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\markboth{\hfil Sturmian comparison method \hfil EJDE--2002/31}
{EJDE--2002/31\hfil Leonid Berezansky \& Yury Domshlak \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 31, pp. 1--18. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Differential equations with several deviating arguments:
 Sturmian comparison method in oscillation theory, II 
 %
\thanks{ {\em Mathematics Subject Classifications:} 34K11.
\hfil\break\indent
{\em Key words:} mixed differential equations,
oscillation, non-oscillation,  \hfil\break\indent
Sturmian comparison method.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted November 29, 2001. Published April 1, 2002.} }
\date{}
%
\author{Leonid Berezansky \& Yury Domshlak}
\maketitle

\begin{abstract} 
  We study the oscillation of solutions to the differential
  equation
  $$
  \dot{x}(t) +a_1(t)x[r(t)]+a_2(t)x[p(t)]=0, \quad t\geq t_0
  $$
  which has a retarded argument $r(t)$ and an advanced argument 
  $p(t)$. We obtain oscillation and non-oscillation conditions 
  which are closed to be necessary. We provide examples to show 
  that our results are best possible and compare them with 
  known results.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

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\section{introduction}

This paper is a continuation of the investigation on equations with two
deviating arguments started in \cite{BD}.  We are concerned with
the following two basic problems in the oscillation theory:\\
1) Sufficient conditions for the existence
of a non-oscillatory solution.\\
2) Sufficient conditions for all solutions to be oscillatory.

Methods of investigation differ significantly for these two problems.
For the first problem, it is enough to prove the
existence of a sign preserving solution. In this case, various fixed point
methods are applied, or a monotone sequence defined which converges
to a non-oscillatory solution.

The investigation of the second oscillation problem can not employ
methods that characterize only some solutions of the equation.
Thus the proof is usually done by contradiction, i.e. the assumption
that there exists a non-oscillatory solution is inconsistent
with the constraints on the equation parameters.

In the works \cite{D1,D2,D3, DA1, DA2,DS1,DS2}, a constructive
method for
proving that
all the solutions are oscillatory  was proposed and was called Sturmian
comparison method.
It is based on the generalization of the classical Sturm comparison theorem
to functional-differential equations and inequalities .
This theorem was originally formulated for
the second order ODE. The Sturmian comparison method was described in
detail in \cite{BD}. We give here just a brief description of the method.

The main idea of the method is to obtain the widest possible set of
functional differential inequalities (the ``testing" equations)
associated with the  equation, such that at
least one of the solutions is a so called ``slowly oscillating".
This together with the Sturmian Comparison Theorem (see Theorem \ref{thmA}
in this paper) yields that all solutions oscillate.

The proposed method is constructive. Indeed,
it is enough to find {\em only one} solution which enjoys certain properties
instead of checking the fact that {\em all} the solutions of the ``test"
equation oscillate.
Note that the results usually called Sturmian comparison oscillation
theorems \cite{Dom1,Dom2,DK,I1} are  not a part of Sturmian comparison
theorem.


The Sturmian comparison method has been applied to the first order
delay differential equations
\cite{D3,D4,DA1,DA2,DS1,DS2}, neutral
differential equations \cite{D1,D2,DS1}, the second
order delay differential equations  \cite{D1,D2,D3,DS1},
impulsive delay differential equation  \cite{BDS1,BDS2,BS} and difference equations
\cite{D5,D6,D7,D8,DC}.

Note the search for the ``slowly oscillating" solutions can be treated
as the search for ``big half-cycles"  \cite{M} of a given length.
Thus this method incidentally solves a rather difficult problem of the
evaluation of the length of sign preserving intervals
\cite{D3,DA2,DS1,DS2}.

This method turned out to be efficient for the investigation of so called
``critical cases" as well. This means that the oscillation of all solutions of
functional DE with almost constant coefficients is studied in the case when
the limit equation has a non-oscillatory solution (see \cite{DSt1,DSt2}
for the application of Sturmian comparison method in such cases).

The present paper is a continuation of \cite{BD}, where the basic ideas of Sturmian comparison method
were described for a rather general functional DE and this method was applied to
DE of the first order with two retarded arguments.

This paper develops the Sturmian comparison method
for the mixed differential equation
\begin{equation}
\label{1}
\dot{x}(t) +a_1(t)x[r(t)]+a_2(t)x[p(t)]=0, \quad t\geq t_0,
\end{equation}
with nonnegative coefficients $a_i(t)$, one delayed argument ($r(t)\leq t$ )
and one advanced argument ($p(t)\geq t$).
To the best of our knowledge, oscillation properties of such equations has not been
studied before except special cases of the  autonomous equations
 \cite{GS,LS,LSt,R,S} and equations with
constant delays
 \cite{Y}.

Sufficient conditions for all solutions of (\ref{1})
to be oscillating are obtained here. The examples presented here show that
these conditions are rather sharp.
Further we obtain sufficient conditions for the existence
of a non-oscillatory solution of (\ref{1}).
Similar non-oscillation results are obtained for (\ref{1}) with non-positive
coefficients.

\section{Sturmian comparison theorem}

Suppose $-\infty<\alpha,\beta<\infty$, $k=1,2$,
and the following assumptions hold:
\begin{enumerate}
\item[(A1)] The functions $a_k(t)$ are continuous on $(\alpha,\beta) $,
$a_k(t)\geq 0$.

\item[(A2)] The functions $r(t), p(t)$ are monotone increasing on
$(\alpha,\beta) $
with continuous derivatives and $r(t)\leq t , p(t)\geq t$.
\end{enumerate}
We can extend functions $r(t), p(t)$, without loss of monotonicity and
differentiability, in such a way that the range of these functions
include $[\alpha,\beta]$.
Then  there exist continuously differentiable functions $g(t), s(t)$ such that
$r[g(t)]=t$ and $p[s(t)]=t$ for $t\in(\alpha,\beta)$.

In this and the next sections, we assume that conditions (A1) and (A2)
hold for (\ref{1}).
We define the following differential operators
on the set of continuous functions on $[s(\alpha),q(\beta)]$
with continuous derivatives on $(\alpha,\beta)$:
\begin{gather}
\label{3}
 (lx)(t):= x'(t)+a_1(t)x[r(t)]+a_2(t)x[p(t)],\quad t\in(\alpha,\beta), \\
\label{4}
(\tilde{l}y)(t):=-y'(t)+ q'(t)\tilde{a}_1[q(t)]y[q(t)]
+s'(t)\tilde{a}_2[s(t)]y[s(t)],\quad t\in(\alpha,\beta).
\end{gather}
Here  $\tilde{a}_k(t)$, $k=1,2$ are continuous  functions
on $(r(\alpha),p(\beta))$.
Let us define the following differential inequalities and equations:
\begin{gather}
\label{5}
(lx)(t)\leq 0,\quad t\in(\alpha,\beta), \\
\label{5a}
(lx)(t)=0,\quad t\in(\alpha,\beta), \\
\label{6}
(\tilde{l}y)(t)\geq 0,\quad t\in(\alpha,\beta).
\end{gather}

\paragraph{Definition} (see \cite{D3})
An interval $(\alpha,\beta)$ is called
{\em a regular half-cycle}  for (\ref{6}) if
$$
r(\beta)>\alpha,\quad \beta>p(\alpha),
$$
and there exist a solution $y(t)$ of (\ref{6}) such that
\begin{equation}
\label{7} \begin{gathered}
y(\alpha)=y(\beta)=0,\quad y(t)>0, \quad t\in (\alpha,\beta),\\
y(t)\leq 0,\quad t\in (s(\alpha),\alpha) \cup (\beta,q(\beta)).
\end{gathered}\end{equation}
The definition of regular half-cycle for  (\ref{5}) and (\ref{5a}) is similar.

\paragraph{Definition}  (see \cite{D3})
a) A solution $x(t)$, $t_0\leq t<\infty$ of a differential equation or
inequality  is called {\em non-oscillatory} if there exists
$T$ such that $x(t)\neq 0$, $t\geq T$ and {\em oscillatory} otherwise.\\
b) An oscillatory solution $x$ of a differential equation or inequality
is called {\em regular oscillatory} if for every $T$ there exists its
regular half-cycle $(\alpha,\beta), \alpha>T$
and {\em quick oscillatory} otherwise.

The following theorem is a
 general Sturmian comparison theorem  \cite[Theorem 1]{BD}
which is formulated for (\ref{1}).

\begin{theorem} \label{thmA}
 Suppose  $(\alpha,\beta)$ is a regular half-cycle for (\ref{6}) ,
\begin{equation}
\label{8}
\tilde{a}_1(t)\geq 0,\quad t\in (\beta,q(\beta));
\quad \tilde{a}_2(t)\geq 0, \quad t\in (s(\alpha),\alpha)),
\end{equation}
\begin{equation}
\label{9}
a_1(t)\geq \left\{ \begin{array}{ll} 0,& t\in (\alpha,q(\alpha)),\\
\tilde{a}_1(t),& t\in  (q(\alpha),\beta),
\end{array}\right. \quad
a_2(t)\geq \left\{ \begin{array}{ll} \tilde{a}_2(t),& t\in (\alpha,s(\beta)),\\
0,& t\in  (s(\beta),\beta),
\end{array}\right.
\end{equation}
and at least one of (\ref{8})-(\ref{9}) is strict on some subinterval of
$(\alpha,\beta)$.
Then (\ref{5})  has no positive solutions on $(s(\alpha),q(\beta))$.
\end{theorem}

\begin{corollary} \label{coroA.1}
Suppose (\ref{6}) has a regular oscillatory solution on $(t_0,\infty)$ and
$$
a_k(t)\geq \tilde{a}_k(t)\geq 0,\quad k=1,2,\quad t\geq t_0.
$$
Then (\ref{5}) has no positive solution on $(t_0,\infty)$.
\end{corollary}

\paragraph{Remark} %1
Theorem \ref{thmA} is concerned with the behavior
of solutions of mixed differential equations and inequalities
on a {\em finite interval} and not on a half-line.
Therefore one can obtain from Theorem \ref{thmA} not only explicit conditions
of oscillation but also estimations of the length of the
sign-preserving intervals of solutions. In this paper we will not consider
this problem.

\section{Construction of the testing equations}

\begin{lemma} \label{lm1}
Let  $r(\beta)>\alpha$, $p(\alpha) <\beta$ and for continuous
functions $\varphi(t)$, $m(t)$ assume the following conditions hold:
\begin{equation}
\label{10}
0< \int_{\alpha}^t \varphi(s)ds <\pi,~t\in (\alpha,\beta);\quad
0< \int_t^{\alpha} \varphi(s)ds <\pi,~t\in (s(\alpha),\alpha);
\end{equation}
\begin{equation}
\label{11}
\int_{\alpha}^{\beta} \varphi(s)ds =\pi;\quad
0< \int_{s(t)}^{q(t)} \varphi(s)ds <\frac{\pi}{2},\quad t\in (r(\alpha),\beta);
\end{equation}
\begin{equation}
\label{12}
m(t)\geq \left\{\begin{array}{ll}
-\varphi(t)\cot\int_{s(t)}^{t} \varphi(s)ds,& t\in (r(\beta),\beta),\\
\varphi(t)\cot\int_{t}^{q(t)} \varphi(s)ds,& t\in (\alpha, p(\alpha)).
\end{array}\right.
\end{equation}
Then the interval $(\alpha,\beta)$ is regular half-cycle for (\ref{6})
where $\tilde{a}_k(t)$ defined by
\begin{multline*}
q'(t)\tilde{a}_1[q(t)]:=
\csc\int_{s(t)}^{q(t)} \varphi(s)ds\exp\big(-\int_{t}^{q(t)} m(s)ds\big)\\
\times\Big\{m(t)\sin\int_{s(t)}^{t} \varphi(s)ds+\varphi(t)\cos\int_{s(t)}^{t}
\varphi(s)ds\Big\},
\end{multline*}
\begin{multline} \label{13}
s'(t)\tilde{a}_2[s(t)]:=
\csc \int_{s(t)}^{q(t)} \varphi(s)ds\exp\big(\int_{s(t)}^{t} m(s)ds\big)\\
\times\Big\{m(t)\sin\int_{t}^{q(t)} \varphi(s)ds-\varphi(t)\cos\int_{t}^{q(t)}
\varphi(s)ds\Big\}.
\end{multline}
\end{lemma}

\paragraph{Proof.} By direct calculations one can check that the function
$$
y(t):=\sin\int_{\alpha}^t \varphi(s)ds\times\exp\int_{\alpha}^t m(s)ds
$$
is a solution of the equation $(\tilde{l}y)(t)=0$.
Conditions (\ref{10})-(\ref{12}) and $p(\beta)>\alpha$
make $ (\alpha,\beta)$ an regular half-cycle for (\ref{6}).

From Theorem \ref{thmA} and Lemma \ref{lm1}  we obtain the following
theorem.

\begin{theorem} \label{thm1}
Suppose for some functions $\varphi(t), m(t)$ conditions of Lemma \ref{lm1}
and (\ref{8})-(\ref{9}) hold,
where $\tilde{a}_i(t)$ are given by (\ref{13})
and at least one of (\ref{8})-(\ref{9}) is strict
on some subinterval. Then (\ref{5})
has no positive solution  on $(r(\alpha), p(\beta))$.
\end{theorem}

\begin{corollary} \label{coro1.1}
Suppose the conditions of Theorem \ref{thm1} hold for a sequence of intervals
$(\alpha_j,\beta_j)$, $\alpha_j\rightarrow\infty$.
 Then all solutions of (\ref{1}) are oscillatory.
\end{corollary}

\paragraph{Remark.} %  2
No limitations are imposed on the coefficients
$a_k(t)$ of (\ref{1}) outside the set
$\cup_{j=1}^{\infty}(\alpha_j,\beta_j)$ in Corollary \ref{coro1.1}.

\begin{corollary} \label{coro1.2}
Let $ r(t)\rightarrow\infty$, $\int_{t_0}^{\infty}\varphi(s)ds =\infty$,
and
\begin{gather}
\label{15}
0< \int_{s(t)}^{q(t)} \varphi(s)ds <\frac{\pi}{2},~t\geq t_0, \\
\label{16}
m(t)\geq \max\Big\{-\varphi(t)\cot\int_{s(t)}^{t} \varphi(s)ds;
\varphi(t)\cot\int_{t}^{q(t)} \varphi(s)ds\Big\},t>t_0.
\end{gather}
Assume also that for every $T$ there exists $\alpha>T$ such that
$$
0< \int_{t}^{\alpha} \varphi(s)ds <\frac{\pi}{2},\;t\in (s(\alpha),\alpha),
\quad
0< \int_{\alpha}^{t} \varphi(s)ds <\pi, \;t\in (\alpha,\beta),
$$
where $\beta$ is defined by $\int_{\alpha}^{\beta} \varphi(s)ds =
\pi$.
If
\begin{equation}
\label{16a}
a_k(t)\geq \tilde{a}_k(t),\quad k=1,2,\; t\geq t_0,
\end{equation}
where $\tilde{a}_i$ are defined by (\ref{13}),
then all  solutions of
(\ref{1}) are oscillatory.
\end{corollary}

On the one hand the following statement gives a new clear
proof of the following well-known \cite{LSt}
fundamental oscillation criterion for autonomous equation with mixed deviations
\begin{equation}
\label{17}
x'(t)+a_1x(t-\tau)+a_2x(t+\sigma)=0,\quad \tau >0,\; \sigma >0,\;
a_k>0,\; k=1,2.
\end{equation}
On the other hand this statement demonstrates the sharpness of
Theorem \ref{thm1},
which in particular case (an autonomous equation)
enables one to obtain necessary and sufficient conditions  of oscillation of all
solutions of (\ref{1}).

\begin{corollary} \label{coro1.3}
Suppose for the characteristic quasi-polynomial of (\ref{17})
$$
F(\lambda):= \lambda+a_1\exp\{-\lambda \tau\}+a_2\exp\{\lambda \sigma\}
$$
 the following condition holds
\begin{equation}
\label{18}
\{F(\lambda)>0,\quad\mbox{for all }\lambda \in (-\infty,\infty)\}.
\end{equation}
Then all solutions of (\ref{17}) are oscillatory.
\end{corollary}

(It is obvious that (\ref{18}) is also necessary for
oscillation of all solutions as well).

\paragraph{Proof.} It is easy to see that the  equation
\begin{equation}
\label{19}
F'(\lambda) :=1-a_1\tau e^{-\lambda\tau}+a_2\sigma e^{\lambda\sigma}=0
\end{equation}
has a unique root $\lambda_0$ and
 $\inf_{-\infty<\lambda<\infty}F(\lambda)=F(\lambda_0).$
Indeed, for every $\lambda$ we have
$$
\lim_{\lambda\rightarrow +\infty}F'(\lambda)=+\infty,\quad
\lim_{\lambda\rightarrow-\infty}F'(\lambda)=-\infty, \quad F^{''}(\lambda)>0.
$$
Equation (\ref{17}) implies
\begin{equation} \label{20} \begin{gathered}
\mbox{Cond. (\ref{18})}
\Longleftrightarrow \left\{F(\lambda_0)>0\right\}\Longleftrightarrow
\sigma(\lambda_0+a_1e^{-\lambda_0\tau}+a_2e^{\lambda_0\sigma})>0 \\
\Longleftrightarrow \sigma\lambda_0+\sigma a_1 e^{-\lambda_0\sigma}+
(a_1\tau e^{-\lambda_0\tau}-1)>0
\Longleftrightarrow a_1>\frac{-\sigma\lambda_0+1}{\tau+\sigma}e^{\lambda_0\tau}.
\end{gathered}\end{equation}
Similarly we have
\begin{equation} \label{21}
\mbox{Cond. (\ref{18})}\Longleftrightarrow \tau F(\lambda_0)>0\Longleftrightarrow
 a_2>\frac{-\tau\lambda_0-1}{\tau+\sigma}e^{-\lambda_0\sigma}.
\end{equation}
Consider two possible cases for $a_1$. Assume first $a_1>\frac{1}{\tau e}$.
Then the equation
$$
y'(t)+a_1y(t-\tau)=0
$$
has no eventually positive solutions and therefore the inequality
$$
y'(t)+a_1y(t-\tau)\leq 0
$$
has no such  solutions. Then (\ref{17}) has no eventually positive solutions.
Assume now that $a_1\leq \frac{1}{\tau e}$. Then
$$
F'(-\frac{1}{\tau})=1-a_1\tau e+a_2\sigma
\exp\big\{-\frac{\sigma}{\tau}\big\}>0.
$$
Since $F'(\lambda_0)=0$ and $F^{''}(\lambda)>0$ for all $\lambda$, we obtain
\begin{equation}
\label{22}
\lambda_0<-\frac{1}{\tau}.
\end{equation}
Put $\varphi(t):=\nu,\quad m(t):=-\lambda_0$ in Theorem \ref{thm1},
where $\nu>0$ is a sufficiently small number
which will be chosen below.
Then (\ref{22}) implies
$$
m(t)=-\lambda_0>\frac{1}{\tau}>\frac{\nu}{\tan\nu\tau}.
$$
Hence (\ref{12}) holds.
Thus, Theorem \ref{thm1} implies that under the conditions
\begin{equation}
\label{23} \begin{gathered}
a_1>\frac{e^{\lambda_0\tau}}{\sin \nu(\tau+\sigma)}\{-\lambda_0\sin \nu\sigma
+\nu\cos\nu\sigma\}:=P_1(\nu)\\
a_2>\frac{e^{-\lambda_0\sigma}}{\sin \nu(\tau+\sigma)}\{-\lambda_0\sin \nu\tau
-\nu\cos\nu\tau\}:=P_2(\nu),
\end{gathered}
\end{equation}
all solutions of (\ref{17}) are oscillatory.

It is easy to see that
$$
\lim_{\nu\rightarrow 0}P_1(t)
=e^{\lambda_0\tau}\frac{-\lambda_0\sigma+1}{\tau+\sigma},\quad
\lim_{\nu\rightarrow 0}P_2(t)
=e^{-\lambda_0\sigma}\frac{-\lambda_0\tau-1}{\tau+\sigma}.
$$
Inequalities (\ref{20}) and (\ref{21}) are sharp and so  (\ref{23})
holds for $\nu \in (0,\nu_0)$. Then Corollary \ref{coro1.3} is proven.
 \smallskip


Next we will obtain from Theorem \ref{thm1} an explicit  condition for oscillation
not only in  terms of pointwise estimations but in terms of
{\em the integral average estimations} as well.
To avoid unwieldy formulations we will omit
the estimation for the length of sign-preserving
intervals of solutions.

\begin{theorem} \label{thm2}
Let $r(t)\rightarrow\infty$ and assume that there exist functions
$b_j(t)$, $j=1,2$, such that:
\begin{equation}
\label{24}
a_j(t)\geq b_j(t)\geq 0, ~j=1,2,~t\geq t_0;
\end{equation}
the following limits exist and are finite:
\begin{equation}
\label{25}
B_{1j}:=\lim_{t\rightarrow\infty}\int_{r(t)}^t b_j(s)ds,\quad
B_{2j}:=\lim_{t\rightarrow\infty}\int_{t}^{p(t)} b_j(s)ds,\quad
j=1,2,
\end{equation}
with
\begin{equation}
\label{26}
B_{11}+B_{22}>0;
\end{equation}
the following system has a positive solution $\{x_1;x_2\}$:
\begin{equation}
\label{27}\begin{gathered}
-(B_{11}B_{22}-B_{12}B_{21})x_1x_2-B_{11}x_1+B_{22}x_2+1=0\\
\ln x_1-B_{11}x_1-B_{12}x_2<0\\
\ln x_2+B_{21}x_1+B_{22}x_2<0\,.
\end{gathered}
\end{equation}
Then all  solutions of (\ref{1}) are oscillatory.
\end{theorem}

\paragraph{Proof.} In view of the first inequality in (\ref{27}) the system
\begin{equation}
\label{28} \begin{gathered}
(x_1B_{11}-1)\alpha_1-x_2B_{12}\alpha_2=0\\
-x_1B_{21}\alpha_1+(x_2B_{22}+1)\alpha_2=0\\
\alpha_1+\alpha_2=1
\end{gathered}
\end{equation}
 has a solution $\{\alpha_1; \alpha_2\}$, $\alpha_j>0,~j=1,2$ (we
omit the details).

In Theorem \ref{thm1},  denote functions $m(t)>0$ and $\varphi(t)$ as follows:
\begin{gather}
\label{29}
\varphi(t):= \nu\alpha_1 x_1 q'(t)b_1[ q(t)]-
\nu\alpha_2 x_2 s'(t)b_2[ s(t)], \\
\label{30}
m(t):= x_1q'(t)b_1[ q(t)]+ x_2 s'(t)b_2[ s(t)],
\end{gather}
where $\nu>0$ is a sufficiently small number
which will be chosen below.
Note that  $\varphi(t)$ defined by  (\ref{29}) is {\em not necessarily}
nonnegative for all $t$.
Nevertheless,
from (\ref{29})-(\ref{30}) we obtain
\begin{gather}
\lim_{t\rightarrow\infty} \int_t^{q(t)} \varphi(s)ds =
\nu\alpha_1 x_1 B_{11}-\nu\alpha_2 x_2 B_{12}=\nu\alpha_1>0, \nonumber\\
\label{31}
\lim_{t\rightarrow\infty} \int_{s(t)}^{t} \varphi(s)ds =
\nu\alpha_1 x_1 B_{21}-\nu\alpha_2 x_2 B_{22}=\nu\alpha_2>0,\\
\label{32}
\lim_{t\rightarrow\infty} \int_{t}^{q(t)}m(s)ds= x_1 B_{11}+x_2 B_{12},\quad
\lim_{t\rightarrow\infty} \int_{s(t)}^{t}m(s)ds= x_1 B_{21}+x_2 B_{22}.
\end{gather}
Furthermore, $\int^{\infty}\varphi(s)ds =\infty$ and
(\ref{10}), (\ref{11}) hold for $ t>T$, where $T$
is sufficiently large.

The inequality
$$
\nu\alpha_1x_1q'(t)b_1[ q(t)]- \nu\alpha_2x_2 s'(t)b_2[ s(t)]
\leq \nu\alpha_1\left[x_1q'(t)b_1[ q(t)]+x_2 s'(t)b_2[ s(t)]\right]
$$
implies that for sufficiently large $t$
\begin{align*}
\varphi(t)\cot \int_t^{q(t)}\varphi(s)ds\leq& \nu\alpha_1 m(t)
\cot \int_t^{q(t)}\varphi(s)ds \\
\leq& \frac{\nu\alpha_1}{\tan \nu\alpha_1}
\frac{\tan \nu\alpha_1}{\tan \int_t^{q(t)}\varphi(s)ds}\leq m(t).
\end{align*}
Similarly, the inequality
$$
\nu\alpha_1x_1q'(t)b_1[ q(t)]- \nu\alpha_2x_2 s'(t)b_2[ s(t)]
\geq -\nu\alpha_2\left[x_1q'(t)b_1[ q(t)]+x_2 s'(t)b_2[ s(t)]\right]
$$
implies
\begin{align*}
\varphi(t)\cot \int_{s(t)}^{t}\varphi(s)ds\geq &-\nu\alpha_2 \cot
\int_{s(t)}^{t}\varphi(s)ds\\
\geq &-\frac{\nu\alpha_2}{\tan \nu\alpha_2}
\frac{\tan \nu\alpha_2}{\tan \int_{s(t)}^t\varphi(s)ds}\geq -m(t).
\end{align*}
Hence $m(t)\geq -\varphi(t)\cot \int_{s(t)}^{t}\varphi(s)ds$.
Therefore (\ref{16}) holds.
Denote
\begin{gather*}
D_1(t,\nu):=\csc \int_{s(t)}^{q(t)}\varphi(s)ds\exp\big\{-\int_t^{q(t)}m(s)ds
\big\}\nu x_1, \\
D_2(t,\nu):=\csc \int_{s(t)}^{q(t)}\varphi(s)ds\exp\big\{\int_{s(t)}^{t}m(s)ds
\big\}\nu x_2.
\end{gather*}
Then
\begin{equation} \label{33} \begin{gathered}
G_1(\nu):=\lim_{t\rightarrow\infty} D_1(t,\nu)=\exp(-x_1B_{11}-x_2B_{12})\frac{\nu x_1}{\sin
\nu}, \\
G_2(\nu):=\lim_{t\rightarrow\infty} D_2(t,\nu)=\exp(x_1B_{21}+x_1B_{22})\frac{\nu x_2}{\sin
\nu}.
\end{gathered} \end{equation}
Condition (\ref{27}) implies $G_k(\nu)<1$ $\nu\in (0,\nu_0)$, $k=1,2$.
Hence
$$
D_k(t,\nu)<1, \quad \nu\in (0,\nu_0),\quad k=1,2,\quad t>T\gg 1
$$
and therefore
\begin{equation}\label{34} \begin{gathered}
q'(t)a_1[q(t)]>D_1(t,\nu)q'(t))b_1[q(t)]\\
s'(t)a_2[q(t)]>D_2(t,\nu)s'(t))b_2[s(t)].
\end{gathered}\end{equation}
for $\nu\in (0,\nu_0)$, $t>T$. The
right-hand sides of  (\ref{13}) can be rewritten in the form
$H_1(t,\nu)+D_1(t,\nu)q'(t))b_1[q(t)]$, and $H_2(t,\nu)+D_2(t,\nu)s'(t))b_2[s(t)]$,
where
\begin{multline*}
H_1(t,\nu):=\csc  \int_{s(t)}^{q(t)} \varphi(s)ds
\exp\big(-\int_{t}^{q(t)} m(s)ds\big)\\
\times\Big[m(t)\big\{\int_{s(t)}^{t}
\varphi(s)ds-\nu\alpha_2\big\}
+\varphi(t)\big\{\cos\int_{s(t)}^{t}\varphi(s)ds -1\big\}\Big]
\end{multline*}
and
\begin{multline*}
H_2(t,\nu):=\csc  \int_{s(t)}^{q(t)} \varphi(s)ds
\exp\int_{s(t)}^{t} m(s)ds \\
\times\Big[m(t)\big\{\int_{t}^{q(t)}
\varphi(s)ds-\nu\alpha_1\big\}-\varphi(t)\big\{cos \int_{t}^{q(t)}
\varphi(s)ds-1\big\}\Big].
\end{multline*}
It is easy to check that
$\lim_{t\rightarrow\infty}H_i(t,\nu)=0$, $i=1,2$, $\nu\in(0,\nu_0)$.
Hence  (\ref{16a}) holds and therefore all solutions of  (\ref{1}) are
oscillatory.

\paragraph{Example} %1
Consider  the equation
\begin{equation}
\label{35}
x'(t)+\frac{a_1}{t}x\left(\frac{t}{\mu}\right)+\frac{a_2}{t}x(t+\tau)=0,~t\geq t_0>0,
\end{equation}
where $\mu>1$, $\tau >0$, $a_1,a_2 >0$.
Put $b_1(t):=a_1(t)=a_1/t$ and $b_2(t):=a_2(t)=a_2/t$ in Theorem \ref{thm2}.
Then $B_{11}=a_1\ln\mu,~B_{12}=a_2\ln\mu,~ B_{21}=B_{22}=0.$
System (\ref{27}) turns into the system
\begin{gather*}
-a_1x_1\ln\mu +1=0\\
\ln x_1-a_1x_1\ln\mu -a_2x_2\ln\mu <0\\
\ln x_2<0
\end{gather*}
which is equivalent to the system
\begin{gather*}
x_1=\frac{1}{a_1\ln\mu}\\
-\ln[a_1\ln\mu]-1<x_2a_2\ln\mu \\
\ln x_2<0
\end{gather*}
and this in turn is equivalent to the system
\begin{gather*}
x_1=\frac{1}{a_1\ln\mu}\\
\frac{-\ln[a_1\ln\mu]-1}{a_2\ln\mu}<x_2<1
\end{gather*}
The last system has a solution if and only if
\begin{equation}
\label{36}
\frac{-\ln[a_1\ln\mu]-1}{a_2\ln\mu}<1\Longleftrightarrow
 a_1\mu^{a_2} >\frac{1}{e\ln\mu}.
\end{equation}
Thus, (\ref{36}) is sufficient for oscillation of all solutions of (\ref{35}).
Note that (\ref{36}) does not depend on $\tau$.

\paragraph{Remark.} % 3.
It will be shown in Section 4 that if
$a_1\mu^{a_2} < 1/(e\ln\mu)$ then (\ref{35}) has a non-oscillatory solution.

\paragraph{Example} % 2
Consider the equation
\begin{equation}
\label{37}
x'(t)+\frac{a_1}{t}x\left(\frac{t}{\mu}\right)
+\frac{a_2}{t^{\beta}}x(t+t^{\alpha})=0,~t\geq t_0>0,
\end{equation}
where $a_1,a_2>0$, $\mu>1$, $0\leq \alpha<1$, $0\leq \beta<1$.
In Theorem \ref{thm2}, Let
$$
b_1(t):=a_1(t)=\frac{a_1}{t},\quad
b_2(t):=\frac{A}{t}\leq a_2(t)=\frac{a_2}{t^{\beta}}, \quad t>T,
$$
where $T$ is sufficiently large, $A$ is an arbitrarily large positive constant.
Then
$ B_{11}=a_1\ln\mu,~ B_{12}=A\ln\mu,~B_{21}=B_{22}=0.$
One can repeat now all calculations in the previous example. Cond.(\ref{36}) is
 $a_1\mu^{A}>\frac{1}{e\ln\mu}$ which holds for every $a_1>0$ if $A$ is
 sufficiently large.
Therefore, {\em all solutions of (\ref{37}) are oscillatory for any
$\{a_1>0$, $a_2>0\}$.}

This result  is rather unexpected.
Indeed, for $a_2 =0$  the condition
$ a_1>\frac{1}{e\ln\mu}$
is necessary and sufficient for oscillation of all solutions for the
``reduced"  equations
$$
y'(t)+\frac{a_1}{t}y\left(\frac{t}{\mu}\right)=0.
$$
For $a_1=0$ the "reduced" equation
$$
z'(t)+\frac{a_2}{t^{\beta}}z(t+t^{\alpha})=0
$$
has a non-oscillatory solution for every $a_2>0$,
$0\leq \alpha<1$, $0\leq \beta<1$.

\paragraph{Remark} % 4.
In \cite[Theorem 1]{N} the same oscillation condition (\ref{36}) for
(\ref{35}) is implied as in our Theorem \ref{thm2}. However,
 for (\ref{37}) one can derive from \cite{N} only condition (\ref{36})
for oscillation of all solutions. Further, our condition $\{a_1>0,\; a_2>0\}$
covers all cases.

\section{Non-oscillation conditions}

In this section we apply some well known results \cite{EKZ,GL,LLZ}
to the delay differential equation
\begin{equation}
\label{40}
x'(t) +\sum_{k=1}^n a_k(t)x[r_k(t)]=0,\quad t\geq t_0,
\end{equation}
and to the advanced differential equation
\begin{equation}
\label{41}
x'(t)-\sum_{k=1}^n b_k(t)x[p_k(t)]=0,~\quad t\geq t_0.
\end{equation}
We assume for (\ref{40}), ( \ref{41}) conditions (A1), and (A2) hold.

\begin{lemma} \label{lm2}
1.)  Equation (\ref{40}) has a non-oscillatory solution if and only if
there exists a function
 $u(t)\geq 0$ and $t_1\geq t_0$ such that
$$
u(t)\geq \sum_{k=1}^m a_k(t)\exp\big\{\int_{r_k(t)}^t
u(s)ds\big\},\quad t\geq t_1.
$$
2.) If $m=1$ and
$\lim_{t\rightarrow\infty}\sup\int_{r(t)}^t a(s)ds <1/e$,
then (\ref{40}) has a non-oscillatory solution.
\end{lemma}

\begin{lemma} \label{lm3}
1.) Equation (\ref{41}) has a non-oscillatory solution if and only if
there exists a function
$u(t)\geq 0$ and $t_1\geq t_0$ such that
$$
u(t)\geq \sum_{k=1}^m b_k(t)\exp\big\{\int_{t}^{p_k(t)} u(s)ds
\big\},\quad t\geq t_1.
$$
2.) If $m=1$ and
$\lim_{t\rightarrow\infty}\sup \int_{t}^{p(t)} b(s)ds <1/e$,
then (\ref{41}) has a non-oscillatory solution.
\end{lemma}

\begin{theorem} \label{thm3}
Suppose $a_1(t), a_2(t), r(t), p(t)$ are uniformly continuous on
the interval $[t_0, \infty)$,
\begin{equation}
\label{42a}
\lim_{t\rightarrow\infty}\sup [t-r(t)]=r<\infty,
\quad \lim_{t\rightarrow\infty}\sup [p(t)-t)]=p<\infty,
\end{equation}
and there exists a non-oscillatory solution of the delay differential equation
\begin{equation}
\label{42}
 y'(t)+a_1(t)y[r(t)]+a_2(t)y(t)=0.
\end{equation}
Then there exists a non-oscillatory solution of (\ref{1}).
\end{theorem}

\paragraph{Proof.}
Lemma \ref{lm2} implies  the existence of  a function $u_0(t)\geq 0$
and $t_1\geq t_0$ such that
\begin{equation}
\label{43}
u_0(t)\geq a_1(t)\exp\big\{\int_{r(t)}^t u_0(s)ds\big\}+a_2(t),\quad
t\geq t_1.
\end{equation}
Consider the space ${\bf C}[t_0,\infty)$ of all continuous and bounded functions on $[t_0,\infty)$
 with supremum norm $\| \cdot \|$ and consider the operator
$$
(Au)(t):= a_1(t)\exp\big\{\int_{r(t)}^t
u(s)ds\big\}+a_2(t)\exp\big\{-\int_{t}^{p(t)} u(s)ds\big\}.
$$
Let $ S=\{u\in {\bf C}:0\leq u(t)\leq u_0(t)\}.$ Equation (\ref{43}) implies
 $0\leq (Au)(t)\leq u_0(t)$.
For $u\in S$, denote the integral operators
$$
(Hu)(t):=\int_{r(t)}^t u(s)ds,\quad (Ru)(t):=\int_{t}^{p(t)} u(s)ds.
$$
Inequalities (\ref{42a}) imply
$$|(Hu)(t)|\leq r \|u_0\|,\quad |(Ru)(t)|\leq p \|u_0\|.$$
Hence the sets $HS$ and $RS$ are bounded in the space
${\bf C}[t_0,\infty)$.
For $u\in S$, we obtain
\begin{align*}
|(Hu)(t_2)-(Hu)(t_1)|\leq& |\int_{r(t_1)}^{r(t_2)} u(s)ds|
+|\int_{t_1}^{t_2}u(s)ds| \\
\leq &\|u_0\|(|r(t_2)-r(t_1)|+|t_2-t_1|)
\end{align*}
and
$$
|(Ru)(t_2)-(Ru)(t_1)|\leq \|u_0\|(|p(t_2)-p(t_1)|+|t_2-t_1|).
$$
Hence the families $HS$ and $RS$ are equicontinuous.
Then the sets  $HS$ and $RS$ are compact.
Therefore, the set $AS$ is also compact.

Schauder's Fix Point Theorem implies that there exists a solution $u\in S$
of the equation $u=Au$. Therefore, the function
$x(t)=x(t_1)\exp\{-\int_{t_1}^t u(s)ds\}$, $t\geq t_1,$ is a positive solution
of (\ref{1}).

\begin{corollary} \label{coro3.1}
Suppose $a_1(t)$, $a_2(t)$, $r(t)$, and $p(t)$ are uniformly continuous on 
$[t_0, \infty)$, (\ref{42a}) holds and
\begin{equation}
\label{44}
\lim_{t\rightarrow\infty}\sup \int_{r(t)}^t a_1(s)\exp\big\{
\int_{r(s)}^sa_2(\tau)d\tau\big\} ds<\frac{1}{e}.
\end{equation}
Then (\ref{1}) has a non-oscillatory solution.
\end{corollary}

\paragraph{Proof.} Substituting $y(t)$ by $z(t)\exp\{-\int_{t_0}^t a_2(s)ds\}$
in (\ref{42}), we obtain
$$
\dot{z}(t)+a_1(t)\exp\big\{ \int_{r(t)}^t a_2(s) ds \big\} z(r(t))=0.
$$
Lemma 4.2 and (\ref{44}) imply that (\ref{42}) and therefore (\ref{1})  has a
 non-oscillatory solution.

\paragraph{Remark.} % 5.
Corollary \ref{coro3.1} implies that if
$$
\frac{-\ln[a_1\ln\mu]-1}{a_2\ln\mu}<1\Longleftrightarrow
 a_1\mu^{a_2} <\frac{1}{e\ln\mu}
$$
then (\ref{35}) has a non-oscillatory solution.
This implies that conditions of Theorems \ref{thm2} and \ref{thm3}
are sharp for oscillation and non-oscillation of (\ref{1}).

\paragraph{Remark.} % 6.
Corollary \ref{coro3.1} improves some results in \cite{G}.

\begin{corollary} \label{coro3.2}
Suppose $a_1(t)$, $a_2(t)$, $r(t)$, and $p(t)$ are uniformly continuous on 
$[t_0, \infty)$, (\ref{42a}) holds and
\begin{equation}
\label{45}
\int_{t_0}^{\infty} a_1(s) ds <\infty.
\end{equation}
Then (\ref{1}) has a  non-oscillatory solution.
\end{corollary}

\paragraph{Proof.}
Condition (\ref{42a}) implies
\begin{multline*}
\lim_{t\rightarrow\infty}\sup \int_{r(t)}^t a_1(s)\exp\big\{
\int_{r(s)}^sa_2(\tau)d\tau\big\}ds\\
\leq e^{r\|a_2\|}\lim_{t\rightarrow\infty}\sup \int_{r(t)}^t a_1(s)ds =0
<\frac{1}{e}.
\end{multline*}
Hence (\ref{1})  has a  non-oscillatory solution.

\begin{theorem} \label{thm4}
Suppose
 $\int_{t_0}^{\infty} a_1(s) ds =\infty$
and $x$ is a non-oscillatory solution of (\ref{1}).
Then $\lim_{t\rightarrow\infty} x(t)=0$.
\end{theorem}

\paragraph{Proof.}
Suppose $x(t)>0$, $t\geq t_1$ and $r(t)\geq t_1,~t\geq t_2$.
Then $\dot{x}(t)\leq 0,~t\geq t_2$.
Denote $u(t)=-\frac{\dot{x}(t)}{x(t)}$, $t\geq t_2$.
Then $u(t)\geq 0$, $t\geq t_2$.
After substituting
\begin{equation}
\label{46}
x(t)=x(t_2)\exp\big\{ -\int_{t_2}^t u(s) ds \big\}, \quad t\geq t_2,
\end{equation}
into  (\ref{1}) we have
\begin{equation}
\label{47}
u(t)=a_1(t)\exp\big\{ \int_{r(t)}^t
u(s)ds \big\} +a_2(t)\exp\big\{ -\int_t^{p(t)}u(s)ds \big\},\quad t\geq t_2.
\end{equation}
Hence $u(t)\geq a_1(t)$ and therefore $\int_{t_0}^{\infty} u(s) ds =\infty$.
Equality (\ref{46}) implies, that $\lim_{t\rightarrow\infty} x(t)=0$.

Consider now the mixed differential equation
\begin{equation}
\label{48}
 x'(t)-a_1(t)x[r(t)]-a_2(t)x[p(t)]=0,~t\geq t_0.
\end{equation}

\begin{theorem} \label{thm5}
Suppose  $a_1(t)$, $a_2(t)$, $r(t)$, and $p(t)$ are uniformly continuous on 
$[t_0, \infty)$, (\ref{42a})  holds and
\begin{equation}
\label{49}
 y'(t)-a_1(t)y(t)-a_2(t)y[p(t)]=0,~t\geq t_0,
\end{equation}
has a non-oscillatory solution.
Then (\ref{48}) has a non-oscillatory solution.
\end{theorem}

\paragraph{Proof.}
In the space ${\bf C}[t_0,\infty)$, consider the operator
$$
(Bu)(t):= a_1(t)\exp\big\{-\int_{r(t)}^t
u(s)ds\big\}+a_2(t)\exp\big\{\int_{t}^{p(t)} u(s)ds\big\}.
$$
Lemma \ref{lm3} implies that there exists a nonnegative solution $u_0(t)$
of the inequality
$$
u(t)\geq a_1(t)+a_2(t)\exp\big\{\int_{t}^{p(t)} u(s)ds\big\}, \quad t\geq t_1.
$$
Let $ S=\{u:0\leq u(t)\leq u_0(t)\}$.
As in the proof of Theorem \ref{thm3}, we obtain $BS\subset S$, and the set
$BS$ is a compact. Therefore,
the equation $u=Bu$ has a nonnegative solution $u$. Hence a function
$$
x(t)=x(t_1)\exp\big\{\int_{t_1}^t u(s) ds\big\},\quad t\geq t_0,
$$
is a positive solution of  (\ref{48}).

\begin{corollary} \label{coro5.1}.
Suppose $a_1(t), a_2(t), r(t), p(t)$ are uniformly continuous on
$[t_0, \infty)$,  (\ref{42a}) holds and
$$
\lim_{t\rightarrow\infty}\sup \int_{t}^{p(t)} a_2(s)\exp\big\{
\int_{s}^{p(s)} a_1(\tau)d\tau\big\}
ds<\frac{1}{e}.
$$
Then  (\ref{48}) has a non-oscillatory solution.
\end{corollary}

The proof of this corollary is similar to the 
proof of Corollary \ref{coro3.1}.

\begin{corollary} \label{coro5.2}
Suppose $a_1(t)$, $a_2(t)$, $r(t)$, and $p(t)$ are uniformly continuous on
$[t_0, \infty)$, (\ref{42a}) holds and
$\int_{t_0}^{\infty} a_2(s) ds <\infty$.
Then (\ref{48})  has a  non-oscillatory solution.
\end{corollary}

\begin{theorem} \label{thm6}
Suppose
$\int_{t_0}^{\infty} a_2(s) ds =\infty$
and $x$ is a non-oscillatory solution of (\ref{48}).
Then $\lim_{t\rightarrow\infty} x(t)=\infty$.
\end{theorem}

The proof of this theorem is similar to the proof of Theorem \ref{thm4}.

\paragraph{Acknowledgement}
The authors are very grateful to the referee whose valuable remarks
helped us improving this article.


\begin{thebibliography}{00} \frenchspacing

\bibitem{BDS1}
 D. Bainov; Yu. Domshlak and  P. S. Simeonov,
On the oscillation properties of
first-order impulsive differential
equations with a deviating argument.
{\em Israel J. Math.} {\bf 98} (1997), 167--187.

\bibitem{BDS2}
D. Bainov; Yu. Domshlak and P. S. Simeonov,
Sturmian comparison theory for
impulsive differential inequalities and equations.
{\em Arch. Math.} (Basel) {\bf 67} (1996), no. 1, 35--49.

\bibitem{BS}
D. Bainov  and P.  Simeonov,
Oscillation Theory of Impulsive Differential Equations.
International Publications,
Orlando, FL, 1998.

\bibitem{BD}
L. Berezansky and Yu. Domshlak,
Differential equations with several deviating arguments:
Sturmian Comparison Method in oscillation theory, I
{\em Elelctron. J. Diff. Eqns.}, {\bf 2001} (2001), no. 40,  1--19.

\bibitem{Dom1}
  A. Domoshnitsky,
Sturm's theorem for equations with delayed argument.
{\em Proc. Georgian Acad. Sci. Math.} {\bf 1} (1993), no. 3, 299--309.

\bibitem{Dom2}
 A. Domoshnitsky,
Extension of the Sturm theorem to equations with retarded argument. (Russian)
{\em Differential'nye Uravneniya} {\bf 19} (1983), no. 9, 1475--1482.

\bibitem{D1}
Yu. Domshlak,
Comparison theorems of Sturm type for first- and second-order
differential equations with
sign-variable deviations of the argument. (Russian)
{\em Ukrain. Mat. Zh.} {\bf 34} (1982), no. 2, 158--163, 267.

\bibitem{D2}
Yu. Domshlak,
Distribution of zeros and oscillation of solutions of first-order
equations of neutral type.(Russian)
{\em Izv. Akad. Nauk Azerba\u\i dzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk}
{\bf 5} (1984), no. 2, 24--30.

\bibitem{D3}
Yu. Domshlak,
Sturmian Comparison Method in Investigation of the Behavior of Solutions for
Differential-Operator Equations,
Elm, Baku, USSR, 1986 (Russian).


\bibitem{D4}
Yu. Domshlak,
On oscillation properties of delay differential equations with
oscillating coefficients.
{\em Funct. Differential Equations} {\bf 2} (1994), 59--68 (1995).

\bibitem{D5}
Yu. Domshlak,
Oscillation properties of discrete difference inequalities and
equations: the new approach.
{\em Funct. Differential Equations} {\bf 1} (1993), 60--82.

\bibitem{D6}
Yu. Domshlak,
 Sturmian comparison method in oscillation study for discrete difference
equations. I, II.
{\em Differential Integral Equations} {\bf 7} (1994), no. 2, 571--582, 583--592.

\bibitem{D7}
Yu. Domshlak,
Delay-difference equations with periodic coefficients: sharp results in oscillation
theory.
{\em Math. Inequal. Appl.} {\bf 1} (1998), no. 3, 403--422.

\bibitem{D8}
Yu. Domshlak,
The Riccati difference equations near "extremal" critical states.
{\em J. Differ. Equations Appl.} {\bf 6} (2000), no. 4, 387--416.

\bibitem{DA1}
Yu. Domshlak and  A. I. Aliev,
Oscillation properties of solutions of first-order
differential equations with retarded
argument and with coefficients that do not preserve the sign. (Russian)
{\em Izv. Akad. Nauk Azerba\u\i dzhan.
SSR Ser. Fiz.-Tekhn. Mat.
Nauk} {\bf 8} (1987), no. 4, 19--24.

\bibitem{DA2}
Yu. Domshlak and A.I. Aliev,
On oscillatory properties of the first order differential equations with one or two arguments.
{\em Hiroshima Math. J.} {\bf 18} (1988), 31-46.

\bibitem{DC}
Yu. Domshlak and Sui Sun Cheng,
Sturmian theorems for a partial difference equation.
{\em Funct. Differ. Equ.} {\bf 3}
(1995), no. 1-2, 83--97.

\bibitem{DS1}
Yu. Domshlak and  L. A. She\u\i khzamanova,
Distribution of zeros of solutions of second-order
functional-differential inequalities of neutral type. I. (Russian)
{\em Izv. Akad. Nauk Azerba\u\i dzhan. SSR
Ser. Fiz.-Tekhn. Mat. Nauk } {\bf 6} (1985), no. 4, 23--29

\bibitem{DS2}
Yu. Domshlak and L. A. She\u\i khzamanova,
Estimates for sign preservation domains in terms of solutions of a
heat equation with time delay. (Russian)
{\em Izv. Akad. Nauk Azerba\u\i dzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk}
{\bf 9} (1988), no. 4, 38--43.

\bibitem{DSt1}
Yu. Domshlak and I. P. Stavroulakis ,
Oscillations of first-order delay differential equations in a
critical state.
{\em Appl. Anal.} {\bf 61} (1996), no. 3-4, 359--371.

\bibitem{DSt2}
Yu. Domshlak and I. P. Stavroulakis ,
Oscillations of differential equations with deviating arguments in a
critical state.
{\em Dynam. Systems Appl.} {\bf 7} (1998), no. 3, 405--414.

\bibitem{DK}
 Michael S. Du and Man Kam Kwong,
 Sturm comparison theorems for second-order delay equations.
{\em J. Math. Anal. Appl.} {\bf 152} (1990), no. 2, 305--323.

\bibitem{Dz1}
Jozef D\v zurina,
Comparison theorems of Sturm's type.
{\em Mathematika} {\bf 41} (1994), no. 2,
312--321.

\bibitem{Dz2}
Jozef D\v zurina,
Oscillation of second-order differential equations with mixed argument.
{\em J. Math. Anal. Appl.} {\bf 190} (1995), no. 3, 821--828.

\bibitem{EKZ}
L. N. Erbe, Q. Kong and B. G. Zhang,
Oscillation Theory for
Functional Differential Equations,
Marcel Dekker, New York, Basel, 1995.

\bibitem{G}
K. Gopalsamy,
Nonoscillatory differential equations with retarded and advanced arguments.
{\em Quart. Appl. Math.} {\bf 43} (1985), no. 2, 211--214.

\bibitem{GS}
  M. K. Grammatikopoulos and  I. P. Stavroulakis,
Oscillations of neutral differential equations.
{\em Rad. Mat.} {\bf 7} (1991), no.1, 47--71.

\bibitem{GL}
I. Gy\"{o}ri and G. Ladas,
Oscillation Theory of Delay Differential Equations,
Clarendon Press, Oxford, 1991.

\bibitem{I1}
 A. F. Ivanov,
Some comparison theorems for second-order differential equations with
deviating arguments.(Russian)
{\em Ukrain. Mat. Zh.} {\bf 35} (1983), no. 5, 626--630.

\bibitem{I2}
 A. F. Ivanov; Y. Kitamura; T. Kusano and  V. N. Shevelo,
Oscillatory solutions of functional-differential equations
generated by deviation of arguments of mixed type.
{\em Hiroshima Math. J.} {\bf 12} (1982), no. 3, 645--655.

\bibitem{K}
Taka\^si Kusano,
On even-order functional-differential equations with advanced and
retarded arguments.
{\em J. Differential Equations} {\bf 45} (1982), no. 1, 75--84.

\bibitem{LS}
G. Ladas, G. and S. W. Schults,
On oscillations of neutral equations with mixed arguments.
{\em Hiroshima Math. J.} {\bf 19}
(1989), no. 2, 409--429.

\bibitem{LSt}
G. Ladas and I. P. Stavroulakis,
Oscillations of differential equations of mixed type.
{\em J. Math. Phys. Sci.} {\bf 18} (1984), no. 3, 245--262.

\bibitem{LLZ}
G.S.Ladde, V. Lakshmikantham and B.G. Zhang,
Oscillation Theory of Differential Equations with Deviating Arguments,
Marcel Dekker, New York, 1987.

\bibitem{M}
A. D. Myshkis,
Linear Differential Equations with Retarded Argument,
Nauka, Moscow, 1972 (in Russian).

\bibitem{N}
V.A. Nadareishvili,
Oscillation and nonoscillation of first order linear differential equations with deviating arguments.
{\em Differential Equations} {\bf 25} (1989), 412-417.

\bibitem{R}
 Aldo Rustichini,
Functional-differential equations of mixed type:
the linear autonomous case.
{\em J. Dynam. Differential Equations} {\bf 1} (1989), no. 2, 121--143

\bibitem{S}
 I. P. Stavroulakis,
Oscillations of mixed neutral equations.
{\em Hiroshima Math. J.} {\bf 19} (1989), no. 3, 441--456.

\bibitem{Y}
Ju Rang Yan,
Comparison theorems for differential equations of mixed type.
{\em Ann. Differential Equations} {\bf 7} (1991), no. 3, 316--322.

\end{thebibliography}


\noindent\textsc{Leonid Berezansky} (e-mail: brznsky@cs.bgu.ac.il)\\
\textsc{Yury Domshlak} (e-mail: domshlak@cs.bgu.ac.il)\\[3pt]
Department of Mathematics, \\
Ben-Gurion University of the Negev, \\
Beer-Sheva 84105, Israel


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