\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 47, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/47\hfil Oscillation for equations with \dots ]
{Oscillation for equations with positive and negative coefficients
and distributed delay II: Applications}

\author[Leonid Berezansky \& Elena Braverman\hfil EJDE--2003/47\hfilneg]
{Leonid Berezansky \& Elena Braverman}

\address{Leonid Berezansky \hfill\break
Department of Mathematics, Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel}
\email{brznsky@cs.bgu.ac.il}

\address{Elena Braverman \hfill\break
Department of Mathematics and Statistics, University of Calgary,
2500 University Drive N. W., Calgary,
Alberta, Canada, T2N 1N4 \hfill\break
Fax: (403)-282-5150,  phone: (403)-220-3956}
\email{maelena@math.ucalgary.ca}


\date{}
\thanks{Submitted February 14, 2003. Published April 24, 2003.}
\thanks{L. Berezansky was partially supported by the Israeli Ministry of
Absorption}
\thanks{E. Braverman was partially supported by an URGC Research Grant}
\subjclass[2000]{34K11, 34K15}
\keywords{Oscillation, non-oscillation, distributed delay,
equations with \hfill\break\indent
several delays, integrodifferential equations}

\begin{abstract}
  We apply the results of our previous paper ``Oscillation of equations
  with positive and negative coefficients and distributed delay I:
  General results" to the study of oscillation properties of equations
  with several delays and positive and negative coefficients
  $$
  \dot{x}(t) + \sum_{k=1}^n a_k(t) x(h_k(t)) -
  \sum_{l=1}^m b_l(t) x(g_l(t)) = 0, \quad a_k(t) \geq 0, b_l(t) \geq 0,
  $$
  integrodifferential equations with oscillating kernels
  and mixed equations combining two above equations.
  Comparison theorems, explicit non-oscillation and oscillation results
  are  presented.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}{Proposition}[section]
\numberwithin{equation}{section}
\allowdisplaybreaks


\section{Introduction}

The study of oscillation properties of delay differential
equations with positive and negative coefficients  began in the eighties
\cite{kulfirst,p1} and was inspired by
the study of equations with oscillating coefficients.
This research was later developed in \cite{l2,l1},
neutral equations with positive and negative coefficients were
studied in \cite{p4,p9,p10}, see also recent publications \cite{p800,p801,p8}.
In \cite{p1,p2,l2} the first order equation with two constant
concentrated delays and a positive and a negative coefficient was studied,
while paper \cite{p901} considered
oscillation of integrodifferential equations  with oscillatory kernels.

In \cite{p2} for the equation
\begin{equation}
\label{01}
\dot{x}(t)+a(t)x(t-\tau)-b(t)x(t-\sigma)=0, \quad t\geq t_0,
\end{equation}
where $a(t)\geq 0,~b(t)\geq 0$ are continuous functions,
$\tau>\sigma>0$, the following result was obtained:  Suppose
\begin{gather}
\label{02}
\int _{t-\tau+\sigma}^t b(s)ds \leq 1,\quad a(t)\geq b(t-\tau+\sigma),\\
\label{03}
\liminf_{t\to\infty} \int_{t-\tau}^t
[a(s)- b(s-\tau+\sigma)]ds >\frac{1}{e}.
\end{gather}
Then all solutions of (\ref{01}) are oscillatory.

In \cite{p10} the inequality (\ref{03}) was improved:
\begin{equation}
\label{03imp}
\liminf_{t\to\infty} \Big( \int_{t-\tau}^t
[a(s)- b(s-\tau+\sigma)]ds +\frac{1}{e}\int_{t-\tau+\sigma}^t
b(s-\tau)ds \Big) > \frac{1}{e}.
\end{equation}

Recently numerous publications on the oscillation of delay equations with
positive and negative coefficients have appeared (in addition to
\cite{p800,jmaa,p11,p801,p7,p8} see \cite{EHS} and references therein).
However all the publications except \cite{p13,jmaa} consider equations with
constant delays only. Paper \cite{jmaa} deals with a more general case
when the delays are not constant.

Our previous paper \cite{previous} gave a general insight into the problem.
In \cite{previous} we considered the equation with a distributed delay
\begin{equation}
\label{1}
\dot{x}(t)+ \int_{0}^t x(s)\,d_s R(t,s)- \int_{0}^t x(s)\,d_s T(t,s)=
0,~~t\geq 0,
\end{equation}
where both $R(t,s)$ and $T(t,s)$ are nondecreasing in $s$ for each $t$.

As special cases, (\ref{1}) includes delay differential equations
with variable concentrated delays, integrodifferential equations
and mixed differential equations. The basic result of the paper
\cite{previous}  was the relation between the following properties
for (\ref{1}): the existence of a nonoscillatory solution of
(\ref{1}), the existence of an eventually positive solution of the
corresponding differential inequality and the existence of a
nonnegative solution of some nonlinear integral inequality which
is explicitly constructed by (\ref{1}). Theorems of this kind are
well known and widely applied for delay differential equations
with positive coefficients.

In the present paper we apply general results obtained in \cite{previous} to
specific classes of equations.
Section \ref{section2} contains preliminaries and relevant results from paper
\cite{previous}. In Section \ref{section3}  equations with positive and negative coefficients
and several concentrated delays are considered.
In Section \ref{section4} oscillation and nonoscillation results 
are deduced for
integrodifferential equations with oscillatory kernels. Section \ref{section5}
deals with mixed equations containing both several concentrated delays and an
integral term.

\section{Preliminaries and General Results}
\label{section2}

We consider a scalar delay differential equation (\ref{1}) under
the following conditions: \begin{itemize}

\item[(a1)] $R(t,\cdot)$, $T(t,\cdot)$ are
left continuous functions of bounded variation and for each $s$
their variations on the segment $[0,s]$
\begin{equation} \label{P}
P_R(t,s) = var_{\tau \in [0,s]} R(t,\tau), ~P_T(t,s) = var_{\tau \in
[0,s]} T(t,\tau)
\end{equation}
are locally integrable functions in $t$, $R(t,s) = R(t,t+)$,
$T(t,s) = T(t,t+)$, $t<s$;

\item[(a2)] $R(t,\cdot)$, $T(t,\cdot)$ are  nondecreasing functions
for each $t$, $R(t,s) \geq T(t,s)$ for each $t,s$;

\item[(a3)] For each $t_1$ there exist $s_1=s(t_1) \leq t_1$, $r_1=r(t_1)\leq t_1$, such that
$R(t,s) = 0$ for $s <s_1$, $t>t_1$, $T(t,s) = 0$ for $s <r_1$, $t>t_1$;
in addition, functions $s(t)$, $r(t)$ satisfy
$$\lim_{t \to \infty}
s(t) = \infty, \lim_{t \to \infty} r(t) = \infty.$$
\end{itemize}
If (a3) holds then we can introduce the following functions
\begin{equation}
\label{h}
h(t) = \inf_s \big\{s : R(t,s) \neq 0   \big\}, \quad
g(t)= \inf_s \big\{s : T(t,s) \neq 0   \big\},
\end{equation}
such that $\lim_{t \to \infty} h(t) = \infty$,
$\lim_{t \to \infty} g(t) = \infty$, and
(\ref{1}) can be rewritten as
\begin{equation}
\label{1star}
\dot{x}(t)+ \int_{h(t)}^t x(s)\,d_s R(t,s)- \int_{g(t)}^t x(s)\,d_s T(t,s)=
0,~~t\geq 0.
\end{equation}
If (a2) and (a3) hold, then obviously $h(t) \leq g(t)$.

Together with (\ref{1star}) we consider for each $t_0\geq 0$ an
initial-value problem
\begin{equation}
\label{2}
\dot{x}(t)+  \int_{h(t)}^t x(s)\,d_s R(t,s)
- \int_{g(t)}^t x(s)\,d_s T(t,s) = 0,~~t\geq t_0.
\end{equation}
\begin{equation}
\label{3}
x(t)=\varphi(t), ~ t<t_0,~ x(t_0)=x_0,
\end{equation}
where $\varphi(t)$ is a Borel measurable bounded function.

\noindent{\bf Definition.} An absolutely continuous on each interval
$[t_0,c]$ function $x:R\to R$ is called {\em a solution of
problem} (\ref{2}), (\ref{3}), if it satisfies  (\ref{2}) for
almost all $t\in [t_0,\infty)$ and equalities (\ref{3}) for $t\leq
t_0$.

\noindent{\bf Definition.} For each $s\geq 0$ solution $X(t,s)$ of the problem
\begin{equation}
\label{3a}
\dot{x}(t)+\int_{h(t)}^t x(s)\,d_s R(t,s)- \int_{g(t)}^t x(s)\,d_s T(t,s) = 0,~x(t)=0,~t<s,~x(s)=1,
\end{equation}
is called {\em a fundamental function} of (\ref{1star}).

We assume $X(t,s)=0, ~0\leq t<s$.


\noindent{\bf Definition.} We will say that equation (\ref{1star}) {\em has a nonoscillatory
solution}
if for some  $t_0, \varphi(t)$ and $x_0$ the solution of (\ref{2})-(\ref{3}) is  eventually
positive or  eventually negative.
Otherwise all solutions of this equation are {\em oscillatory}.

Below we present the results obtained in \cite{previous}
on oscillation of equation (\ref{1star}) with a distributed delay.

Consider together with (\ref{1star}) the following delay
differential inequality
\begin{equation}
\label{6}
\dot{y}(t)+ \int_{h(t)}^t y(s)\,d_s R(t,s)
- \int_{g(t)}^t y(s)\,d_s T(t,s) \leq 0,~~t\geq 0.
\end{equation}

The following theorem establishes sufficient nonoscillation conditions.


\begin{lemma}
\label{theorem1}{\rm \cite{previous}}
Suppose (a1)-(a3) hold. Consider  the following hypotheses:
\begin{enumerate}
\item There exists $t_1\geq 0$ such that for $t\geq t_1$ the following inequality
\begin{equation}
\label{7}
u(t)\geq  \int_{h(t)}^t \exp\big\{\int_{s}^t
u(\tau)d \tau \big\}\,d_s R(t,s) -
\int_{g(t)}^t \exp\big\{\int_{s}^t u(\tau) d\tau \big\}\,d_s T(t,s)
\end{equation}
has a nonnegative locally integrable  solution (we assume $u(t)=0$ for $t<t_1$);

\item There exists $t_2\geq 0$ such that $X(t,s)>0, ~t\geq s\geq t_2$;

\item Equation (\ref{1star}) has a nonoscillatory
solution;

\item Inequality (\ref{6}) has an eventually positive
solution.
\end{enumerate}
Then the implication $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (4)$
is valid.
\end{lemma}


Necessary nonoscillation
(sufficient oscillation)  conditions require some more constraints on
$R,T$. Let
\begin{itemize}
\item[(a4)] for any $t$
$R(t,s) - T(t,s-h(t)+g(t))$ is nondecreasing in $s$ and
$$ \limsup_{t \to \infty}
T(t,t+)[g(t)-h(t)] \leq l < 1.$$
\end{itemize}

\begin{lemma}[\cite{previous}] \label{theorem2}
Suppose (a1)--(a4) hold.
Then hypotheses (1)--(4) of Theorem 1 are equivalent.
\end{lemma}


To obtain other necessary oscillation conditions we consider the following
form of equation (\ref{1star}):
\begin{equation}
\label{1starnew}
\dot{x}(t)+ \sum_{k=1}^n \int_{h_k(t)}^t x(s)\,d_s R_k(t,s)-
\sum_{l=1}^m \int_{g_l(t)}^t x(s)\,d_s T_l(t,s) = 0,~~t\geq 0,
\end{equation}
where $R_k,T_l,h_k,g_l$ satisfy the following conditions:
\begin{itemize}
\item[(a1$^\star$)] $R_k(t,\cdot)$, $T_l(t,\cdot)$ are  left
continuous functions of bounded variation and for each $s$ their
variations on the segment $[0,s]$ $P_{R_k}(t,s)$, $P_{T_l}(t,s)$
are locally integrable functions in $t$, $R_k(t,s) = R_k(t,t+)$,
$T_l(t,s) = T_l(t,t+)$, $t<s$;

\item[(a2$^\star$)]
 $R_k(t,\cdot)$, $T_l(t,\cdot)$ are  nondecreasing functions
for each $t$, \\ $\sum_k R_k(t,s) \geq \sum_l T_l(t,s)$ for each $t,s$;

\item[(a3$^\star$)] For each $k,l$  $\lim_{t \to \infty}
h_k(t)= \infty$, $\lim_{t \to \infty} g_l(t) = \infty$. 

Denote
\begin{gather}
\label{121star}
R(t,s) = \sum_{k=1}^n R_k(t,s) ,T(t,s)=\sum_{l=1}^m T_l(t,s),\\
\label{hgnotation}
 h(t) = \max_k h_k(t), g(t) = \min_l g_l(t).
\end{gather}
\end{itemize}
Let us also introduce the following additional constraints:
\begin{itemize}
\item[(add1)] $m=n$, $R_k(t,s) \geq T_k(t,s)$ for each $t,s$, $k=1, \dots,n$;
for any $t,k$, function
$R_k(t,s) - T_k(t,s-h_k(t)+g_k(t))$ is nondecreasing in $s$ and
$$ \limsup_{t \to \infty}
\sum_{k=1}^n T_k(t,t+)[g_k(t)-h_k(t)]  < 1.$$

\item[(add2)] $h(t) \leq g(t), R(t,s)  >   T(t,s),
R(t,s) - T(t,s-h(t)+g(t))$ is nondecreasing in
$s$ for any $t$ and
\begin{equation} \label{add2formula}
\begin{split}
\limsup_{t\to \infty}  \big[& T(t,t+)[g(t)-h(t)]
+ \sum_{k=1}^n R_k(t,t+)  \big( h(t)-h_k(t)\big)\\
&+ \sum_{l=1}^m  T_l(t,t+) \big( g_l(t)-g(t) \big)  \big]  < 1.
\end{split}
\end{equation}
\end{itemize}
Consider together with (\ref{1starnew}) the  delay
differential inequality
\begin{equation}
\label{6starnew}
\dot{y}(t)+ \sum_{k=1}^n \int_{h_k(t)}^t y(s)\,d_s R_k(t,s)-
\sum_{l=1}^m \int_{g_l(t)}^t
y(s)\,d_s T_l(t,s) \leq
0,\; t\geq 0.
\end{equation}

The next lemma establishes non-oscillation criteria for
(\ref{1starnew}).

\begin{lemma}
\label{theorem3} {\rm \cite{previous}}
Suppose $R_k,T_l,h_k,g_k$ satisfy (a1$^\star$)-(a3$^\star$) and at least
one
of conditions (add1), (add2) hold. Then  the following
hypotheses are equivalent:
\begin{enumerate}
\item There exists $t_1\geq 0$ such that the inequality
\begin{equation} \label{7starnew}
\begin{split}
u(t)&\geq  \sum_{k=1}^n \int_{h_k(t)}^t \exp\big\{\int_{s}^t
u(\tau)d \tau \big\}\,d_s R_k(t,s)\\
&\quad  -  \sum_{l=1}^m \int_{g_l(t)}^t \exp\big\{\int_{s}^t u(\tau) d\tau \big\}
\,d_s T_l(t,s),\quad t\geq t_1,
\end{split}
\end{equation}
has a nonnegative locally integrable  solution (we assume $u(t)=0$ for
$t<t_1$);

\item There exists $t_2\geq 0$ such that $X(t,s)>0, ~t\geq s\geq t_2$;

\item Equation (\ref{1starnew}) has a nonoscillatory
solution;

\item Inequality (\ref{6starnew}) has an eventually positive
solution.
\end{enumerate}
\end{lemma}

Lemmas \ref{theorem1}--\ref{theorem3} yield the following comparison
result.
Let us compare the oscillation properties of the equation
\begin{equation}
\label{ampersand}
\dot{x}(t)+ \sum_{k=1}^n \int_{\tilde{h}_k(t)}^t x(s)\,d_s L_k(t,s)-
\sum_{l=1}^m \int_{\tilde{g}_l(t)}^t
x(s)\,d_s D_l(t,s) = 0,
\end{equation}
to the oscillation properties of (\ref{1starnew}).

\begin{lemma}[\cite{previous}] \label{theorem4}
\begin{enumerate}
\item  If
$(a1^{\star})-(a3^{\star})$ and anyone of the conditions
(add1), (add2) hold for (\ref{ampersand}) (where $R_k,T_l$ are
changed by $L_k,D_l$), $L_k(t,s) \geq R_k(t,s)$, $D_l(t,s) \leq
T_l(t,s)$ and (\ref{ampersand}) has a nonoscillatory solution,
then (\ref{1starnew}) has a nonoscillatory solution.

\item  If $(a1^{\star})-(a3^{\star})$ and any one of the
conditions (add1),(add2) hold for (\ref{1starnew}), $L_k(t,s) \leq
R_k(t,s)$, $D_l(t,s) \geq T_l(t,s)$ and all solutions of
(\ref{ampersand}) are oscillatory, then all solutions of
(\ref{1starnew}) are oscillatory.
\end{enumerate}
\end{lemma}

Lemma \ref{theorem5} describes the asymptotic behavior of
nonoscillatory solutions of (\ref{1starnew}).

\begin{lemma}[\cite{previous}] \label{theorem5}
Suppose (a1$^\star$)-(a3$^\star$) and
anyone of the following conditions holds:
\begin{enumerate}
\item (add1) is satisfied and for some $k$
\begin{equation} \label{int1}
\int_{0}^{\infty}
\big[ R_k(t,t+) - T_k(t,t+) \big] dt = \infty;
\end{equation}

\item (add2) is satisfied and
\begin{equation}\label{int2}
\int_{0}^{\infty}  \big[
R(t,t+) - T(t,t+) \big] dt = \infty.
\end{equation}
\end{enumerate}
Then any nonoscillatory solution
 $x$ of (\ref{1starnew}) satisfies
$\lim_{t \to \infty} x(t) = 0$.
\end{lemma}

Consider the following two equations
\begin{equation} \label{compare1}
\begin{aligned}
\dot{x}(t) + \int_{h(t)}^t x(s) \,d_s
\big[ R(t,s) - T(t,s-h(t)+g(t)) \big]& \\
+ \int_{g(t)}^{t} x(s)
\big( \exp\big\{\int_{s+h(t)-g(t)}^s  [R(\tau,\tau+)- T(\tau,\tau+)]
d \tau \big\} - 1
\big)  \,d_s T(t,s) &= 0
\end{aligned}
\end{equation}
and
\begin{equation} \label{compare2}
\begin{aligned}
\dot{x}(t) + \int_{g(t)}^t x(s)
\Big( \exp\big\{\int_{s-g(t)+h(t)}^s [R(\tau,\tau+)- T(\tau,\tau+)]
d\tau \big\} - 1 \Big)& \\
\times d_s R(t,s-g(t)+h(t)) + \int_{g(t)}^t x(s)
\,d_s \big[ R(t,s-g(t)+h(t)) - T(t,s)  \big]& = 0.
\end{aligned}
\end{equation}
If (a1)-(a4) hold, then equations
(\ref{compare1}),(\ref{compare2}) contains terms with positive
coefficients only. The oscillation properties of these equations
will be compared to the properties of (\ref{1star}).

\begin{lemma}[\cite{previous}] \label{theorem6}
Suppose (a1)-(a4) hold for
(\ref{1star}). If all solutions of either (\ref{compare1}) or
(\ref{compare2}) are oscillatory, then all solutions of
 (\ref{1star}) are also oscillatory.
\end{lemma}

\begin{corollary}
\label{cortheorem61} Suppose (a1)-(a4) hold for   (\ref{1star})
and at least one of the following four inequalities is satisfied:
\begin{multline*}
\llap{\rm (1)}\quad \liminf_{t \to \infty}\Big\{ \int_{h(t)}^t
\big[ R(t,\tau) - T(t,\tau-h(t)+g(t)) \big]d \tau \\
+\int_{g(t)}^{t} d \tau \int_{h(\tau)}^{\tau}
\Big( \exp\big\{ \int_{s+h(t)-g(t)}^s  [R(u,u+)- T(u,u+)]
d u \big\} - 1\Big) \,d_s T(t,s) \Big\}  > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm (2)}\quad \liminf_{t \to \infty}\Big\{ \int_{h(t)}^t
\big[ R(t,\tau+) - T(t,\tau-h(t)+g(t)) \big] \,d\tau\\
 +\int_{g(t)}^{t} d \tau \int_{h(\tau)}^{\tau}
\Big(\int_{s+h(t)-g(t)}^s  [R(u,u+)- T(u,u+)]
d u \Big) \,ds T(t,s) \Big\}  > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm (3)}\quad \liminf_{t \to \infty} \Big\{ \int_{g(t)}^t d\tau
\int_{g(\tau)}^{\tau}
\Big( \exp\big\{\int_{s-g(t)+h(t)}^s [R(u,u+)- T(u,u+)] du \big\} - 1\Big)\\
\times d_s R(t,s-g(t)+h(t))+ \int_{g(t)}^t
\big[  R(t,\tau-g(t)+h(t)) - T(t,\tau)  \big]  \,d \tau \Big\}  >
\frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm (4)}\quad \liminf_{t \to \infty} \Big\{ \int_{g(t)}^t d\tau
\int_{g(\tau)}^{\tau}
\Big( \int_{s-g(t)+h(t)}^s [R(u,u+)- T(u,u+)]d u \Big) \\
\times d_s R(t,s-g(t)+h(t)) + \int_{g(t)}^t
 \big[ R(t,\tau-g(t)+h(t)) - T(t,\tau)  \big] \big\}  > \frac{1}{e}
\end{multline*}
Then all solutions of  (\ref{1star}) are oscillatory.
\end{corollary}

Similar results can be obtained for   (\ref{1starnew}).

\begin{lemma}[\cite{previous}] \label{theorem7}
Suppose $R_k,T_l,h_k,g_l$ satisfy (a1$^\star$)-(a3$^\star$) and
(add1) holds.
If all solutions of either
\begin{equation}\label{compare5}
\begin{split}
&\dot{x}(t) + \sum_{k=1}^n \int_{h_k(t)}^t x(s) \,d_s
\big[ R_k(t,s) - T_k(t,s-h_k(t)+g_k(t))    \big]\\
&+ \sum_{k=1}^n \int_{g_k(t)}^{t} x(s)
 \Big( \exp\big\{\int_{s+h_k(t)-g_k(t)}^s [R_k(\tau,\tau+)-
T_k(\tau,\tau+)] d \tau \big\} - 1\Big)\\
&\times d_s T_k(t,s) = 0
\end{split}
\end{equation}
or
\begin{equation} \label{compare6}
\begin{split}
&\dot{x}(t) +\sum_{k=1}^n \int_{g_k(t)}^t  x(s)
\Big( \exp\big\{\int_{s-g_k(t)+h_k(t)}^s
 [R_k (\tau,\tau+) -  T_k(\tau,\tau+)]
 \times d\tau \big\} - 1 \Big)\\
&\quad \times d_s R_k(t,s-g_k(t)+h_k(t)) \\
&+ \sum_{k=1}^n \int_{g_k(t)}^t x(s)
\,d_s \Big[ R_k(t,s-g_k(t)+h_k(t)) - T_k(t,s)  \Big] = 0
\end{split}
\end{equation}
are oscillatory, then all solutions of  (\ref{1starnew}) are also
oscillatory.
\end{lemma}

\begin{corollary}
\label{cortheorem71} Suppose (a1$^\star$)-(a3$^\star$) and (add1)
hold for   (\ref{1starnew}) and at least one of the following
inequalities is satisfied:
\begin{multline*}
\llap{\rm(1)}\quad \liminf_{t \to \infty} \sum_{k=1}^n
\big\{ \int_{h_k(t)}^t\big[ R_k(t,\tau) - T(t,\tau-h_k(t)+g_k(t)) \big]
d \tau
+  \int_{g_k(t)}^{t} d \tau \\
\times \int_{h_k(\tau)}^{\tau}
\big( \exp\big\{\int_{s+h_k(t)-g_k(t)}^s  [R_k(u,u+)- T_k(u,u+)]d u \big\} - 1
\big) \,d_s T_k(t,s) \big\}  > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm(2)}\quad \liminf_{t \to \infty} \sum_{k=1}^n \big\{ \int_{h_k(t)}^t
\big[ R_k(t,\tau+) - T_k(t,\tau-h_k(t)+g_k(t)) \big] \,d\tau \\
+  \int_{g_k(t)}^{t} d \tau \int_{h_k(\tau)}^{\tau}
\big( \int_{s+h_k(t)-g_k(t)}^s  [R_k(u,u+)- T_k(u,u+)]
d u \big) \,ds T_k(t,s) \big\}  > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm(3)}\quad  \liminf_{t \to \infty} \sum_{k=1}^n  \big\{ \int_{g_k(t)}^t
 d\tau \int_{g_k(\tau)}^{\tau}
\Big( \exp\big\{\int_{s-g_k(t)+h_k(t)}^s [R_k(u,u+)- T_k(u,u+)] \\
\times du \big\} - 1 \Big) d_s R_k(t,s-g_k(t)+h_k(t))\\
+ \int_{g_k(t)}^t
\big[ R_k(t,\tau-g_k(t)+h_k(t)) - T_k(t,\tau)  \big]  \,d \tau \big\}
> \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm(4)}\quad \liminf_{t \to \infty}  \sum_{k=1}^n \big\{ \int_{g_k(t)}^t d\tau
\int_{g_k(\tau)}^{\tau}
\big( \int_{s-g_k(t)+h_k(t)}^s [R_k(u,u+)- T_k(u,u+)] d u \big)\\
\times d_s  R_k(t,s-g_k(t)+h_k(t)) + \int_{g_k(t)}^t
 \big[ R_k(t,\tau-g_k(t)+h_k(t)) - T_k(t,\tau)  \big] \big\}  >
\frac{1}{e}
\end{multline*}
Then all solutions of  (\ref{1star}) are oscillatory.
\end{corollary}

Let us proceed with  nonoscillation conditions.

\begin{lemma}[\cite{previous}]\label{theorem8}
Suppose (a1)-(a4) hold for
(\ref{1star}) and there exists $\lambda,~0<\lambda<1$, such that
\begin{gather}
\label{17new}
\limsup_{t\to \infty} \int_{h(t)}^{g(t)}
[R(s,s+)-\lambda T(s,s+)] \,ds <\frac{1}{e}\ln \frac{1}{\lambda},\\
\label{18new}
\limsup_{t\to\infty} \int_{h(t)}^t [R(s,s+)-\lambda
T(s,s+)]ds<\frac{1}{e}.
\end{gather}
Then  (\ref{1star}) has a nonoscillatory solution.
\end{lemma}

\begin{lemma}[\cite{previous}]\label{theorem9}
Suppose $n=m$, conditions (a1$^{\star}$)-(a3$^{\star}$), (add1) and
the following inequality
\begin{equation}
\label{eq26}
\limsup_{t\to\infty} \sum_{k=1}^n \int_{h_k(t)}^t
\big[ R_k(s,s+)- \frac{1}{e} T_k(s,s+)\big] ds <\frac{1}{e}
\end{equation}
hold. Then  (\ref{1starnew}) has a nonoscillatory solution.
\end{lemma}


\section{Equations with Concentrated Delays}
\label{section3}

Let us study oscillation properties of
a delay differential equation with several variable concentrated delays
\begin{equation} \label{3zaa}
\dot{x}(t) + \sum_{k=1}^n a_k(t) x(h_k(t)) -
\sum_{l=1}^m b_l(t) x(g_l(t)) = 0, t \geq 0.
\end{equation}

This equation is a special case of   (\ref{1}) when we assume
\begin{equation}
\label{4zaa}
R(t,s) = \sum_{k=1}^n a_k (t) \chi_{[h_k(t),\infty)} (s), \quad
T(t,s) = \sum_{l=1}^m b_l(t) \chi_{[g_l(t),\infty)} (s),
\end{equation}
where $\chi_{[c,d]}$ is a characteristic function
of segment $[c,d]$.

An initial value problem, definitions of a solution, the fundamental solution,
oscillatory and nonoscillatory solutions for equation (\ref{3zaa}) are the same as for
  (\ref{1starnew}).


The hypotheses of Lemma \ref{theorem1}  are satisfied for the delay
equation (\ref{3zaa}) if the following conditions hold:
\begin{itemize}
\item[(C1)] $a_k(t) \geq 0$, $b_l(t) \geq 0$ are Lebesgue measurable
essentially locally bounded functions;

\item[(C2)] For any $t\geq s\geq 0$
$$
\sum_{k=1}^n a_k (t) \chi_{[h_k(t),\infty)} (s)\geq
\sum_{l=1}^m b_l(t) \chi_{[g_l(t),\infty)} (s).
$$

\item[(C3)] $h_k(t),g_l(t):[0,\infty) \rightarrow {\bf R}$ are Lebesgue
measurable functions, $h_k(t) \leq t, g_l(t) \leq t$,
$\lim_{t \to \infty} h_k(t) =\infty$,
$\lim_{t \to \infty} g_l(t) =\infty$.
\end{itemize}

Consider the inequality
\begin{equation}
\label{6anewstar}
\dot{y}(t) + \sum_{k=1}^n a_k(t) y(h_k(t)) -
\sum_{l=1}^m b_l(t) y(g_l(t)) \leq 0, t \geq 0.
\end{equation}
The following proposition is an immediate consequence of Lemma \ref{theorem1}.

\begin{proposition}
\label{prop31}
Suppose (C1)--(C3) hold. Consider the following hypotheses:
\begin{enumerate}
\item There exists $t_1 \geq 0$ such that the inequality
\begin{equation}\label{20starnew}
u(t) \geq \sum_{k=1}^n a_k(t) \exp \big\{  \int_{h_k(t)}^t u(s)ds \big\}
- \sum_{l=1}^m b_l(t) \exp \big\{  \int_{g_l(t)}^t u(s)ds \big\}, ~t
\geq t_1
\end{equation}
has a nonnegative locally integrable  solution (we assume $u(t)=0$ for
$t<t_1$);

\item There exists $t_2 \geq 0$ such that the fundamental function
$X(t,s)>0$, $t\geq s \geq t_2$;

\item  Equation (\ref{3zaa}) has a nonoscillatory solution;

\item Inequality (\ref{6anewstar}) has an eventually positive solution.
\end{enumerate}
Then the implications $1) \Rightarrow 2) \Rightarrow 3) \Rightarrow 4)$ are
 valid.
\end{proposition}

Oscillation criteria for equations with several concentrated delays can be obtained as
corollaries of Lemma \ref{theorem3}. To this end denote
$$h(t) = \max_k h_k(t), \quad g(t) = \min_l g_l(t), \quad
 a(t) = \sum_{k=1}^n a_k(t), \quad b(t)) = \sum_{l=1}^m b_l(t).
 $$
Proposition \ref{prop32} provides  oscillation criteria.

\begin{proposition} \label{prop32}
Suppose (C1), (C3) and anyone of the following conditions
\begin{itemize}
\item[(C4)] $m=n$, $a_k \geq b_k$, $h_k(t) \leq g_k(t)$ for $k=1, \dots, n$, $t \geq 0$,
$$\limsup_{t \to \infty} \big\{ \sum_{k=1}^n b_k(t) [g_k(t) - h_k(t) ]
\big\}< 1$$
\item[(C5)] $h(t) \leq g(t)$, $a(t) \geq b(t)$,
$$\limsup_{t \to \infty} \big\{ b(t)[g(t) - h(t) ]
+ \sum_{k=1}^n a_k(t) [h(t) - h_k(t)] + \sum_{l=1}^m b_l(t) [g_l(t) - g(t)]
\big\} < 1 $$
\end{itemize}
hold. Then all four hypotheses of Proposition \protect{\ref{prop31}} are equivalent.
\end{proposition}

\noindent{\bf Remark.} It is to be noted that (C5) is a special case of (C4), if we rewrite
the left hand side of (\ref{3zaa}) as a sum of $(n+m+1)$ positive and $(n+m+1)$
negative terms:
\begin{align*}
&\dot{x}(t) + \sum_{k=1}^n a_k(t) x(h_k(t)) -
\sum_{l=1}^m b_l(t) x(g_l(t))\\
&= \dot{x}(t) + \sum_{k=1}^n a_k(t) x(h_k(t))
-  \sum_{k=1}^n a_k(t) x(h(t)) +  \sum_{k=1}^n a_k(t) x(h(t)) \\
&\quad -\sum_{l=1}^m b_l(t) x(g(t)) + \sum_{l=1}^m b_l(t) x(g(t))
 - \sum_{l=1}^m b_l(t) x(g_l(t))\\
&= \dot{x}(t) + a(t) x(h(t)) - b(t) x(g(t))  + \sum_{k=1}^n a_k(t)
\big[ x(h_k(t)) - x(h(t))\big] \\
&\quad + \sum_{l=1}^m b_l(t) \big[ x(g(t)) - x(g_l(t)) \big]
\end{align*}

We proceed with comparison results which are deduced from Lemma \ref{theorem4}.
To this end consider the equation
\begin{equation}
\label{6comp}
\dot{x}(t) + \sum_{k=1}^n \tilde{a}_k(t) x(\tilde{h}_k(t)) -
\sum_{l=1}^m \tilde{b}_l(t) x(\tilde{g}_l(t)) = 0, t \geq 0.
\end{equation}

\begin{proposition}\label{prop34}
\begin{enumerate}
\item  Suppose (C1), (C3) and either (C4) or (C5) hold
for (\ref{6comp}), where $a_k,b_l,h_k,g_l$ are changed by
$\tilde{a}_k,\tilde{b}_l,\tilde{h}_k,\tilde{g}_l$, respectively.
If $\tilde{a}_k (t) \geq a_k(t)$, $\tilde{b}_l (t) \leq b_l(t)$,
$\tilde{h}_k (t) \leq h_k(t)$, $\tilde{g}_l(t)\geq g_l(t)$ and
(\ref{6comp}) has a nonoscillatory
solution, then   (\ref{3zaa}) also has a nonoscillatory solution.

\item Suppose (C1), (C3) and either (C4) or (C5) hold. If $\tilde{a}_k
(t) \leq a_k(t)$, $\tilde{b}_l (t) \geq b_l(t)$, $\tilde{h}_k (t)
\geq h_k(t)$, $\tilde{g}_l(t)\leq g_l(t)$ and all solutions of
(\ref{6comp})  are oscillatory, then all solutions of
(\ref{3zaa}) are also oscillatory.
\end{enumerate}
\end{proposition}

To apply Proposition \ref{prop34} consider the autonomous delay equation
\begin{equation}
\label{autonom}
\dot{y} + \sum_{i=1}^n c_i y(t-\delta_i) - \sum_{l=1}^m d_l y(t-\sigma_l) =0,
~t \geq 0,
\end{equation}
where the following conditions hold:
\begin{itemize}
\item[(A1)] $c_i\geq 0, d_l\geq 0, \delta_i\geq 0, \sigma_l\geq 0$
\end{itemize}
and one of the two following conditions satisfied:
\begin{itemize}
\item[(A2)] $ n=m, c_i\geq d_i, \delta_i\geq \sigma_i,
\sum_{i=1}^n  d_i(\delta_i-\sigma_i)<1$;

\item[(A3)] $c= \sum_{i=1}^n c_i\geq d=\sum_{l=1}^m d_l,
\delta=\min \delta_i\geq \sigma=\max \sigma_l$, $$
d(\delta-\sigma)+\sum_{i=1}^n c_i(\delta_i-\delta)+\sum_{l=1}^m
d_l( \sigma- \sigma_l)<1;
$$

\end{itemize}
\begin{corollary} \begin{enumerate}
\item  Suppose (C1)-(C3), (A1) and at least one of (A2), (A3) hold.
If $a_k(t) \leq c_k, h_k(t) \geq t-\delta_k, b_l(t) \geq d_l, g_l(t)
\leq t-\sigma_l$ and
(\ref{autonom}) has a nonoscillatory
solution, then (\ref{3zaa}) also has a nonoscillatory solution.

\item  Suppose (C1), (C3), (A1) and at least one of (C4),
(C5) hold. If $a_k(t) \geq c_k, h_k(t) \leq t-\delta_k,  b_l(t)
\leq d_l, g_l(t) \geq t-\sigma_l$ and all solutions of
(\ref{autonom}) are oscillatory, then all solutions of
 (\ref{3zaa}) are also oscillatory.
\end{enumerate}
\end{corollary}

Lemma \ref{theorem5} immediately implies the following
result on the asymptotic behaviour of solutions.

\begin{proposition} \label{prop35}
Suppose(C1),(C3) and one of the following two conditions is satisfied
\begin{enumerate}
\item (C4) holds and there exists such $k$ that
${ \int_{0}^{\infty} [ a_k(t) - b_k(t)] dt =
\infty}$.

\item  (C5) holds and  ${ \int_{0}^{\infty} [ a(t) - b(t)] dt =
\int_{0}^{\infty}  \big[ \sum_{k=1}^n a_k(t) - \sum_{l=1}^m
b_l(t) \big] dt  = \infty}$.
\end{enumerate}
Then any nonoscillatory solution of (\ref{3zaa}) satisfies
$\lim_{t\to \infty} y(t) =0$.
\end{proposition}

\begin{proposition}\label{prop36}
Suppose the hypotheses (C1), (C3), (C4) hold.
If all solutions of anyone of the following
equations are oscillatory
\begin{gather} \label{24new}
\begin{aligned}
&\dot{x}(t) + \sum_{k=1}^n [a_k(t)-b_k(t)]x(h_k(t))\\
&+ \sum_{k=1}^n b_k(t)
\Big( \exp \big\{ \int_{h_k(t)}^{g_k(t)} [a_k(s)-b_k(s)] ds \big\} - 1
\Big) x(g_k(t))=0,
\end{aligned}\\
\label{25new}
\dot{x}(t) + \sum_{k=1}^n \big[ a_k(t)-b_k(t) + a_k(t)\Big( \exp \big\{
\int_{h_k(t)}^{g_k(t)} [a_k(s)-b_k(s)] ds \big\} - 1 \Big)
\big] x(g_k(t))=0, \\
\label{24newstop}
\dot{x}(t) + \sum_{k=1}^n [a_k(t)-b_k(t)]x(h_k(t))
+ \sum_{k=1}^n b_k(t) x(g_k(t)) \int_{h_k(t)}^{g_k(t)} [a_k(s)-b_k(s)] ds = 0,\\
\label{25newstop}
\dot{x}(t) + \sum_{k=1}^n \Big( a_k(t)-b_k(t) + a_k(t)
\int_{h_k(t)}^{g_k(t)}  [a_k(s)-b_k(s)] ds \Big) x(g_k(t)) = 0,
\end{gather}
then all solutions of   (\ref{3zaa}) are also oscillatory.
\end{proposition}

\noindent{\bf Remark.} Since (C5) is a special case of (C4), similar equations can be presented if
(C1),(C3),(C5) are satisfied.


\begin{corollary} \label{corprop361}
Suppose (C1), (C3), (C4) are satisfied and at least one of the following
conditions holds:
\begin{multline*}
\llap{\rm (1)}\quad
\liminf_{t \to \infty} \Big\{ \sum_{k=1}^n [a_k(t)-b_k(t)](t-h_k(t))\\
+ \sum_{k=1}^n b_k(t) \Big(  \exp \big\{ \int_{h_k(t)}^{g_k(t)}
[a_k(s)-b_k(s)] ds \big\}- 1 \Big)(t-g_k(t)) \Big\} > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm (2)}\quad \liminf_{t \to \infty} \Big\{
\sum_{k=1}^n [a_k(t)-b_k(t)](t-h_k(t))\\
 + \sum_{k=1}^n b_k(t)(t-h_k(t)) \int_{h_k(t)}^{g_k(t)}
[a_k(s)-b_k(s)] ds \Big\} > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm (3)}\quad \liminf_{t \to \infty} \Big\{
\sum_{k=1}^n \int_{g_k(t)}^t \big[ a_k(s)-b_k(s)\\
+ a_k(s)\big(  \exp \big\{ \int_{h_k(s)}^{g_k(s)} [a_k(\tau)-b_k(\tau)] d\tau
\big\}- 1 \big) \big] ds \Big\} > \frac{1}{e}
\end{multline*}
\[
\llap{\rm (4)} \quad \liminf_{t \to \infty} \Big\{
\sum_{k=1}^n \int_{g_k(t)}^t\big[ a_k(s)-b_k(s) + a_k(s)
\int_{h_k(s)}^{g_k(s)} [a_k(\tau)-b_k(\tau)] d\tau \big] ds
\Big\} > \frac{1}{e},
\]
then all solutions of   (\ref{3zaa}) are oscillatory.
\end{corollary}

\begin{corollary}
Suppose (A1), (A2) and at least one of the following conditions hold:
\begin{enumerate}
\item $ \sum_{k=1}^n \big[ (a_k-b_k)\delta_k + b_k \big(
e^{(a_k-b_k)(\delta_k-\sigma_k)} - 1 \big) \sigma_k \big] > 1/e$;

\item $\sum_{k=1}^n (a_k-b_k) \big[ \delta_k + b_k \sigma_k
(\delta_k-\sigma_k)\big] > 1/e$;

\item $\sum_{k=1}^n \big[a_k-b_k+a_k \big(e^{(a_k-b_k)(\delta_k-\sigma_k)} -
1 \big) \big] \sigma_k  > 1/e$;

\item  $\sum_{k=1}^n (a_k-b_k)\big[ 1+ a_k (\delta_k-\sigma_k)
\big] \sigma_k  > 1/e$.
\end{enumerate}
Then all solutions of   (\ref{autonom}) are oscillatory.
\end{corollary}

Now we proceed with explicit nonoscillation conditions.

\begin{proposition} \label{prop38}
Suppose either (C1), (C3), (C4) hold and the following inequality is satisfied
$$
\limsup_{t \to \infty} \sum_{k=1}^n \int_{h_k(t)}^t \big[
a_k(s) - \frac{1}{e} b_k(s)  \big] ds <\frac{1}{e}.
$$
or (C1), (C3), (C5) hold and
\begin{align*}
&\limsup_{t \to \infty} \Big\{\big( 1 - \frac{1}{e} \big)
\sum_{k=1}^n \int_{h_k(t)}^t a_k(s)\,ds\\
& + \int_{h(t)}^t  \big[ \sum_{k=1}^n a_k(s) - \frac{1}{e} \sum_{l=1}^m
b_k(s)  \big] ds + \big( 1 - \frac{1}{e} \big) \sum_{l=1}^m
\int_{g(t)}^t b_l(s)\,ds \Big\} < \frac{1}{e}.
\end{align*}
Then   (\ref{3zaa}) has a nonoscillatory solution.
\end{proposition}

Denote
$$H(t)=\min_k h_k(t),\quad  G(t)=\max_l g_l(t).$$

\begin{proposition} \label{prop39}
Suppose there exist $\tilde{a}(t)$, $\tilde{b}(t)$, such that
$$\tilde{b}(t) \leq b(t) \leq a(t) \leq \tilde{a}(t),\quad\mbox{where}\quad
a(t)= \sum_{k=1}^n a_k(t), ~b(t) = \sum_{l=1}^m b_l(t),$$
there exist finite limits
\begin{gather} \label{system1a}
B_{11}=  \lim_{t \to \infty}  \int_{H(t)}^t \tilde{a}(s)\,ds, \quad
{ B_{12}=  \lim_{t \to \infty}  \int_{H(t)}^t \tilde{b}(s)\,ds,}\\
\label{system1b}
B_{21}=  \lim_{t \to \infty}  \int_{G(t)}^t \tilde{a}(s)\,ds, \quad
{ B_{22}=  \lim_{t \to \infty}  \int_{G(t)}^t \tilde{b}(s)\,ds,}
\end{gather}
and (C1)--(C3) hold.
Suppose, in addition, that the system
\begin{gather}
\label{system2a}
\ln x_1  >  x_1 B_{11} - x_2 B_{12} \\
\label{system2b}
\ln x_2 <  x_1 B_{21} - x_2 B_{22}
\end{gather}
has a positive solution $(x_1,x_2)$ such that eventually $x_1\tilde{a}(t)
\geq x_2\tilde{b}(t)$.
Then   (\ref{3zaa}) has a nonoscillatory solution.
\end{proposition}

\begin{proof}
Consider the function $u(t) = x_1\tilde{a}(t) - x_2\tilde{b}(t)$.
Then $u(t)$ is nonnegative and
the system (\ref{system2a})-(\ref{system2b}) yields
$$x_1  > \exp \{  x_1 B_{11} - x_2 B_{12} \},
\quad x_2  < \exp \{  x_1 B_{21} - x_2 B_{22} \}.
$$
Thus  by definitions (\ref{system1a})-(\ref{system1b}) there exists
$t_1 > 0$, such that for $t \geq t_1$
\begin{gather*}
x_1  \geq  \exp \big\{ x_1 \int_{H(t)}^t \tilde{a}(s)\,ds -
x_2 \int_{H(t)}^t \tilde{b}(s)\,ds \big\}  = \exp \big\{ \int_{H(t)}^t
u(s)\,ds \big\} \\
-x_2  \geq  - \exp \big\{ x_1 \int_{G(t)}^t \tilde{a}(s)\,ds -
x_2 \int_{G(t)}^t \tilde{b}(s)\,ds \big\}  = -\exp \big\{ \int_{G(t)}^t
u(s)\,ds \big\}
\end{gather*}
After multiplying the first inequality by $\tilde{a}(t)$, the second one by
$\tilde{b}(t)$ and summation we have
\begin{align*}
 u(t) &= x_1\tilde{a}(t) - x_2\tilde{b}(t)\\
&\geq \tilde{a}(t) \exp \big\{ \int_{H(t)}^t u(s)\,ds \big\}
- \tilde{b}(t)\exp \big\{\int_{G(t)}^t u(s)\,ds \big\} \\
&\geq a(t)\exp \big\{ \int_{H(t)}^t u(s)\,ds \big\} - b(t)\exp \big\{
\int_{G(t)}^t u(s)\,ds \big\} \\
&= \sum_{k=1}^n a_k(t) \exp \big\{ \int_{H(t)}^t u(s)\,ds \big\} -
\sum_{l=1}^m b_l(t)\exp \big\{ \int_{G(t)}^t u(s)\,ds \big\}\\
&\geq \sum_{k=1}^n a_k(t) \exp \big\{ \int_{h_k(t)}^t u(s)\,ds \big\} -
\sum_{l=1}^m b_l(t)\exp \big\{ \int_{g_l(t)}^t u(s)\,ds \big\}.
\end{align*}
By Proposition \ref{prop31},   (\ref{3zaa}) has a nonoscillatory
solution. \end{proof}

\noindent
{\bf Example 1.} Consider the equation
\begin{equation}
\label{ex1}
\dot{x}(t) + \sum_{k=1}^n \frac{a_k}{t}x \big( \frac{t}{\mu_k}  \big)
- \sum_{k=1}^n \frac{b_k}{t}x \big( \frac{t}{\nu_k}  \big) =0, ~t \geq
t_0>0,
\end{equation}
where $a_k \geq b_k \geq 0$, $\mu_k \geq \nu_k > 1$. We apply
Corollary \ref{corprop361} (Ineq. 4):
\begin{align*}
&\liminf_{t \to \infty} \big\{ \sum_{k=1}^n \int_{t/\nu_k}^t
\big( \frac{a_k-b_k}{s} + \frac{a_k}{s} \int_{s/\mu_k}^{s/\nu_k}
\frac{a_k-b_k}{\tau}\,d\tau \big) ds \big\} \\
&= \liminf_{t \to \infty} \big\{ \sum_{k=1}^n \int_{t/\nu_k}^t
\big( \frac{a_k-b_k}{s} + \frac{a_k}{s} \big[  \ln \frac{s}{\nu_k} -
\frac{s}{\mu_k} \big] (a_k-b_k)  \big) ds \big\}\\
&= \liminf_{t \to \infty} \big\{ \sum_{k=1}^n  (a_k-b_k) \int_{t/\nu_k}^t
\big( \frac{1}{s} + \frac{a_k}{s} \ln \frac{\mu_k}{\nu_k}\big) ds\big\}\\
&= \liminf_{t \to \infty} \big\{ \sum_{k=1}^n (a_k-b_k) \big(
1+a_k\ln\frac{\mu_k}{\nu_k}\big) \big[ \ln t - \ln\frac{t}{\nu_k} \big]
\big\} \\
&= \sum_{k=1}^n (a_k-b_k)\big( 1+a_k\ln\frac{\mu_k}{\nu_k}\big) \ln\nu_k
\end{align*}
Thus if
$$
\sum_{k=1}^n (a_k-b_k)\big( 1+a_k\ln\frac{\mu_k}{\nu_k}\big) \ln\nu_k >
\frac{1}{e},
$$
then all solutions of   (\ref{ex1}) are oscillatory.
For nonoscillation results, we apply Proposition \ref{prop38}.
\begin{align*}
\limsup_{t \to \infty} \big\{ \sum_{k=1}^n \int_{t/\mu_k}^t
\big[ \frac{a_k}{s} - \frac{1}{e}  \frac{b_k}{s}\big] ds \big\}
&=\limsup_{t \to \infty} \big\{ \sum_{k=1}^n \big(
a_k - \frac{1}{e}  b_k \big) \big( \ln t - \ln \frac{t}{\mu_k} \big)
\big\}\\
&= \sum_{k=1}^n \big( a_k - \frac{1}{e}  b_k \big)\ln \mu_k
\end{align*}
Thus if
$\sum_{k=1}^n \big( a_k - \frac{1}{e}  b_k \big)\ln \mu_k < 1/e$,
 then   (\ref{ex1}) has a nonoscillatory solution. \smallskip

\noindent {\bf Example 2.} Consider the equation
\begin{equation}
\label{ex2}
\dot{x}(t) + \sum_{k=1}^n \frac{a_k}{t}x \big( \frac{t}{\mu_k}  \big)
- \frac{b}{t}x \big( \frac{t}{\nu}  \big) =0, \quad t \geq t_0>0,
\end{equation}
where ${ \sum_{k=1}^n a_k \geq b \geq 0}$, $\mu_k \geq \nu > 1$.
Unlike in Example 1, here the number of positive terms is not equal to the number
of negative terms.
This equation can be rewritten in one of the following forms:
\begin{enumerate}
\item ${\dot{x}(t) + \sum_{k=1}^n \frac{a_k}{t}x \big( \frac{t}{\mu_k}  \big)
- \sum_{k=1}^n \frac{a_k}{t}x \big( \frac{t}{\nu}  \big)
+ \big( \sum_{k=1}^n \frac{a_k}{t} \big) x \big( \frac{t}{\nu}  \big)
- \frac{b}{t}x \big( \frac{t}{\nu}  \big) =0}$

\item ${\dot{x}(t) + \sum_{k=1}^n \frac{a_k}{t}x \big( \frac{t}{\mu_k}
\big)- \sum_{k=1}^n \frac{b_k}{t}x \big( \frac{t}{\nu}  \big) =0}$,
where $b_1=b, ~ b_k=0, k>1$ and $a_1 \geq b$.

\item ${\dot{x}(t) + \sum_{k=1}^n \frac{a_k}{t}x \big( \frac{t}{\mu_k}  \big)
- \sum_{k=1}^n \frac{\lambda_k b}{t}x \big( \frac{t}{\nu}  \big) =0}$,
where $\lambda_k>0$, $ \sum_{k=1}^n \lambda_k=1$ and $a_k \geq \lambda_k b$.
\end{enumerate}
A computation similar to Example 1 yields that if anyone of the
following three conditions holds \begin{enumerate}
\item ${ \big( \sum_{k=1}^n a_k-b \big) \ln \nu >\frac{1}{e}}$

\item $a_1 \geq b$ and ${(a_1-b)\big( 1+a_1 \ln \frac{\mu_1}{\nu} \big) \ln \nu
+ \big( \sum_{k=2}^n a_k \big) \big( 1+a_k\ln \frac{\mu_k}{\nu} \big)
\ln \nu > 1/e}$

\item For each $k$ $a_k \geq \lambda_k b$ and
$\sum_{k=1}^n \big( a_k - \lambda_k b \big)
\big( 1+ a_k\ln \frac{\mu_k}{\nu} \big) \ln \nu > 1/e$.
\end{enumerate}
then all solutions of  (\ref{ex2}) are oscillatory.

If anyone of the following three conditions holds \begin{enumerate}
\item $\sum_{k=1}^n a_k \big( 1 - \frac{1}{e} \big)\ln \mu_k
+ \big( \sum_{k=1}^n a_k-b \big) \ln \nu  < 1/e$,
\item $a_1 \geq b$ and
${ \big( a_1 - \frac{1}{e} b \big) \ln \mu_1 + \sum_{k=2}^n
a_k \ln \mu_k < \frac{1}{e}}$,
\item For each $k$ $a_k \geq \lambda_k b$ and
$ \sum_{k=1}^n \big( a_k - \frac{1}{e} \lambda_k b \big)
\ln \mu_k < 1/e$,
\end{enumerate}
then   (\ref{ex2}) has a nonoscillatory solution.


\section{Integrodifferential Equations}
\label{section4}

In this section we will study the following integrodifferential equation
\begin{equation}
\label{5zaa}
\dot{x}(t) + \int_{0}^t K(t,s) x(s) ~ ds
- \int_{0}^t M(t,s) x(s)~ ds = 0,
\end{equation}
  (\ref{5zaa}) is a special case of   (\ref{1}) if we assume
\begin{equation}
\label{6zaa}
R(t,s) = \int_{0}^s K(t,\zeta) \,d\zeta, \quad
T(t,s) = \int_{0}^s M(t,\zeta) \,d\zeta.
\end{equation}
The hypotheses of Lemma \ref{theorem1} are satisfied for the delay
equation (\ref{5zaa}) if the following conditions hold:
\begin{itemize}
\item[(I1)] $K(t,s)$, $M(t,s)$ are Lebesgue integrable over each finite square
$[0,b] \times [0,b]$ functions;

\item[(I2)] There exist finite functions $h(t),g(t)$ such that
$h(t) = \inf\{ s| K(t,s) \geq 0 \}$,
$g(t) = \inf\{ s| M(t,s) \geq 0 \}$ and
$\lim_{t \to \infty} h(t) = \infty$,
$\lim_{t \to \infty} g(t) = \infty$;

\item[(I3)] For each $t,s$ $K(t,s) \geq 0$, $M(t,s) \geq 0$ and
$$ \int_{h(s)}^s K(t,\tau)\,d \tau \geq \int_{g(s)}^s
M(t,\tau)\,d \tau .$$
\end{itemize}
For $t_0\geq 0$ consider an initial-value problem
\begin{gather}
\label{*}
\dot{x}(t) + \int_{h(t)}^t K(t,s) x(s)\, ds
- \int_{g(t)}^t M(t,s)x(s) \, ds = 0, \quad t > t_{0}. \\
\label{initsec4}
x(t)=\varphi(t), \quad  t<t_0,\;  x(t_0)=x_0.
\end{gather}
As in the general case we will say that (\ref{5zaa}) has a nonoscillatory
solution if for some
$t_0\geq 0$, $\varphi(t)$ and $x_0$ the solution of
(\ref{*})-(\ref{initsec4}) is eventually positive or
eventually negative. Otherwise all solutions of (\ref{5zaa}) are oscillatory.

Consider in addition the inequality
\begin{equation}
\label{44newstar}
u(t)\geq  \int_{h(t)}^t K(t,s) \exp\big\{\int_{s}^t u(\tau)d \tau
\big\}\,d s
- \int_{g(t)}^t M(t,s) \exp\big\{\int_{s}^t u(\tau)d \tau \big\}\,ds.
\end{equation}
Lemma \ref{theorem1} immediately implies the following proposition.

\begin{proposition}
\label{prop41}
Suppose (I1)-(I3) hold. Consider the following hypotheses
\begin{enumerate}
\item There exists $t_1\geq 0$ such that inequality (\ref{44newstar})
has a nonnegative locally integrable  solution;

\item There exists $t_2\geq 0$ such that $X(t,s)>0$, $t\geq s\geq t_2$,
where $X(t,s)$
is a fundamental function of (\ref{5zaa});

\item  (\ref{5zaa}) has a nonoscillatory solution;

\item Inequality
\begin{equation}
\label{5zaaineq}
\dot{y}(t) + \int_{t_0}^t K(t,s) y(s) ~ ds
- \int_{t_0}^t M(t,s)y(s) ~ ds \leq 0
\end{equation}
has an eventually positive solution.
\end{enumerate}
Then the implications $1) \Rightarrow 2) \Rightarrow 3) \Rightarrow 4)$ are
valid.
\end{proposition}

Now let us apply Lemma \ref{theorem2} to   (\ref{5zaa}). Let us
introduce the following additional assumption
\begin{itemize}
\item[(I4)] $K(t,s) \geq M(t,s-h(t)+g(t))$ for each $t,s$ and
$$ \limsup_{t \to \infty}  [g(t)-h(t)] \int_{g(t)}^t M(t,s) \,ds
< 1.
$$
\end{itemize}

\begin{proposition} \label{prop42}
Suppose (I1)--(I4) hold. Then all four hypotheses of Proposition
\protect{\ref{prop41}} are equivalent.
\end{proposition}


In addition to (\ref{5zaa}) consider the integrodifferential
equation
\begin{equation}
\label{intdif}
\dot{x}(t) + \sum_{i=1}^n \int_{h_i(t)}^t K_i(t,s)x(s)\,ds - \sum_{l=1}^m
\int_{g_l(t)}^t M_l(t,s)x(s)\,ds =0,
\end{equation}
where the following conditions hold:
\begin{itemize}
\item[(I1$^\star$)] $K_i(t,s)$, $M_l(t,s)$ are Lebesgue integrable over each
finite square $[0,b] \times [0,b]$ functions;

\item[(I2$^\star$)] There exist finite functions
$h_i(t) = \inf\{ s| K_i(t,s) \geq 0 \}$, \\
$g_l(t) = \inf\{ s| M_l(t,s) \geq 0 \}$ and
$\lim_{t \to \infty} h_i(t) = \infty$,
$\lim_{t \to \infty} g_l(t) = \infty$;

\item[(I3$^\star$)] For each $t,s,i,l$ ~~~  $K_i(t,s) \geq 0$, $M_l(t,s) \geq 0$,
$$ \sum_{i=1}^n \int_{h_i(t)}^t K_i(t,\tau)\,d \tau \geq
\sum_{l=1}^m \int_{g_l(t)}^t M_l(t,\tau)\,d \tau $$
and one of two following conditions is satisfied:

\item[(I4$^\star$)] $m=n$, $K_i(t,s) \geq M_i(t,s)$ for each $t,s$, $i=1, \dots,n$,
for any $t,i$
$K_i(t,s) \geq M_i(t,s-h_i(t)+g_i(t))$  and
$$ \limsup_{t \to \infty}  \sum_{i=1}^n
[g_i(t)-h_i(t)] \int_{g_i(t)}^t M_i(t,s) \,ds  < 1.$$

\item[(I5$^\star$)] For each $i,l,t,s$ $h(t) \leq g(t)$, where $h,g$ are
defined in (\ref{hgnotation}) and
$$ K(t,s) = \sum_{i=1}^m K_i(t,s) \geq  \sum_{l=1}^l M_l(t,s) = M(t,s),
$$
we have $K(t,s) \geq M(t,s-h(t)+g(t))$ and
\begin{equation} \label{add2formulaint}
\begin{aligned}
&\limsup_{t\to \infty} \Big[ (g(t)-h(t)) \int_{g(t)}^t
M(t,s) \,ds
+ \sum_{i=1}^n (h(t)-h_i(t)) \int_{h_i(t)}^{t} K_i(t,s)\,ds \\
&+ \sum_{l=1}^m (g_l(t)-g(t)) \int_{g_l(t)}^{t} M_l(t,s)\,ds
\Big]  < 1.
\end{aligned}
\end{equation}
\end{itemize}

\begin{proposition} \label{prop43}
Suppose (I1$^\star$)--(I3$^\star$) and one of (I4$^\star$),(I5$^\star$) hold.
Then the following hypotheses are equivalent:
\begin{enumerate}
\item  Inequality
\begin{equation}
\label{intdifineq}
\dot{y} + \sum_{i=1}^n  \int_{h_i(t)}^t K_i(t,s)y(s)\,ds
- \sum_{l=1}^m
\int_{g_l(t)}^t   M_l(t,s)y(s)\,ds  \leq 0, \quad t \geq 0,
\end{equation}
has an eventually positive solution;

\item  There exists $t_2\geq 0$ such that $X(t,s)>0, ~t\geq s\geq t_2$,
where $X(t,s)$
is a fundamental function of (\ref{intdif});

\item  (\ref{intdif}) has a nonoscillatory solution.

\item There exists $t_3\geq 0$ such that for $t \geq t_3$ the inequality
\begin{equation} \label{intdif44star}
\begin{aligned}
u(t)&\geq \sum_{i=1}^n \int_{h_i(t)}^t K_i(t,s) \exp
\big\{\int_{s}^t u(\tau)d \tau \big\}\,ds\\
&\quad - \sum_{l=1}^m \int_{g_l(t)}^t M_l(t,s) \exp\big\{\int_{s}^t u(\tau)d
\tau \big\}\,ds
\end{aligned}
\end{equation}
has a nonnegative locally integrable  solution.
\end{enumerate}
\end{proposition}

We proceed with comparison results which are deduced from Lemma \ref{theorem4}.
To this end consider the equation
\begin{equation}
\label{intdifcomp}
\dot{x}(t) + \sum_{i=1}^n \int_{\tilde{h}_k(t)}^t \tilde{K}_i(t,s) x(s) ~ ds
- \sum_{l=1}^m \int_{\tilde{g}_l(t)}^t \tilde{M}_l(t,s) x(s) ~ ds = 0.
\end{equation}
For   (\ref{intdifcomp}) $h_i$,$g_l$ are changed by $\tilde{h}_i,
\tilde{g}_l$, respectively.

\begin{proposition} \label{prop44}
\begin{enumerate}
\item Suppose (I1$^{\star}$)--(I3$^{\star}$) and at
least one of (I4$^{\star}$),(I5$^{\star}$) hold, where
$K_i$,$M_l$,$h$,$g$ are changed by $\tilde{K}_i, \tilde{M}_l,
\tilde{h}, \tilde{g}$, respectively. If $\tilde{K}_i(t,s) \geq
K_i(t,s)$, $\tilde{M}_l(t,s) \leq M_l(t,s)$ and (\ref{intdifcomp})
has a nonoscillatory solution, then (\ref{intdif}) also has a
nonoscillatory solution.

\item Suppose (I1$^{\star}$)-(I3$^{\star}$) and at least one of
(I4$^{\star}$),(I5$^{\star}$) hold. If $\tilde{K}_i(t,s) \leq
K_i(t,s)$, $\tilde{M}_l(t,s) \geq M_l(t,s)$ and all solutions of
(\ref{intdifcomp}) are oscillatory, then all solutions of
(\ref{intdif}) are also oscillatory.
\end{enumerate}
\end{proposition}

\begin{corollary} \label{corprop441}
\begin{enumerate}
\item Suppose (I1$^\star$)--(I3$^\star$), (A1) and at least one of (A2), (A3) hold.
If for each $t,i,l$
$$
\int_{h_i(t)}^t K_i(t,s) \,ds \leq c_i,~~ \int_{g_l(t)}^t M_l(t,s)
\,ds \geq d_l$$ and   (\ref{autonom}) has a nonoscillatory
solution, then   (\ref{intdif}) also has a nonoscillatory
solution.

\item  Suppose (I1$^\star$), (I3$^\star$), (A1) and at least on of
(I4$^\star$), (I5$^\star$) hold.
If for each $t,i,l$
$$
\int_{h_i(t)}^t K_i(t,s) \,ds \geq c_i,~~ \int_{g_l(t)}^t M_l(t,s)
\,ds \leq d_l$$ and all solutions of   (\ref{autonom}) are
oscillatory, then all solutions of  (\ref{intdif}) are also
oscillatory.
\end{enumerate}
\end{corollary}

Lemma \ref{theorem5} immediately implies the following result
on the asymptotic behaviour of solutions.

\begin{proposition} \label{prop45}
Suppose (I1$^{\star}$)--(I3$^{\star}$) is satisfied and anyone of the
following conditions holds:
\begin{enumerate}
\item (I4$^{\star}$) is satisfied and for some $l$,
$\int_{0}^{\infty} \big[ \int_{t_0}^t (K_l(t,s)- M_l(t,s))\,ds \big]\,dt
= \infty$
\item (I5$^{\star}$) is satisfied and
$\int_{0}^{\infty} \big[ \int_{t_0}^t (K(t,s)-M(t,s))\,ds \big]\,dt =
\infty$.
\end{enumerate}
Then any nonoscillatory solution $y$ of (\ref{intdif}) satisfies
${ \lim_{t \to \infty} y(t) = 0}$.
\end{proposition}

Lemma \ref{theorem7} yields oscillation conditions for (\ref{intdif}).

\begin{proposition} \label{prop46}
Suppose hypotheses (I1$^{\star}$)--(I4$^{\star}$) hold.
If all solutions of anyone of the following two
equations are oscillatory
\begin{equation} \label{24intdif}
\begin{aligned}
&\dot{x}(t) + \sum_{l=1}^n \int_{h_l(t)}^t \big[ K_l(t,s) -
M_l(t,s-h_l(t) + g_l(t)) \big] x(s)\,ds
+ \sum_{l=1}^n \int_{g_l(t)}^t M_l(t,s) x(s) \\
&\times \Big( \exp \big\{ \int_{s+h_l(t) -g_l(t)}^s \!\!\!\! d\tau
\int_{h_l(\tau)}^{\tau} \big[ K_l(\tau,\zeta) - M_l(\tau,\zeta) \big]
d\zeta \big\} - 1 \Big) ds=0\,,
\end{aligned}
\end{equation}
\begin{equation} \label{25intdif}
\begin{aligned}
&\dot{x}(t) + \sum_{l=1}^n \int_{g_l(t)}^t K_l(t,s+h_l(t) -g_l(t))x(s)\\
&\times
\Big( \exp \big\{ \int_{s+h_l(t) -g_l(t)}^s d\tau
\int_{h_l(\tau)}^{\tau} \big[ K_l(\tau,\zeta) - M_l(\tau,\zeta) \big]
d\zeta \big\} - 1 \Big) \\
&+ \sum_{l=1}^n \int_{g_l(t)}^t \big[ K_l(t,s+h_l(t) -g_l(t)) - M_l(t,s)
\big] x(s)\,ds =0\,,
\end{aligned}
\end{equation}
then all solutions of   (\ref{intdif}) are also oscillatory.
\end{proposition}

\noindent{\bf Remark.}
Since (I5$^{\star}$) is a special case of (I4$^{\star}$), similar
equations can be presented if
(I1$^{\star}$)-(I3$^{\star}$),(I5$^{\star}$) are satisfied.
\smallskip

The oscillation conditions for integrodifferential equations with nonnegative
kernels \cite{ZAA} lead to the following result.

\begin{corollary}\label{corprop461}
Suppose the hypotheses (I1)-(I4) hold for (\ref{5zaa}) and at least one
of the following inequalities is valid:
\begin{multline*}
\llap{\rm 1)}\quad \liminf_{t \to \infty}\Big\{ \int_{h(t)}^t
\big[ \int_{h(s)}^s K(t,\tau)\,d\tau - \int_{g(s)}^s M(t,\tau-h(t)+g(t))\,d\tau
\big] ds \\
+ \int_{g(t)}^{t}  d \tau \int_{h(\tau)}^{\tau}
\big( \exp\big\{\int_{s+h(t)-g(t)}^s   \big[ \int_{h(u)}^u  K(u,\tau)d\tau\\
 - \int_{g(u)}^u  M(u,\tau)d\tau \big] d u \big\} - 1 \big) M(t,s)ds \Big\}
 > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm 2)}\quad  \liminf_{t \to \infty} \Big\{ \int_{h(t)}^t
\big[ \int_{h(s)}^s K(t,\tau)\,d\tau - \int_{g(s)}^s
M(t,\tau-h(t)+g(t))\,d\tau \big] \,ds\\
+  \int_{g(t)}^{t} d \tau \int_{h(\tau)}^{\tau}
\big( \int_{s+h(t)-g(t)}^s  \big[ \int_{h(u)}^u K(u,\tau)\,d\tau\\
- \int_{g(u)}^u M(u,\tau)\,d\tau \big]
d u \big) M(t,s) \,ds  \Big\}  > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm 3)}\quad  \liminf_{t \to \infty} \Big\{ \int_{g(t)}^t d\tau
\int_{g(\tau)}^{\tau}  \Big( \exp\big\{\int_{s-g(t)+h(t)}^s
\big[ \int_{h(u)}^u  K(u,\tau)d\tau \\
 - \int_{g(u)}^u  M(u,\tau)d\tau \big] du \big\} - 1\Big)K(t,s-g(t)+h(t))ds \\
+  \int_{g(t)}^t \big[ \int_{h(s)}^s  K(t,\tau-g(t)+h(t))\,d\tau
- \int_{g(s)}^s M(t,\tau-h(t)+g(t))\,d\tau \big] ds
\Big\}  > \frac{1}{e}
\end{multline*}
\begin{multline*}
\llap{\rm 4)}\quad \liminf_{t \to \infty} \Big\{ \int_{g(t)}^t  d\tau
\int_{g(\tau)}^{\tau}\Big(  \int_{s-g(t)+h(t)}^s
\Big[  \int_{h(u)}^u  K(u,\tau)d\tau\\
-\int_{g(u)}^u M(u,\tau)d\tau \Big] d u \Big) K(t,s-g(t)
+h(t))ds \\
+ \int_{g(t)}^t \big[
\int_{h(s)}^s K(t,\tau-g(t)+h(t))\,d\tau - \int_{g(s)}^s
M(t,\tau-h(t)+g(t))\,d\tau \big] ds  \Big\}  > \frac{1}{e}
\end{multline*}
Then all solutions of  (\ref{5zaa}) are oscillatory.
\end{corollary}

Lemma \ref{theorem9} implies the following nonoscillation results
for (\ref{intdif}).

\begin{proposition}\label{prop47}
Suppose (I1$^{\star}$)--(I4$^{\star}$) and the following inequality
\begin{equation}\label{eq25}
\limsup_{t \to \infty} \sum_{l=1}^n \int_{h_l(t)}^t
\big[ \int_{h_l(s)}^s K_l (s, \tau)\,d\tau
- \frac{1}{e} \int_{g_l(s)}^s M_l (s, \tau)\,d\tau \big] ds < \frac{1}{e}
\end{equation}
hold. Then   (\ref{intdif}) has a nonoscillatory solution.
\end{proposition}

\begin{proposition} \label{prop48}
Suppose (I1$^{\star}$)--(I3$^{\star}$),(I5$^{\star}$) and the following inequality
\begin{align*}
&\limsup_{t \to \infty} \big\{ \sum_{l=1}^n \big( 1 - \frac{1}{e} \big)
\int_{h_l(t)}^t ds \int_{h_l(s)}^s K_l (s, \tau)\,d\tau
 \int_{0}^t ds \big[ \int_{0}^s K(s, \tau)\,d\tau\\
&- \frac{1}{e} \int_{0}^s M(s, \tau)\,d\tau \big]
+\big( 1 - \frac{1}{e} \big) \int_{g_l(t)}^t ds \int_{g_l(s)}^s
M_l (s, \tau)\,d\tau \big\} < \frac{1}{e}
\end{align*}
hold. Then   (\ref{intdif}) has a nonoscillatory solution.
\end{proposition}


Similar to Proposition \ref{prop39} the following result can be obtained.
Let $H(t) = \min_i h_i(t), G(t) = \max_l g_l (t)$.

\begin{proposition} \label{prop49}
Suppose there exist $\tilde{K}(t,s)$, $\tilde{M}(t,s)$, such that
$$
\tilde{M}(t,s) \leq M(t,s) \leq K(t,s) \leq \tilde{K}(t,s),
$$
where $K(t,s)= \sum_{i=1}^n K_i(t,s)$, $M(t,s) = \sum_{l=1}^m M_l(t,s)$,
the following limits  exist and are finite:
\begin{gather} \label{system1aintdif}
B_{11}=  \lim_{t \to \infty}  \int_{H(t)}^t ds
\int_{H(s)}^s  \tilde{K}(s,\tau)\,d\tau,
{ B_{12}=  \lim_{t \to \infty}  \int_{H(t)}^t
ds \int_{H(s)}^s  \tilde{M}(s,\tau)\,d\tau,}  \\
\label{system1bintdif}
B_{21}=  \lim_{t \to \infty}  \int_{G(t)}^t  ds
\int_{G(s)}^s  \tilde{K}(s,\tau)\,d\tau,
{ B_{22}=  \lim_{t \to \infty}  \int_{G(t)}^t
ds \int_{G(s)}^s  \tilde{M}(s,\tau)\,d\tau,}
\end{gather}
and (I1)--(I4) hold for
$\tilde{K}(t,s)$, $\tilde{M}(t,s)$. Suppose, in addition, that
the system
\begin{gather}
\label{system2aintdif}
\ln x_1  >  x_1 B_{11} - x_2 B_{12} \\
\label{system2bintdif}
\ln x_2  <  x_1 B_{21} - x_2 B_{22}
\end{gather}
has a positive solution $(x_1,x_2)$ such that eventually
$$
x_1 \int_{h(t)}^t \tilde{K}(t,s)\,ds \geq x_2 \int_{g(t)}^t \tilde{M}(t,s)\,ds.
$$
Then   (\ref{intdif}) has a nonoscillatory solution.
\end{proposition}

\noindent{\bf Example 3.}
Consider the integrodifferential equation
\begin{equation} \label{newloc1}
\dot{x}(t) + \int_0^t L(t,s) x(s)\,ds=0.
\end{equation}
Let $\alpha >0$,
\begin{gather*}
 L(t,s) = \begin{cases}
\alpha \sin(s-t), & 0 \leq t-s \leq 2\pi, \\
0, & \mbox{otherwise}, \end{cases} \\
K(t,s) = L^+(t,s)=\frac{1}{2} (|L(t,s)|+L(t,s))=\alpha \sin(s-t)
\chi_{[t-2\pi,t-\pi]}(s),\\
M(t,s) = L^-(t,s)=\frac{1}{2} (|L(t,s)|-L(t,s))=
-\alpha \sin(s-t)\chi_{[t-\pi,t]}(s).
\end{gather*}
Then $h(t)=t-2\pi, ~g(t)= t-\pi, M(t,s+\pi)=K(t,s)$ and
\begin{align*}
&\limsup_{t \to \infty} \big\{ [g(t)-h(t)] \int_{g(t)}^t M(t,s)\,ds
\big\}\\
&= \limsup_{t \to \infty} \big\{  [g(t)-h(t)] \int_{t-\pi}^t
\alpha (-\sin(s-t))\,ds \big\} \\
& = \limsup_{t \to \infty} \big\{ \pi \alpha \big( \cos 0 - \cos(-\pi)
\big)  \big\}
= 2\pi \alpha.
\end{align*}
Thus the hypothesis (I4) holds for (\ref{newloc1})
if $\alpha < 1/(2\pi)$.
Let us proceed to nonoscillation conditions for this equation.
To this end we will apply Proposition \ref{prop47}.
We have
\begin{gather*}
 \int_{h(t)}^t K(t,s)\,ds =
\int_{t-2\pi}^{t-\pi} \alpha \sin(s-t)\,ds = 2\alpha, \\
\int_{g(t)}^t M(t,s)\,ds = - \int_{t-\pi}^{t} \alpha \sin(s-t)\,ds
= 2 \alpha,
\end{gather*}
which after the substitution in
(\ref{eq25}) yields
\begin{align*}
\limsup_{t \to \infty} \int_{h(t)}^t
\Big[ \int_{h(s)}^s K (s, \tau)\,d\tau
- \frac{1}{e} \int_{g(s)}^s M (s, \tau)\,d\tau \Big] ds
&= [t-h(t)] 2\alpha \big(1 - \frac{1}{e} \big)\\
&= 2 \pi 2\alpha \frac{e-1}{e} < \frac{1}{e},
\end{align*}
which is satisfied when $4\pi \alpha (e-1) < 1$. Consequently, if $\alpha <
\frac{1}{4\pi (e-1)}$, then (\ref{newloc1}) has a nonoscillatory solution.
\smallskip

\noindent {\bf Example 4.} Let $0<\alpha < \beta$. Consider equation
(\ref{newloc1}) with
$$ L(t,s) = \begin{cases}
\alpha \sin(s-t), & 0 \leq t-s \leq \pi, \\
\beta \sin(s-t), & \pi \leq t-s \leq 2\pi, \\
0, & \mbox{otherwise}, \end{cases}
$$
Then $K(t,s) = \beta \sin(s-t) \chi_{[t-2\pi,t-\pi]}(s)$,
$M(t,s) = \alpha \sin(s-t) \chi_{[t-\pi,t]}(s)$ and
$$
\limsup_{t \to \infty} \big\{  [g(t)-h(t)] \int_{g(t)}^t
M(t,s)\,ds\big\} = 2\pi \alpha .
$$
The hypothesis (I4) holds for (\ref{newloc1}) if in addition
$\alpha<\frac{1}{2\pi}$. Similarly to Example 1, if $4 \pi (\beta e-\alpha)< 1$,
 we have
\begin{align*}
\int_{h(t)}^t \big[ \int_{h(s)}^s K(s,\tau)\,d\tau
-\frac{1}{e} \int_{g(s)}^s M(s,\tau)\,d\tau \big] \,ds
&= 2\pi 2 \big( \beta - \frac{\alpha}{e} \big) \\
&= \frac{4 \pi}{e} (\beta e-\alpha)< \frac{1}{e}\,.
\end{align*}
Consequently, if $\beta e-\alpha< \frac{1}{4 \pi}$, then
(\ref{newloc1}) has a nonoscillatory solution.

For this kernel we can also obtain oscillation conditions.
Since
$$
\int_{h(s)}^s K(t,\tau)\,d\tau - \int_{g(s)}^s M(t,\tau-h(t)+g(t))\,d\tau
= 2 (\beta - \alpha)
$$
for $t-\pi \leq \tau \leq t$,  we have
$$
\int_{h(u)}^u K(u,\tau)\,d\tau -\int_{g(u)}^u M(u,\tau)\,d\tau  =
2(\beta - \alpha).
$$
Then after substituting these results into the first formula in Corollary
\ref{corprop461} we have
\begin{align*}
&\liminf_{t \to \infty}\Big\{ \int_{h(t)}^t
\Big[ \int_{h(s)}^s K(t,\tau)\,d\tau - \int_{g(s)}^s
M(t,\tau-h(t)+g(t))\,d\tau\Big] ds \\
&+ \int_{g(t)}^{t} d \tau \int_{h(\tau)}^{\tau}
\Big( \exp\Big\{\int_{s+h(t)-g(t)}^s   \big[ \int_{h(u)}^u
K(u,\tau)\,d\tau \\
&- \int_{g(u)}^u M(u,\tau)\,d\tau \big]d u \Big\} - 1 \Big) M(t,s) \,ds\Big\}\\
& \geq \int_{t-\pi}^t \Big[\int_{h(s)}^s K(t,\tau)\,d\tau
- \int_{g(s)}^s M(t,\tau-h(t)+g(t))\,d\tau \Big]ds\\
&\quad + 2(\beta -\alpha) \int_{g(t)}^{t} d \tau \int_{h(\tau)}^{\tau}
\big( e^{2\pi (\beta-\alpha)} -1 \big) \,d_s T(t,s) \\
&= 2\pi(\beta -\alpha) + \big( e^{2\pi (\beta-\alpha)} -1 \big) 2\pi.
\end{align*}
Thus if $2\pi(\beta -\alpha + \exp \{ 2\pi
(\beta-\alpha)  \} -1 ) > 1/e$, then all solutions of
(\ref{newloc1}) are oscillatory.

\section{Mixed Equations} \label{section5}

In this section we will consider mixed equations
\begin{equation} \label{11zaa}
\begin{aligned}
&\dot{x}(t) + \sum_{k=1}^n a_k(t) x(h_k(t)) - \sum_{l=1}^mb_l(t) x(g_l(t))
+ \sum_{i=1}^r \int_{0}^t K_i(t,s) x(s)ds \\
&-\sum_{j=1}^p \int_{0}^t M_j(t,s) x(s)ds =0, \quad t \geq t_0\geq 0,
\end{aligned}
\end{equation}
with the initial conditions
\begin{equation} \label{initsec5}
x(t)=\varphi(t), \quad t<t_0,\; x(t_0)=x_0.
\end{equation}
We assume that (C1)--(C3) and (I1$^{\star}$)--(I3$^{\star}$) hold.
To avoid confusion, in (I2$^{\star}$) instead of
$h_k, g_l$ the following functions are introduced:
$$
\tilde{h}_i = \inf\{ s| K_i(t,s) \geq 0 \}, \quad
\tilde{g}_j = \inf\{ s| M_j(t,s) \geq 0 \}.
$$
Consider in addition the following hypotheses:
\begin{itemize}
\item[(M1)]  $m=n$, $a_k \geq b_k$, $h_k(t) \leq g_k(t)$ for $k,t$,
$t \geq 0$, r=p, $K_i(t,s) \geq M_i(t,s-\tilde{h}_i(t)+\tilde{g}_i(t))$
and
$$\limsup_{t \to \infty} \Big\{ \sum_{k=1}^n b_k(t) [g_k(t) - h_k(t) ]
+\sum_{i=1}^r [\tilde{g}_i(t) - \tilde{h}_i(t)]
\int_{\tilde{g}_i(t)}^t M_i(t,s)ds \Big\} < 1.
$$
\item[(M2)] $ h(t)  \leq g(t)$,
${ a(t) = \sum_{k=1}^n a_k(t) \geq b(t) = \sum_{l=1}^m b_l(t)}$,
$\tilde{h}_i(t) \leq \tilde{h}(t) \leq \tilde{g}(t)\leq \tilde{g}_j(t)$,
for each $i,j,t$, ${ K(t,s)= \sum_{i=1}^r K_i(t,s)
\geq M(t,s)=\sum_{j=1}^p M_j(t,s) }$ for each $t,s$ and
\begin{align*}
&\limsup_{t \to \infty} \Big\{  b(t)[g(t) - h(t) ]
+ \sum_{k=1}^n a_k(t) [h(t) - h_k(t)] + \sum_{l=1}^m
b_l(t) [g_l(t) - g(t)] \\
&+ (\tilde{g}(t)-\tilde{h}(t)) \int_{\tilde{g}(t)}^t M(t,s) \,ds
+ \sum_{i=1}^n (\tilde{h}(t)-\tilde{h}_i(t)) \int_{\tilde{h}_i(t)}^{t}
 K_i(t,s)\,ds \\
&+ \sum_{l=1}^m (\tilde{g}_l(t)-\tilde{g}(t))
\int_{\tilde{g}_l(t)}^{t} M_l(t,s)\,ds \Big\}  < 1.
\end{align*}
\end{itemize}
Obviously two other combinations of (C4), (C5) with
(I4$^{\star}$), (I5$^{\star}$) can be considered.

\begin{proposition} \label{prop51}
Suppose (C1)--(C3), (I1$^{\star}$)--(I3$^{\star}$) and at least
one of hypotheses (M1), (M2) hold. Then the following hypotheses
are equivalent.
\begin{enumerate}
\item There exists $t_1 \geq 0$ such that for $t \geq t_1$ the inequality
\begin{multline*}
u(t) \geq \sum_{k=1}^n a_k(t) \exp \big\{  \int_{h_k(t)}^t u(s)ds \big\}
- \sum_{l=1}^m b_l(t) \exp \big\{  \int_{g_k(t)}^t u(s)ds \big\}\\
+\sum_{i=1}^r \int_{\tilde{h}_i(t)}^t K_i(t,s) \exp\big\{\int_{s}^t
u(\tau)d \tau \big\}\,ds
- \sum_{j=1}^p \int_{\tilde{g}_l(t)}^t M_j(t,s) \exp\big\{\int_{s}^t
u(\tau)d \tau \big\}\,ds
\end{multline*}
has a nonnegative locally integrable  solution
(we assume $u(t)=0$ for $t<t_1$);

\item  There exists $t_2 \geq 0$ such that the fundamental function of
(\ref{11zaa}) $X(t,s)>0$, $t \geq s \geq t_2$;

\item  Equation (\ref{11zaa}) has a nonoscillatory solution;

\item  The inequality
\begin{equation} \label{11zaaineq}
\begin{aligned}
&\dot{y}(t) + \sum_{k=1}^n a_k(t) y(h_k(t)) - \sum_{l=1}^m b_l(t) y(g_l(t))\\
&+ \sum_{i=1}^r \int_{0}^t K_i(t,s) y(s)\,ds -
\sum_{j=1}^p \int_{0}^t M_j(t,s) x(s)\,ds \leq 0
\end{aligned}
\end{equation}
has an eventually positive solution.
\end{enumerate}
\end{proposition}

The comparison result for equation (\ref{11zaa}) combines Propositions
\ref{prop34} and \ref{prop44}.
Consider the comparison equation
\begin{equation} \label{11zaacomp}
\begin{aligned}
&\dot{x}(t) + \sum_{k=1}^n \tilde{a}_k(t) x(\bar{h}_k(t)) - \sum_{l=1}^m
\tilde{b}_l(t) x(\bar{g}_l(t)) \\
&+ \sum_{i=1}^r \int_{0}^t \tilde{K}_i(t,s) x(s)\,ds -
\sum_{j=1}^p \int_{0}^t \tilde{M}_j(t,s) x(s)\,ds = 0.
\end{aligned}
\end{equation}


\begin{proposition}
\label{prop52} 1) Suppose (C1)-(C3),(I1$^{\star}$)-(I3$^{\star}$)
and either (M1) or (M2) hold for (\ref{11zaacomp}), where
$a_k,b_l,h_k,g_l$,$K_i$,$M_l$ are changed by
$\tilde{a}_k,\tilde{b}_l,\bar{h}_k,\bar{g}_l,
\tilde{K}_i,\tilde{M}_l$, respectively. If $\tilde{a}_k (t) \geq
a_k(t)$, $\tilde{b}_l (t) \leq b_l(t)$, $\bar{h}_k (t) \leq
h_k(t)$, $\bar{g}_l(t)\geq g_l(t)$, $\tilde{K}_i(t,s) \geq
K_i(t,s)$, $\tilde{M}_l(t,s) \leq M_l(t,s)$ and
(\ref{11zaacomp}) has a nonoscillatory
solution, then   (\ref{11zaa}) also has a nonoscillatory solution. \\
2) Suppose (C1)-(C3),I1$^{\star}$)-(I3$^{\star}$) and either (M1) or (M2) hold for (\ref{11zaa}). If
$\tilde{a}_k (t) \leq a_k(t)$, $\tilde{b}_l (t) \geq b_l(t)$, $\bar{h}_k (t) \geq
h_k(t)$, $\bar{g}_l(t)\leq g_l(t)$, $\tilde{K}_i(t,s) \leq K_i(t,s)$,
$\tilde{M}_l(t,s) \geq M_l(t,s)$ and all solutions of
  (\ref{11zaacomp})  are
oscillatory, then all solutions of   (\ref{11zaa}) are also
oscillatory.
\end{proposition}

\begin{proposition} \label{prop53}
Suppose (C1)--(C3), (I1$^{\star}$)--(I3$^{\star}$), (M1) hold and either 
there exists such $k$ that
${ \int_{0}^{\infty}  \big[ a_k(t) - b_k(t) \big] dt =\infty }  $
or there exists such $i$ that
$$
\int_{0}^{\infty} \Big[ \int_{\tilde{h}_i(t)}^t K_i(t,s)\,ds -
\int_{\tilde{g}_i(t)}^t M_i(t,s)\,ds \Big] dt = \infty.
$$
Then any nonoscillatory solution of (\ref{11zaa}) tends to zero at infinity.
\end{proposition}

\noindent{\bf Remark.} Similar result can be obtained if (C1)--(C3),
I1$^{\star}$)--(I3$^{\star}$), (M2)
are satisfied.

Proposition \ref{prop54} presents oscillation conditions for
(\ref{11zaa}).

\begin{proposition} \label{prop54}
Suppose (C1)--(C3), (I1$^{\star}$)--(I3$^{\star}$), (M1) and the following 
inequality hold
\begin{align*}
&\liminf_{t \to \infty} \Big\{ \sum_{k=1}^n [a_k(t)-b_k(t)](t-h_k(t)) \\
&+ \sum_{k=1}^n b_k(t) \Big(  \exp \big\{ \int_{h_k(t)}^{g_k(t)}
[a_k(s)-b_k(s)] ds \big\} - 1 \Big)  (t-g_k(t)) \\
&+ \sum_{i=1}^r \int_{\tilde{h}_i(t)}^t
\Big[ \int_{\tilde{h}_i(s)}^s K_i(t,\tau)\,d\tau - \int_{\tilde{g}_i(s)}^s
M_i(t,\tau-\tilde{h}_i(t)+\tilde{g}_i(t))\,d\tau \Big] ds\\
&+ \sum_{i=1}^r \int_{\tilde{g}_i(t)}^{t} \!\!\!\! d \tau
\int_{\tilde{h}_i(\tau)}^{\tau}
\Big( \exp\Big\{\int_{s+\tilde{h}_i(t)-\tilde{g}_i(t)}^s
\Big[ \int_{\tilde{h}_i(u)}^u K_i(u,\tau)d\tau  \\
& -\int_{\tilde{g}_i(u)}^u \!\! M_i(u,\tau)d\tau \Big]
d u \Big\} - 1 \Big) M_i(t,s)ds\Big\}  > \frac{1}{e}
\end{align*}
Then all solutions of  (\ref{11zaa}) are oscillatory.
\end{proposition}

\noindent{\bf Remark.} Similarly to inequality 1) in Corollary
\ref{cortheorem71} other oscillation conditions for
(\ref{11zaa}) can be deduced using inequalities 2)-4) of this
corollary.

Proposition \ref{prop55} present nonoscillation conditions.

\begin{proposition} \label{prop55}
Suppose (C1)--(C3), (I1$^{\star}$)--(I3$^{\star}$), (M1) and the following
inequality holds
\begin{align*}
&\limsup_{t \to \infty} \Big\{
\sum_{k=1}^n \int_{h_k(t)}^t \big[ a_k(s)- \frac{1}{e} b_k(s) \big]ds\\
&+ \sum_{i=1}^r \int_{\tilde{h}_i(t)}^{t} ds
\Big[\int_{\tilde{h}_i(s)}^s K_i(s,\tau)\,d\tau -
\frac{1}{e}\int_{\tilde{g}_i(s)}^s M_i(s,\tau)\,d\tau   \Big] \Big\} <
\frac{1}{e}.
\end{align*}
Then  (\ref{11zaa}) has a nonoscillatory solution.
\end{proposition}

\noindent {\bf Remark.} Similar results are obtained (see Propositions \ref{prop38},
\ref{prop47} and \ref{prop48}) if (M2) is satisfied instead of (M1).

As a final example, consider the following equation of the mixed type
\begin{equation}
\label{11zaastar}
\dot{x}(t) + \sum_{k=1}^n a_k(t)x(h_k(t)) - \int_{0}^t K(t,s)x(s)\,ds =0,
\end{equation}
under the following conditions:
\begin{itemize}
\item[(m1)] $a_k \geq 0$ are Lebesgue measurable bounded functions, $K$ is Lebesgue
integrable over each finite square $[0,b] \times [0,b]$;

\item[(m2)] There exists finite function $g(t) = \inf \{ s| K(t,s) > 0 \}$ and
 $\lim_{t \to \infty} g(t) = \infty$;
 $\lim_{t \to \infty} h_k(t)= \infty$ for each $k$;

\item[(m3)] For any $t\geq s\geq 0$,
$\sum_{k=1}^n a_k(t) \chi_{[h_k(t), \infty) (s) \geq \int_{g(s)}^s K(t,\tau)
\,d\tau }$.
\end{itemize}
Consider also the following hypothesis
\begin{itemize}
\item[(m4)] There exist constants $c_1, \dots, c_n$, ${ \sum_{k=1}^n
c_k=1} $ such that
$$  a_k(t) \chi_{[h_k(t), \infty)} (s) \geq
c_k \int_{s-h_k(t)}^{s-h_k(t)+g(t)} K(t,\tau)\,d\tau
$$
and
$$ \limsup_{t \to \infty} \big\{  \big[ \sum_{k=1}^n c_k(g(t)-h_k(t))
\big] \int_{g(t)}^t K(t,s)\,ds \big\} < 1  .
$$
\end{itemize}

\begin{proposition} \label{prop56}
Suppose (m1)--(m3) hold. Consider the following hypotheses
\begin{enumerate}
\item There exists $t_1 \geq 0$ such that the inequality
$$
u(t) \geq \sum_{k=1}^n a_k(t) \exp \big\{  \int_{h_k(t)}^t u(s)ds \big\}
- \int_{g(t)}^t K(t,s)  \exp \big\{  \int_{s}^t u(\tau)d\tau \big\}, ~t
\geq t_1
$$
has a nonnegative locally integrable  solution (we assume $u(t)=0$ for $t<t_1$);

\item There exists $t_2 \geq 0$ such that the fundamental function of
(\ref{11zaastar}) $X(t,s)>0$, $t\geq s \geq t_2$;

\item Equation (\ref{11zaastar}) has a nonoscillatory solution;

\item The inequality
\begin{equation} \label{11zaastarineq}
\dot{y}(t) + \sum_{k=1}^n a_k(t)y(h_k(t)) - \int_{0}^t K(t,s)y(s)\,ds  \leq 0
\end{equation}
has an eventually positive solution.
\end{enumerate}
Then the implications $1) \Rightarrow 2) \Rightarrow 3) \Rightarrow 4)$ are
valid. If in addition (m4) holds then hypotheses 1)--4) are equivalent.
\end{proposition}

To deduce a comparison result we introduce the equation
\begin{equation}
\label{11starcomp}
\dot{x}(t) + \sum_{k=1}^n b_k(t)x(\tilde{h}_k(t)) - \int_{0}^t M(t,s)x(s)\,ds =0.
\end{equation}

\begin{proposition} \label{prop57}
1) Suppose (m1)-(m4) hold, where $a_k,h_k,K$ are
changed by $b_k,\tilde{h}_k, M$, respectively. If $b_k(t) \geq
a_k(t), \tilde{h}_k(t) \leq h_k(t), M(t,s) \geq K(t,s)$ for each
$t,s,k$ and   (\ref{11starcomp}) has a nonoscillatory solution,
then   (\ref{11zaastar}) also has a nonoscillatory solution.\\
2) Suppose (m1)-(m4) hold. If $b_k(t) \leq a_k(t), \tilde{h}_k(t)
\geq h_k(t), M(t,s) \leq K(t,s)$ for each $t,s,k$  and all
solutions of   (\ref{11starcomp}) are oscillatory, then all
solutions of   (\ref{11zaastar}) are  also oscillatory.
\end{proposition}

\begin{proposition} \label{prop58}
Suppose (m1)--(m4) hold and
$$
\int_{0}^{\infty} \Big[ \sum_{k=1}^n a_k(t) - \int_{g(t)}^t K(t,s)\,ds
\Big] dt = \infty.
$$
Then any nonoscillatory solution $x$ of
(\ref{11zaastar}) satisfies ${ \lim_{t \to \infty}
x(t) = 0} $.
\end{proposition}

Note that Corollary \ref{cortheorem71}, 1) implies the following result.

\begin{proposition} \label{prop59}
Suppose (m1)--(m4) and the following inequality hold
\begin{align*}
&\liminf_{t \to \infty} \Big\{ \sum_{k=1}^n \int_{h_k(t)}^t
\big[ a_k(s) - c_k \int_{s-h_k(t)}^{s-h_k(t)+g(t)} K(t,\tau)\,d\tau \big]
ds  + \sum_{k=1}^n c_k \int_{g(t)}^t  d\tau \\
&\times \int_{h_k(\tau)}^{\tau}
 \Big( \exp\big\{ \int_{s+h_k(t)-g(t)}^s \big[ a_k(u)
- \int_{g(u)}^u  K(u,\zeta)\,d\zeta \big] du \big\} - 1 \Big) K(t,s)ds
 \Big\} > \frac{1}{e}.
\end{align*}
Then all solutions of  (\ref{11zaastar}) are oscillatory.
\end{proposition}

\noindent{\bf Remark.} Similarly inequalities 2)-4) in Corollary
\ref{cortheorem71} can be rewritten for  (\ref{11zaastar}).

\begin{proposition}\label{prop510}
Suppose (m1)--(m4) and the inequality
$$ \limsup_{t \to \infty} \Big\{
\sum_{k=1}^n  \int_{h_k(t)}^t \big[ a_k(s) - \frac{c_k}{e}
\int_{g(s)}^s K(s,\tau)\,d\tau \big] ds   \Big\} < \frac{1}{e}
$$
holds. Then  (\ref{11zaastar}) has a nonoscillatory solution.
\end{proposition}


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