t)+q(t)h\bigl(x(t)\bigr)=0. $$
(1.1)
Later, Tiryaki and Zafer [2], Zhao and Feng [3], Zhao et al. [4] and Huang and Meng [5] obtained several oscillation criteria for solutions of (1.1), which extended and improved the results in [1]. In this work, the conditions in terms of the coefficients involving integral averages over the whole half-line \([t_{0},\infty)\) are used. However, oscillation is an interval property as pointed out early in [6, 7]. So, it is more reasonable to investigate solutions on an infinite set of bounded intervals. In 2005, Tiryaki and Zafer [8] investigated the interval oscillation of (1.1). Later, some interval oscillation criteria were also given by Shi [9] and Huang and Meng [10] for a forced equation of the form
$$ \bigl(r(t)k_{1}\bigl(x(t),x'(t)\bigr) \bigr)'+p(t)k_{2}\bigl(x(t),x'(t) \bigr)x'(t)+q(t)h\bigl(x(t)\bigr)=f(t). $$
(1.2)
In recent years, interval oscillation of impulsive differential equations also aroused the interest of many researchers; see, for example, [1116]. In 2009, under impulse effects, Özbekler and Zafer [17] first considered the special case of equation (1.2) as follows:
$$ \left \{ \textstyle\begin{array}{l} (r(t)\varphi_{\alpha}(x'))'+p(t)\varphi_{\alpha}(x')+q(t)\varphi _{\beta}(x)=f(t), \quad t\neq\theta_{i}, \\ \Delta(r(t)\varphi_{\alpha}(x'))+q_{i}\varphi_{\beta}(x)=f_{i}, \quad t=\theta_{i}, i\in\mathbb{N}. \end{array}\displaystyle \right . $$
(1.3)
They improved and extended the earlier results for the equations without impulse or damping of [1821]. Recently, Li et al. [22] investigated equation (1.2) with impulses and obtained several interval oscillation theorems by introducing an auxiliary function of one variable and generalized and extended some known results in [9, 17, 23].
In this article, motivated mainly by [810, 17, 22], we continue to study the interval oscillation of (1.2) under impulse effects of the form
$$ \left \{ \textstyle\begin{array}{l} (r(t)k_{1}(x(t),x'(t)))'+p(t)k_{2}(x(t),x'(t))x'(t)+q(t)g(x(t))=f(t), \quad t\neq t_{i},\\ \Delta(r(t)k_{1}(x(t),x'(t)))+q_{i}g(x(t))=f_{i}, \quad t= t_{i}, i=1,2,\ldots, \end{array}\displaystyle \right . $$
(1.4)
where \(\Delta y(t)=y(t^{+})-y(t^{-})\), \(y(t^{\pm})=\lim_{t\rightarrow t^{\pm}}y(t)\), \(\{t_{i}\}\) is an impulse moments sequence with \(0\leq t_{0}< t_{1}< t_{2}<\cdots<t_{i}<t_{i+1}<\cdots\) and \(\lim_{i\rightarrow\infty}t_{i}=\infty\).

By introducing an auxiliary function of two variables and using a Riccati transformation, we establish some Kamenev type (cf. Philos [24]) interval oscillation criteria. Our methods are different from those of Özbekler and Zafer [17] and Li et al. [22] because we use an auxiliary function of two variables and divide the considered interval into two parts to study oscillation of (1.4). The results obtained here are the generalization and extension of some known ones, such as those of Huang and Feng [10]. Moreover, we also give two examples to illustrate the effectiveness and non-emptiness of our results.

The organization of this paper is as follows. In the next section, we will give four theorems of Kamenev type interval oscillation criteria. Section 3 is devoted to the proofs of the theorems. Two examples will be illustrated in Section 4.

2 The main results

We first introduce a set \(\operatorname{PLC}(I,\mathbb{R})\) and a function class \(\mathscr{H}_{D}\) of bivariate functions \(H(t,s)\) on a set D.

Let \(I\subset\mathbb{R}\) be an interval, \(\operatorname{PLC}(I,\mathbb{R})\) := {\(y: I\rightarrow\mathbb{R} | y\) is continuous on \(I\setminus\{t_{i}\}\) and at each \(t_{i}\), \(y(t_{i}^{+})\) and \(y(t_{i}^{-})\) exist, and the left continuity of y is assumed, i.e., \(y(t_{i}^{-})=y(t_{i})\), \(i\in\mathbb{N}\)}.

Let \(D=\{(t,s): t\geq s\geq t_{0}\}\), \(\mathscr{H}_{D}\) is a class of functions \(H(t,s)\in C(D,\mathbb{R})\) which satisfy the following assumptions:
  1. (i)

    \(H(t,t)=0\), \(H(t,s)>0\), for \(t>s\geq t_{0}\);

     
  2. (ii)

    \(\frac{\partial}{\partial t}H(t,s)=h_{1}(t,s)\sqrt{H(t,s)}\), \(\frac{\partial}{\partial s}H(t,s)=-h_{2}(t,s)\sqrt{H(t,s)}\),

     
for some \(h_{1},h_{2}\in L_{\mathrm{loc}}(D,\mathbb{R})\).

For any given \(t_{0}\), we say a continuous function \(x(t)\) defined on \([t_{0},\infty)\) is a solution of (1.4) with initial value \(x(t_{0})=x_{0}\) and \(x'(t_{0})=x'_{0}\) if \(x'(t), (r(t)k_{1}(x(t),x'(t)))'\in \operatorname{PLC}([t_{0},\infty),\mathbb{R})\) and \(x(t)\) satisfies (1.4) for \(t\geq t_{0}\).

The following preparatory lemma will be useful to prove our theorems.

Lemma 2.1

Let \(\eta_{0}, \eta_{1}, \eta_{2} \in C([t_{0},\infty),\mathbb{R})\) with \(\eta_{2}>0\), and \(w\in C^{1}([t_{0},\infty),\mathbb{R})\), for \(t\neq t_{i}\), together with \(w(t_{i}^{\pm})\) exist. If there exist \([a,b]\subset[t_{0},\infty)\), \(t_{i}\in(a,b)\), \(w_{i}>0\) for \(i\in\mathbb{N}\), and \(\alpha>0\) such that
$$ w'(t)\leq-\eta_{0}(t)+\eta_{1}(t)w(t)- \eta_{2}(t)\bigl|w(t)\bigr|^{1+1/\alpha}, \quad t\in(a,b), t\neq t_{i}, $$
(2.1)
and
$$ \Delta w(t) \leq-w_{i}, \quad t= t_{i}\in(a,b), $$
(2.2)
then, for every \(H\in\mathscr{H}_{(a,b)}\) and \(c\in(a,b)\setminus\{t_{i}\}\),
$$\begin{aligned}& \frac{1}{H^{\alpha+1}(c,a)} \int^{c}_{a} \bigl[\tilde{\eta}_{2}(s) \tilde{H}_{1}^{\alpha+1}(s,a)-\eta_{0}(s) H^{\alpha+1}(s,a) \bigr]\,\mathrm{d}s \\& \qquad {}+\frac{1}{H^{\alpha+1}(b,c)} \int^{b}_{c} \bigl[\tilde{\eta}_{2}(s) \tilde{H}_{2}^{\alpha+1}(b,s)-\eta_{0}(s) H^{\alpha+1}(b,s) \bigr] \,\mathrm{d}s \\& \quad \geq \frac{1}{H^{\alpha+1}(c,a)}\sum_{a< t_{i}< c}H^{\alpha+1}(t_{i},a)w_{i} +\frac{1}{H^{\alpha+1}(b,c)}\sum_{c< t_{i}< b}H^{\alpha+1}(b,t_{i})w_{i} , \end{aligned}$$
(2.3)
where
$$\begin{aligned}& \tilde{\eta}_{2}(s)=\frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha +1}}\eta_{2}^{-\alpha}(s), \\& \tilde{H}_{1}(s,a)=\bigl\vert (\alpha+1)h_{1}(s,a) \sqrt{H(s,a)}+\eta _{1}(s)H(s,a)\bigr\vert , \\& \tilde{H}_{2}(b,s)=\bigl\vert (\alpha+1)h_{2}(b,s) \sqrt{H(b,s)}-\eta _{1}(s)H(b,s)\bigr\vert . \end{aligned}$$

Proof

Multiplying (2.1) by \(H^{\alpha+1}(t,s)\) and integrating it (with t replaced by s) over \([c,t)\) for \(t\in[c,b)\), we have
$$\begin{aligned} \int^{t}_{c} H^{\alpha+1}(t,s)w'(s) \,\mathrm{d}s \leq & - \int^{t}_{c} \eta_{0}(s) H^{\alpha+1}(t,s)\,\mathrm{d}s+ \int^{t}_{c} \eta _{1}(s)w(s)H^{\alpha+1}(t,s) \,\mathrm{d}s \\ &{} - \int^{t}_{c} \eta_{2}(s)\bigl\vert w(s) \bigr\vert ^{1+1/\alpha}H^{\alpha+1}(t,s)\,\mathrm{d}s. \end{aligned}$$
(2.4)
In view of (2.2) and the assumptions (i) and (ii) for \(H(t,s)\), we get
$$\begin{aligned}& \int^{t}_{c} \eta_{0}(s) H^{\alpha+1}(t,s)\,\mathrm{d}s+\sum_{c< t_{i}< t}H^{\alpha+1}(t,t_{i})w_{i} - H^{\alpha+1}(t,c)w(c) \\& \quad \leq \int^{t}_{c} \bigl[ \bigl\vert ( \alpha+1)h_{2}(t,s)\sqrt{H(t,s)}-\eta _{1}(s)H(t,s) \bigr\vert H^{\alpha}(t,s)\bigl\vert w(s)\bigr\vert \\& \qquad {}- \eta_{2}(s)H^{\alpha+1}(t,s) \bigl\vert w(s)\bigr\vert ^{1+1/\alpha} \bigr]\,\mathrm{d}s. \end{aligned}$$
(2.5)
Defining the integrand function in right-hand side of (2.5) by
$$ \Phi(w)=\bigl\vert (\alpha+1)h_{2}(t,s)\sqrt{H(t,s)}- \eta_{1}(s)H(t,s) \bigr\vert H^{\alpha}(t,s)\bigl\vert w(s) \bigr\vert - \eta_{2}(s)H^{\alpha+1}(t,s)\bigl\vert w(s)\bigr\vert ^{1+1/\alpha}, $$
we easily see that
$$ \max_{w\in\mathbb{R}}\Phi(w)= \frac{\alpha^{\alpha}}{(\alpha +1)^{\alpha+1}}\eta_{2}^{-\alpha}(s) \bigl\vert (\alpha+1)h_{2}(t,s)\sqrt{H(t,s)}-\eta_{1}(s)H(t,s) \bigr\vert ^{\alpha+1}. $$
Therefore,
$$\begin{aligned}& \int^{t}_{c} \eta_{0}(s) H^{\alpha+1}(t,s)\,\mathrm{d}s+\sum_{c< t_{i}< t}H^{\alpha+1}(t,t_{i})w_{i} - H^{\alpha+1}(t,c)w(c) \\& \quad \leq \int^{t}_{c} \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}\eta _{2}^{-\alpha}(s) \bigl\vert (\alpha+1)h_{2}(t,s) \sqrt{H(t,s)}-\eta_{1}(s)H(t,s)\bigr\vert ^{\alpha +1}\, \mathrm{d}s. \end{aligned}$$
(2.6)
Letting \(t\rightarrow b^{-}\) in (2.6)
$$\begin{aligned}& \int^{b}_{c} \eta_{0}(s) H^{\alpha+1}(b,s)\,\mathrm{d}s+\sum_{c< t_{i}< b}H^{\alpha+1}(b,t_{i})w_{i} - H^{\alpha+1}(b,c)w(c) \\& \quad \leq \int^{b}_{c} \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}\eta _{2}^{-\alpha}(s) \bigl\vert (\alpha+1)h_{2}(b,s) \sqrt{H(b,s)}-\eta_{1}(s)H(b,s)\bigr\vert ^{\alpha +1}\, \mathrm{d}s. \end{aligned}$$
(2.7)
Similarly, if (2.1) (with t replaced by s) is multiplied by \(H^{\alpha+1}(s,t)\) and integrated over \((t,c]\) for \(t\in (a,c]\), we have
$$\begin{aligned} \int^{c}_{t} H^{\alpha+1}(s,t)w'(s) \,\mathrm{d}s \leq& - \int^{c}_{t} \eta_{0}(s) H^{\alpha+1}(s,t)\,\mathrm{d}s + \int^{c}_{t} \eta _{1}(s)w(s)H^{\alpha+1}(s,t) \,\mathrm{d}s \\ &{} - \int^{c}_{t} \eta_{2}(s)\bigl\vert w(s) \bigr\vert ^{1+1/\alpha}H^{\alpha+1}(s,t)\,\mathrm{d}s. \end{aligned}$$
(2.8)
In view of (2.2) and using the assumptions (i) and (ii), we get
$$\begin{aligned}& \int^{c}_{t} \eta_{0}(s) H^{\alpha+1}(s,t)\,\mathrm{d}s+\sum_{t< t_{i}< c}H^{\alpha+1}(t_{i},t)w_{i} + H^{\alpha+1}(c,t)w(c) \\& \quad \leq \int^{c}_{t} \bigl[\bigl\vert ( \alpha+1)h_{1}(s,t)\sqrt{H(s,t)}+\eta _{1}(s)H(s,t)\bigr\vert H^{\alpha}(s,t)\bigl\vert w(s)\bigr\vert \\& \qquad {}- \eta_{2}(s)H^{\alpha+1}(s,t) \bigl\vert w(s)\bigr\vert ^{1+1/\alpha}\bigr]\,\mathrm{d}s \\& \quad \leq \int^{c}_{t} \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}\eta _{2}^{-\alpha}(s) \bigl\vert (\alpha+1)h_{1}(s,t) \sqrt{H(s,t)}+\eta_{1}(s)H(s,t)\bigr\vert ^{\alpha +1}\, \mathrm{d}s. \end{aligned}$$
(2.9)
Letting \(t\rightarrow a^{+}\) in (2.9)
$$\begin{aligned}& \int^{c}_{a} \eta_{0}(s) H^{\alpha+1}(s,a)\,\mathrm{d}s+\sum_{a< t_{i}< c}H^{\alpha+1}(t_{i},a)w_{i} + H^{\alpha+1}(c,a)w(c) \\& \quad \leq \int^{c}_{a} \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}\eta _{2}^{-\alpha}(s) \bigl\vert (\alpha+1)h_{1}(s,a) \sqrt{H(s,a)}+\eta_{1}(s)H(s,a)\bigr\vert ^{\alpha +1}\, \mathrm{d}s. \end{aligned}$$
(2.10)

Finally, dividing (2.7) and (2.10) by \(H^{\alpha+1}(b,c)\) and \(H^{\alpha+1}(c,a)\), respectively, and then adding them, we get (2.3). □

2.1 Interval oscillation when \(g(x)\) is differentiable

In this section we need the following conditions:
(A1): 

\(r(t)\in C([t_{0},\infty),(0,\infty))\) and is nondecreasing, \(p(t), q(t), f(t)\in \operatorname{PLC}([t_{0},\infty),\mathbb{R})\);

(A2): 

\(vk_{1}(u,v)\geq\mu|k_{1}(u,v)|^{1+1/\alpha}\) for some \(\mu>0\), \(\alpha >0\) and for all \((u,v)\in\mathbb{R}^{2}\);

(A3): 

\(vk_{2}(u,v)\varphi_{\frac{1}{\alpha}}(g(u))\geq\tau |k_{1}(u,v)|^{1+1/\alpha}\) for some \(\tau>0\), \(\alpha>0\), and for all \((u,v)\in\mathbb{R}^{2}\), where \(\varphi_{*}(g)=|g|^{*-1}g\);

(A4): 

\(g(u)\) is differentiable, \(ug(u)>0\) for \(u\neq0\) and \(g'(u)\geq \varsigma|g(u)|^{1-1/\alpha}\) for some \(\varsigma>0\), \(\alpha>0\), and for all \(u\in\mathbb {R}\setminus\{0\}\).

Theorem 2.1

Assume that (A1)-(A4) hold. For any given \(T_{0}\geq t_{0}\), there exist intervals \(I_{j}=[a_{j},b_{j}]\subset [T_{0},\infty)\), \(j=1,2\), such that
(A5): 

\(p(t)\geq0\), \(\forall t\in\{I_{1}\cup I_{2} \}\);

(A6): 

\(q(t)\geq0\), \(\forall t\in\{I_{1}\cup I_{2} \}\setminus\{t_{i}\}\), \(q_{i}\geq0\), \(t_{i}\in\{I_{1}\cup I_{2} \}\), \(\forall i\in\mathbb{N}\);

(A7): 

\(f(t)\bigl \{\scriptsize{ \begin{array}{l@{\quad}l} \leq0, & t\in I_{1}\setminus\{t_{i}\}, \\ \geq0, & t\in I_{2}\setminus\{t_{i}\}, \end{array}} \bigr.\) \(f_{i}\bigl \{\scriptsize{ \begin{array}{l@{\quad}l} \leq0, t_{i}\in I_{1}, \\ \geq0, t_{i}\in I_{2}, \end{array}} \bigr.\) \(\forall i\in\mathbb{N}\).

If there exist \(H_{j}\in\mathscr{H}_{(a_{j},b_{j})}\) and \(c_{j}\in(a_{j},b_{j})\setminus\{t_{i}\}\), \(j=1,2\), such that
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})} \int^{c_{j}}_{a_{j}} \bigl[\tilde {p}(s) \tilde{H}_{j1}^{\alpha+1}(s,a_{j}) -q(s) H^{\alpha+1}_{j}(s,a_{j}) \bigr]\,\mathrm{d}s \\& \qquad {}+\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})} \int^{b_{j}}_{c_{j}} \bigl[\tilde {p}(s) \tilde{H}_{j2}^{\alpha+1}(b_{j},s) -q(s) H^{\alpha+1}_{j}(b_{j},s) \bigr]\,\mathrm{d}s \\& \quad < \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})}\sum_{a_{j}< t_{i}< c_{j}}H^{\alpha +1}_{j}(t_{i},a_{j}) q_{i} +\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})}\sum_{c_{j}< t_{i}< b_{j}}H^{\alpha +1}_{j}(b_{j},t_{i}) q_{i} , \end{aligned}$$
(2.11)
where
$$\begin{aligned}& \tilde{p}(s)=\frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}} \frac{r^{\alpha+1}(s)}{(\tau p(s)+\mu\varsigma r(s))^{\alpha}}, \\& \tilde{H}_{j1}(s,a_{j})= \bigl\vert (\alpha+1)h_{j1}(s,a_{j}) \sqrt {H_{j}(s,a_{j})} \bigr\vert , \\& \tilde{H}_{j2}(b_{j},s)= \bigl\vert (\alpha+1)h_{j2}(b_{j},s) \sqrt {H_{j}(b_{j},s)} \bigr\vert , \end{aligned}$$
(2.12)
then (1.4) is oscillatory.

Proof

Suppose that there exists a nonoscillatory solution \(x(t)\) of (1.4). Then there exists a \(T_{0}\geq t_{0}\) such that \(x(t)\neq0\) on \([T_{0},\infty)\). Define
$$ w(t)=:\frac{r(t)k_{1}(x(t),x'(t))}{g(x(t))},\quad t\geq T_{0}. $$
(2.13)
Differentiating (2.13) and using (1.4) for \(t\geq T_{0}\), we have
$$\begin{aligned} w'(t) =& -q(t)-\frac{p(t)k_{2}(x(t),x'(t))x'(t)}{g(x(t))} \\ &{} - \frac{r(t)k_{1}(x(t),x'(t))x'(t)g'(x(t))}{g^{2}(x(t))}+ \frac{f(t)}{g(x(t))},\quad t\neq t_{i}, \end{aligned}$$
(2.14)
and
$$ \Delta w(t)=-q_{i}+\frac{f_{i}}{g(x(t))}, \quad t= t_{i}. $$
(2.15)
By using assumptions (A1)-(A6) and from (2.14), we obtain, for \(t\in\{I_{1}\cup I_{2}\}\),
$$ w'(t) \leq-q(t)-\frac{\tau p(t)+\mu\varsigma r(t)}{ r^{1+1/\alpha}(t)} \bigl|w(t)\bigr|^{1+1/\alpha}+\frac{ f(t)}{g(x(t))},\quad t\neq t_{i}. $$
(2.16)
If \(x(t)>0\), we choose the interval \(I_{1}\) to consider. From (2.16) and (2.15) and in view of condition (A7), we get
$$ w'(t) \leq- q(t)-\frac{\tau p(t)+\mu\varsigma r(t)}{r^{1+1/\alpha}(t)} \bigl\vert w(t) \bigr\vert ^{1+1/\alpha},\quad t \in I_{1}, t\neq t_{i}, $$
(2.17)
and
$$ \Delta w(t)\leq-q_{i}, \quad t \in I_{1}, t= t_{i}. $$
(2.18)
Applying Lemma 2.1 to (2.17) and (2.18), for every \(H_{1}\in\mathscr{H}_{(a_{1},b_{1})}\) and \(c_{1}\in(a_{1},b_{1})\setminus\{t_{i}\}\), we have
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})} \int^{c_{1}}_{a_{1}} \bigl[\tilde{p}(s) \tilde{H}_{11}^{\alpha+1}(s,a_{1})-q(s) H^{\alpha+1}_{1}(s,a_{1}) \bigr]\,\mathrm{d}s \\& \qquad {} +\frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})} \int^{b_{1}}_{c_{1}} \bigl[\tilde{p}(s) \tilde{H}_{12}^{\alpha+1}(b_{1},s)-q(s) H^{\alpha+1}_{1}(b_{1},s) \bigr]\,\mathrm{d}s \\& \quad \geq \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})}\sum_{a_{1}< t_{i}< c_{1}}H^{\alpha+1}_{1}(t_{i},a_{1}) q_{i} +\frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})}\sum_{c_{1}< t_{i}< b_{1}}H^{\alpha +1}_{1}(b_{1},t_{i}) q_{i} , \end{aligned}$$
where \(\tilde{p}(s)\), \(\tilde{H}_{11}(s,a_{1})\) and \(\tilde{H}_{12}(b_{1},s)\) are defined by (2.12). This contradicts (2.11) for \(j=1\).

If \(x(t)<0\), we choose the interval \(I_{2}\) to consider. Similar to the above proof we can also obtain a contradiction to (2.11) for \(j=2\). The proof is complete. □

Remark 2.1

In many published papers on the subject of interval oscillation criteria, the authors have insisted on the use of the function \(\rho(t)\) as a multiplier in the Riccati transformation (2.13). This function has no meaning, i.e., it can be taken as \(\rho(t)=1\).

Remark 2.2

When the impulses in (1.4) disappear, i.e., \(q_{i}=f_{i}=0\), for all \(i\in\mathbb{N}\), (1.4) reduces to the equation (1.2) studied by Huang and Feng in [10]. Therefore Theorem 2.1 provides an extension of Theorem 2.1 of [10], to the impulsive differential equations.

Remark 2.3

If the inequality in condition (A3) is strict, then the sign condition \(p(t)\) can be replaced by
$$ \tau p(t)+\mu\varsigma r(t)>0 $$
in the above theorem.
In the next theorem the sign condition \(p(t)\geq0\) will be sufficiently eliminated by replacing (A3) with
(\(\mathrm{A}'_{3}\)): 

\(vk_{2}(u,v)= \zeta k_{1}(u,v)\) for some \(\zeta>0\) and all \((u,v)\in\mathbb{R}^{2}\).

Theorem 2.2

Assume that (A1), (A2), (\(\mathrm{A}'_{3}\)), and (A4) hold. For any given \(T_{0}\geq t_{0}\), there exist intervals \(I_{j}=[a_{j},b_{j}]\subset[T_{0},\infty)\), \(j=1,2\) such that (A6) and (A7) hold. If there exist \(H_{j}\in \mathscr{H}_{(a_{j},b_{j})}\) and \(c_{j}\in(a_{j},b_{j})\setminus\{t_{i}\}\), \(j=1,2\), such that
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})} \int^{c_{j}}_{a_{j}} \bigl[\tilde{r}(s) \tilde{H}_{j1}^{\alpha+1}(s,a_{j})- q(s) H^{\alpha+1}_{j}(s,a_{j}) \bigr]\,\mathrm{d}s \\& \qquad {}+\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})} \int^{b_{j}}_{c_{j}} \bigl[\tilde{r}(s) \tilde{H}_{j2}^{\alpha+1}(b_{j},s)- q(s) H^{\alpha+1}_{j}(b_{j},s) \bigr] \,\mathrm{d}s \\& \quad < \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})}\sum_{a_{j}< t_{i}< c_{j}}H^{\alpha +1}_{j}(t_{i},a_{j}) q_{i} +\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})}\sum_{c_{j}< t_{i}< b_{j}}H^{\alpha +1}_{j}(b_{j},t_{i}) q_{i} , \end{aligned}$$
(2.19)
where
$$\begin{aligned}& \tilde{r}(s)=\frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}\frac {r(s)}{(\mu\varsigma r(s))^{\alpha}}, \\& \tilde{H}_{j1}(s,a_{j})=\biggl\vert (\alpha+1)h_{j1}(s,a_{j}) \sqrt{H_{j}(s,a_{j})} -\frac{\zeta p(s)}{r(s)}H_{j}(s,a_{j}) \biggr\vert , \\& \tilde{H}_{j2}(b_{j},s)=\biggl\vert (\alpha+1)h_{j2}(b_{j},s) \sqrt{H_{j}(b_{j},s)} +\frac{\zeta p(s)}{r(s)} H_{j}(b_{j},s) \biggr\vert , \end{aligned}$$
then (1.4) is oscillatory.

Proof

Suppose that there exists a nonoscillatory solution \(x(t)\) of (1.4). Then there exists a \(T_{0}\geq t_{0}\) such that \(x(t)\neq0\) on \([T_{0},\infty)\). Define
$$ w(t)=: \frac{r(t)k_{1}(x(t),x'(t))}{g(x(t))}, \quad t\geq T_{0}. $$
(2.20)
Similar to the proof of Theorem 2.1, we can obtain (2.14) and (2.15). By using assumptions (A1), (A2), (\(\mathrm{A}'_{3}\)), (A4), and from (2.14), we obtain for \(t\in\{I_{1}\cup I_{2}\}\)
$$ w'(t) \leq-q(t)-\frac{\zeta p(t)}{r(t)}w(t)- \frac{\mu\varsigma}{r^{1/\alpha}(t)} \bigl\vert w(t)\bigr\vert ^{1+1/\alpha}+ \frac{ f(t)}{g(x(t))},\quad t\neq t_{i}. $$
(2.21)
If \(x(t)>0\), we choose the interval \(I_{1}\) to consider. From (2.21) and (2.15) and in view of condition (A7), we obtain
$$ w'(t) \leq-q(t)-\frac{\zeta p(t)}{r(t)}w(t)- \frac{\mu\varsigma}{r^{1/\alpha}(t)} \bigl\vert w(t)\bigr\vert ^{1+1/\alpha},\quad t\neq t_{i}, $$
(2.22)
and
$$ \Delta w(t)\leq-q_{i},\quad t= t_{i}. $$
(2.23)
Using the same method as in the proof of Theorem 2.1, we can obtain a contradiction to the condition (2.19) for \(j=1\).

If \(x(t)<0\), we choose the interval \(I_{2}\) to consider. Similar to the above proof we can also obtain a contradiction to (2.19) for \(j=2\).

Therefore the proof is complete. □

2.2 Interval oscillation when \(g(x)\) is not differentiable

In this section we assume that the conditions (A1), (A2), (A5)-(A7) in Section 2.1 and the following conditions are satisfied:
(A8): 

\(uvk_{2}(u,v)\geq\xi|k_{1}(u,v)|^{1+1/\alpha}\) for some \(\xi>0\), \(\alpha>0\), and all \((u,v)\in\mathbb{R}^{2}\);

(A9): 

\(g(u)/\varphi_{\alpha}(u)\geq\vartheta|u|^{\beta-\alpha}\) for some \(\vartheta>0\), \(\beta\geq\alpha>0\), and all \(u\in\mathbb {R}\setminus\{0\}\), where \(\varphi_{*}(u)=|u|^{*-1}u\).

Theorem 2.3

Assume that (A1), (A2), (A8), and (A9) hold. For any given \(T_{0}\geq t_{0}\), there exist intervals \(I_{j}=[a_{j},b_{j}]\subset[T_{0},\infty)\), \(j=1,2\), such that (A5)-(A7) hold. If there exist \(H_{j}\in \mathscr{H}_{(a_{j},b_{j})}\) and \(c_{j}\in(a_{j},b_{j})\setminus\{t_{i}\}\), \(j=1,2\), such that
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})} \int^{c_{j}}_{a_{j}} \bigl[\tilde{p}(s) \tilde{H}_{j1}^{\alpha+1}(s,a_{j})-\tilde{q}(s) H^{\alpha +1}_{j}(s,a_{j}) \bigr]\,\mathrm{d}s \\& \qquad {} +\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})} \int^{b_{j}}_{c_{j}} \bigl[\tilde{p}(s) \tilde{H}_{j2}^{\alpha+1}(b_{j},s)-\tilde{q}(s) H^{\alpha +1}_{j}(b_{j},s) \bigr] \,\mathrm{d}s \\& \quad < \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})}\sum_{a_{j}< t_{i}< c_{j}}H^{\alpha +1}_{j}(t_{i},a_{j}) \tilde{ q}_{i} +\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})}\sum_{c_{j}< t_{i}< b_{j}}H^{\alpha +1}_{j}(b_{j},t_{i}) \tilde{ q}_{i} , \end{aligned}$$
(2.24)
where
$$\begin{aligned}& \tilde{p}(s)=\frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}} \frac{r^{\alpha+1}(s)}{(\xi p(s)+\alpha\mu r(s))^{\alpha}}, \\& \tilde{H}_{j1}(s,a_{j})=\bigl\vert (\alpha+1)h_{j1}(s,a_{j}) \sqrt {H_{j}(s,a_{j})}\bigr\vert , \\& \tilde{H}_{j2}(b_{j},s)=\bigl\vert (\alpha+1)h_{j2}(b_{j},s) \sqrt {H_{j}(b_{j},s)}\bigr\vert , \end{aligned}$$
(2.25)
and
$$ \begin{aligned} &\tilde{q}(s)=\beta\alpha^{-\alpha/\beta}(\beta- \alpha)^{\alpha /\beta-1}\bigl(\vartheta q(s)\bigr)^{\alpha/\beta} \bigl\vert f(s) \bigr\vert ^{1-\alpha/\beta}, \\ &\tilde{q}_{i}=\beta\alpha^{-\alpha/\beta}(\beta-\alpha)^{\alpha /\beta-1}( \vartheta q_{i})^{\alpha/\beta} \vert f_{i}\vert ^{1-\alpha/\beta}, \end{aligned} $$
(2.26)
then (1.4) is oscillatory.

Proof

Suppose that there exists a nonoscillatory solution \(x(t)\) of (1.4). Then there exists a \(T_{0}\geq t_{0}\) such that \(x(t)\neq0\) on \([T_{0},\infty)\). Define
$$ w(t)=: \frac{r(t)k_{1}(x(t),x'(t))}{\varphi_{\alpha}(x(t))}, \quad t\geq T_{0}. $$
(2.27)
Differentiating (2.27) and using (1.4) for \(t\geq T_{0}\), we have, for \(t\in\{I_{1}\cup I_{2}\}\) and \(t\neq t_{i}\),
$$\begin{aligned} w'(t) =& -\frac{p(t)k_{2}(x(t),x'(t))x'(t)}{\varphi_{\alpha}(x(t))} -q(t)\frac{g(x(t))}{\varphi_{\alpha}(x(t))} \\ &{}- \frac{\alpha r(t)k_{1}(x(t),x'(t))x'(t)}{x(t)\varphi_{\alpha}(x(t))} +\frac{f(t)}{\varphi_{\alpha}(x(t))}, \end{aligned}$$
(2.28)
and, for \(t= t_{i}\),
$$ \Delta w(t)=-q_{i}\frac{g(x(t))}{\varphi_{\alpha}(x(t))} + \frac{f_{i}}{\varphi_{\alpha}(x(t))}. $$
(2.29)
If \(x(t)>0\), we choose the interval \(I_{1}\) to consider. By using assumptions (A1), (A2), (A5)-(A7), (A9), (A10), and from (2.28), we obtain, for \(t\in I_{1}\),
$$ w'(t) \leq - \bigl(\vartheta q(t)\bigl\vert x(t) \bigr\vert ^{\beta-\alpha} +\bigl\vert f(t)\bigr\vert \bigl\vert x(t)\bigr\vert ^{-\alpha} \bigr) -\frac{\xi p(t)+\alpha\mu r(t)}{(r(t))^{1+1/\alpha}} \bigl\vert w(t)\bigr\vert ^{1+1/\alpha},\quad t\neq t_{i}. $$
(2.30)
Defining the functions
$$ Q(x):=\vartheta q|x|^{\beta-\alpha}+|f||x|^{-\alpha} $$
and
$$ \tilde{q}(t):=\beta\alpha^{-\alpha/\beta}(\beta-\alpha)^{\alpha /\beta-1}\bigl( \vartheta q(t)\bigr)^{\alpha/\beta} \bigl\vert f(t)\bigr\vert ^{1-\alpha/\beta}, $$
we easily see that, when \(\beta>\alpha\),
$$ Q(x)\geq\min_{x\in\mathbb{R}\setminus\{0\}}Q(x)=\tilde{q}(t) $$
(2.31)
and when \(\beta=\alpha\),
$$ Q(x)=\vartheta q(t)=\tilde{q}(t). $$
(2.32)
Applying (2.31) and (2.32) into (2.30), we have
$$ w'(t) \leq -\tilde{q}(t)-\frac{\xi p(t)+\alpha\mu r(t)}{(r(t))^{1+1/\alpha}} \bigl\vert w(t)\bigr\vert ^{1+1/\alpha}, \quad t\neq t_{i}. $$
(2.33)
On the other hand, from (2.29) we have, for \(t \in I_{1}\),
$$ \Delta w(t) \leq- \bigl(\vartheta q_{i}\bigl\vert x(t) \bigr\vert ^{\beta-\alpha}+\vert f_{i}\vert \bigl\vert x(t)\bigr\vert ^{-\alpha } \bigr) \leq-\tilde{q}_{i}, \quad t= t_{i}, $$
(2.34)
where \(\tilde{q}_{i}=\beta\alpha^{-\alpha/\beta}(\beta-\alpha)^{\alpha /\beta-1}(\vartheta q_{i})^{\alpha/\beta} |f_{i}|^{1-\alpha/\beta}\).
Using Lemma 2.1 to (2.33) and (2.44), for \(H_{1}\in\mathscr{H}_{(a_{1},b_{1})}\) and \(c_{1}\in(a_{1},b_{1})\setminus\{t_{i}\}\), we have
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})} \int^{c_{1}}_{a_{1}} \bigl[\tilde{p}(s) \tilde{H}_{11}^{\alpha+1}(s,a_{1})-\tilde{q}(s) H^{\alpha +1}_{1}(s,a_{1}) \bigr]\,\mathrm{d}s \\& \qquad {} +\frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})} \int^{b_{1}}_{c_{1}} \bigl[\tilde{p}(s) \tilde{H}_{12}^{\alpha+1}(b_{1},s)-\tilde{q}(s) H^{\alpha +1}_{1}(b_{1},s) \bigr] \,\mathrm{d}s \\& \quad \geq \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})}\sum_{a_{1}< t_{i}< c_{1}}H^{\alpha+1}_{1}(t_{i},a_{1}) \tilde{ q}_{i} +\frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})}\sum_{c_{1}< t_{i}< b_{1}}H^{\alpha +1}_{1}(b_{1},t_{i}) \tilde{ q}_{i} , \end{aligned}$$
where \(\tilde{p}(s)\), \(\tilde{H}_{11}(s,a_{1})\) and \(\tilde{H}_{12}(b_{1},s)\) are defined by (2.25). This contradicts (2.24) for \(j=1\).

If \(x(t)<0\), we choose the interval \(I_{2}\) to consider. Similar to the above proof we can also obtain a contradiction to (2.24) for \(j=2\). The proof is complete. □

In the next theorem we will consider the special case of (1.4) of the following form:
$$ \left \{ \textstyle\begin{array}{l} (r(t)k_{1}(x(t),x'(t)))'+p(t)k_{1}(x(t),x'(t))+q(t)g(x(t))=f(t), \quad t\neq t_{i}, \\ \Delta(r(t)k_{1}(x(t),x'(t)))+q_{i}g(x(t))=f_{i}, \quad t= t_{i}, i=1,2,\ldots. \end{array}\displaystyle \right . $$
(2.35)
In this theorem the sign condition for \(p(t)\) will be eliminated.

Theorem 2.4

Assume that (A1), (A2), (A9) hold. For any given \(T_{0}\geq t_{0}\), there exist intervals \(I_{j}=[a_{j},b_{j}]\subset [T_{0},\infty)\), \(j=1,2\), such that (A6) and (A7) hold. If there exist \(H_{j}\in\mathscr{H}_{(a_{j},b_{j})}\) and \(c_{j}\in(a_{j},b_{j})\setminus\{t_{i}\}\), \(j=1,2\) such that
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})} \int^{c_{j}}_{a_{j}} \bigl[\tilde{r}(s) \tilde{H}_{j1}^{\alpha+1}(s,a_{j})-\tilde{q}(s) H^{\alpha +1}_{j}(s,a_{j}) \bigr] \,\mathrm{d}s \\& \qquad {}+\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})} \int^{b_{j}}_{c_{j}} \bigl[\tilde{r}(s) \tilde{H}_{j2}^{\alpha+1}(b_{j},s)-\tilde{q}(s) H^{\alpha +1}_{j}(b_{j},s) \bigr] \,\mathrm{d}s \\& \quad < \frac{1}{H^{\alpha+1}_{j}(c_{j},a_{j})}\sum_{a_{j}< t_{i}< c_{j}}H^{\alpha +1}_{j}(t_{i},a_{j}) \tilde{q}_{i} +\frac{1}{H^{\alpha+1}_{j}(b_{j},c_{j})}\sum_{c_{j}< t_{i}< b_{j}}H^{\alpha +1}_{j}(b_{j},t_{i}) \tilde{q}_{i} , \end{aligned}$$
(2.36)
where
$$\begin{aligned}& \tilde{r}(s)=\frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}\frac {r(s)}{(\alpha\mu)^{\alpha}}, \\& \tilde{H}_{j1}(s,a_{j})= \biggl\vert ( \alpha+1)h_{j1}(s,a_{j})\sqrt{H_{j}(s,a_{j})}- \frac {p(s)}{r(s)}H_{j}(s,a_{j})\biggr\vert , \\& \tilde{H}_{j2}(b_{j},s)= \biggl\vert ( \alpha+1)h_{j2}(b_{j},s)\sqrt{H_{j}(b_{j},s)}+ \frac {p(s)}{r(s)}H_{j}(b_{j},s)\biggr\vert , \end{aligned}$$
(2.37)
and
$$\begin{aligned}& \tilde{q}(s)=\beta\alpha^{-\alpha/\beta}(\beta-\alpha)^{\alpha /\beta-1}\bigl( \vartheta q(s)\bigr)^{\alpha/\beta} \bigl\vert f(s)\bigr\vert ^{1-\alpha/\beta}, \\& \tilde{q}_{i}=\beta\alpha^{-\alpha/\beta}(\beta-\alpha)^{\alpha /\beta-1}( \vartheta q_{i})^{\alpha/\beta} \vert f_{i}\vert ^{1-\alpha/\beta}, \end{aligned}$$
then (1.4) is oscillatory.

Proof

Suppose that there exists a nonoscillatory solution \(x(t)\) of (1.4). Then there exists a \(T_{0}\geq t_{0}\) such that \(x(t)\neq0\) on \([T_{0},\infty)\). Similar to the proof of Theorem 2.3 we have
$$\begin{aligned} w'(t) =&-q(t)\frac{g(x(t))}{\varphi_{\alpha}(x(t))}-\frac {p(t)k_{1}(x(t),x'(t))}{\varphi_{\alpha}(x(t))} \\ &{}- \frac{\alpha r(t)k_{1}(x(t),x'(t))x'(t)}{x(t)\varphi_{\alpha}(x(t))}+\frac{f(t)}{\varphi_{\alpha}(x(t))},\quad t\neq t_{i}, \end{aligned}$$
(2.38)
and
$$ \Delta w(t)=-q_{i}\frac{g(x(t))}{\varphi_{\alpha}(x(t))} + \frac{f_{i}}{\varphi_{\alpha}(x(t))}, \quad t= t_{i}, $$
(2.39)
where \(w(t)\) is defined as of (2.27). If \(x(t)>0\), we choose the interval \(I_{1}\) to consider. By using assumptions (A1), (A2), (A6), (A7), and (A9) and from (2.38), we obtain, for \(t\in I_{1}\),
$$\begin{aligned} w'(t) \leq&- \bigl(\vartheta q(t)\bigl\vert x(t) \bigr\vert ^{\beta-\alpha}+\bigl\vert f(t)\bigr\vert \bigl\vert x(t)\bigr\vert ^{-\alpha} \bigr) \\ &{}-\frac{p(t)}{r(t)}w(t) -\frac{\alpha\mu}{r^{1/\alpha}(t)} \bigl\vert w(t)\bigr\vert ^{1+1/\alpha},\quad t\neq t_{i}, \end{aligned}$$
(2.40)
and
$$ \Delta w(t) \leq- \bigl(\vartheta q_{i}\bigl\vert x(t) \bigr\vert ^{\beta-\alpha}+\vert f_{i}\vert \bigl\vert x(t)\bigr\vert ^{-\alpha } \bigr),\quad t=t_{i}. $$
(2.41)
Then we have
$$ w'(t) \leq -\tilde{q}(t)-\frac{p(t)}{r(t)}w(t) - \frac{\alpha\mu}{r^{1/\alpha}(t)} \bigl\vert w(t)\bigr\vert ^{1+1/\alpha}, \quad t\neq t_{i}, $$
(2.42)
and
$$ \Delta w(t) \leq-\tilde{q}_{i}, \quad t= t_{i}, $$
(2.43)
where
$$ \begin{aligned} &\tilde{q}(t)=\beta\alpha^{-\alpha/\beta}(\beta- \alpha)^{\alpha /\beta-1}\bigl(\vartheta q(t)\bigr)^{\alpha/\beta} \bigl\vert f(t) \bigr\vert ^{1-\alpha/\beta}, \\ &\tilde{q}_{i}=\beta\alpha^{-\alpha/\beta}(\beta-\alpha)^{\alpha /\beta-1}( \vartheta q_{i})^{\alpha/\beta} \vert f_{i}\vert ^{1-\alpha/\beta}. \end{aligned} $$
(2.44)
Using Lemma 2.1 in (2.42) and (2.43) we obtain a contradiction to (2.36) for \(t\in I_{1}\).

If \(x(t)<0\), we choose the interval \(I_{2}\) to consider. Similar to the above proof we can also obtain a contradiction to (2.36) for \(j=2\). The proof is complete. □

3 Examples

In this section, we give two examples to illustrate the effectiveness and non-emptiness of our results. We will see that if there is no impulse then no oscillation conclusion can be drawn.

Example 3.1

Consider second-order nonlinear impulsive differential equations
$$ \left \{ \textstyle\begin{array}{l} (k_{1}(x(t),x'(t)))'+(\cos t) k_{2}(x(t),x'(t))x'(t)+(\cos t) g(x(t))=\sin (3t), \quad t\neq t_{i}, \\ \Delta k_{1}(x(t),x'(t))+\gamma(\cos t) g(x(t))= \lambda\sin(3t), \quad t= t_{i}, \end{array}\displaystyle \right . $$
(3.1)
where γ, λ are positive real numbers, and
$$ k_{1}(u,v)=\frac{u^{2/3}\varphi_{1/3}(v)}{(1+u^{2})^{1/3}},\qquad k_{2}(u,v)= \frac{u^{5/3}\varphi_{1/3}(v)}{(1+u^{2})^{4/3}},\qquad g(u)=\varphi_{1/3}(u). $$
Then it is easy to see \(\mu=\tau=1\), \(\varsigma=\alpha=1/3\). For any given \(T_{0}\geq0\) we may choose \(n\in\mathbb{N}\) sufficiently large so that \(n\pi\geq T_{0}\). If we let \(I_{1}=[a_{1},b_{1}]=[2n\pi-\pi/3,2n\pi]\), \(I_{2}=[a_{2},b_{2}]=[2n\pi,2n\pi+\pi/3]\), \(c_{1}=2n\pi-5\pi/24\), \(c_{2}=2n\pi+5\pi/24\), impulsive points \(t_{i,j}=2n\pi+(-1)^{j} i\pi/12\) (\(i=1,2,3\), \(j=1,2\)), then the conditions (A1)-(A6) hold, \(t_{i,1}\in[a_{1},b_{1}]\) and \(t_{i,2}\in[a_{2},b_{2}]\) for \(i=1,2,3\). Letting \(\rho(t)=1\) and \(H_{j}(t,s)=(t-s)^{2}\) for \(j=1,2\), we have
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})} \int^{c_{1}}_{a_{1}} \bigl[\tilde {p}(s) \tilde{H}_{11}^{\alpha+1}(s,a_{1}) -\rho(s)q(s) H^{\alpha+1}_{1}(s,a_{1}) \bigr]\,\mathrm{d}s \\& \quad = \biggl(\frac{8}{\pi} \biggr)^{8/3} \int^{2n\pi-5\pi/24}_{2n\pi-\pi/3} \biggl[\frac{2^{4/3}(s-2n\pi+\pi/3)^{4/3}}{(3\cos s +1)^{1/3}} -(s-2n\pi+ \pi/3)^{8/3}\cos s \biggr] \,\mathrm{d}s \\& \quad = \biggl(\frac{8}{\pi} \biggr)^{8/3} \int^{\pi/8}_{0} \biggl[\frac{(2s)^{4/3}}{(3\cos(s-{\pi}/3) +1)^{1/3}} -s^{8/3}\cos (s-{\pi}/3) \biggr] \,\mathrm{d}s \approx0.929, \\& \frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})} \int^{b_{1}}_{c_{1}} \bigl[\tilde {p}(s) \tilde{H}_{12}^{\alpha+1}(b_{1},s) -\rho(s)q(s) H^{\alpha+1}_{1}(b_{1},s) \bigr] \,\mathrm{d}s \\& \quad = \biggl(\frac{24}{5\pi} \biggr)^{8/3} \int^{5\pi/24}_{0} \biggl[\frac{(2s)^{4/3}}{(3\cos s +1)^{1/3}} -s^{8/3}\cos s \biggr] \,\mathrm{d}s\approx0.653, \\& \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})}\sum_{a_{1}< t_{i,1}< c_{1}}H^{\alpha +1}_{1}(t_{i,1},a_{1}) \rho(t_{i,1}) q_{i,1} = \biggl(\frac{8}{\pi} \biggr)^{8/3}(t_{i,1}-a_{1})^{8/3}\gamma\cos t_{i,1} \\& \quad = \gamma \biggl(\frac{8}{\pi} \biggr)^{8/3} \biggl( \frac{\pi}{12} \biggr)^{8/3} \cos\biggl(-\frac{\pi}{4}\biggr) \approx0.240\gamma, \\& \frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})}\sum_{c_{1}< t_{i,1}< b_{1}}H^{\alpha +1}_{1}(t_{i,1},a_{1}) \rho(t_{i,1}) q_{i,1} \\& \quad = \gamma \biggl(\frac{24}{5\pi} \biggr)^{8/3} \biggl[ \biggl( \frac{\pi }{12} \biggr)^{8/3} \cos\biggl(\frac{\pi}{12}\biggr)+ \biggl(\frac{2\pi}{12} \biggr)^{8/3}\cos \biggl(\frac{2\pi}{12} \biggr) \biggr]\approx0.562\gamma. \end{aligned}$$
From (2.11) for \(j=1\), we have
$$ \gamma>1.905. $$
(3.2)
For \(j=2\), by similarly calculating we also obtain the above condition (3.2). It follows from Theorem 2.1 that (3.1) is oscillatory if \(\gamma>1.905\).

Example 3.2

Consider the second-order nonlinear impulsive differential equations
$$ \left \{ \textstyle\begin{array}{l} ((2+\sin t)k_{1}(x(t),x'(t)))'+(2-\sin t)k_{2}(x(t),x'(t))x'(t)+|\tan t|g(x(t)) \\ \quad =\sin t, \quad t\neq t_{i}, \\ \Delta((2+\sin t)k_{1}(x(t),x'(t)))+\gamma(2-\sin t)g(x(t))=\lambda \sin t, \quad t=t_{i}, \end{array}\displaystyle \right . $$
(3.3)
where γ, λ are positive real numbers, and
$$ k_{1}(u,v)=\frac{u^{2}v}{1+u^{2}},\qquad k_{2}(u,v)= \frac{u^{3}v(1+u^{2}+v^{2})}{(1+u^{2})^{2}},\qquad g(u)=10^{-3}\varphi_{2}(u). $$
If we let \(I_{1}=[(2n-1)\pi,2n\pi]\), \(I_{2}=[2n\pi,(2n+1)\pi]\), \(t_{i}=(2n+(-1)^{i}/2)\pi\) (for \(i=1,2\)), and \(H(t,s)=t-s\), it is easy to see \(\mu=\xi=1\), \(\vartheta=10^{-2}\), \(\alpha=1\), \(\beta=2\), and the conditions (A1), (A2), (A5)-(A9) hold. For any given \(T_{0}\geq0\) we may choose \(n\in\mathbb{N}\) sufficiently large so that \((2n-1)\pi\geq T_{0}\). Letting \(\rho(t)=1\), \(c_{1}=(2n-1/4)\pi\), we have
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})} \int^{c_{1}}_{a_{1}} \bigl[\tilde {p}(s) \tilde{H}_{11}^{\alpha+1}(s,a_{1}) -\rho(s)\tilde{q}(s) H^{\alpha+1}_{1}(s,a_{1}) \bigr]\,\mathrm{d}s \\& \qquad {}+ \frac{1}{H^{\alpha+1}_{1}(b_{1},c_{1})} \int^{b_{1}}_{c_{1}} \bigl[\tilde {p}(s) \tilde{H}_{12}^{\alpha+1}(b_{1},s) -\rho(s)\tilde{q}(s) H^{\alpha+1}_{1}(b_{1},s) \bigr] \,\mathrm{d}s \\& \quad = \biggl(\frac{4}{3\pi} \biggr)^{2} \int^{3\pi/4}_{0} \biggl[\frac{(2-\sin s)^{2}}{4}- \frac{2 \sin s}{10\sqrt{\cos {s}}} s^{2} \biggr]\,\mathrm{d}s \\& \qquad {}+ \biggl( \frac{4}{\pi} \biggr)^{2} \int^{\pi/4}_{0} \biggl[\frac{(2-\sin s)^{2}}{4}- \frac{2 \sin s}{10\sqrt{\cos {s}}} s^{2} \biggr]\,\mathrm{d}s \approx 0.927, \\& \frac{1}{H^{\alpha+1}_{2}(c_{2},a_{2})} \int^{c_{2}}_{a_{2}} \bigl[\tilde {p}(s) \tilde{H}_{21}^{\alpha+1}(s,a_{1}) -\rho(s)\tilde{q}(s) H^{\alpha+1}_{2}(s,a_{2}) \bigr]\,\mathrm{d}s \\& \qquad {} + \frac{1}{H^{\alpha+1}_{2}(b_{2},c_{2})} \int^{b_{2}}_{c_{2}} \bigl[\tilde {p}(s) \tilde{H}_{22}^{\alpha+1}(b_{2},s) -\rho(s)\tilde{q}(s) H^{\alpha+1}_{2}(b_{2},s) \bigr]\,\mathrm{d}s \\& \quad = \biggl(\frac{4}{\pi} \biggr)^{2} \int^{\pi/4}_{0} \biggl[\frac{(2+\sin s)^{2}}{4}- \frac{2 \sin s}{10\sqrt{\cos {s}}} s^{2} \biggr]\,\mathrm{d}s \\& \qquad {}+ \biggl( \frac{4}{3\pi} \biggr)^{2} \int^{3\pi/4}_{0} \biggl[\frac{(2+\sin s)^{2}}{4}- \frac{2 \sin s}{10\sqrt{\cos {s}}} s^{2} \biggr]\,\mathrm{d}s \approx 2.492, \\& \frac{1}{H^{\alpha+1}_{1}(c_{1},a_{1})}\sum_{a_{1}< t_{i}< c_{1}}H^{\alpha +1}_{1}(t_{i},a_{1}) \rho(t_{i}) \tilde{q}_{i} \\& \quad = \biggl(\frac{4}{3\pi} \biggr)^{2} \biggl( \frac{\pi}{2} \biggr)^{2}\frac{2}{10}\sqrt{3\gamma}\sqrt{ \lambda} \approx 0.154\sqrt{\gamma\lambda}, \end{aligned}$$
and
$$\begin{aligned}& \frac{1}{H^{\alpha+1}_{2}(b_{2},c_{2})}\sum_{c_{2}< t_{i}< b_{2}}H^{\alpha +1}_{2}(b_{2},t_{i}) \rho(t_{i}) \tilde{q}_{i} \\& \quad = \biggl(\frac{4}{3\pi} \biggr)^{2} \biggl( \frac{\pi}{2} \biggr)^{2}\frac{2}{10}\sqrt{\gamma}\sqrt{ \lambda} \approx 0.889\sqrt{\gamma\lambda}. \end{aligned}$$
It follows from Theorem 2.3 that (3.3) is oscillatory if \(\gamma\lambda>36.228\).

Declarations

Acknowledgements

The authors thank the main editor and anonymous referees for their valuable comments and suggestions, leading to improvement of this paper. This work has been supported by the NNSF of China (Grant 11561019), the Innovation and Developing School Project (Grant 2014KZDXM065) and the Science Innovation Project (Grant 2013KJCX0125) of Department of Education of Guangdong Province, the Training Program Project for OYTCU in Guangdong Province (Grant Yq2014116), and the NSFP of Lingnan Normal University (Grant ZL1303).

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)
Department of Mathematics, Lingnan Normal University, Zhanjiang, Guangdong, 524048, P.R. China
(2)
Department of Mathematics, Hechi University, Yizhou, Guangxi, 546300, P.R. China

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