\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 49, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/49\hfil Oscillation criteria]
{Oscillation criteria for third-order nonlinear differential equations
 with functional arguments}

\author[Y. Shoukaku \hfil EJDE-2013/49\hfilneg]
{Yutaka Shoukaku}  % in alphabetical order

\address{Yutaka Shoukaku \newline
Faculty of Engineering, Kanazawa University,
 Kanazawa 920-1192, Japan}
\email{shoukaku@t.kanazawa-u.ac.jp}

\thanks{Submitted February 1, 2013. Published February 18, 2013.}
\subjclass[2000]{34K11, 34C10}
\keywords{Oscillation criteria; third order; Riccati inequality}

\begin{abstract}
 In this article, we consider the third-order nonlinear differential
 equations with functional arguments. By using the Riccati inequality,
 we find conditions for all solutions to be oscillatory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We are concerned with the oscillation of solutions to the  nonlinear
third-order functional differential equation
\begin{equation} \label{eE}
y'''(t) + a(t)y''(t) + b(t)y'(t) + \sum_{i=1}^{m}c_i(t)\varphi_i(y(\sigma_i(t)))
 = 0, \quad t > 0.
\end{equation}
Throughout this paper we assume the following conditions:
\begin{itemize}
\item[(H1)]
$a(t),  b(t),  c_i(t) \in C((0,\infty);[0,\infty))$, $(i=1,2,\dots,m)$;

\item[(H2)]
$\sigma_i(t) \in C([0,\infty);\mathbb{R})$,
$\lim_{t \to \infty}\sigma_i(t) = \infty \ (i=1,2,\dots,m)$,
there exists a positive constant $\sigma$ such that
$$
\sigma'_j(t) \ge \sigma \quad \text{and} \quad t \ge \sigma_j(t)
$$
for some $j \in \{1,2,\dots,m\}$;
\item[(H3)]
$\varphi_i(s) \in C^1(\mathbb{R};\mathbb{R})$ $(i=1,2,\dots,m)$,
$\varphi_i(-s)=-\varphi_i(s)$ for $s \ge 0$, $\varphi'_j(s) > 0$,
$\varphi'_j(s)$ is nondecreasing for $s >0$ and some $j \in \{1,2,\dots,m\}$.
\end{itemize}


\begin{definition} \label{def1} \rm
By a  solution of \eqref{eE} we mean a function $y(t) \in C^3([T_y,\infty);\mathbb{R})$
satisfying $\sup\{|y(t)|:t>T_y\} > 0$ for any $T_y \ge t_y$, where
$$
t_y = \min\Big\{0,\min_{1 \le i \le m}\big\{\inf_{t \ge 0}\sigma_i(t)\big\}\Big\}.
$$
A solution of \eqref{eE} is said to be  oscillatory if it has arbitrarily
large zeros, otherwise it is non-oscillatory.
\end{definition}


\begin{definition} \label{def3} \rm
A function $H$ belongs to the class $\mathbb{H}$,
if $H$ is in $ C(D;[0,\infty))$; $H$ satisfies
$$
H(t,t)=0, \quad  H(t,s) > 0 \quad \text{for }  t > s > t_1,
$$
where $D=\{(t,s) \in \mathbb{R}^2 : t \ge s \ge t_1\}$; there exists a constant
 $k_0 >0$ such that
$$
\lim_{t \to \infty}\frac{H(t,s)}{H(t,t_1)} = k_0 \quad \text{for  all }
 t \ge t_1;
$$
and the partial derivative $\partial H/\partial s$ exists on
$D_0 = \{(t,s) \in \mathbb{R}^2 ; t > s \ge t_1\}$ and satisfies
$$
\frac{\partial H}{\partial s}(t,s) = -h(t,s) H(t,s),
$$
for some function  $h$ in $C(D_0;\mathbb{R})$.
\end{definition}

Since the work by Hanan \cite{h1} was published,
oscillation of solutions to third-order differential equations in
special cases have been widely studied by many authors
\cite{a1,a2,b1,e1,h1,n1,s1,s2,t1}.
This maybe because third-order differential equations have applications
in mechanical, physical and biological problems \cite{s1},
and because \eqref{eE} plays an important role in control theory. 

In the mid-nineteenth century,  Maxwell analyzed the stability problem of the
Watt's governor, and obtained conditions for stability which are based
on third-order linear differential equations with constant coefficients.
Later,  Routh and Hurwitz derived more general stability conditions
which are known today as the Routh-Hurwitz stability criteria.
In 1976, Erbe \cite{e1} studied the oscillatory and asymptotic behavior
of solutions of the  equation
\begin{equation} \label{eA1}
 y'''(t) + a(t)y''(t) + b(t)y'(t) + c(t)y^{\alpha}(t) = 0,
\end{equation}
where $\alpha $ is the quotient of positive odd integers.

\begin{theorem}[{\cite[Theorem 4.9]{e1}}] \label{thmA}
Let $a(t)b(t) + b'(t) \le 0$ and $y(t)$ be a nontrivial solution
of \eqref{eA1} with $F[y(t_0)] \le 0$ for some $t_0 > 0$, where
$$
F[y(t)] = e^{A(t)}[2y''(t)y(t) - y'^2(t) + b(t)y^2(t)].
$$
If the equation
\begin{equation} \label{eA2}
 \left(e^{A(t)}z'(t)\right)' + e^{A(t)}\left\{b(t)z(t)
+ \lambda^{\alpha}t^{\alpha}c(t)z^{\alpha}(t)\right\} = 0
\end{equation}
is oscillatory (that is, all solutions of \eqref{eA2} are oscillatory)
for some $0 < \lambda < \frac{1}{2}$, then $y(t)$ is oscillatory.
\end{theorem}

Tiryki and Aktas \cite{t1}, Agarwal et al \cite{a1}, and
 Aktas et al \cite{a2}  studied third-order nonlinear differential
equations of the form
\begin{equation} \label{eB1}
 \left(r_2(t)\left(r_1(t)y'(t)\right)'\right)' + p(t)y'(t)
+ q(t)\varphi(y(\sigma(t))) = 0.
\end{equation}
Aktas et al \cite{a2} established the following results which ensures
that every solution  is oscillatory or converges to zero.


\begin{theorem}[{\cite[Theorem 3.1]{a2}}] \label{thmB}
Assume that
$$
R_1(t,t_0)=\int_{t_0}^{t}\frac{ds}{r_1(s)} \to \infty, \quad
R_2(t,t_0)=\int_{t_0}^{t}\frac{ds}{r_2(s)} \to \infty \quad
 \text{as }  t \to \infty,
$$
 that there exist functions $\phi(t)$ and $\rho_1(t)$ in 
$C([0,\infty);(0,\infty))$ such that
\begin{gather*}
\rho_1'(t) \ge 0, \quad
\phi(t) = \left(r_2(t)\rho'_1(t)\right)'r_1(t) + \rho_1(t)p(t) \ge 0, \\
\phi'(t) \le 0, \quad \int^{\infty}\rho_1(t)q(t)dt = \infty,
\end{gather*}
and that the equation
$$
\left(r_2(t)z'(t)\right)' + \frac{p(t)}{r_1(t)}z(t) = 0
$$
is non-oscillatory. If there exists a function
$\rho_2(t) \in C^1([0,\infty);(0,\infty))$ such that
$$
\limsup_{t \to \infty}\int_{T}^{t}
\big\{\rho_2(s)q(s) - \frac{\beta^2(s)}{4\alpha(s)}\big\}ds = \infty,
$$
then every solution  of \eqref{eB1} either oscillates or converges
to zero as $t \to \infty$. Here
$$
\alpha(t)=\frac{K_0 R_2(\sigma(t),t)\sigma'(t)}{r_1(\sigma(t))\rho_2(t)}, \quad
\beta(t)=\frac{\rho'_2(t)}{\rho_2(t)} - p(t)\frac{R_2(\sigma(t),t)}{r_1(t)}.
$$
\end{theorem}

For the case when \eqref{eE} has constant coefficients, it is easy to see
that neither $a(t)b(t) + b (t) \leq 0$ in Theorem \ref{thmA}, nor
$R_2 (t, t_0 )\to \infty$ in Theorem \ref{thmB} is satisifed.
The natural question to ask is:
\begin{quote}
 Is it possible to find oscillation criteria for equation \eqref{eE},
which include the case of constant coefficients?
\end{quote}
In this article we obtain an affirmative answer to this question.


\section{Preliminaries}

First we sate an assumption to be used in the next lemma, which
is needed for proving our main results.
\begin{itemize}
\item[(H4)] $a(t) \ge b(t) + 1$.
\end{itemize}

\begin{lemma} \label{lem1}
Assume that {\rm (A4)} holds, 
\begin{equation}\label{lm1}
\int_0^{\infty}\pi(t)e^{A(t)} \sum_{i=1}^{m}c_i(t)dt = \infty,
\end{equation}
where
$$
A(t)=\int_{0}^{t}a(s)ds, \quad 
\pi(t)=\int_{t}^{\infty}e^{-A(s)}ds,
$$
and $y(t)$ is a non-oscillatory solution of  \eqref{eE}.
Then there exists a $t_0 >0$ such that
\begin{equation}\label{lm2}
y(t)y'(t) > 0, \quad \forall t \ge t_0.
\end{equation}
\end{lemma}

\begin{proof}
Suppose that $y(t)$ is a non-oscillatory solution of \eqref{eE}.
Without loss of generality, we assume that $y(t) > 0$ and $y(\sigma_i(t)) > 0$ 
$(i=1,2,\dots,m)$.
Note that if $y(t)$ is a negative solution, then $-y(t)$ is a positive solution
of \eqref{eE}.

We claim that  $y'(t)$ is non-oscillatory. If $y'(t)$ is socillatory,
then  $x(t) = -y'(t)$ is oscillatory and satisfies
\begin{equation}\label{le11}
\Big(e^{A(t)}x'(t)\Big)' + b(t)e^{A(t)}x(t)
= \sum_{i=1}^{m}c_i(t)e^{A(t)}\varphi_i(y(\sigma_i(t))) \ge 0.
\end{equation}
Let $x(t)$ have consecutive zeros at $\alpha$ and $\beta$
$(t_0 < \alpha < \beta)$ such that $x'(\alpha) \ge 0$, $x'(\beta) \le 0$
and $x(t) \ge 0$ for $t \in (\alpha,\beta)$. 
Multiplying \eqref{le11} by $e^{-t}$
and integrating over $[\alpha,\beta]$, we obtain
$$
\int_{\alpha}^{\beta}e^{-t}\left(e^{A(t)}x'(t)\right)'dt
+ \int_{\alpha}^{\beta}b(t)e^{A(t)}x(t)e^{-t}dt \ge 0.
$$
Integrating by parts,
\[
e^{A(\beta)-\beta}x'(\beta) - e^{A(\alpha)-\alpha}x'(\alpha)
 + \int_{\alpha}^{\beta}e^{A(t)-t}x'(t)dt
+ \int_{\alpha}^{\beta}b(t)e^{A(t)-t}x(t)dt \ge 0.
\]
Integrating by parts again and using that $x(\alpha)=x(\beta)=0$,
\[
e^{A(\beta)-\beta}x'(\beta) - e^{A(\alpha)-\alpha}x'(\alpha)
\ge \int_{\alpha}^{\beta}e^{A(t)-t}\{a(t) - 1 - b(t)\}x(t)dt \ge 0.
\]
Since $x'(\alpha) \ge 0$ and $x'(\beta) \le 0$, the above inequality
is a contradiction. Therefore, $x(t)$ is non-oscillatory, and there 
are two possible cases:

Case 1: $x(t)<0$ for all $t$ large enough. By definition $x(t) = -y'(t)$.
So $y'(t)>0$ while $y(t)>0$ for all $t$ large enough.
Therefore, \eqref{lm2} is satisfied.

Case 2: $x(t)>0$ for all $t$ large enough. Then
$y'(t)<0$ while $y(t) >0$. From the continuity of $\varphi_i$,
there is a positive constant $K_0$ such that
$$
\varphi_i(y(\sigma_i(t))) \le K_0\,.
$$
From \eqref{le11},
\begin{equation}\label{le12}
\Big(e^{A(t)} x'(t)\Big)' + b(t)e^{A(t)}x(t)
 \le K_0 e^{A(t)}\sum_{i=1}^{m}c_i(t)\,.
\end{equation}
Let
\begin{equation}\label{le13}
v(t) = x(t) + K_0 \int_{t}^{\infty}e^{-A(s)}\int_{t_0}^{s}e^{A(\xi)}
\sum_{i=1}^{m}c_i(\xi)d\xi ds,.
\end{equation}
Then \eqref{le12} implies 
$$
\Big(e^{A(t)} v'(t)\Big)' \le - b(t)e^{A(t)} x(t) \le 0.
$$
From this inequality, either $v'(t) \ge 0$ or $v'(t) < 0$ for all
$t$ large enough. Differentiating \eqref{le13},
we have
$$
v'(t) = x'(t) - K_0 e^{-A(t)}\int_{t_0}^{t}e^{A(s)}\sum_{i=1}^{m}c_i(s)ds
\le x'(t) = -y''(t).
$$
If $v'(t) \ge 0$, then $y''(t)= -v'(t)\leq 0$. Since $y'(t)<0$ and
$y''(t) \le 0$, we have $\lim_{t \to \infty}y(t)=-\infty$
which contradicts $y(t)\geq0$.
Therefore, $v'(t)<0$. From $v(t)>0$ and $v'(t)<0$ it follows
that there is a constant $K_1 > 0$ such that
\begin{align*}
K_1 > v(t)
&> K_0 \int_{t}^{\infty}e^{-A(s)}\int_{t_0}^{s}e^{A(\xi)}\sum_{i=1}^{m}c_i(\xi)d\xi ds \\
&= K_0 \int_{t}^{\infty}(-\pi(s))'
 \Big(\int_{t_0}^{s}e^{A(\xi)}\sum_{i=1}^{m}c_i(\xi)d\xi\Big) ds \\
&\ge K_0 \int_{t_0}^{t}\pi(s)e^{A(s)}\sum_{i=1}^{m}c_i(s)ds,
\end{align*}
which contradicts the assumption \eqref{lm1}. Therefore, Case 2 can not happen.
The proof is complete.
\end{proof}

For the next lemma we use the assumption
\begin{itemize}
\item[(H5)]
there exists $a(t) \in C^1((0,\infty);[0,\infty))$ such that
$$
b(t) \ge a'(t).
$$
\end{itemize}

\begin{lemma} \label{lem2}
Assume that {\rm (H5)} and \eqref{lm1} hold.
If  $y(t)$ is a nonoscillatory solution of  \eqref{eE},
then there exists a $t_0 > 0$ such that  \eqref{lm2} is satisfied.
\end{lemma}

\begin{proof}
Suppose that $y(t)$ is a non-oscillatory solution of \eqref{eE}.

without loss of generality, we assume that $y(t) > 0$ and $y(\sigma_i(t)) > 0$ 
$(i=1,2,\dots,m)$. 
Note that if  $y(t)$ is a negative solution, then $-y(t)$ is a positive 
solution of \eqref{eE}.

 We claim that $y'(t)$ is non-oscillatory. If 
$y'(t)$ is oscillatory, then $x(t) = -y'(t)$ is oscillatory and satisfies
\begin{equation}\label{le21}
x''(t) + a(t)x'(t) + b(t)x(t) \ge 0.
\end{equation}
Let $x(t)$ be a consecutive zeros at $\alpha$ and $\beta$
$(t_0 < \alpha < \beta)$ such that $x'(\alpha) \ge 0$ and $x'(\beta) \le 0$.
 Multiplying \eqref{le21} by
 $\frac{1}{a(t)}e^{\int_{t_0}^{t}\frac{b(s)}{a(s)}ds}$ and integrating over
$[\alpha,\beta]$, we obtain
$$
\int_{\alpha}^{\beta}\Big\{\frac{1}{a(t)}e^{\int_{t_0}^{t}\frac{b(s)}{a(s)}ds}
 x''(t) + \Big(e^{\int_{t_0}^{t}\frac{b(s)}{a(s)}ds} x(t)\Big)'\Big\}dt \ge 0.
$$
Integrating by parts, 
\[
\frac{1}{a(\beta)}e^{\int_{t_0}^{\beta}\frac{b(s)}{a(s)}ds}x'(\beta)
- \frac{1}{a(\alpha)}e^{\int_{t_0}^{\alpha}\frac{b(s)}{a(s)}ds}x'(\alpha) \\
\ge \int_{\alpha}^{\beta}\Big(\frac{1}{a(t)}
 e^{\int_{t_0}^{t}\frac{b(s)}{a(s)}ds}\Big)' x'(t)dt.
\]
Integrating by parts again and using that $x(\alpha)=x(\beta)=0$,
\[
0 \ge \frac{1}{a(\beta)}e^{\int_{t_0}^{\beta}\frac{b(s)}{a(s)}ds}x'(\beta)
 - \frac{1}{a(\alpha)}e^{\int_{t_0}^{\alpha}\frac{b(s)}{a(s)}ds}x'(\alpha) \\
\ge \int_{\alpha}^{\beta}\frac{\left(b(t) - a'(t)\right)}
 {a^2(t)}e^{\int_{t_0}^{t}\frac{b(s)}{a(s)}ds} x'(t)dt,
\]
which implies that $x'(t) \le 0$ on $[\alpha,\beta]$.
The rest of the proof is the same as in
Lemma \ref{lem1},  and hence is omitted.
\end{proof}

\begin{theorem} \label{thm1}
Assume that {\rm (H1)-- (H4)} or {\rm (H1)}--{\rm (H3), (H5)} are satisfied.
 If  \eqref{lm1} holds and the Riccati inequalities
$$
z'(t) + \frac{1}{2}\frac{1}{P_i(t)}z^2(t) \le -Q_i(t) \quad (i=1,2)
$$
have no solution on intervals $[T,\infty)$ for all large $T > 0$, then every
solution  of  \eqref{eE} is oscillatory.
Here 
\begin{gather*}
P_1(t) = 1, \quad Q_1(t) = -\frac{1}{2}a^2(t) + b(t) + K_1 c_j(t), \\
P_2(t) = \frac{1}{\sigma K_2 A_e(\sigma_j(t))}, \quad
Q_2(t) = c_j(t)e^{A(t)}
  - \frac{1}{2}\Big(\frac{b^2(t)A_e(\sigma_j(t))}{\sigma K_2}\Big), \\
A_e(\sigma_j(t)) = \int_{0}^{\sigma_j(t)}e^{-A(s)}ds
\end{gather*}
for some $K_i > 0$  $(i=1,2)$, and some $i \in \{1,2,\dots,m\}$.
\end{theorem}

\begin{proof}
Suppose that $y(t)$ is a nonoscillatory solution of \eqref{eE} on
$[t_0,\infty)$ for some $t_0 \ge T > 0$. Then there exists a $t_1 \ge t_0$
such that $y(t) > 0 $ and $y(\sigma_i(t)) > 0 \ (i=1,2,\dots,m)$
 for $t \ge t_1$. We shall consider only this case, because the proof
when $y(t) < 0$ is similar. From \eqref{eE}, for each $j\in\{1,2,\dots,m\}$,
we have 
$$
\left(e^{A(t)}y''(t)\right)' + b(t)e^{A(t)}y'(t)
+ c_j(t)e^{A(t)}\varphi_j(y(\sigma_j(t))) \le 0.
$$
According Lemma \ref{lem1} or Lemma \ref{lem2}, 
$y'(t) \ge 0$. Then from the above inequality,
$$
\Big(e^{A(t)}y''(t)\Big)' \le 0.
$$
Hence $y''(t) \ge 0$ or $y''(t) < 0$. First we assume that $y''(t) < 0$.
Letting
$$
w_1(t) = \frac{e^{A(t)}y''(t)}{y'(t)},
$$
we have
\begin{align*}
w'_1(t)
&= \frac{\left(e^{A(t)}y''(t)\right)'}{y'(t)} - e^{-A(t)}w^2_1(t) \\
&\le -b(t)e^{A(t)} -c_j(t)e^{A(t)}\frac{\varphi_j(y(\sigma_j(t)))}{y'(t)}
 - e^{-A(t)}w^2_1(t).
\end{align*}
On the other hand, there exist constants $K_0$ and $K_1$ such that
$$
y(t) \ge K_0 \quad \text{and} \quad y'(t) \le K_1.
$$
It is easy to see  that
$$
w'_1(t) \le -\Big(b(t) + \frac{K_0}{K_1}c_j(t)\Big)e^{A(t)} - e^{-A(t)} w^2_1(t).
$$
Multiplying this by $e^{-A(t)}$, we obtain
\begin{equation} \label{th11}
 \Big(e^{-A(t)}w_1(t)\Big)' + a(t)e^{-A(t)}w_1(t)
\le -\Big(b(t) + \frac{K_0}{K_1}c_j(t)\Big) - \Big(e^{-A(t)}w_1(t)\Big)^2.
\end{equation}
By H\"older's inequality we have
\begin{equation}\label{th12}
|a(t)e^{-A(t)}w_1(t)| \le \frac{1}{2}
\Big(a^2(t) + \Big(e^{-A(t)}w_1(t)\Big)^2\Big)
\end{equation}
Substituting \eqref{th12} into \eqref{th11} yields
\begin{equation}\label{th13}
\Big(e^{-A(t)}w_1(t)\Big)' + \frac{1}{2}\Big(e^{-A(t)}w_1(t)\Big)^2
\le -\Big(-\frac{1}{2}a^2(t) + b(t) + \frac{K_0}{K_1}c_j(t)\Big),
\end{equation}
which clearly imply that $e^{-A(t)}w_1(t)$ is a solution of \eqref{th13}.
Next we assume that $y''(t) \ge 0$. Setting
$$
w_2(t) = \frac{e^{A(t)}y''(t)}{\varphi_j(y(\sigma_j(t)))},
$$
we obtain
\begin{equation} \label{th14}
\begin{aligned}
w'_2(t)
&= \frac{\left(e^{A(t)}y''(t)\right)'}{\varphi_j(y(\sigma_j(t)))}
 - e^{A(t)}y''(t) \frac{\varphi'(y(\sigma_j(t))) y'(\sigma_j(t))
 \sigma'_j(t)}{\varphi^2_j(y(\sigma_j(t)))} \\
&\le -\frac{b(t)e^{A(t)}y'(t)}{\varphi_j(y(\sigma_j(t)))} - c_j(t)e^{A(t)}
 - e^{A(t)}y''(t)\frac{\sigma K_2 y'(\sigma_j(t))}{\varphi^2_j(y(\sigma_j(t)))}
 \\
&\le -\frac{b(t)y'(t)}{\varphi_j(y(\sigma_j(t)))} - c_j(t)e^{A(t)}
 - e^{A(t)}y''(t)\frac{\sigma K_2 y'(\sigma_j(t))}{\varphi^2_j(y(\sigma_j(t)))}
\end{aligned}
\end{equation}
Since $(e^{A(t)}y''(t))' \le 0$, we see that
\begin{align*}
y'(t) &\ge y'(\sigma_j(t))
\ge \int_{t_0}^{\sigma_j(t)}e^{-A(s)}\left(e^{A(s)}y''(s)\right)ds \\
&\ge e^{A(\sigma_j(t))}y''(\sigma_j(t))\int_{t_0}^{\sigma_j(t)}e^{-A(s)}ds \\
&\ge e^{A(t)}y''(t)\int_{t_0}^{\sigma_j(t)}e^{-A(s)}ds = e^{A(t)}y''(t)A_e(\sigma_j(t)).
\end{align*}
By using this relation, \eqref{th14} is rewritten as
\[
w'_2(t)
\le -b(t)A_e(\sigma_j(t))w_2(t) - c_j(t)e^{A(t)}
- \sigma K_2 A_e(\sigma_j(t))w^2_2(t).
\]
Applying H\"older's inequality, 
\[
|b(t)A_e(\sigma_j(t))w_2(t)|
\le \frac{1}{2}\Big(\frac{\left(b(t)A_e(\sigma_j(t))\right)^2}
{\left(\sigma K_2 A_e(\sigma_j(t))\right)}
 + \sigma K_2 A_e(\sigma_j(t))w^2_2(t)\Big).
\]
It is easy to establish the  inequality
\begin{equation}\label{th15}
w'_2(t) \le \frac{1}{2}\left(\frac{b^2(t)A_e(\sigma_j(t))}{\sigma K_2}\right)
- c_j(t)e^{A(t)} - \frac{1}{2}\sigma K_2 A_e(\sigma_j(t))w^2_2(t),
\end{equation}
and then, $w_2(t)$ is a solution of \eqref{th15}.
This contradicts the hypothesis and completes the proof.
\end{proof}


\section{Main results}

In this section, we establish some new oscillatory criteria for \eqref{eE}.
First, we state following useful lemmas.

\begin{lemma}[{\cite[Theorem 4]{u1}}] \label{lem3}
If there is a function $\phi(t) \in C^{1}([T_0,\infty );(0,\infty ))$
 such that
\begin{gather*}
\int_{T_{1}}^{\infty}
\Big(\frac{\bar{p}(t)|\phi'(t)|^{\beta }}{\phi(t)}\Big)^{1/(\beta - 1)}dt
< \infty, \quad
\int_{T_{1}}^{\infty}\frac{1}{\bar{p}(t)(\phi(t))^{\beta -1}}dt = \infty, \\
\int_{T_{1}}^{\infty}\phi (t)\bar{q}(t)dt = \infty
\end{gather*}
for some $T_1 \ge T_0$, then the Riccati inequality
\begin{equation}\label{eq1}
x'(t) + \frac{1}{\beta }\frac{1}{\bar{p}(t)}|x(t)|^{\beta } \le -\bar{q}(t),
\end{equation}
where $\beta >1$, $\bar{p}(t) \in C([T_0,\infty );(0,\infty ))$ and
$\bar{q}(t)\in C([T_{0},\infty );{\mathbb{R}})$, has no solution on
intervals $[T,\infty)$ for all large $T$.
\end{lemma}

Let $\rho(s) \in C^1([0,\infty);(0,\infty))$, and define an integral operator
$A_{\tau}^{\rho}$ by
$$
A_{\tau}^{\rho}(v;t) = \int_{\tau}^{t}H(t,s)v(s)\rho(s)ds, \quad t \ge \tau \ge T,
$$
where $v \in ([\tau,\infty);\mathbb{R})$. It is easy to see that $A_{\tau}^{\rho}$
is linear and positive, and in fact satisfies the following conditions:
\begin{itemize}
\item[(H6)]
$A_{\tau}^{\rho}(k_1 v_1 + k_2 v_2;r) = k_1 A_{\tau}^{\rho}(v_1;r)
+ k_2 A_{\tau}^{\rho}(v_2;r)$ for $k_1, k_2 \in \mathbb{R}$;
\item[(H7)]
$A_{\tau}^{\rho} \ge 0$ for $v \ge 0$;
\item[(H8)]
$A_{\tau}^{\rho}(v';r) = -H(r,\tau)v(\tau)\rho(\tau)
+ A_{\tau}^{\rho}((h-\frac{\rho'}{\rho})v;r)$.
\end{itemize}

\begin{lemma}[{\cite[Theorem 1]{x1}}]  \label{lem4}
If
$$
\limsup_{t \to \infty}\frac{1}{H(t,T)}A_{T}^{\rho}
\Big(\bar{q}-\frac{\beta-1}{\beta}\bar{p}^{\frac{1}{\beta-1}}
|h-\frac{\rho'}{\rho}|^{\beta/(\beta-1)};t\Big) = \infty,
$$
then  \eqref{eq1} has no solution on $[T,\infty)$ for all large $T$.
\end{lemma}


\begin{theorem} \label{thm2}
Assume that {\rm (H1)}--{\rm (H4)} or {\rm (H1)}--{\rm (H3)}, {\rm (H5)}
are satisfied.
If \eqref{lm1} holds, and there exists functions
$\phi_i(t) \in C^{1}([T_0,\infty );(0,\infty))$ $(i=1,2)$ such that
\begin{gather*}
 \int_{T}^{\infty}\left(\frac{P_i(t) \phi'_i(t)^2}{\phi_i(t)}\right)dt < \infty,
\quad
\int_{T}^{\infty}\frac{1}{P_i(t)\phi_i(t)}dt = \infty, \\
 \int_{T}^{\infty}\phi_i(t)Q_i(t)dt = \infty \quad (i=1,2),
\end{gather*}
then every solution $y(t)$ of  \eqref{eE} is oscillatory.
\end{theorem}


\subsection*{An application}

The flow of chemically reacting mixtures of gases plays a functional role
in studying such diverse problems as the solar atmosphere,  the atmosphere
of other stars, and the gas flow in the combustion chamber of a rocket engine.
It can be shown that for certain types of gases the propagation of small
disturbances through the gas as time $t$ varies is described by the
DE $y''' + ay'' + by' + cy = 0$, where the given constants $a$, $b$ and
$c$ are all positive. The independent variable $y(t)$ is proportional to
the gas pressure. The coefficient $a$, $b$ and $c$ are related to physical
properties and the temperature of the gas. In particular, the constants
 $b$ and $c$ are usually called the frozen and equilibrium sound speeds
of the gas, respectively. From the physical properties, it is known that $b > c$.
If the DE is asymptotically stable, then all disturbances to the gas will
eventually disappear because they are dissipated by the chemical reactions.
If the DE is not asymptotically stable, then there are disturbances which
do not decay as $t \to \infty$. Then shock waves may form in the gas
 (see,  \cite{r1}). Thus we consider the equation
\begin{equation}\label{ex1}
y'''(t) + \frac{3}{4}y''(t) + \frac{1}{4}y'(t) + \frac{3}{16}y(t) = 0, \quad t >0.
\end{equation}
Here $a(t)=3/4$, $b(t)=1/4$ and $c(t)=3/16$. It is easy to check that
$a b = c$,
which implies that \eqref{ex1} is not asymptotically stable.
Since $b(t) > c(t)$ and
$$
a(t) = \frac{3}{4} < \frac{5}{4} = b(t) + 1,
$$
Assumption (H4) is not satisfied, but (H5) is satisfied.
 A straightforward computation yields
$$
\int^{\infty}\pi(t)e^{A(t)}c(t)dt
= \int^{\infty}\Big(\frac{4}{3}e^{-3t/4}\Big)
 \left(e^{3t/4}\right)  \Big(\frac{3}{16}\Big)dt = \infty.
$$
By choosing $\phi_1(t)=t^{1/2}$ and $\phi_2(t)=e^{-t/2}$,
 we can show that
\begin{gather*}
\int^{\infty}\frac{P_1(t)\phi'_1(t)^2}{\phi_1(t)}dt
= \int^{\infty}\Big(\frac{1 \cdot \big(\frac{1}{2}t^{-\frac{1}{2}}\big)^2}
{t^{1/2}}\Big)dt < \infty, \\
\int^{\infty}\frac{1}{P_1(t)\phi_1(t)}dt
  = \int^{\infty}\Big(\frac{1}{1 \cdot t^{1/2}}\Big)dt = \infty, \\
\int^{\infty}\phi_1(t)Q_1(t)dt = \int^{\infty}\left(t^{1/2}\right)
\Big(\frac{5}{32}\Big)dt = \infty,
\end{gather*}
and
\begin{gather*}
 \int^{\infty}\frac{P_2(t)\phi'_2(t)^2}{\phi_2(t)}dt
 = \int^{\infty}\Big(\frac{\frac{3}{4 (1-e^{-3t/4})}
 \left(-\frac{1}{2}e^{-t/2}\right)^2}
 {e^{-t/2}}\Big)dt < \infty, \\
 \int^{\infty}\frac{1}{P_2(t)\phi_2(t)}dt
= \int^{\infty}\Big(\frac{1}{\frac{3}{4 (1-e^{-3t/4})}
 e^{-t/2}}\Big)dt = \infty, \\
 \int^{\infty}\phi_2(t)Q_2(t)dt
= \int^{\infty}e^{-t/2}\big\{\frac{3}{16}e^{3t/4}
- \frac{1}{24}(1-e^{-3t/4})\big\}dt = \infty.
\end{gather*}
So every solution of \eqref{ex1} is oscillatory by Theorem \ref{thm2}.
Moreover, we note that $y(t)=\sin{\frac{t}{2}}$ is a solution of \eqref{ex1},
which is oscillatory.


\begin{theorem} \label{thm3}
Assume that {\rm (H1)}--{\rm (H4)} or {\rm (H1)}--{\rm (H3)}, {\rm (H5)}
are satisfied. If
$$
\limsup_{t \to \infty}\frac{1}{H(t,T)}A_{T}^{\rho}
\Big(Q_i-P_{i}|h-\frac{\rho'}{\rho}|;t\Big) = \infty,
$$
then every solution  of  \eqref{eE} is oscillatory.
\end{theorem}

Now, we consider the linear case of equation \eqref{eE}:
\begin{equation} \label{eEL}
 y'''(t) + a(t)y''(t) + b(t)y'(t) + \sum_{i=1}^{m}c_i(t)y(\sigma_i(t)) = 0, \quad
 t > 0,
\end{equation}
where $\sigma_i(t) \ge t$  $(i=1,2,\dots,m)$.


\begin{corollary} \label{coro1}
Assume that {\rm (H1)}--{\rm (H4)} or {\rm (H1)}--{\rm (H3)}, {\rm (H5)}
are satisfied. If \eqref{lm1} holds and
\[
\int^{\infty}\Big\{\frac{2}{27}a^3(t) - \frac{1}{3}a(t)b(t) + c_j(t) \
 -\frac{2}{3\sqrt{3}}\Big(\frac{a^2(t)}{3} - (b(t)-a'(t))\Big)^{3/2}\Big\}dt
 = \infty,
\]
then every solution  of \eqref{eEL} is oscillatory.
\end{corollary}


\begin{proof}
Suppose that $y(t)$ is a nonoscillatory solution of \eqref{eEL}.
It follows from Lemma \ref{lem1} or Lemma \ref{lem2} that $y(t)y'(t) > 0$ holds.
Now we define
$$
u(t) = \frac{y'(t)}{y(t)} > 0,
$$
then we see that
\begin{align*}
u''(t)
&= \frac{y'''(t)}{y(t)} - \frac{y'(t)y''(t)}{y^2(t)} - 2u'(t)u(t) \\
&\le -a(t)u'(t) - 3u'(t)u(t) - \{u^3(t) + a(t)u^2(t) + b(t)u(t)
+ c_j(t)\},
\end{align*}
and so,
\begin{equation} \label{c11}
\begin{aligned}
&\big[u'(t) + \frac{3}{2}u^2(t) + a(t)u(t)\big]' \\
&\le  -\big\{u^3(t) + a(t)u^2(t) + \left(b(t)-a'(t)\right)u(t) + c_j(t)\big\}
\equiv -F(u(t),t).
\end{aligned}
\end{equation}
Clearly, $F(u(t),t)$ has a minimum value for $u(t) >0$ at
$$
u(t) = \frac{-a(t) + \sqrt{a^2(t)-3(b(t)-a'(t))}}{3}.
$$
This, together with \eqref{c11}, implies that
\begin{align*}
& \big[u'(t) + \frac{3}{2}u^2(t) + a(t)u(t)\big]' \\
&\le - \Big\{\frac{2}{27}a^3(s) - \frac{1}{3}a(s)b(s) + c_j(s)
- \frac{2}{3\sqrt{3}}\Big(\frac{a^2(s)}{3}-(b(s)-a'(s))\Big)^{3/2}\Big\}.
\end{align*}
Integrating this over $[t_0,t]$ yields
\begin{align*}
u'(t)
&\le u'(t_0) + \frac{3}{2}u^2(t_0) + a(t_0)u(t_0)
- \int_{t_0}^{t}\Big\{\frac{2}{27}a^3(s) - \frac{1}{3}a(s)b(s) + c_j(s) \\
&\quad - \frac{2}{3\sqrt{3}}\Big(\frac{a^2(s)}{3}-(b(s)-a'(s))\Big)^{3/2}\Big\}ds,
\end{align*}
which implies that $u(t) < 0$ for large $t$.
This  contradiction completes the proof.
\end{proof}

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\end{document}