Theorem 3.1
Let conditions (1.2) and (1.6) be satisfied and
$$ g\in\mathrm{ C}^{1}(\mathbb{I},\mathbb{R}) \quad\textit{and}\quad g'(t)>0. $$
(3.1)
Assume that there exists a function
\(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that, for all constants
\(M>0\),
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[\rho(s)q(s)-\frac {2^{p-1}}{p^{p}} \frac{ \rho(s)a(s)(h_{+}(s))^{p}}{(Mg'(s)g^{n-2}(s))^{p-1}} \biggr]\,\mathrm{ d}s=\infty. $$
(3.2)
If, for some constant
\(\lambda_{0}\in(0,1)\),
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[q(s) \biggl( \frac {\lambda_{0}}{(n-2)!}g^{n-2}(s)\delta(s) \biggr)^{p-1} E(t_{0},s)-\varphi(s) \biggr]\,\mathrm{ d}s=\infty, $$
(3.3)
then every solution
x
of (1.1) is either oscillatory or satisfies condition
\(\lim_{t\rightarrow\infty}x(t)=0\).
Proof
Assume that (1.1) has a nonoscillatory solution x which is eventually positive and such that
$$ \lim_{t\rightarrow\infty}x(t)\neq0. $$
(3.4)
Modifying the proof in Zhang et al. ([3], Lemma 2.3), we can show that, for all \(t\geq t_{1}\), there exist two possible cases:
-
(1)
\(x(t)>0\), \(x'(t)>0\), \(x^{(n-1)}(t)>0\), \(x^{(n)}(t)<0\);
-
(2)
\(x(t)>0\), \(x^{(n-2)}(t)>0\), \(x^{(n-1)}(t)<0\),
where \(t_{1}\geq t_{0}\) is sufficiently large. We consider each of the two cases separately.
Case I. Assume first that case (1) holds. For \(t\geq t_{1}\), we define the function ω by
$$ \omega(t):=\rho(t)\frac{a(t)(x^{(n-1)}(t))^{p-1}}{x^{p-1} (\frac{g(t)}{2} )}. $$
(3.5)
Then \(\omega(t)>0\) for all \(t\geq t_{1}\) and
$$\begin{aligned} \omega'(t)={}&\rho'(t)\frac{a(t)(x^{(n-1)}(t))^{p-1}}{x^{p-1} (\frac{g(t)}{2} )} +\rho(t) \frac{(a(t)(x^{(n-1)}(t))^{p-1})'}{x^{p-1} (\frac {g(t)}{2} )} \\ &{}-\frac{p-1}{2}\rho(t)g'(t)\frac{a(t)(x^{(n-1)}(t))^{p-1} x' (\frac{g(t)}{2} )}{x^{p} (\frac{g(t)}{2} )}. \end{aligned}$$
Let \(u:=x'\). It follows from Lemma 2.1 that, for some constant \(M>0\) and for all sufficiently large t,
$$x' \biggl(\frac{g(t)}{2} \biggr)\geq Mg^{n-2}(t)x^{(n-1)} \bigl(g(t)\bigr)\geq Mg^{n-2}(t)x^{(n-1)}(t). $$
Thus, we deduce that
$$\begin{aligned} \omega'(t)\leq{}&\rho'(t)\frac {a(t)(x^{(n-1)}(t))^{p-1}}{x^{p-1} (\frac{g(t)}{2} )} +\rho(t) \frac{(a(t)(x^{(n-1)}(t))^{p-1})'}{x^{p-1} (\frac {g(t)}{2} )} \\ &{}-\frac{p-1}{2}M\rho(t)g'(t)g^{n-2}(t) \frac{a(t)(x^{(n-1)}(t))^{p} }{x^{p} (\frac{g(t)}{2} )}. \end{aligned}$$
From (1.1) and (3.5), we obtain
$$\begin{aligned} \omega'(t)\leq{}&-\rho(t)q(t)+ \biggl[ \frac{\rho'(t)}{\rho (t)}-\frac{r(t)}{a(t)} \biggr]\omega(t) -\frac{p-1}{2}M g'(t)g^{n-2}(t)\frac{\omega^{p/(p-1)}(t)}{(\rho (t)a(t))^{1/(p-1)}} \\ ={}&-\rho(t)q(t)+h(t)\omega(t)- \frac{p-1}{2}M g'(t)g^{n-2}(t) \frac{\omega^{p/(p-1)}(t)}{(\rho(t)a(t))^{1/(p-1)}}. \end{aligned}$$
(3.6)
Hence, we have
$$\omega'(t)\leq-\rho(t)q(t)+h_{+}(t)\omega(t)-\frac{p-1}{2}M g'(t)g^{n-2}(t)\frac{\omega^{p/(p-1)}(t)}{(\rho(t)a(t))^{1/(p-1)}}. $$
Let
$$y:=\omega(t),\qquad D:=h_{+}(t),\quad \text{and}\quad C:=\frac{(p-1)Mg'(t)g^{n-2}(t)}{2(\rho(t)a(t))^{1/(p-1)}}. $$
Using the inequality
$$ Dy-Cy^{p/(p-1)}\leq \frac{(p-1)^{p-1}}{p^{p}}\frac{D_{+}^{p}}{C^{p-1}}, $$
(3.7)
where \(C>0\), \(y\geq0\), and \(D_{+}:=\max(0,D)\) (see Fišnarová and Mařík ([8], Lemma 1) for details), we get
$$\omega'(t)\leq-\rho(t)q(t)+\frac{2^{p-1}}{p^{p}}\frac{ \rho(t)a(t)(h_{+}(t))^{p}}{(Mg'(t)g^{n-2}(t))^{p-1}}. $$
Integrating this inequality from \(t_{1}\) to t, we obtain
$$\int_{t_{1}}^{t} \biggl[\rho(s)q(s)-\frac{2^{p-1}}{p^{p}} \frac{ \rho(s)a(s)(h_{+}(s))^{p}}{(Mg'(s)g^{n-2}(s))^{p-1}} \biggr]\,\mathrm{ d}s\leq\omega(t_{1}), $$
which contradicts (3.2).
Case II. Assume now that case (2) is satisfied. For \(t\geq t_{1}\), we define another function v as follows:
$$ v(t):=-\frac{a(t)(-x^{(n-1)}(t))^{p-1}}{(x^{(n-2)}(t))^{p-1}}. $$
(3.8)
Then \(v(t)<0\) for all \(t\geq t_{1}\). Since
$$ \bigl(-a(t) \bigl(-x^{(n-1)}(t)\bigr)^{p-1}E(t_{0},t) \bigr)'=-q(t)x^{p-1}\bigl(g(t)\bigr)E(t_{0},t)< 0, $$
(3.9)
we conclude that \(-a(t)(-x^{(n-1)}(t))^{p-1}E(t_{0},t)\) is decreasing. Thus, for all \(s\geq t\geq t_{1}\),
$$-a(s) \bigl(-x^{(n-1)}(s)\bigr)^{p-1}E(t_{0},s)\leq -a(t) \bigl(-x^{(n-1)}(t)\bigr)^{p-1}E(t_{0},t). $$
Hence, for all \(s\geq t\geq t_{1}\),
$$x^{(n-1)}(s)\leq \frac{(a(t)E(t_{0},t))^{1/(p-1)}}{(a(s)E(t_{0},s))^{1/(p-1)}}x^{(n-1)}(t). $$
Integrating this inequality from t to ι, we obtain
$$x^{(n-2)}(\iota)\leq x^{(n-2)}(t)+ \bigl(a(t)E(t_{0},t) \bigr)^{1/(p-1)}x^{(n-1)}(t) \int_{t}^{\iota}\frac{\,\mathrm{ d}s}{(a(s)E(t_{0},s))^{1/(p-1)}}. $$
Taking \(\iota\rightarrow\infty\) and using the fact that \(\lim_{\iota\rightarrow\infty}x^{(n-2)}(\iota)\geq0\) and the definition of δ, we have
$$ 0\leq x^{(n-2)}(t)+\bigl(a(t)E(t_{0},t) \bigr)^{1/(p-1)}x^{(n-1)}(t)\delta(t). $$
(3.10)
Inequality (3.10) implies that
$$ -\frac{x^{(n-1)}(t)}{x^{(n-2)}(t)}\bigl(a(t)E(t_{0},t) \bigr)^{1/(p-1)}\delta (t)\leq1. $$
(3.11)
Hence, by (3.8) and (3.11), we get
$$ -v(t)\delta^{p-1}(t)E(t_{0},t)\leq1. $$
(3.12)
Differentiation of (3.8) yields
$$v'(t)=\frac{(-a(t)(-x^{(n-1)}(t))^{p-1})'}{( x^{(n-2)}(t))^{p-1}}-(p-1)\frac{a(t)(-x^{(n-1)}(t))^{p}}{( x^{(n-2)}(t))^{p}}. $$
From (1.1) and (3.8), it follows that
$$ v'(t)=-r(t)\frac{v(t)}{a(t)}-q(t) \frac{x^{p-1}(g(t))}{( x^{(n-2)}(t))^{p-1}}-(p-1)\frac{(-v(t))^{p/(p-1)}}{a^{1/(p-1)}(t)}. $$
(3.13)
On the other hand, by Lemma 2.2, we have, for every \(\lambda\in(0,1)\) and for all sufficiently large t,
$$ x(t)\geq\frac{\lambda}{(n-2)!}t^{n-2}x^{(n-2)}(t). $$
(3.14)
Using (3.12) in (3.13), we have
$$\begin{aligned} v'(t)\leq{}&\frac{r(t)}{a(t)\delta^{p-1}(t)E(t_{0},t)}-q(t)\frac {x^{p-1}(g(t))}{( x^{(n-2)}(g(t)))^{p-1}} \frac{(x^{(n-2)}(g(t)))^{p-1}}{(x^{(n-2)}(t))^{p-1}} \\ &{}-(p-1)\frac{(-v(t))^{p/(p-1)}}{a^{1/(p-1)}(t)}. \end{aligned}$$
(3.15)
It follows from (3.14) and (3.15) that
$$\begin{aligned} v'(t)\leq\frac{r(t)}{a(t)\delta^{p-1}(t)E(t_{0},t)}-q(t) \biggl( \frac {\lambda}{(n-2)!}g^{n-2}(t) \biggr)^{p-1} -(p-1)\frac{(-v(t))^{p/(p-1)}}{a^{1/(p-1)}(t)}. \end{aligned}$$
(3.16)
Multiplying (3.16) by \(\delta^{p-1}(t)E(t_{0},t)\) and integrating the resulting inequality from \(t_{1}\) to t, we obtain
$$\begin{aligned} &\delta^{p-1}(t)E(t_{0},t)v(t)-\delta^{p-1}(t_{1})E(t_{0},t_{1})v(t_{1})- \int _{t_{1}}^{t}\frac{r(s)}{a(s)}\,\mathrm{ d}s \\ &\qquad{}+(p-1) \int_{t_{1}}^{t}a^{-1/(p-1)}(s)\delta^{p-2}(s)E(t_{0},s) \phi _{+}(s)v(s)\,\mathrm{ d}s \\ &\qquad{}+ \int_{t_{1}}^{t}q(s) \biggl(\frac{\lambda}{(n-2)!}g^{n-2}(s) \biggr)^{p-1}\delta^{p-1}(s)E(t_{0},s)\,\mathrm{ d}s \\ &\qquad{}+(p-1) \int_{t_{1}}^{t}\frac{(-v(s))^{p/(p-1)}}{a^{1/(p-1)}(s)}\delta ^{p-1}(s)E(t_{0},s)\,\mathrm{ d}s\leq0. \end{aligned}$$
Let
$$y:=-v(s),\qquad D:=a^{-1/(p-1)}(s)\delta^{p-2}(s)E(t_{0},s) \phi_{+}(s), $$
and
$$C:={\delta^{p-1}(s)}E(t_{0},s)/{a^{1/(p-1)}(s)}. $$
Using inequalities (3.7), (3.12), and the definition of φ, we have
$$\begin{aligned} \int_{t_{1}}^{t} \biggl[q(s) \biggl( \frac{\lambda }{(n-2)!}g^{n-2}(s) \biggr)^{p-1}\delta^{p-1}(s)E(t_{0},s)- \varphi (s) \biggr]\,\mathrm{ d}s \leq\delta^{p-1}(t_{1})E(t_{0},t_{1})v(t_{1})+1, \end{aligned}$$
which contradicts (3.3). This completes the proof. □
Assume \(n=2\) and let the definition of ω in (3.5) be replaced by
$$\omega(t):=\rho(t)\frac{a(t)(x'(t))^{p-1}}{x^{p-1} (g(t) )},\quad t\geq t_{1}. $$
Then we have the following result.
Theorem 3.2
Let conditions (1.2), (1.6), and (3.1) be satisfied and
\(n=2\). Suppose that there exists a function
\(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[\rho(s)q(s)-\frac {1}{p^{p}} \frac{ \rho(s)a(s)(h_{+}(s))^{p}}{(g'(s))^{p-1}} \biggr]\,\mathrm{ d}s=\infty. $$
(3.17)
If
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \bigl[q(s)\delta^{p-1}(s) E(t_{0},s)-\varphi(s) \bigr]\,\mathrm{ d}s=\infty, $$
(3.18)
then (1.1) is oscillatory.
Example 3.3
For \(t\geq1\), consider a second-order delay differential equation with damping
$$ \bigl(t^{2}x'(t) \bigr)'+ \frac{t}{2}x'(t)+q_{0}x \biggl(\frac {t}{2} \biggr)=0, $$
(3.19)
where \(q_{0}>0\) is a constant. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=t/2\), and \(\rho(t)=1\). Then \(h_{+}(t)=0\) and thus condition (3.17) is satisfied. It is easy to see that \(E(t_{0},t)=t^{1/2}\), \(\delta(t)=2t^{-3/2}/3\), \(\phi(t)=2t^{-1/2}/3\), and \(\varphi (t)=2t^{-1}/3\). Then condition (3.18) holds for \(q_{0}>1\). Therefore, by Theorem 3.2, equation (3.19) is oscillatory provided that \(q_{0}>1\). Observe, however, that if \(\gamma=1\), then
$$\begin{aligned} &\int_{t_{0}}^{\infty}\biggl[\frac{1}{a(t)\exp (\int_{t_{0}}^{t}\frac {r(s)}{a(s)}\,\mathrm{ d}s )} \int_{t_{0}}^{t} q(s)\exp \biggl( \int_{t_{0}}^{s} \frac{r(u)}{a(u)}\,\mathrm{ d}u \biggr) \\ &\qquad{}\times \biggl( \int_{g(s)}^{\infty}\frac{\,\mathrm{ d}u}{ (a(u)\exp (\int_{t_{0}}^{u}\frac{r(v)}{a(v)}\,\mathrm{ d}v ) )^{1/\gamma}} \biggr)^{\gamma}\,\mathrm{ d}s \biggr]^{1/\gamma}\,\mathrm{ d}t \\ &\quad= \frac{2^{5/2}}{3}q_{0} \int_{1}^{\infty}\frac{\ln t}{t^{5/2}}\,\mathrm{ d}t< \infty \end{aligned}$$
and
$$\begin{aligned} &\int_{t_{0}}^{\infty}\biggl[\frac{1}{a(t)} \int_{t_{0}}^{t}\exp \biggl(- \int _{s}^{t}\frac{r(\tau)}{a(\tau)}\,\mathrm{ d}\tau \biggr) \biggl( \int_{s}^{\infty}\biggl(\frac{1}{a(u)} \biggr)^{1/\gamma}\,\mathrm{ d}u \biggr)^{\gamma}q(s)\,\mathrm{ d}s \biggr]^{1/\gamma}\,\mathrm{ d}t \\ &\quad \leq q_{0} \int_{1}^{\infty}\frac{\ln t}{t^{2}}\,\mathrm{ d}t< \infty, \end{aligned}$$
which mean that the results obtained in [7, 20] fail to apply in equation (3.19).
Theorem 3.4
Let conditions (1.6) and (1.7) hold. Assume that there exists a function
\(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that, for all constants
\(M>0\),
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[\rho(s)q(s)-\frac {2^{p-1}}{p^{p}} \frac{ \rho(s)a(s)(h_{+}(s))^{p}}{(Ms^{n-2})^{p-1}} \biggr]\,\mathrm{ d}s=\infty. $$
(3.20)
If there exists a function
\(m\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that
$$ \frac{m(t)}{(a(t)E(t_{0},t))^{1/(p-1)}\delta(t)}+m'(t)\leq0 $$
(3.21)
and, for some constant
\(\lambda_{0}\in(0,1)\),
$$\begin{aligned} \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[q(s) \biggl( \frac {\lambda_{0}}{(n-2)!}g^{n-2}(s)\frac{m(g(s))}{m(s)}\delta(s) \biggr)^{p-1} E(t_{0},s) -\varphi(s) \biggr]\,\mathrm{ d}s=\infty, \end{aligned}$$
(3.22)
then the conclusion of Theorem
3.1
remains intact.
Proof
Assume that x is an eventually positive solution of (1.1) that satisfies (3.4). Similar analysis to that in Zhang et al. ([3], Lemma 2.3) leads to the conclusion that, for all \(t\geq t_{1}\), there exist two possible cases (1) and (2) (as those in the proof of Theorem 3.1), where \(t_{1}\geq t_{0}\) is sufficiently large. Assume first that case (1) holds. We define the function ω by
$$\omega(t):=\rho(t)\frac{a(t)(x^{(n-1)}(t))^{p-1}}{x^{p-1} (\frac{t}{2} )},\quad t\geq t_{1}. $$
With a similar proof as that of Case I in Theorem 3.1, one arrives at a contradiction with condition (3.20). Assume, instead, that case (2) holds. Define the function v as in (3.8). As in the proof of Theorem 3.1, we obtain (3.11), (3.12), (3.14), and (3.15). On the other hand, we derive from (3.11) that
$$\frac{x^{(n-1)}(t)}{x^{(n-2)}(t)}\geq-\frac {1}{(a(t)E(t_{0},t))^{1/(p-1)}\delta(t)}. $$
Using the latter inequality and (3.21), we have
$$\begin{aligned} \biggl(\frac{x^{(n-2)}(t)}{m(t)} \biggr)'&=\frac {x^{(n-1)}(t)m(t)-x^{(n-2)}(t)m'(t)}{m^{2}(t)} \\ &\geq-\frac{x^{(n-2)}(t)}{m^{2}(t)} \biggl[\frac {m(t)}{(a(t)E(t_{0},t))^{1/(p-1)}\delta(t)}+m'(t) \biggr] \geq0, \end{aligned}$$
which implies that \(x^{(n-2)}/m\) is nondecreasing. Hence, it follows from (1.7) that
$$\frac{x^{(n-2)}(g(t))}{x^{(n-2)}(t)}\geq\frac{m(g(t))}{m(t)}. $$
Thus, by (3.14) and (3.15), we have
$$\begin{aligned} v'(t)\leq\frac{r(t)}{a(t)\delta^{p-1}(t)E(t_{0},t)}-q(t) \biggl(\frac {\lambda g^{n-2}(t)}{(n-2)!} \biggr)^{p-1} \biggl(\frac{m(g(t))}{m(t)} \biggr)^{p-1} -(p-1)\frac{(-v(t))^{p/(p-1)}}{a^{1/(p-1)}(t)}. \end{aligned}$$
The remaining proof is similar to that of Case II in Theorem 3.1, and hence is omitted. □
Remark 3.5
The optional function m satisfying condition (3.21) exists and can be constructed by taking \(m(t):=\delta(t)\).
Assume \(n=2\) and let ω be as follows:
$$\omega(t):=\rho(t)\frac{a(t)(x'(t))^{p-1}}{x^{p-1}(t)},\quad t\geq t_{1}. $$
Then we obtain the following result that leads to the conclusion of Theorem 3.2.
Theorem 3.6
Let conditions (1.6) and (1.7) be satisfied and
\(n=2\). Assume that there exists a function
\(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[\rho(s)q(s)-\frac{ \rho(s)a(s)(h_{+}(s))^{p}}{p^{p}} \biggr]\,\mathrm{ d}s=\infty. $$
(3.23)
If there exists a function
\(m\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that (3.21) is satisfied and
$$ \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[q(s) \biggl(\frac {m(g(s))}{m(s)} \delta(s) \biggr)^{p-1}E(t_{0},s)-\varphi(s) \biggr]\,\mathrm{ d}s=\infty, $$
(3.24)
then the conclusion of Theorem
3.2
remains intact.
Example 3.7
For \(t\geq1\), consider a second-order advanced differential equation with damping
$$ \bigl(t^{2}x'(t) \bigr)'+ \frac{t}{2}x'(t)+q_{0}x(2t)=0, $$
(3.25)
where \(q_{0}>0\) is a constant. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=2t\), \(\rho(t)=1\), and \(m(t)=\delta(t)=2t^{-3/2}/3\). Similar analysis to that in Example 3.3 implies that condition (3.23) holds and condition (3.24) is satisfied for \(q_{0}>2\sqrt{2}\). Thus, by Theorem 3.6, equation (3.25) is oscillatory if \(q_{0}>2\sqrt{2}\). Observe that the results reported in [7, 20] cannot be applied to equation (3.25) since \(g(t)>t\).
In the next theorem, we consider equation (1.1) under the assumptions that (1.7) holds and
$$ A(t):= \int_{t}^{\infty}\frac{\,\mathrm{ d}s}{a^{1/(p-1)}(s)} \quad\text{and}\quad A(t_{0})< \infty. $$
(3.26)
Note that condition (1.6) is also satisfied in this case.
Theorem 3.8
Let conditions (1.7) and (3.26) hold. Assume that there exists a function
\(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that (3.20) holds for all constants
\(M>0\). If there exists a function
\(\xi\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that
$$ \frac{\xi(t)}{a^{1/(p-1)}(t)A(t)}+\xi'(t)\leq0 $$
(3.27)
and, for some constant
\(\lambda_{0}\in(0,1)\),
$$ r(t)\leq q(t) \biggl(\frac{\lambda_{0} g^{n-2}(t)\delta(g(t))}{(n-2)!} \biggr)^{p-1}a(t)E(t_{0},t) $$
(3.28)
and
$$\begin{aligned} & \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[q(s) \biggl(\frac {\lambda_{0} g^{n-2}(s)\xi(g(s))A(s)}{(n-2)!\xi(s)} \biggr)^{p-1}-\frac{r(s)}{a(s)} \\ &\qquad{}-\frac{(p-1)^{p}}{p^{p}}\frac{1}{A(s)a^{1/(p-1)}(s)} \biggr]\,\mathrm{ d}s=\infty, \end{aligned}$$
(3.29)
then the conclusion of Theorem
3.1
remains intact.
Proof
Assuming again that x is an eventually positive solution of (1.1) that satisfies (3.4) and proceeding as in the proof of Theorem 3.4, we end up having to show case (2) (as the corresponding case in Theorem 3.1). As in the proof of Case II in Theorem 3.1, one arrives at the inequalities (3.9), (3.10), and (3.14) which holds for all \(\lambda\in(0,1)\). Inequalities (3.10) and (3.14) yield, for all \(\lambda_{0}\in(0,1)\) and for all sufficiently large t,
$$ x(t)\geq-\frac{\lambda _{0}}{(n-2)!}t^{n-2}\bigl(a(t)E(t_{0},t) \bigr)^{1/(p-1)}x^{(n-1)}(t)\delta(t). $$
(3.30)
From (1.1) and (3.30), we obtain
$$\begin{aligned} & \bigl(-a(t) \bigl(-x^{(n-1)}(t)\bigr)^{p-1} \bigr)' \\ &\quad\leq r(t) \bigl(-x^{(n-1)}(t)\bigr)^{p-1} -a\bigl(g(t)\bigr)E\bigl(t_{0},g(t)\bigr) \bigl(-x^{(n-1)} \bigl(g(t)\bigr)\bigr)^{p-1} q(t) \biggl(\frac{\lambda_{0} g^{n-2}(t)\delta(g(t))}{(n-2)!} \biggr)^{p-1}. \end{aligned}$$
Using (3.9) and condition (1.7), we have
$$-a\bigl(g(t)\bigr)E\bigl(t_{0},g(t)\bigr) \bigl(-x^{(n-1)} \bigl(g(t)\bigr)\bigr)^{p-1} \leq -a(t)E(t_{0},t) \bigl(-x^{(n-1)}(t)\bigr)^{p-1}. $$
Thus, by (3.28), we get
$$\begin{aligned} & \bigl(-a(t) \bigl(-x^{(n-1)}(t)\bigr)^{p-1} \bigr)' \\ &\quad\leq\bigl(-x^{(n-1)}(t)\bigr)^{p-1} \biggl[r(t)-q(t) \biggl( \frac{\lambda_{0} g^{n-2}(t)\delta(g(t))}{(n-2)!} \biggr)^{p-1}a(t)E(t_{0},t) \biggr] \leq0, \end{aligned}$$
which implies that, for all \(s\geq t\geq t_{1}\),
$$x^{(n-1)}(s)\leq\frac{a^{1/(p-1)}(t)}{a^{1/(p-1)}(s)}x^{(n-1)}(t). $$
Integrating this inequality from t to ι, we obtain
$$x^{(n-2)}(\iota)\leq x^{(n-2)}(t)+ a^{1/(p-1)}(t)x^{(n-1)}(t) \int_{t}^{\iota}\frac{\,\mathrm{ d}s}{a^{1/(p-1)}(s)}. $$
Letting \(\iota\rightarrow\infty\) and using the definition of A, we get
$$0\leq x^{(n-2)}(t)+a^{1/(p-1)}(t)x^{(n-1)}(t)A(t), $$
which yields
$$ -\frac{x^{(n-1)}(t)}{x^{(n-2)}(t)}a^{1/(p-1)}(t)A(t)\leq1. $$
(3.31)
Now, we define the function v by (3.8). From (3.8) and (3.31), we see that
$$ -v(t)A^{p-1}(t)\leq1. $$
(3.32)
Differentiating (3.8) and using (1.1), we have (3.13). On the other hand, by (3.27) and (3.31), we obtain
$$\begin{aligned} \biggl(\frac{x^{(n-2)}(t)}{\xi(t)} \biggr)'=\frac{x^{(n-1)}(t)\xi (t)-x^{(n-2)}(t)\xi'(t)}{\xi^{2}(t)} \geq-\frac{x^{(n-2)}(t)}{\xi^{2}(t)} \biggl[\frac{\xi (t)}{a^{1/(p-1)}(t)A(t)}+\xi'(t) \biggr]\geq0, \end{aligned}$$
which shows that \(x^{(n-2)}/\xi\) is nondecreasing. Hence, using condition (1.7), we get
$$ \frac{x^{(n-2)}(g(t))}{x^{(n-2)}(t)}\geq\frac{\xi(g(t))}{\xi(t)}. $$
(3.33)
Thus, from (3.13), (3.14), (3.32), and (3.33), it follows that
$$\begin{aligned} v'(t)\leq{}&\frac{r(t)}{a(t)A^{p-1}(t)}-q(t)\frac{x^{p-1}(g(t))}{( x^{(n-2)}(g(t)))^{p-1}} \frac{(x^{(n-2)}(g(t)))^{p-1}}{(x^{(n-2)}(t))^{p-1}} -(p-1)\frac{(-v(t))^{p/(p-1)}}{a^{1/(p-1)}(t)} \\ \leq{}&\frac{r(t)}{a(t)A^{p-1}(t)}-q(t) \biggl(\frac{\lambda_{0} g^{n-2}(t)\xi(g(t))}{(n-2)!\xi(t)} \biggr)^{p-1} -(p-1)\frac{(-v(t))^{p/(p-1)}}{a^{1/(p-1)}(t)}. \end{aligned}$$
(3.34)
Multiplying (3.34) by \(A^{p-1}(t)\) and integrating the resulting inequality from \(t_{1}\) to t, we have
$$\begin{aligned} &A^{p-1}(t)v(t)-A^{p-1}(t_{1})v(t_{1})- \int_{t_{1}}^{t}\frac{r(s)}{a(s)}\,\mathrm{ d}s +(p-1) \int_{t_{1}}^{t}a^{-1/(p-1)}(s)A^{p-2}(s)v(s) \,\mathrm{ d}s \\ &\qquad{}+ \int_{t_{1}}^{t}q(s) \biggl(\frac{\lambda_{0} g^{n-2}(s)\xi(g(s))}{(n-2)!\xi(s)} \biggr)^{p-1}A^{p-1}(s)\,\mathrm{ d}s \\ &\qquad{}+(p-1) \int_{t_{1}}^{t}\frac {(-v(s))^{p/(p-1)}}{a^{1/(p-1)}(s)}A^{p-1}(s) \,\mathrm{ d}s\leq0. \end{aligned}$$
Let
$$y:=-v(s),\qquad D:=a^{-1/(p-1)}(s)A^{p-2}(s),\quad \text{and}\quad C:={A^{p-1}(s)}/{a^{1/(p-1)}(s)}. $$
Using inequality (3.7), we derive from (3.32) that
$$\begin{aligned} &\int_{t_{1}}^{t} \biggl[q(s) \biggl(\frac{\lambda_{0} g^{n-2}(s)\xi (g(s))A(s)}{(n-2)!\xi(s)} \biggr)^{p-1}-\frac{r(s)}{a(s)}-\frac{(p-1)^{p}}{p^{p}} \frac{1}{A(s)a^{1/(p-1)}(s)} \biggr]\,\mathrm{ d}s\\ &\quad \leq A^{p-1}(t_{1})v(t_{1})+1, \end{aligned}$$
which contradicts (3.29). This completes the proof. □
Remark 3.9
The optional function ξ satisfying assumption (3.27) can reasonably be constructed by taking \(\xi(t):=A(t)\).
Similarly, we have the following criterion for (1.1) in the case when \(n=2\).
Theorem 3.10
Let (1.7) and (3.26) be satisfied and
\(n=2\). Suppose that there exists a function
\(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that (3.23) holds. If
$$ r(t)\leq q(t)\delta^{p-1}\bigl(g(t)\bigr)a(t)E(t_{0},t) $$
(3.35)
and there exists a function
\(\xi\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\)
such that (3.27) holds and
$$\begin{aligned} & \limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[q(s) \biggl(\frac {\xi(g(s))A(s)}{\xi(s)} \biggr)^{p-1}-\frac{r(s)}{a(s)}-\frac{(p-1)^{p}}{p^{p}}\frac{1}{A(s)a^{1/(p-1)}(s)} \biggr]\,\mathrm{ d}s=\infty, \end{aligned}$$
(3.36)
then the conclusion of Theorem
3.2
remains intact.
Example 3.11
For \(t\geq1\) and \(q_{0}>0\), consider the second-order advanced differential equation (3.25). Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=2t\), \(\rho(t)=1\), and \(\xi(t)=A(t)=t^{-1}\). Then \(E(t_{0},t)=t^{1/2}\), \(\delta(t)=2t^{-3/2}/3\), and \(h_{+}(t)=0\). It is not difficult to verify that all conditions of Theorem 3.10 are satisfied for \(q_{0}\geq3\sqrt{2}/2\). Therefore, using Theorem 3.10, equation (3.25) is oscillatory provided that \(q_{0}\geq3\sqrt{2}/2\), whereas Theorem 3.6 implies that equation (3.25) is oscillatory if \(q_{0}>2\sqrt {2}\). Hence, Theorem 3.10 improves Theorem 3.6 in some cases. However, to achieve such improvement, an additional assumption (3.35) is required. Therefore, we observe that Theorems 3.4, 3.6, 3.8, and 3.10 are of independent interest.
The following example is provided to show that our results are sharp for the second-order Euler differential equation \((t^{2}x'(t) )'+q_{0}x(t)=0\), \(q_{0}>0\).
Example 3.12
For \(t\geq1\), consider a second-order differential equation with damping
$$ \bigl(t^{2}x'(t) \bigr)'+r_{0}x'(t)+q_{0}x(t)=0, $$
(3.37)
where \(r_{0}\geq0\) and \(q_{0}>0\) are constants. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=r_{0}\), \(q(t)=q_{0}\), \(g(t)=t\), and \(\rho(t)=1\). Then \(h_{+}(t)=0\) and so condition (3.23) is satisfied. It is not hard to verify that \(1\leq E(t_{0},t)\leq\mathrm{ e}^{r_{0}}\), \(\mathrm{ e}^{-r_{0}}t^{-1}\leq\delta(t)\leq t^{-1}\), and \(A(t)=1/t\). Then condition (3.35) is satisfied for all sufficiently large t and, for \(q_{0}>1/4\),
$$\begin{aligned} &\limsup_{t\rightarrow\infty} \int_{t_{0}}^{t} \biggl[q(s) \biggl(\frac {\xi(g(s))A(s)}{\xi(s)} \biggr)^{p-1}-\frac{r(s)}{a(s)} -\frac{(p-1)^{p}}{p^{p}}\frac{1}{A(s)a^{1/(p-1)}(s)} \biggr]\,\mathrm{ d}s \\ &\quad= \limsup_{t\rightarrow\infty} \int_{1}^{t} \biggl[\frac{q_{0}}{s}- \frac {r_{0}}{s^{2}}-\frac{1}{4s} \biggr]\,\mathrm{ d}s=\infty. \end{aligned}$$
Hence, by Theorem 3.10, equation (3.37) is oscillatory provided that \(q_{0}>1/4\) (it is well known that \(q_{0}>1/4\) is the best possible for the oscillation of equation (3.37) when \(r_{0}=0\)). Observe, however, that if \(\gamma=1\), then
$$\begin{aligned} &\int_{t_{0}}^{\infty}\biggl[\frac{1}{a(t)\exp (\int_{t_{0}}^{t}\frac {r(s)}{a(s)}\,\mathrm{ d}s )} \int_{t_{0}}^{t} q(s)\exp \biggl( \int_{t_{0}}^{s} \frac{r(u)}{a(u)}\,\mathrm{ d}u \biggr) \\ &\qquad{}\times \biggl( \int_{g(s)}^{\infty}\frac{\,\mathrm{ d}u}{ (a(u)\exp (\int_{t_{0}}^{u}\frac{r(v)}{a(v)}\,\mathrm{ d}v ) )^{1/\gamma}} \biggr)^{\gamma}\,\mathrm{ d}s \biggr]^{1/\gamma}\,\mathrm{ d}t \\ &\quad\leq q_{0}\mathrm{ e}^{r_{0}} \int_{1}^{\infty}\frac{\ln t}{t^{2}}\,\mathrm{ d}t< \infty \end{aligned}$$
and
$$\begin{aligned} &\int_{t_{0}}^{\infty}\biggl[\frac{1}{a(t)} \int_{t_{0}}^{t}\exp \biggl(- \int _{s}^{t}\frac{r(\tau)}{a(\tau)}\,\mathrm{ d}\tau \biggr) \biggl( \int_{s}^{\infty}\biggl(\frac{1}{a(u)} \biggr)^{1/\gamma}\,\mathrm{ d}u \biggr)^{\gamma}q(s)\,\mathrm{ d}s \biggr]^{1/\gamma}\,\mathrm{ d}t \\ &\quad\leq q_{0} \int_{1}^{\infty}\frac{\ln t}{t^{2}}\,\mathrm{ d}t< \infty, \end{aligned}$$
which mean that the results reported in [7, 20] cannot be applied to equation (3.37).
Finally, the following example is given to present an open problem of this paper.
Example 3.13
For \(t\geq1\), consider the second-order Euler differential equation
$$ \bigl(t^{2}x'(t) \bigr)'+ \frac{t}{2}x'(t)+q_{0}x(t)=0, $$
(3.38)
where \(q_{0}>0\) is a constant. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=t/2\), \(\rho(t)=1\), \(m(t)=\delta(t)=2t^{-3/2}/3\), and \(\xi (t)=A(t)=t^{-1}\). It is easy to see that \(h_{+}(t)=0\), \(E(t_{0},t)=t^{1/2}\), \(\phi(t)=2t^{-1/2}/3\), and \(\varphi(t)=2t^{-1}/3\). Applications of Theorems 3.2 and 3.6 imply that equation (3.38) is oscillatory if \(q_{0}>1\), whereas Theorem 3.10 yields oscillation of equation (3.38) for \(q_{0}>3/4\). Similar analysis to that in Example 3.3 shows that the results obtained in [7, 20] fail to apply in equation (3.38). However, it is well known that equation (3.38) is oscillatory if and only if \(q_{0}>9/16\). How to extend this sharp result to equation (1.1) remains open at the moment.