1. Introduction
In this work, we study asymptotic properties of solutions of differential
equation of the fourth order and with middle term
\begin{align}
\left( r\left( \ell \right) y^{\prime \prime \prime }\left( \ell \right)
\right) ^{\prime }+p\left( \ell \right) y^{\prime \prime \prime }\left( \ell
\right) +q\left( \ell \right) y\left( \sigma \left( \ell \right) \right) =0,
\ell \geq \ell _{0}. \label{1-y}
\end{align}
(1)
We assume \(r\left( \ell \right) \in C\left( \left[ \ell _{0},\infty \right) ,\mathbb{R}\right) ,\) \(r\left( \ell \right) >0\ \)for all \(\ell \geq \ell _{0},\ p,\) \(
q,\) \(\sigma \in C\left( \left[ \ell _{0},\infty \right) ,\mathbb{R}\right) \) such that \(p\left( \ell \right) \geq 0,\ q>0\),\(\ \sigma \left(
\ell \right) \leq \ell \) and \(\underset{\ell \rightarrow \infty }{\lim }
\sigma \left( \ell \right) =\infty .\)
We say that a function \(y\left( \ell \right) \)is a solution of (1), we mean a non-trivial real function \(y(\ell ) \in C\left( [\ell
_{y},\infty )\right) \), \(\ell _{y}\geq \ell _{0}\), satisfying (1)
on \([\ell _{y},\infty )\) and moreover, having the properties: \(y(\ell )\), \(
y^{\prime }(\ell )\), \(y^{\prime \prime }(\ell )\) and \(r\left( \ell \right)
\left[ y^{\prime \prime \prime }\left( \ell \right) \right] \) are
continuously differentiable for all \(\ell \in \lbrack \ell _{y},\infty )\).
We consider only those solutions \(y\left( \ell \right) \) of (1)
which satisfy \(\sup \{\left\vert y(\ell )\right\vert :\ell \geq \ell \}>0,\)
for any \(\ell \geq \ell _{y}\). A solution of (1) is called
oscillatory if it has arbitrary large zeros, otherwise it is called
nonoscillatory.
The oscillations of higher and fourth order differential equations have been
studied by several authors and several techniques have been proposed for
obtaining oscillatory criteria for higher and fourth order differential
equations. For treatments on this subject, we refer the reader to the texts
[1, 2, 3, 4, 5, 6] and the
articles [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In what follows, we review some results that have provided
the background and the motivation, for the present work.
Hou and Cheng [22] studied the oscillation of differential equation
with middle term
\begin{align*}
y^{\left( 4\right) }\left( \ell \right) +p\left( \ell \right) y^{\prime
}\left( \ell \right) +q\left( \ell \right) f\left( y\left( \sigma \left(
\ell \right) \right) \right) =0, \ell \geq \ell _{0}
\end{align*}
under the condition: third-order differential equation \(z^{\prime \prime
\prime }\left( \ell \right) +p\left( \ell \right) z\left( \ell \right) =0,\)
is nonoscillatory.
Džurina and Jadlovská [11] have considered the
differential equation of the fourth order with a negative term
\begin{align*}
y^{\left( 4\right) }\left( \ell \right) +p\left( \ell \right) y^{\prime
}\left( \ell \right) +q\left( \ell \right) y\left( \sigma \left( \ell
\right) \right) =0,\ell \geq \ell _{0}
\end{align*}
under the condition: all solutions of the contributory third-order
differential equation are nonoscillatory.
Moaaz
et al. [
19] studied the oscillatory behavior of the
third-order nonlinear differential equation with middle term of the form
\begin{align*}
\left[ r_{2}\left( \ell \right) \left( r_{1}\left( \ell \right) \left(
y^{\prime }\left( \ell \right) \right) ^{\alpha }\right) ^{\prime }\right]
^{\prime }+\phi \left( \ell ,y^{\prime }\left( \sigma \left( \ell \right)
\right) \right) +q\left( \ell \right) f\left( y\left( \sigma \left( \ell
\right) \right) \right) =0,\ell \geq \ell _{0}
\end{align*}
under
\begin{align}
\int_{\ell _{0}}^{\infty }\frac{1}{r^{\frac{1}{\alpha }}\left( \ell \right) }
d\ell =\infty . \label{2-y}
\end{align}
(2)
Our aim, in the present work, is to use the generalized Riccati
transformations and new comparison principles to obtain a new conditions for
the oscillation of every solutions of (1) under condition
\begin{align}
R\left( \ell \right) =\int_{\ell _{0}}^{\ell }\dfrac{1}{r\left( s\right) }
\exp \left( -\int_{\ell _{0}}^{s}\dfrac{p\left( u\right) }{r\left( u\right) }
du\right) ds\ \text{and }R\left( \ell \right) \rightarrow \infty \text{as }
\ell \rightarrow \infty . \label{2y}
\end{align}
(3)
An example is included to illustrate the main results.
2. Main results
In the next section, we shall establish some oscillation criteria
for equation (1). The proof of our main results are essentially
based on the following lemmas.
Lemma 1.
(See [20], Lemma 1) Let \(\beta \geq 1\ \)be\(\
\)a ratio of two numbers\(,\) where \(U\) and \(V\) are constants. Then
\begin{equation*}
Uz-Vz^{\frac{\beta +1}{\beta }}\leq \frac{\beta ^{\beta }}{(\beta +1)^{\beta
+1}}\frac{U^{\beta +1}}{V^{\beta }},\ V>0.
\end{equation*}
Lemma 2.
(See [6], Lemma 13) If the
function \(u\) satisfies \(u^{(\kappa )}>0,\) \(\kappa =0,1,…,m,\) and \(
u^{\left( m+1\right) }< 0,\) then
\begin{equation*}
\frac{u\left( \ell \right) }{\ell ^{m}/m!}\geq \frac{u^{\prime }\left( \ell
\right) }{\ell ^{m-1}/\left( m-1\right) !}.
\end{equation*}
Lemma
(See [1], Lemma 2.2.3) Let \(h\in C^{n}\left(
\left[ \ell _{0},\infty \right) ,\left( 0,\infty \right) \right) .\) Let \(
h^{\left( n\right) }\left( \ell \right) \) is of a fixed sign and on the
interval \(\left[ \ell _{0},\infty \right) \), \(h^{\left( n\right) }\left(
\ell \right) \) not identically zero, and that there exists a \(\ell _{1}\geq
\ell _{0}\) such that \(h^{\left( n-1\right) }\left( \ell \right) h^{\left(
n\right) }\left( \ell \right) \leq 0\) for all \(\ell \geq \ell _{1}\). If \(
\underset{\ell \rightarrow \infty }{\lim }h\left( \ell \right) \neq 0\) then
for every \(\lambda \in \left( 0,1\right) \) there exists \(\ell _{\lambda
}\geq \ell _{0}\) such that
\begin{equation*}
h\left( \ell \right) \geq \frac{\lambda }{\left( n-1\right) !}\ell
^{n-1}\left\vert h^{\left( n-1\right) }\left( \ell \right) \right\vert \text{
for }\ell \geq \ell _{\lambda }.
\end{equation*}
For convenience, we denote
\begin{equation*}
\rho _{+}^{\prime }\left( \ell \right) :=\max \left\{ 0,\ \rho _{+}^{\prime
}\left( \ell \right) \right\} \text{ and\ }\delta _{+}^{\prime }\left(
\ell \right) :=\max \left\{ 0,\ \delta ^{\prime }\left( \ell \right)
\right\} .
\end{equation*}
Theorem 4.
Assume that (4) holds . If there exist positive
functions \(\rho ,\delta \in C^{1}\left( \left[ \ell _{0},\infty \right)
\right) \ \)such that
\begin{equation}
\int_{\ell _{0}}^{\infty }\left( \rho \left( s\right) q\left( s\right) \frac{
\mu }{2}\sigma ^{2}\left( s\right) -\frac{1}{4\rho \left( s\right) r\left(
s\right) }\left[ \frac{\rho _{+}^{\prime }\left( s\right) }{\rho \left(
s\right) }-\frac{p\left( s\right) }{r\left( s\right) }\right] ^{2}\right)
ds=\infty , \label{3-y}
\end{equation}
(4)
for some \(\mu \in \left( 0,1\right) ,\ \)and
\begin{equation}
\int_{\ell _{0}}^{\infty }\left[ \delta \left( s\right) \int_{s}^{\infty }
\left[ \frac{1}{r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left(
\nu \right) \left( \dfrac{\sigma ^{2}\left( \nu \right) }{\nu ^{2}}\right)
d\nu \right] d\upsilon -\dfrac{\left( \delta ^{\prime }\left( s\right)
\right) ^{2}}{4\delta \left( s\right) }\right] ds=\infty , \label{4-y}
\end{equation}
(5)
then all solutions of (1) are oscillatory.
Proof.
Assume that (1) has a solution \(y\) is nonoscillatory. Without loss
of generality, we may assume that there exists a \(\ell _{1}\in \left[ \ell
_{0},\infty \right) \) such that \(y(\ell )>0\), \(y(\sigma (\ell ))>0\) for all \(
\ell \in \left[ \ell _{0},\infty \right) \). It follows from (1)
and (4) that \(\left( r\left( \ell \right) y^{\prime \prime \prime
}\left( \ell \right) \right) ^{\prime }< 0\) and there exist two possible
cases for\(\ \ell \geq \ell _{1},\ \)where\(\ \ell _{1}\geq \ell _{0}\ \)is
sufficiently large:
\begin{eqnarray*}
&&\left. \left( C_{1}\right) y^{\left( \kappa \right) }\left( \ell
\right) >0 \;\; \text{for} \;\;\kappa =0,1,2,3;\right. \\
&&\left. \left( C_{2}\right) \;\; y^{\left( \kappa \right) }\left( \ell
\right) >0\;\; \text{for}\;\; \kappa =0,1,3,\text{ and }y^{\prime \prime }\left(
\ell \right) < 0.\right.
\end{eqnarray*}
Assume that we have Case \(\left( C_{1}\right) \). We define a
generalized Riccati substitution by
\begin{equation}
\omega \left( \ell \right) :=\rho \left( \ell \right) \frac{r\left( \ell
\right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) }{
y^{\prime \prime }}. \label{y-2}
\end{equation}
(6)
Then \(\omega \left( \ell \right) >0\). Differentiating, we obtain
\begin{eqnarray*}
\omega ^{\prime }\left( \ell \right) &=&\rho \left( \ell \right) \left(
\frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell
\right) \right) }{y^{\prime \prime }}\right) ^{\prime }+\rho ^{\prime
}\left( \ell \right) \frac{r\left( \ell \right) \left( y^{\prime \prime
\prime }\left( \ell \right) \right) }{y^{\prime \prime }\left( \ell \right) }
, \\
&=&\rho \left( \ell \right) \frac{\left( r\left( \ell \right) \left(
y^{\prime \prime \prime }\left( \ell \right) \right) \right) ^{\prime }}{
y^{\prime \prime }}-\rho \left( \ell \right) \frac{r\left( \ell \right)
\left( y^{\prime \prime \prime }\left( \ell \right) \right) ^{2}}{\left(
y^{\prime \prime }\right) ^{2}}+\rho ^{\prime }\left( \ell \right) \frac{
r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right)
\right) }{y^{\prime \prime }}.
\end{eqnarray*}
By virtue of (1), we have.
\begin{equation}
\left( r\left( \ell \right) y^{\prime \prime \prime }\left( \ell \right)
\right) ^{\prime }+p\left( \ell \right) y^{\prime \prime \prime }\left( \ell
\right) =-q\left( \ell \right) y\left( \sigma \left( \ell \right) \right).
\label{y-1}
\end{equation}
(7)
From (7),we see that
\begin{equation*}
\omega ^{\prime }\left( \ell \right) \leq \rho _{+}^{\prime }\left( \ell
\right) \frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left(
\ell \right) \right) }{y^{\prime \prime }}-\rho \left( \ell \right) \frac{
p\left( \ell \right) y^{\prime \prime \prime }\left( \ell \right) }{
y^{\prime \prime }}-\frac{\rho \left( \ell \right) \left( q\left( \ell
\right) y\left( \sigma \left( \ell \right) \right) \right) }{y^{\prime
\prime }}-\rho \left( \ell \right) \frac{r\left( \ell \right) \left(
y^{\prime \prime \prime }\left( \ell \right) \right) ^{2}}{\left( y^{\prime
\prime }\right) ^{2}}.
\end{equation*}
Hence, by (7), we obtain
\begin{equation}
\omega ^{\prime }\left( \ell \right) \leq \frac{\rho _{+}^{\prime }\left(
\ell \right) }{\rho \left( \ell \right) }\omega \left( \ell \right) -\frac{
p\left( \ell \right) }{r\left( \ell \right) }\omega \left( \ell \right)
-\rho \left( \ell \right) q\left( \ell \right) \frac{y\left( \sigma \left(
\ell \right) \right) }{y^{\prime \prime }}-\frac{1}{\rho \left( \ell \right)
r\left( \ell \right) }\omega \left( \ell \right) ^{2}. \label{6-y}
\end{equation}
(8)
From Lemma 2, we have that \(y\left( \ell \right) \geq \frac{\ell }{2
}y^{\prime }\left( \ell \right) \) and hence,
\begin{equation*}
\frac{y^{\prime }\left( \ell \right) }{y\left( \ell \right) }\leq \frac{2}{
\ell }.
\end{equation*}
Integrating from \(\ell \) to \(\sigma \left( \ell \right) \) we find
\begin{equation*}
\frac{y\left( \sigma \left( \ell \right) \right) }{y\left( \ell \right) }
\geq \frac{\sigma ^{2}\left( \ell \right) }{\ell ^{2}}.
\end{equation*}
This implies that
\begin{equation}
y\left( \sigma \left( \ell \right) \right) \geq \frac{\sigma ^{2}\left( \ell
\right) }{\ell ^{2}}y\left( \ell \right) . \label{8-y}
\end{equation}
(9)
It follows from Lemma 3 that
\begin{equation}
y\left( \ell \right) \geq \frac{\mu }{2}\ell ^{2}y^{\prime \prime }\left(
\ell \right) , \label{9-y}
\end{equation}
(10)
for all \(\mu \in \left( 0,1\right) \) and every sufficiently large \(\ell \).
Thus, by (8), (9) and (10), we get
\begin{eqnarray*}
\omega ^{\prime }\left( \ell \right) &\leq &\frac{\rho _{+}^{\prime }\left(
\ell \right) }{\rho \left( \ell \right) }\omega \left( \ell \right) -\frac{
p\left( \ell \right) }{r\left( \ell \right) }\omega \left( \ell \right)
-\rho \left( \ell \right) q\left( \ell \right) \frac{\sigma ^{2}\left( \ell
\right) }{\ell ^{2}}\frac{y\left( \ell \right) }{y^{\prime \prime }}-\frac{1
}{\rho \left( \ell \right) r\left( \ell \right) }\omega \left( \ell \right)
^{2}, \\
&\leq &\frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell
\right) }\omega \left( \ell \right) -\frac{p\left( \ell \right) }{r\left(
\ell \right) }\omega \left( \ell \right) -\rho \left( \ell \right) q\left(
\ell \right) \left( \frac{\sigma \left( \ell \right) }{\ell }\right) ^{2}
\frac{\mu }{2}\ell ^{2}-\frac{1}{\rho \left( \ell \right) r\left( \ell
\right) }\omega \left( \ell \right) ^{2}.
\end{eqnarray*}
This implies that
\begin{equation}
\omega ^{\prime }\left( \ell \right) \leq \left[ \frac{\rho _{+}^{\prime
}\left( \ell \right) }{\rho \left( \ell \right) }-\frac{p\left( \ell \right)
}{r\left( \ell \right) }\right] \omega \left( \ell \right) -\rho \left( \ell
\right) q\left( \ell \right) \frac{\mu }{2}\sigma ^{2}\left( \ell \right) –
\frac{1}{\rho \left( \ell \right) r\left( \ell \right) }\omega \left( \ell
\right) ^{2}. \label{10-y}
\end{equation}
(11)
Using Lemma 1 with \(U=\left[ \frac{\rho _{+}^{\prime }\left( \ell
\right) }{\rho \left( \ell \right) }-\frac{p\left( \ell \right) }{r\left(
\ell \right) }\right] ,V=\frac{1}{\rho \left( \ell \right) r\left( \ell
\right) }\) and \(z=\omega \), we get
\begin{equation*}
\left[ \frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell
\right) }-\frac{p\left( \ell \right) }{r\left( \ell \right) }\right] \omega
\left( \ell \right) -\frac{1}{\rho \left( \ell \right) r\left( \ell \right) }
\omega \left( \ell \right) ^{2}\leq \frac{1}{4\rho \left( \ell \right)
r\left( \ell \right) }\left[ \frac{\rho _{+}^{\prime }\left( \ell \right) }{
\rho \left( \ell \right) }-\frac{p\left( \ell \right) }{r\left( \ell \right)
}\right] ^{2}.
\end{equation*}
Hence, we obtain
\begin{equation*}
\omega ^{\prime }\left( \ell \right) \leq -\rho \left( \ell \right) q\left(
\ell \right) \frac{\mu }{2}\sigma ^{2}\left( \ell \right) +\frac{1}{4\rho
\left( \ell \right) r\left( \ell \right) }\left[ \frac{\rho _{+}^{\prime
}\left( \ell \right) }{\rho \left( \ell \right) }-\frac{p\left( \ell \right)
}{r\left( \ell \right) }\right] ^{2}.
\end{equation*}
Which implies that
\begin{equation*}
\int_{\ell _{1}}^{\ell }\left( \rho \left( s\right) q\left( s\right) \frac{
\mu }{2}\sigma ^{2}\left( s\right) -\frac{1}{4\rho \left( s\right) r\left(
s\right) }\left[ \frac{\rho _{+}^{\prime }\left( s\right) }{\rho \left(
s\right) }-\frac{p\left( s\right) }{r\left( s\right) }\right] ^{2}\right)
ds\leq \omega \left( \ell _{1}\right) ,
\end{equation*}
for some \(\mu \in \left( 0,1\right) \ \)which contradicts (4).
Assume that we have Case \(\left( C_{2}\right) \) holds. Define
\begin{equation}
\psi \left( \ell \right) :=\delta \left( \ell \right) \frac{y^{\prime
}\left( \ell \right) }{y\left( \ell \right) },\ \ell \geq \ell _{1}.
\label{15-A}
\end{equation}
(12)
Then \(\psi \left( \ell \right) >0\ \)for\(\ \ell \geq \ell _{1}\ \)and
\begin{equation}
\psi ^{\prime }\left( \ell \right) =\delta ^{\prime }\left( \ell \right)
\frac{y^{\prime }\left( \ell \right) }{y\left( \ell \right) }-\delta \left(
\ell \right) \frac{y^{\prime \prime }\left( \ell \right) y\left( \ell
\right) -\left( y^{\prime }\right) ^{2}\left( \ell \right) }{y^{2}\left(
\ell \right) }, \label{16-A}
\end{equation}
(13)
\begin{equation}
=\delta \left( \ell \right) \frac{y^{\prime \prime }\left( \ell \right) }{
y\left( \ell \right) }+\frac{\delta ^{\prime }\left( \ell \right) }{\delta
\left( \ell \right) }\psi \left( \ell \right) -\frac{\psi ^{2}\left( \ell
\right) }{\delta \left( \ell \right) }. \label{y-4}
\end{equation}
(14)
Integrating (1) from \(\ell \) to \(u\) we find
\begin{equation*}
r\left( u\right) \left( y^{\prime \prime \prime }\right) \left( u\right)
-r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell
\right) +\int_{\ell }^{u}q\left( s\right) y\left( \sigma \left( s\right)
\right) ds\leq 0.
\end{equation*}
By virtue of\(\ y^{\prime }\left( \ell \right) >0, y(\ell )>0 \)and\(\
y^{\prime \prime }\left( \ell \right) 0,\) that
\begin{equation*}
r\left( u\right) \left( y^{\prime \prime \prime }\right) \left( u\right)
-r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell
\right) +y\left( \ell \right) \int_{\ell }^{u}q\left( s\right) \left( \frac{
\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\leq 0.
\end{equation*}
Letting \(u\rightarrow \infty \), we arrive at the inequality
\begin{equation*}
-r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell
\right) +y\left( \ell \right) \int_{\ell }^{\infty }q\left( s\right) \left(
\frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\leq 0,
\end{equation*}
we get
\begin{equation}
y^{\prime \prime }\left( \ell \right) +y\left( \ell \right) \int_{\ell
}^{\infty }\left[ \frac{1}{r\left( \ell \right) }\int_{\upsilon }^{\infty
}q\left( \ell \right) \left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}
\right) ds\right] d\upsilon \leq 0, \label{17-A}
\end{equation}
(15)
we see
\begin{equation}
\frac{y^{\prime \prime }\left( \ell \right) }{y\left( \ell \right) }\leq –
\left[ \frac{1}{r\left( \ell \right) }\int_{\upsilon }^{\infty }q\left( \ell
\right) \left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\right]
d\upsilon . \label{y-5}
\end{equation}
(16)
Hence, by (16) in (14), we find
\begin{equation}
\psi ^{\prime }\left( \ell \right) \leq -\delta \left( \ell \right)
\int_{\ell }^{\infty }\left[ \frac{1}{r\left( \upsilon \right) }
\int_{\upsilon }^{\infty }q\left( \ell \right) \left( \frac{\sigma
^{2}\left( s\right) }{s^{2}}\right) ds\right] d\upsilon +\frac{\delta
^{\prime }\left( \ell \right) }{\delta \left( \ell \right) }\psi \left( \ell
\right) -\frac{\psi ^{2}\left( \ell \right) }{\delta \left( \ell \right) }.
\label{y-6}
\end{equation}
(17)
Thus, we have
\begin{equation}
\psi ^{\prime }\left( \ell \right) \leq -\delta \left( \ell \right)
\int_{\ell }^{\infty }\left[ \frac{1}{r\left( \upsilon \right) }
\int_{\upsilon }^{\infty }q\left( s\right) \left( \frac{\sigma ^{2}\left(
s\right) }{s^{2}}\right) ds\right] d\upsilon +\frac{\left( \delta ^{\prime
}\left( \ell \right) \right) ^{2}}{4\delta \left( \ell \right) }.
\label{18-A}
\end{equation}
(18)
Integrating from \(\ell _{1}\) \ to \(\ell \), we get
\begin{equation*}
\left. \int_{\ell _{1}}^{\ell }\left[ \delta \left( s\right)
\int_{s}^{\infty }\left[ \frac{1}{r\left( \upsilon \right) }\int_{\upsilon
}^{\infty }q\left( \nu \right) \left( \tfrac{\sigma ^{2}\left( \nu \right) }{
\nu ^{2}}\right) d\nu \right] d\upsilon -\tfrac{\left( \delta ^{\prime
}\left( s\right) \right) ^{2}}{4\delta \left( s\right) }\right] ds\leq \psi
\left( \ell _{1}\right) ,\right.
\end{equation*}
which contradicts (5).
The proof of the theorem is complete.
Another criteria for oscillation of (1) can be establish by
comparison with ordinary equations of the lower order. We extend a
comparison theorem that fasten properties of solutions of (1) with
those of second-order differential equations. It is well known (see [
1]Agarwal}) that the differential equation
\begin{equation}
\left[ a\left( \ell \right) \left( y^{\prime }\left( \ell \right) \right)
\right] ^{\prime }+q\left( \ell \right) y\left( \ell \right) =0\text{
}\ell \geq \ell _{0,} \label{28-y}
\end{equation}
(19)
where \(a\) ,\(\ q\in C[\ell _{0}\) \(\infty ),\ a\left( \ell
\right) ,\ q\left( \ell \right) >0\), and the necessary and sufficient
condition for nonoscillatory of this equation is to there exist a number \(\ell \geq \ell _{0}\),\ and a function \(\upsilon \) \(\in C^{1}[\ell \)
\(\infty ),\) satisfying
\begin{equation*}
\upsilon ^{\prime }\left( \ell \right) +a^{-1}\left( \ell \right) \upsilon
^{2}\left( \ell \right) +q\left( \ell \right) \leq 0\text{ on\ }[\ell \text{ }\infty ).
\end{equation*}
In the following, we compare the behavior of oscillatory of (1) with
the half-linear differential equations of type (19).
Lemma 5.
(see[1]) Let
\begin{equation*}
\int_{\ell _{0}}^{\infty }\frac{1}{a\left( s\right) }ds=\infty .
\end{equation*}
Then the condition
\begin{equation*}
\underset{\ell \rightarrow \infty }{\lim \inf }\left( \int_{\ell
_{0}}^{\infty }\frac{1}{a\left( s\right) }ds\right) \int_{\ell }^{\infty
}q\left( s\right) ds>\frac{1}{4},
\end{equation*}
guarantees oscillation of (19).
Theorem 6.
Let (4) and assume that the equation
\begin{equation}
\left[ r\left( \ell \right) y^{\prime }\left( \ell \right) \right] ^{\prime
}+q\left( \ell \right) \frac{\mu }{2}\sigma ^{2}\left( \ell \right) y\left(
\ell \right) =0 \label{29-y}
\end{equation}
(20)
and
\begin{equation}
y^{\prime \prime }\left( \ell \right) +\left( \int_{\ell }^{\infty }\left[
\frac{1}{r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left( s\right) \left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\right]
d\upsilon \right) y\left( \ell \right) =0, \label{30-y}
\end{equation}
(21)
are oscillatory, then every solution of (1) is oscillatory.
Proof.
Proceeding as in proof of the Theorem 4. If we set \(\rho \left(
\ell \right) =1\) in (11), then we get
\begin{equation*}
\omega ^{\prime }\left( \ell \right) +\frac{p\left( \ell \right) }{r\left(
\ell \right) }\omega \left( \ell \right) +q\left( \ell \right) \frac{\mu }{2}
\sigma ^{2}\left( \ell \right) +\frac{1}{r\left( \ell \right) }\omega \left(
\ell \right) ^{2}\leq 0,
\end{equation*}
for every constant \(\mu \in \left( 0,\ 1\right) . \)Thus, we can see that
equation (20) is nonoscillatory for every constant \(\mu \in \left(
0,\ 1\right) ,\ \)which is a contradiction. If we now set \(\delta \left( \ell
\right) =1\) in (17), then we find
\begin{equation*}
\psi ^{\prime }\left( \ell \right) +\int_{\ell }^{\infty }\left[ \frac{1}{
r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left( s\right)
\left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\right] d\upsilon
+\psi ^{2}\left( \ell \right) \leq 0.
\end{equation*}
Hence, equation (21) is nonoscillatory, which is a contradiction.
Theorem 6 is proved.
3. Example
In this section, we give the following example to illustrate our main
results.
Example 1.
Consider the differential equation
\begin{equation}
\left( \frac{1}{\ell }y^{\prime \prime \prime }\left( \ell \right) \right)
^{\prime }+\left( 1\backslash 2\ell ^{2}\right) y^{\prime \prime \prime
}\left( \ell \right) +\frac{\beta }{\ell }y\left( \frac{\ell }{2}\right)
=0,\ \ell \geq 1, \label{Ex1-y}
\end{equation}
(22)
where \(\beta >0\) is a constant. Let
\begin{equation*}
r\left( \ell \right) =\frac{1}{\ell },\ p\left( \ell \right) =1\backslash
2\ell ^{2},\ \sigma \left( \ell \right) =\frac{\ell }{2},\ p\left( \ell
\right) =1\backslash 2\ell ^{2},\ q\left( \ell \right) =\frac{\beta }{\ell },
\end{equation*}
we get
\begin{equation*}
R\left( \ell \right) =2\backslash 3\left( \ell ^{3\backslash 2}-1\right)
\rightarrow \infty \text{as }\ell \rightarrow \infty .
\end{equation*}
If we now set \(\rho \left( s\right) =\delta \left( s\right) =1\) then
\begin{equation*}
\int_{\ell _{0}}^{\infty }\left( \rho \left( s\right) q\left( s\right) \frac{
\mu }{2}\sigma ^{2}\left( s\right) -\frac{1}{4\rho \left( s\right) r\left(
s\right) }\left[ \frac{\rho _{+}^{\prime }\left( s\right) }{\rho \left(
s\right) }-\frac{p\left( s\right) }{r\left( s\right) }\right] ^{2}\right) ds,
\end{equation*}
\begin{equation*}
=\left( \frac{\beta \mu }{8}-\frac{1}{4}\right) \int_{\ell _{0}}^{\infty
}\ell d\ell =\infty ,\ \text{if\ }\beta >\frac{2}{\mu }\ \text{for some
constant }\mu \in \left( 0,\ 1\right)
\end{equation*}
and
\begin{equation*}
\int_{\ell _{0}}^{\infty }\left[ \delta \left( s\right) \int_{s}^{\infty }
\left[ \frac{1}{r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left(
\nu \right) \left( \dfrac{\sigma ^{2}\left( \nu \right) }{\nu ^{2}}\right)
d\nu \right] d\upsilon -\dfrac{\left( \delta ^{\prime }\left( s\right)
\right) ^{2}}{4\delta \left( s\right) }\right] ds=\infty .
\end{equation*}
Thus, by Theorem 4, every solution of equation (22) is
oscillatory, provided \(\beta >\frac{2}{\mu }\).
Conclusion
In this work, by using the generalized Riccati transformations technique and
new comparison principles, we offer some new sufficient conditions which
ensure that any solution of Equation (1) oscillates under the condition (
4).\(\ \)Further, we can try to get some oscillation criteria of Equation (
1) if \(y\left( \ell \right) =x\left( \ell \right) +p\left( \ell
\right) x\left( \tau \left( \ell \right) \right) \) in the future work.
Acknowledgments
The authors thank the reviewers for for their useful comments, which led to the improvement of the content of the paper.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
