We need the following auxiliary lemmas.

**Lemma 2.1** *Assume that* (1.2)

*is satisfied and let* $x(t)$ *be an eventually positive solution of* (1.1).

*Then there exists a sufficiently large* ${t}_{1}\ge {t}_{0}$ *such that*,

*for all* $t\ge {t}_{1}$,

$x(t)>0,\phantom{\rule{2em}{0ex}}{x}^{(n-1)}(t)>0,\phantom{\rule{2em}{0ex}}{x}^{(n)}(t)<0.$

(2.1)

*Proof* Let

$x(t)$ be an eventually positive solution of (1.1). Then there exists a

${T}_{0}\ge {t}_{0}$ such that

$x(t)>0$ and

$x(g(t))>0$ for all

$t\ge {T}_{0}$. By virtue of (1.1),

${(r(t){({x}^{(n-1)}(t))}^{\gamma})}^{\prime}+p(t){({x}^{(n-1)}(t))}^{\gamma}<0.$

Thus,

${(exp\left({\int}_{{t}_{0}}^{t}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)r(t){({x}^{(n-1)}(t))}^{\gamma})}^{\prime}<0,$

(2.2)

which means that the function

$exp\left({\int}_{{t}_{0}}^{t}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)r(t){({x}^{(n-1)}(t))}^{\gamma}$

is decreasing for $t\ge {T}_{0}$. Therefore, ${x}^{(n-1)}(t)$ does not change sign eventually, that is, there exists a ${t}_{1}\ge {T}_{0}$ such that either ${x}^{(n-1)}(t)>0$ or ${x}^{(n-1)}(t)<0$ for all $t\ge {t}_{1}$.

We claim that

${x}^{(n-1)}(t)>0$ for all

$t\ge {t}_{1}$. Otherwise, there should exist a

$T\ge {t}_{1}$ such that

$exp\left({\int}_{{t}_{0}}^{T}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)r(T){({x}^{(n-1)}(T))}^{\gamma}=Mexp\left({\int}_{{t}_{0}}^{T}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)<0$

and, for all

$t\ge T$,

$exp\left({\int}_{{t}_{0}}^{t}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)r(t){({x}^{(n-1)}(t))}^{\gamma}\le Mexp\left({\int}_{{t}_{0}}^{T}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)<0,$

(2.3)

where

$M:=r(T){({x}^{(n-1)}(T))}^{\gamma}.$

Inequality (2.3) yields

${x}^{(n-1)}(t)\le {M}^{1/\gamma}{[\frac{1}{r(t)}exp(-{\int}_{T}^{t}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau )]}^{1/\gamma}.$

Integrating this inequality from

${T}_{1}$ to

*t*,

${T}_{1}\ge T$, we conclude that

${x}^{(n-2)}(t)\le {x}^{(n-2)}({T}_{1})+{M}^{1/\gamma}{\int}_{{T}_{1}}^{t}{[\frac{1}{r(s)}exp(-{\int}_{T}^{s}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau )]}^{1/\gamma}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$

Passing to the limit as

$t\to \mathrm{\infty}$ and using (1.2), we deduce that

$\underset{t\to \mathrm{\infty}}{lim}{x}^{(n-2)}(t)=-\mathrm{\infty}.$

It follows now from the inequalities

${x}^{(n-1)}(t)<0$ and

${x}^{(n-2)}(t)<0$ that

$x(t)<0$, which contradicts our assumption that

$x(t)>0$. Finally, write (2.2) in the form

$\begin{array}{r}exp\left({\int}_{{t}_{0}}^{t}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right)[{r}^{\prime}(t)+p(t)]{({x}^{(n-1)}(t))}^{\gamma}\\ \phantom{\rule{1em}{0ex}}+\gamma r(t)exp\left({\int}_{{t}_{0}}^{t}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right){({x}^{(n-1)}(t))}^{\gamma -1}{x}^{(n)}(t)<0,\end{array}$

which implies that ${x}^{(n)}(t)<0$. This completes the proof. □

**Lemma 2.2** (Agarwal *et al.* [3])

*Assume that* $u\in {\mathrm{C}}^{n}([{t}_{0},\mathrm{\infty}),{\mathbb{R}}^{+})$,

${u}^{(n)}(t)$ *is non*-

*positive for all large* *t* *and not identically zero on* $[{t}_{0},\mathrm{\infty})$.

*If* ${lim}_{t\to \mathrm{\infty}}u(t)\ne 0$,

*then for every* $\lambda \in (0,1)$,

*there exists a* ${t}_{\lambda}\in [{t}_{0},\mathrm{\infty})$ *such that* $u(t)\ge \frac{\lambda}{(n-1)!}{t}^{n-1}|{u}^{(n-1)}(t)|$

*holds on* $[{t}_{\lambda},\mathrm{\infty})$.

**Lemma 2.3** (Agarwal *et al.* [4])

*The equation*
${(r(t){({x}^{\prime}(t))}^{\gamma})}^{\prime}+a(t){x}^{\gamma}(t)=0,$

*where* $\gamma >0$ *is a quotient of odd natural numbers*,

$r\in {\mathrm{C}}^{1}([{t}_{0},\mathrm{\infty}),(0,\mathrm{\infty}))$,

*and* $a\in \mathrm{C}([{t}_{0},\mathrm{\infty}),\mathbb{R})$ *is non*-

*oscillatory if and only if there exist a number* $T\ge {t}_{0}$ *and a function* $v\in {\mathrm{C}}^{1}([T,\mathrm{\infty}),\mathbb{R})$ *such that*,

*for all* $t\ge T$,

${v}^{\prime}(t)+\gamma \frac{{v}^{(\gamma +1)/\gamma}(t)}{{r}^{1/\gamma}(t)}+a(t)\le 0.$

For a compact presentation of our results, we introduce the following notation:

$\begin{array}{c}E(k,l):=exp\left({\int}_{k}^{l}\frac{p(\tau )}{r(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right),\phantom{\rule{2em}{0ex}}\delta (t):={\int}_{t}^{\mathrm{\infty}}\frac{\mathrm{d}s}{{(r(s)E({t}_{0},s))}^{1/\gamma}},\hfill \\ \phi (t):=\frac{p(t)}{r(t)}+\frac{{\gamma}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}}\frac{{\varphi}_{+}^{\gamma +1}(t)E({t}_{0},t)}{\delta (t){r}^{1/\gamma}(t)},\hfill \\ \varphi (t):=\frac{1}{{E}^{1/\gamma}({t}_{0},t)}-\frac{1}{\gamma}\delta (t)p(t){r}^{(1-\gamma )/\gamma}(t),\phantom{\rule{2em}{0ex}}{\varphi}_{+}(t):=max[0,\varphi (t)].\hfill \end{array}$

**Theorem 2.4**
*Assume that*
$\frac{1}{{((n-1)!)}^{\gamma}}\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}inf}{\int}_{g(t)}^{t}\frac{q(s)}{r(g(s))}{({g}^{n-1}(s))}^{\gamma}E(g(s),s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s>\frac{1}{\mathrm{e}}.$

(2.4)

*Then every solution* $x(t)$ *of* (1.1)

*is either oscillatory or satisfies* $\underset{t\to \mathrm{\infty}}{lim}x(t)=0$

(2.5)

*provided that either*
- (i)

- (ii)
(1.4)

*is satisfied and*,

*for some* ${\lambda}_{1}\in (0,1)$,

$\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{0}}^{t}[q(s){(\frac{{\lambda}_{1}}{(n-2)!}{g}^{n-2}(s)\delta (s))}^{\gamma}E({t}_{0},s)-\phi (s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty}.$

(2.6)

*Proof* Assume that (1.1) has a non-oscillatory solution

$x(t)$ which is eventually positive and such that

$\underset{t\to \mathrm{\infty}}{lim}x(t)\ne 0.$

(2.7)

Case (i) By Lemma 2.1, we conclude that (2.1) holds for all

$t\ge {t}_{1}$, where

${t}_{1}\ge {t}_{0}$ is sufficiently large. It follows from Lemma 2.2 that

$x(t)\ge \frac{\lambda {t}^{n-1}}{(n-1)!}{x}^{(n-1)}(t)=\frac{\lambda {t}^{n-1}}{(n-1)!{r}^{1/\gamma}(t)}{r}^{1/\gamma}(t){x}^{(n-1)}(t),$

for every

$\lambda \in (0,1)$ and for all sufficiently large

*t*. Let

$y(t):=r(t){({x}^{(n-1)}(t))}^{\gamma}.$

By virtue of (1.1), we conclude that

$y(t)$ is a positive solution of a differential inequality

${y}^{\prime}(t)+\frac{p(t)}{r(t)}y(t)+q(t){\left(\frac{\lambda {g}^{n-1}(t)}{(n-1)!{r}^{1/\gamma}(g(t))}\right)}^{\gamma}y(g(t))\le 0.$

However, it follows from the result due to Werbowski [[15], Corollary 1] that the latter inequality does not have positive solutions under the assumption (2.4), which is a contradiction. The proof of part (i) is complete.

Case (ii) Similar analysis to that in Lemma 2.1 leads to the conclusion that a non-oscillatory positive solution with the property (2.7) satisfies, for

$t\ge {t}_{1}$, either conditions (2.1) or

$x(t)>0,\phantom{\rule{2em}{0ex}}{x}^{(n-2)}(t)>0,\phantom{\rule{2em}{0ex}}{x}^{(n-1)}(t)<0,$

(2.8)

where

${t}_{1}\ge {t}_{0}$ is sufficiently large. Assume first that (2.1) holds. As in the proof of the part (i), one arrives at a contradiction with the condition (2.4). Suppose now that (2.8) holds. For

$t\ge {t}_{1}$, define a new function

$v(t)$ by

$v(t):=\frac{r(t){({x}^{(n-1)}(t))}^{\gamma}}{{({x}^{(n-2)}(t))}^{\gamma}}.$

(2.9)

Then

$v(t)<0$ for

$t\ge {t}_{1}$. Since

${(r(t){({x}^{(n-1)}(t))}^{\gamma}E({t}_{0},t))}^{\prime}=-q(t){x}^{\gamma}(g(t))E({t}_{0},t)<0,$

we deduce that the function

$r(t){({x}^{(n-1)}(t))}^{\gamma}E(t,{t}_{0})$ is decreasing. Thus, for

$\phantom{\rule{0.25em}{0ex}}s\ge t\ge {t}_{1}$,

${(r(s)E({t}_{0},s))}^{1/\gamma}{x}^{(n-1)}(s)\le {(r(t)E({t}_{0},t))}^{1/\gamma}{x}^{(n-1)}(t).$

(2.10)

Dividing both sides of (2.10) by

${(r(s)E({t}_{0},s))}^{1/\gamma}$ and integrating the resulting inequality from

*t* to

*T*, we obtain

${x}^{(n-2)}(T)\le {x}^{(n-2)}(t)+{(r(t)E({t}_{0},t))}^{1/\gamma}{x}^{(n-1)}(t){\int}_{t}^{T}\frac{\mathrm{d}s}{{(r(s)E({t}_{0},s))}^{1/\gamma}}.$

Letting

$T\to \mathrm{\infty}$ and taking into account that

${x}^{(n-1)}(t)<0$ and

${x}^{(n-2)}(t)>0$, we conclude that

$\underset{T\to \mathrm{\infty}}{lim}{x}^{(n-2)}(T)\ge 0.$

Hence,

$0\le {x}^{(n-2)}(t)+{(r(t)E({t}_{0},t))}^{1/\gamma}{x}^{(n-1)}(t)\delta (t),$

which yields

$-\frac{{x}^{(n-1)}(t)}{{x}^{(n-2)}(t)}\delta (t){(r(t)E({t}_{0},t))}^{1/\gamma}\le 1.$

Thus, by (2.9), we conclude that

$-v(t){\delta}^{\gamma}(t)E({t}_{0},t)\le 1.$

(2.11)

Differentiation of (2.9) yields

${v}^{\prime}(t)=\frac{{(r(t){({x}^{(n-1)}(t))}^{\gamma})}^{\prime}}{{({x}^{(n-2)}(t))}^{\gamma}}-\gamma \frac{r(t){({x}^{(n-1)}(t))}^{\gamma +1}}{{({x}^{(n-2)}(t))}^{\gamma +1}}.$

It follows now from (1.1) and (2.9) that

${v}^{\prime}(t)=-p(t)\frac{v(t)}{r(t)}-q(t)\frac{{x}^{\gamma}(g(t))}{{({x}^{(n-2)}(t))}^{\gamma}}-\gamma \frac{{v}^{(\gamma +1)/\gamma}(t)}{{r}^{1/\gamma}(t)}.$

On the other hand, it follows from Lemma 2.2 that

$x(t)\ge \frac{\lambda}{(n-2)!}{t}^{n-2}{x}^{(n-2)}(t),$

for every

$\lambda \in (0,1)$ and for all sufficiently large

*t*. Therefore, (2.11) yields

$\begin{array}{rl}{v}^{\prime}(t)& \le \frac{p(t)}{r(t){\delta}^{\gamma}(t)E({t}_{0},t)}-q(t){\left(\frac{x(g(t))}{{x}^{(n-2)}(g(t))}\right)}^{\gamma}{\left(\frac{{x}^{(n-2)}(g(t))}{{x}^{(n-2)}(t)}\right)}^{\gamma}-\gamma {\left(\frac{{v}^{\gamma +1}(t)}{r(t)}\right)}^{1/\gamma}\\ \le \frac{p(t)}{r(t){\delta}^{\gamma}(t)E({t}_{0},t)}-q(t){(\frac{\lambda}{(n-2)!}{g}^{n-2}(t))}^{\gamma}-\gamma {\left(\frac{{v}^{\gamma +1}(t)}{r(t)}\right)}^{1/\gamma}.\end{array}$

(2.12)

Multiplying (2.12) by

${\delta}^{\gamma}(t)E({t}_{0},t)$ and integrating the resulting inequality from

${t}_{1}$ to

*t*, we have

$\begin{array}{r}{\delta}^{\gamma}(t)E({t}_{0},t)v(t)-{\delta}^{\gamma}({t}_{1})E({t}_{0},{t}_{1})v({t}_{1})-{\int}_{{t}_{1}}^{t}\frac{p(s)}{r(s)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}+\gamma {\int}_{{t}_{1}}^{t}{r}^{-1/\gamma}(s){\delta}^{\gamma -1}(s)E({t}_{0},s){\varphi}_{+}(s)v(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{1}}^{t}q(s){(\frac{\lambda}{(n-2)!}{g}^{n-2}(s))}^{\gamma}{\delta}^{\gamma}(s)E({t}_{0},s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{1}}^{t}\gamma {\left(\frac{{v}^{\gamma +1}(s)}{r(s)}\right)}^{1/\gamma}{\delta}^{\gamma}(s)E({t}_{0},s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le 0.\end{array}$

Let

$A:={\delta}^{\gamma}(s)E({t}_{0},s){r}^{-1/\gamma}(s)$ and

$B:={r}^{-1/\gamma}(s){\delta}^{\gamma -1}(s)E({t}_{0},s){\varphi}_{+}(s)$. Using the fact that

${v}^{(\gamma +1)/\gamma}(s)={(-v(s))}^{(\gamma +1)/\gamma}$ and the inequality

$-Bv(s)-A{v}^{(\gamma +1)/\gamma}(s)\le \frac{{\gamma}^{\gamma}}{{(\gamma +1)}^{\gamma +1}}\frac{{B}^{\gamma +1}}{{A}^{\gamma}},\phantom{\rule{1em}{0ex}}A>0$

(see Zhang and Wang [[

19], Lemma 2.3] for details) and the definition of

*φ*, we derive from (2.11) that

${\int}_{{t}_{1}}^{t}[q(s){(\frac{\lambda}{(n-2)!}{g}^{n-2}(s))}^{\gamma}{\delta}^{\gamma}(s)E({t}_{0},s)-\phi (s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le {\delta}^{\gamma}({t}_{1})E({t}_{0},{t}_{1})v({t}_{1})+1,$

which contradicts (2.6). This completes the proof for the part (ii). □

**Remark 2.5** For a result similar to the one established in part (i) in Theorem 2.4, see also Zhang *et al.* [[16], Theorem 5.3].

**Remark 2.6** For $p(t)=0$, Theorem 2.4 includes Theorem 1.1.

In the remainder of this section, we use different approaches to arrive at the conclusion of Theorem 2.4. First, we employ the integral averaging technique to replace assumption (2.6) with a Philos-type condition.

To this end, let

$\mathbb{D}=\{(t,s):t\ge s\ge {t}_{0}\}$. We say that a function

$H\in \mathrm{C}(\mathbb{D},\mathbb{R})$ belongs to the class

${\mathcal{P}}_{\gamma}$ if

$H(t,t)=0,\phantom{\rule{1em}{0ex}}\text{for}t\ge {t}_{0},\phantom{\rule{2em}{0ex}}H(t,s)0,\phantom{\rule{1em}{0ex}}\text{for}ts\ge {t}_{0},$

and

*H* has a non-positive continuous partial derivative

$\partial H/\partial s$ with respect to the second variable satisfying the condition

$-\frac{\partial}{\partial s}H(t,s)=\xi (t,s){H}^{\gamma /(\gamma +1)}(t,s)$

for some function $\xi \in {L}_{\mathrm{loc}}(\mathbb{D},\mathbb{R})$.

**Theorem 2.7** *Let* $\delta (t)$ *be as in Theorem* 2.4

*and suppose that* (1.4)

*and* (2.4)

*hold*.

*Assume that there exists a function* $H\in {\mathcal{P}}_{\gamma}$ *such that* $\begin{array}{r}\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{1}}^{t}[H(t,s)q(s){(\frac{{\lambda}_{1}}{(n-2)!}{g}^{n-2}(s))}^{\gamma}\\ \phantom{\rule{1em}{0ex}}-\frac{H(t,s)p(s)}{r(s){\delta}^{\gamma}(s)E({t}_{0},s)}-\frac{r(s){(\xi (t,s))}^{\gamma +1}}{{(\gamma +1)}^{\gamma +1}}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s>0,\end{array}$

(2.13)

*for all* ${t}_{1}\ge {t}_{0}$ *and for some* ${\lambda}_{1}\in (0,1)$. *Then the conclusion of Theorem* 2.4 *remains intact*.

*Proof* Assuming that

$x(t)$ is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we arrive at the inequality (2.12) which holds for all

$\lambda \in (0,1)$. Multiplying (2.12) by

$H(t,s)$ and integrating the resulting inequality from

${t}_{1}$ to

*t*, we obtain

$\begin{array}{r}{\int}_{{t}_{1}}^{t}H(t,s)[q(s){\left(\frac{\lambda {g}^{n-2}(s)}{(n-2)!}\right)}^{\gamma}-\frac{p(s)}{r(s){\delta}^{\gamma}(s)E({t}_{0},s)}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}\le H(t,{t}_{1})v({t}_{1})+{\int}_{{t}_{1}}^{t}\frac{\partial H(t,s)}{\partial s}v(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s-{\int}_{{t}_{1}}^{t}\gamma H(t,s)\frac{{v}^{(\gamma +1)/\gamma}(s)}{{r}^{1/\gamma}(s)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}=H(t,{t}_{1})v({t}_{1})-{\int}_{{t}_{1}}^{t}\xi (t,s){H}^{\gamma /(\gamma +1)}(t,s)v(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s-{\int}_{{t}_{1}}^{t}\gamma H(t,s)\frac{{v}^{(\gamma +1)/\gamma}(s)}{{r}^{1/\gamma}(s)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\end{array}$

Let

$A:={(\gamma H(t,s)\frac{{(-v(s))}^{(\gamma +1)/\gamma}}{{r}^{1/\gamma}(s)})}^{\gamma /(\gamma +1)}$

and

$B:={\left(\frac{\gamma \xi (t,s){r}^{1/(\gamma +1)}(s)}{(\gamma +1){\gamma}^{\gamma /(\gamma +1)}}\right)}^{\gamma}.$

Using the inequality

$\frac{\gamma +1}{\gamma}A{B}^{1/\gamma}-{A}^{(\gamma +1)/\gamma}\le \frac{1}{\gamma}{B}^{(\gamma +1)/\gamma},$

we obtain

$\begin{array}{r}{\int}_{{t}_{1}}^{t}[H(t,s)q(s){(\frac{\lambda}{(n-2)!}{g}^{n-2}(s))}^{\gamma}-\frac{H(t,s)p(s)}{r(s){\delta}^{\gamma}(s)E({t}_{0},s)}-\frac{r(s){\xi}^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \phantom{\rule{1em}{0ex}}\le H(t,{t}_{1})v({t}_{1})<0,\end{array}$

which contradicts assumption (2.13). This completes the proof. □

Finally, we formulate also a comparison result for (1.1) that leads to the conclusion of Theorem 2.4.

**Theorem 2.8** *Let* $\delta (t)$ *be as above*,

*and assume that* (1.4)

*and* (2.4)

*hold*.

*If a second*-

*order half*-

*linear ordinary differential equation* ${(r(t){({u}^{\prime}(t))}^{\gamma})}^{\prime}+[q(t){(\frac{{\lambda}_{1}}{(n-2)!}{g}^{n-2}(t))}^{\gamma}-\frac{p(t)}{r(t){\delta}^{\gamma}(t)E({t}_{0},t)}]{u}^{\gamma}(t)=0$

(2.14)

*is oscillatory for some* ${\lambda}_{1}\in (0,1)$, *then the conclusion of Theorem* 2.4 *remains intact*.

*Proof* Assuming again that $x(t)$ is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we obtain (2.12) which holds for all $\lambda \in (0,1)$. By virtue of Lemma 2.3, we conclude that (2.14) is non-oscillatory, which is a contradiction. The proof is complete. □