\documentclass[reqno]{amsart}
\usepackage{amsfonts}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 03, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University-San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2004/03\hfil Neutral delay differential equations]
{Asymptotic properties, nonoscillation, and stability for scalar first order
linear autonomous neutral delay differential equations}
\author[Christos G. Philos \& Ioannis K. Purnaras\hfil EJDE--2004/03\hfilneg]
{Christos G. Philos \& Ioannis K. Purnaras}
\address{Department of Mathematics\\
University of Ioannina\\
P.O. Box 1186\\
451 10 Ioannina, Greece}
\email[Christos G. Philos]{cphilos@cc.uoi.gr}
\email[Ioannis K. Purnaras]{ipurnara@cc.uoi.gr}
\date{}
\thanks{Submitted October 6, 2003. Published January 2, 2004.}
\subjclass[2000]{34K11, 34K20, 34K25, 34K40}
\keywords{Neutral differential equation, asymptotic behavior, \hfill\break\indent
nonoscillation, stability}
\begin{abstract}
We study scalar first order linear autonomous neutral delay
differential equations with distributed type delays.
This article presents some new results on the asymptotic behavior,
the nonoscillation and the stability. These results are obtained
via a real root (with an appropriate property) of the characteristic
equation. Applications to the special cases such as (non-neutral)
delay differential equations are also presented.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
\textit{Neutral delay differential equations}
are differential equations depending on
past and present values, which involve derivatives with delays as well as
the unknown function itself. Besides its theoretical interest, the study of
such equations has great importance in various applications in natural
sciences and technology. For the basic theory of neutral delay differential
equations, the reader is referred to the books by Diekmann \textit{et al.}
\cite{d1}, Hale \cite{h1}, and Hale and Verduyn Lunel \cite{h2}.
Driver, Sasser and Slater \cite{d5} have obtained some significant results
on the asymptotic behavior, the nonoscillation and the stability for a first
order linear delay differential equation with constant coefficients and one
constant delay. These results have been improved and extended by Philos \cite
{p1} for first order linear delay differential equations in which the
coefficients are periodic functions with a common period and the delays are
constants and multiples of this period. The results in \cite{d5} have also
been improved and extended by Kordonis, Niyianni and Philos \cite{k1} for
first order linear neutral delay differential equations with constant
coefficients and constant delays. Philos and Purnaras \cite{p2} have studied
the more general case of first order linear neutral delay differential
equations with periodic coefficients and constant delays, where the
coefficients have a common period and the delays are multiples of this
period. The results in \cite{p2} contain especially those in \cite{p1} (in
an improved version) as well as the ones given in \cite{k1}. Moreover, the
results obtained by Graef and Qian \cite{g1} are also motivated by those in
\cite{d5} and are closely related. For some related results we refer to the
papers by Arino and Pituk \cite{a1}, Driver \cite{d3}, and Gy\"{o}ri \cite
{g2}.
In \cite{d2}, Driver studied first order linear autonomous delay
differential equations with infinitely many distributed delays and obtained
some important results on the asymptotic behavior, the nonoscillation and
the stability. For previous related results we refer to the references cited
in \cite{d2}. The results given in this paper are essentially motivated by
the corresponding ones in \cite{d2} and the techniques applied in the
present paper are originated in some of the methods used in \cite{d2}.
This paper deals with the asymptotic behavior, the nonoscillation and the
stability for scalar first order linear autonomous neutral delay
differential equations with distributed type delays. A basic asymptotic
criterion is established. Also, a nonoscillation result is given. Moreover,
a useful estimate of the solutions is obtained and a stability criterion is
derived. Our results are obtained by the use of a real root (with an
appropriate property) of the corresponding characteristic equation. The
results given here can be applied to the corresponding \textit{non-neutral}
equations. An application of our results to the special case of
(non-neutral) delay differential equations leads to an improved version of
some of the results given by Driver in \cite{d2}.
Recently, a very interesting article has been published by Frasson and
Verduyn Lunel \cite{f1} concerning the large time behaviour of linear
functional differential equations. It is shown there that the spectral
theory for linear autonomous as well as periodic functional differential
equations yields explicit formulas for the large time behaviour of
solutions. The results in \cite{f1} are based on resolvent computations and
Dunford calculus. Some known results (see \cite{d5,k1}) can be obtained as
applications of the general results given in \cite{f1}. The work in \cite{f1}
may be viewed as a generalization of previous works for first order scalar
linear autonomous and periodic functional differential equations (see \cite
{d2,d5,k1,p1,p2}). It must be noted that, in \cite{d2,d5,k1,p1,p2} as well
as in the present paper, the method used in obtaining the results is very
simple and is essentially based on elementary calculus.
Consider the neutral delay differential equation
\begin{equation}
\Big[ x(t)+\int_{-\sigma }^0x(t+s)d\zeta (s)\Big]^{\prime}=\int_{-\tau
}^0x(t+s)d\eta (s), \label{E}
\end{equation}
where \textit{$\sigma $ and $\tau $ are positive constants, $\zeta $ and $\eta $ are
real-valued functions of bounded variation on the intervals $[-\sigma ,0]$
and $[-\tau ,0]$ respectively, and the integrals are Riemann-Stieltjes
integrals. It will be supposed that $\eta $ is not constant on $[-\tau ,0]$.}
Set
\begin{equation*}
r=\max \{\sigma ,\tau \}.
\end{equation*}
Clearly, $r$ is a positive constant.
As usual, a continuous real-valued
function $x$ defined on the interval $[-r,\infty )$ is said to be a \textit{solution}
of the neutral delay differential equation \eqref{E} if the function $%
x(t)+\int_{-\sigma}^0x(t+s)d\zeta (s)$ is continuously differentiable for $%
t\geq 0$ and $x$ satisfies \eqref{E} for all $t\geq 0$.
In the sequel, by $C([-r,0],\mathbb{R})$ we will denote the set of all
continuous real-valued functions on the interval $[-r,0]$. This set is a
Banach space endowed with the sup-norm $\left\| \phi \right\| =\sup_{t\in
[-r,0]}| \phi(t)|$.
It is well-known (see, for example, Diekmann \textit{et al}. \cite{d1}, Hale
\cite{h1}, or Hale and Verduyn Lunel \cite{h2}) that, for any given \textit{%
initial function} $\phi $ in $C([-r,0],\mathbb{R})$, there exists a unique
solution $x$ of the differential equation \eqref{E} which satisfies the
\textit{initial condition}
\begin{equation}
x(t)=\phi (t)\quad \text{for }t\in [-r,0]\text{;} \label{C}
\end{equation}
this function $x$ will be called the solution of the \textit{initial problem}
\eqref{E}-\eqref{C} or, more briefly, the solution of \eqref{E}-\eqref{C}.
The \textit{characteristic equation} of \eqref{E} is
\begin{equation}
\lambda \Big[ 1+\int_{-\sigma }^0e^{\lambda s}d\zeta (s)\Big] =\int_{-\tau
}^0e^{\lambda s}d\eta (s). \label{*}
\end{equation}
Throughout this paper, by $V(\zeta )$ we will denote the \textit{total
variation function} of $\zeta $, which is defined on the interval $[-\sigma
,0]$ as follows: $V(\zeta )(-\sigma )=0$,\ and $V(\zeta )(s)$\ is the total
variation of $\zeta $ on $[-\sigma ,s]$\ for each $s$ in $(-\sigma ,0]$.
Also, $V(\eta )$ will stand for the total variation function of $\eta $
defined on the interval $[-\tau ,0]$ by an analogous way: $V(\eta )(-\tau
)=0 $,\ and $V(\eta )(s)$ is equal to the total variation of $\eta $ on $%
[-\tau ,s]$\ for each $s\in (-\tau ,0]$. Note that the functions $V(\zeta )$
and $V(\eta )$\ are nonnegative and increasing on the intervals $[-\sigma
,0] $\ and $[-\tau ,0]$\ respectively. Moreover, it must be noted that $%
V(\zeta ) $ is identically zero on $[-\sigma ,0]$\ if $\zeta $\ is constant
on this interval, and that $V(\eta )$ is not identically zero on the
interval $[-\tau ,0]$\ (and so it is always not constant on $[-\tau ,0]$).
It will be considered that the reader is familiar with the theory of
functions of bounded variation and the theory of Riemann-Stieltjes
integration.
To obtain the main results of this paper, we will make use of a real root $%
\lambda_0$ of the characteristic equation \eqref{*} with the property
\begin{equation}
\int_{-\sigma }^0\left[ 1+\left| \lambda_0\right| (-s)\right]
e^{\lambda_0s}dV(\zeta )(s)+\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s)<1.
\label{Pl0}
\end{equation}
Let us consider the special case of the (\textit{non-neutral}) delay differential
equation
\begin{equation}
x^{\prime}(t)=\int_{-\tau }^0x(t+s)d\eta (s). \label{E0}
\end{equation}
This equation can be obtained (as a special case) from the differential
equation \eqref{E}, by choosing $\sigma $ to be an arbitrary positive
constant with $\sigma \leq \tau $ and considering $\zeta $ to be any
constant real-valued function on $[-\sigma ,0]$.
As it concerns the (non-neutral) delay differential equation \eqref{E0}, we
have the constant $\tau $ in place of $r$.
By a \textit{solution} of \eqref{E0}, we mean a continuous real-valued function $x$
defined on the interval $[-\tau ,\infty )$, which is continuously
differentiable on $[0,\infty )$\ and satisfies \eqref{E0} for $t\geq 0$. In
the special case of \eqref{E0}, the Initial Condition \eqref{C} becomes
\begin{equation}
x(t)=\phi (t)\text{ \ for }t\in [-\tau ,0]. \label{C0}
\end{equation}
The \textit{characteristic equation} of \eqref{E0} is
\begin{equation}
\lambda =\int_{-\tau }^0e^{\lambda s}d\eta (s). \label{*0}
\end{equation}
With respect to the (non-neutral) delay differential equation \eqref{E0}, we
need a real root $\lambda_0$ of the characteristic equation \eqref{*0} with
the property
\begin{equation}
\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s)<1. \label{pl0}
\end{equation}
The notions of the \textit{stability}, \textit{instability}, \textit{uniform stability}, \textit{asymptotic
stability} and \textit{uniform asymptotic stability} of the \textit{trivial solution} of a
neutral (or non-neutral) delay differential equation will be considered in
the usual sense (see, for example, Diekmann \textit{et al.} \cite{d1}, Hale
\cite{h1}, or Hale and Verduyn Lunel \cite{h2}; for the non-neutral case,
see also Driver \cite{d4}). Note that, since the differential equation %
\eqref{E} (and, in particular, the differential equation \eqref{E0}) is
autonomous, the trivial solution of \eqref{E} (and, in particular, of %
\eqref{E0}) is uniformly stable or uniformly asymptotically stable if and
only if it is stable (at 0) or asymptotically stable (at 0) respectively.
Our main results are two theorems and two corollaries of the first of these
theorems. The main results of the paper are stated in Section 2. The proof
of the first theorem is given in Section 3, while the proof of the second
theorem is presented in Section 4. Section 5 is devoted to the application
of the main results to the special case of the (non-neutral) delay
differential equation \eqref{E0}. Sufficient conditions for the
characteristic equation \eqref{*} (and, in particular, for \eqref{*0}) to
have a real root $\lambda_0$ with the property \eqref{Pl0} (and, in
particular, with the property \eqref{pl0}) are obtained in Section 6.
\section{Statement of the main results}
Theorem \ref{thm1} below is a basic asymptotic criterion for the solutions
of the neutral delay differential equation \eqref{E}.
\begin{theorem} \label{thm1}
Let $\lambda_0$ be a real root of the characteristic equation
\eqref{*} with the property \eqref{Pl0} and set
\[
\gamma (\lambda_0)=\int_{-\sigma }^0\left[ 1-\lambda_0(-s)\right]
e^{\lambda_0s}d\zeta (s)+\int_{-\tau }^0(-s)e^{\lambda_0s}d\eta (s).
\]
Then, for every $\phi \in C([-r,0],\mathbb{R})$, the
solution $x$ of \eqref{E}-\eqref{C} satisfies
\[
\lim_{t\rightarrow \infty } \big[ e^{-\lambda_0t}x(t)\big]
=\frac{L(\lambda_0;\phi )}{1+\gamma (\lambda_0)},
\]
where
\begin{align*}
L(\lambda_0;\phi )
&=\phi (0)+\int_{-\sigma }^0\Big[ \phi (s)-\lambda
_0e^{\lambda_0s}\int_{s}^0e^{-\lambda_0u}\phi (u)du\Big] d\zeta
(s)\\
&\quad +\int_{-\tau }^0e^{\lambda_0s}
\Big[\int_{s}^0e^{-\lambda_0u}\phi (u)du\Big] d\eta (s).
\end{align*}
\end{theorem}
\textit{Note}: Property \eqref{Pl0} guarantees that $1+\gamma (\lambda_0)>0$.\smallskip
We immediately see that $\lambda_0=0$\ is a root of the characteristic
equation \eqref{*} with the property \eqref{Pl0} if and only if
\begin{equation*}
\int_{-\tau }^0d\eta (s)=0\quad \text{and}\quad \int_{-\sigma}^0dV(\zeta
)(s)+\int_{-\tau }^0(-s)dV(\eta )(s)<1\,,
\end{equation*}
i.e. if and only if the following condition holds:
\begin{equation}
\eta (-\tau )=\eta (0)\quad \text{and}\quad V(\zeta)(0)+\int_{-\tau
}^0(-s)dV(\eta )(s)<1\,. \label{Q}
\end{equation}
Note that $V(\zeta )(0)$ is the total variation of $\zeta $ on the interval $%
[-\sigma ,0]$. Thus, an application of Theorem \ref{thm1} with $\lambda_0=0$
leads to the following corollary.
\begin{corollary} \label{coro1}
Let Condition \eqref{Q} be satisfied.
Then, for $\phi \in C([-r,0],\mathbb{R})$, the solution $x$
of \eqref{E}-\eqref{C} satisfies
\[
\lim_{t\rightarrow \infty } x(t)
=\frac{\phi (0)+\int_{-\sigma
}^0\phi (s)d\zeta (s)+\int_{-\tau }^0\big[ \int_{s}^0\phi (u)du\big]
d\eta (s)}{1+[\zeta (0)-\zeta (-\sigma )]+\int_{-\tau }^0(-s)d\eta (s)}.
\]
\end{corollary}
\textit{Note}: The second assumption of \eqref{Q} ensures that
\begin{equation*}
1+[\zeta (0)-\zeta (-\sigma )]+\int_{-\tau }^0(-s)d\eta (s)>0.
\end{equation*}
Another immediate consequence of Theorem \ref{thm1} is the following result.
As customary, a solution of \eqref{E} is said to be \textit{nonoscillatory} if it is
either eventually positive or eventually negative.
\begin{corollary} \label{coro2}
Let $\lambda_0$ be a real root of the characteristic equation
\eqref{*} with the property \eqref{Pl0}. Then, for any
$\phi \in C([-r,0],\mathbb{R})$, the solution $x$ of
\eqref{E}-\eqref{C} will be nonoscillatory, except possibly if
$\phi$ is such that $L(\lambda_0;\phi )=0$, where
$L(\lambda_0;\phi )$ is defined as in Theorem \ref{thm1}.
\end{corollary}
Consider a real root $\lambda_0$ of \eqref{*} with the property \eqref{Pl0}
and, for any $\phi \in C([-r,0],\mathbb{R})$, let $L(\lambda_0;\phi)$ be
defined as in Theorem \ref{thm1}. Clearly, the operator $L(\lambda_0;\cdot )$
is linear. Moreover, there exists a function $\phi_0\in C([-r,0],\mathbb{R})$
such that $L(\lambda_0;\phi_0)\neq 0$. Indeed, if we set
\begin{equation*}
\phi_0(t)=e^{\lambda_0t}\quad \text{for }t\in [-r,0],
\end{equation*}
then $\phi_0\in C([-r,0],\mathbb{R})$ and we have
\begin{align*}
L(\lambda_0;\phi_0) &\equiv \phi_0(0)+\int_{-\sigma }^0\Big[ \phi
_0(s)-\lambda_0e^{\lambda_0s}\int_{s}^0e^{-\lambda_0u}\phi _0(u)du\Big] %
d\zeta (s) \\
&\quad +\int_{-\tau }^0e^{\lambda_0s}\Big[ \int_{s}^0e^{-\lambda_0u}%
\phi_0(u)du\Big] d\eta (s) \\
&= 1+\int_{-\sigma }^0\left[ e^{\lambda_0s}-\lambda_0e^{\lambda _0s}(-s)%
\right] d\zeta (s)+\int_{-\tau }^0e^{\lambda_0s}(-s)d\eta (s) \\
&= 1+\int_{-\sigma }^0\left[ 1-\lambda_0(-s)\right] e^{\lambda _0s}d\zeta
(s)+\int_{-\tau }^0(-s)e^{\lambda_0s}d\eta (s) \\
&= 1+\gamma (\lambda_0)>0,
\end{align*}
where $\gamma (\lambda_0)$ is defined as in Theorem \ref{thm1}. So, by the
same method with the one that was used by Driver in \cite{d2} (see, also,
Philos \cite{p1}), one can prove the following result, which can be
considered as a complement of Corollary \ref{coro2}. \smallskip
\textit{Let $\lambda_0$ be a real root of the characteristic equation \eqref{*} with
the property \eqref{Pl0}. Moreover, for any $\phi \in C([-r,0],\mathbb{R})$,
let $L(\lambda_0;\phi )$ be defined as in Theorem \ref{thm1}. Then the set
of all functions $\phi \in C([-r,0],\mathbb{R})$ which satisfy $%
L(\lambda_0;\phi )=0$ is a nowhere dense subset of the Banach space $%
C([-r,0],\mathbb{R})$ (with the sup-norm)}. \smallskip
The following theorem establishes an estimate for the solutions of the
neutral delay differential equation \eqref{E} and, also, a stability
criterion for the trivial solution of \eqref{E}.
\begin{theorem} \label{thm2}
Let $\lambda_0$ be a real root of the characteristic equation
\eqref{*} with the property \eqref{Pl0}.
Consider $\gamma (\lambda_0)$ as in Theorem \ref{thm1} and set
\[
\mu (\lambda_0)=\int_{-\sigma }^0\left[ 1+\left| \lambda_0\right|
(-s)\right] e^{\lambda_0s}dV(\zeta )(s)+\int_{-\tau }^0(-s)e^{\lambda
_0s}dV(\eta )(s).
\]
Then, for any $\phi \in C([-r,0],\mathbb{R})$, the
solution $x$ of \eqref{E}-\eqref{C} satisfies
\[
\left| x(t)\right| \leq N(\lambda_0)\left\| \phi \right\| e^{\lambda
_0t}\quad \text{for all }t\geq 0,
\]
where
\[
N(\lambda_0)=\frac{1+\mu (\lambda_0)}{1+\gamma (\lambda_0)}+\Big[
1+\frac{1+\mu (\lambda_0)}{1+\gamma (\lambda_0)}\Big] \mu (\lambda
_0)\max \{1,e^{\lambda_0r}\}.
\]
Here the constant $N(\lambda_0)$ is greater than 1.
Moreover, the trivial solution of \eqref{E} is uniformly stable
if $\lambda_0=0$, uniformly asymptotically stable if
$\lambda_0<0$, and unstable if $\lambda_0>0$.
\end{theorem}
Note that the criterion for the uniform stability stated in Theorem \ref
{thm2} can equivalently be formulated as follows:
\textit{The trivial solution of \eqref{E} is uniformly stable if Condition \eqref{Q}
holds.}
\section{Proof of Theorem \ref{thm1}}
First of all, let us define $\mu (\lambda_0)$ as in Theorem \ref{thm2}, i.e.
\begin{equation*}
\mu (\lambda_0)=\int_{-\sigma }^0\left[ 1+\left| \lambda_0\right| (-s)\right]
e^{\lambda_0s}dV(\zeta )(s)+\int_{-\tau }^0(-s)e^{\lambda _0s}dV(\eta )(s).
\end{equation*}
Property \eqref{Pl0} implies
\begin{equation}
0<\mu (\lambda_0)<1. \label{3.1}
\end{equation}
We have
\begin{align*}
\left| \gamma (\lambda_0)\right| &\leq \Big| \int_{-\sigma }^0\left[%
1-\lambda_0(-s)\right] e^{\lambda_0s}d\zeta (s)\Big| +\Big|\int_{-\tau
}^0(-s)e^{\lambda_0s}d\eta (s)\Big| \\
&\leq \int_{-\sigma }^0\left| 1-\lambda_0(-s)\right| e^{\lambda _0s}dV(\zeta
)(s)+\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s) \\
&\leq \int_{-\sigma }^0\left[ 1+\left| \lambda_0\right| (-s)\right]
e^{\lambda_0s}dV(\zeta )(s)+\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s),
\end{align*}
that is $|\gamma (\lambda_0)| \leq \mu (\lambda_0)$. So, in view of %
\eqref{3.1}, it holds $\left| \gamma (\lambda_0)\right| <1$. This, in
particular, implies that $1+\gamma (\lambda_0)>0$.
Consider now an arbitrary initial function $\phi $\ in $C([-r,0],\mathbb{R})$
and let $x$ be the solution of \eqref{E}-\eqref{C}. Define
\begin{equation*}
y(t)=e^{-\lambda_0t}x(t)\quad\text{for }t\geq -r.
\end{equation*}
Then, using the fact that $\lambda_0$ is a (real) root of the characteristic
equation \eqref{*}, we obtain for every $t\geq 0$
\begin{align*}
&\Big[ x(t)+\int_{-\sigma }^0x(t+s)d\zeta (s)\Big] ^{\prime}-\int_{-\tau
}^0x(t+s)d\eta (s) \\
&= e^{\lambda_0t}\Big\{ \Big[ y(t)+\int_{-\sigma }^0e^{\lambda
_0s}y(t+s)d\zeta (s)\Big] ^{\prime}+\lambda_0\Big[ y(t)+\int_{-\sigma
}^0e^{\lambda_0s}y(t+s)d\zeta (s)\Big] \\
&\quad -\int_{-\tau }^0e^{\lambda_0s}y(t+s)d\eta (s)\Big\} \\
&= e^{\lambda_0t}\Big\{ \Big[ y(t)+\int_{-\sigma }^0e^{\lambda
_0s}y(t+s)d\zeta (s)\Big] ^{\prime} \\
&\quad +\Big[ -\lambda_0\int_{-\sigma }^0e^{\lambda _0s}d\zeta
(s)+\int_{-\tau }^0e^{\lambda_0s}d\eta (s)\Big] y(t) \\
&\quad +\lambda_0\int_{-\sigma }^0e^{\lambda _0s}y(t+s)d\zeta
(s)-\int_{-\tau }^0e^{\lambda_0s}y(t+s)d\eta (s)\Big\} \\
&= e^{\lambda_0t}\Big\{ \Big[ y(t)+\int_{-\sigma }^0e^{\lambda
_0s}y(t+s)d\zeta (s)\Big] ^{\prime}-\lambda_0\int_{-\sigma }^0e^{\lambda_0s}%
\big[ y(t)-y(t+s)\big] d\zeta (s) \\
&\quad +\int_{-\tau }^0e^{\lambda_0s}\big[y(t)-y(t+s)\big] d\eta (s)\Big\} .
\end{align*}
Thus, since $x$ satisfies \eqref{E} for all $t\geq 0$, it follows that $y$
satisfies
\begin{equation} \label{3.2}
\begin{aligned} &\Big[ y(t)+\int_{-\sigma }^0e^{\lambda_0s}y(t+s)d\zeta
(s)\Big]'\\ &=\lambda_0\int_{-\sigma }^0e^{\lambda_0s}\big[ y(t)-y(t+s)\big]
d\zeta (s) -\int_{-\tau }^0e^{\lambda_0s}\big[ y(t)-y(t+s)\big] d\eta (s), t\geq 0.
\end{aligned}
\end{equation}
On the other hand, the Initial Condition \eqref{C} becomes
\begin{equation}
y(t)=e^{-\lambda_0t}\phi (t)\quad\text{for }t\in [-r,0]. \label{3.3}
\end{equation}
Furthermore, we can see that \eqref{3.2} is equivalently written as
\begin{align*}
&y(t)+\int_{-\sigma }^0e^{\lambda_0s}y(t+s)d\zeta (s) \\
&=\lambda_0\int_{-\sigma }^0e^{\lambda_0s}\Big[ \int_{t+s}^{t}y(u)du\Big] %
d\zeta (s)-\int_{-\tau }^0e^{\lambda_0s} \Big[ \int_{t+s}^{t}y(u)du\Big] %
d\eta (s)+K \quad \text{for }t\geq 0
\end{align*}
for some real constant $K$. But, by taking into account \eqref{3.3} and the
definition of $L(\lambda_0;\phi )$, we have
\begin{align*}
K &= y(0)+\int_{-\sigma }^0e^{\lambda_0s}y(s)d\zeta (s)-\lambda
_0\int_{-\sigma }^0e^{\lambda_0s}\Big[ \int_{s}^0y(u)du\Big] d\zeta (s) \\
&\quad +\int_{-\tau }^0e^{\lambda_0s}\Big[\int_{s}^0y(u)du\Big] d\eta (s) \\
&= \phi (0)+\int_{-\sigma }^0\phi (s)d\zeta (s)-\lambda_0\int_{-\sigma
}^0e^{\lambda_0s}\Big[ \int_{s}^0e^{-\lambda_0u}\phi (u)du\Big] d\zeta (s) \\
&\quad +\int_{-\tau }^0e^{\lambda_0s}\Big[ \int_{s}^0e^{-\lambda_0u}\phi
(u)du\Big] d\eta (s) \\
&= \phi (0)+\int_{-\sigma }^0\left[ \phi (s)-\lambda_0e^{\lambda
_0s}\int_{s}^0e^{-\lambda_0u}\phi (u)du\right] d\zeta (s) \\
&\quad +\int_{-\tau }^0e^{\lambda_0s}\Big[ \int_{s}^0e^{-\lambda_0u}\phi
(u)du\Big] d\eta (s) \\
&\equiv L(\lambda_0;\phi ).
\end{align*}
So, \eqref{3.2} is equivalent to
\begin{equation} \label{3.4}
\begin{aligned} &y(t)+\int_{-\sigma }^0e^{\lambda_0s}y(t+s)d\zeta (s)\\
&=\lambda_0\int_{-\sigma }^0e^{\lambda_0s} \Big[
\int_{t+s}^{t}y(u)du\Big]d\zeta (s) -\int_{-\tau
}^0e^{\lambda_0s}\Big[\int_{t+s}^{t}y(u)du\Big] d\eta (s) +L(\lambda_0;\phi
) \end{aligned}
\end{equation}
for $t\geq 0$.
Next, we set
\begin{equation*}
z(t)=y(t)-\frac{L(\lambda_0;\phi )}{1+\gamma (\lambda_0)}\quad \text{for }%
t\geq -r.
\end{equation*}
Then, using the definition of $\gamma (\lambda_0)$, it is easy to check that %
\eqref{3.4} takes the following equivalent form
\begin{equation} \label{3.5}
\begin{aligned} &z(t)+\int_{-\sigma }^0e^{\lambda_0s}z(t+s)d\zeta (s)\\
&=\lambda_0\int_{-\sigma }^0e^{\lambda_0s}\Big[ \int_{t+s}^{t}z(u)du\Big]
d\zeta (s)-\int_{-\tau}^0e^{\lambda_0s} \Big[ \int_{t+s}^{t}z(u)du\Big]
d\eta (s)\quad \text{for }t\geq 0. \end{aligned}
\end{equation}
Moreover, \eqref{3.3} is written as
\begin{equation}
z(t)=e^{-\lambda_0t}\phi (t)-\frac{L(\lambda_0;\phi )}{1+\gamma (\lambda_0)}%
\quad\text{for\ }t\in [-r,0]. \label{3.6}
\end{equation}
By the definitions of $y$ and $z$, what we have to prove is that
\begin{equation}
\underset{t\rightarrow \infty }{\lim }z(t)=0. \label{3.7}
\end{equation}
In the rest of the proof we will establish \eqref{3.7}. Put
\begin{equation*}
M(\lambda_0;\phi )=\underset{t\in [-r,0]}{\max }\Big| e^{-\lambda _0t}\phi
(t)-\frac{L(\lambda_0;\phi )}{1+\gamma (\lambda_0)}\Big| .
\end{equation*}
Then, in view of \eqref{3.6}, we have
\begin{equation}
\left| z(t)\right| \leq M(\lambda_0;\phi )\quad\text{for }-r\leq t\leq 0.
\label{3.8}
\end{equation}
We will show that $M(\lambda_0;\phi )$ is a bound of $z$ on the whole
interval $[-r,\infty )$, namely that
\begin{equation}
\left| z(t)\right| \leq M(\lambda_0;\phi )\quad\text{for all }t\geq -r.
\label{3.9}
\end{equation}
To this end, let us consider an arbitrary number $\epsilon >0$. We claim
that
\begin{equation}
\left| z(t)\right| 0$ such that
\begin{equation*}
\left| z(t)\right| 0$. Hence, \eqref{3.9} is satisfied. Now, by virtue of \eqref{3.9}, from %
\eqref{3.5} we derive for $t\geq 0$,
\begin{align*}
\left| z(t)\right| &\leq \Big| \int_{-\sigma }^0e^{\lambda_0s}z(t+s)d\zeta
(s)\Big| +| \lambda_0| \Big|\int_{-\sigma }^0e^{\lambda_0s} \Big[ %
\int_{t+s}^{t}z(u)du\Big] d\zeta (s)\Big| \\
&\quad +\Big| \int_{-\tau }^0e^{\lambda_0s}\Big[ \int_{t+s}^{t}z(u)du\Big] %
d\eta (s)\Big| \\
&\leq \int_{-\sigma }^0e^{\lambda_0s}\left| z(t+s)\right| dV(\zeta
)(s)+\left| \lambda_0\right| \int_{-\sigma }^0e^{\lambda_0s}\left|
\int_{t+s}^{t}z(u)du\right| dV(\zeta )(s) \\
&\quad +\int_{-\tau }^0e^{\lambda_0s}\Big| \int_{t+s}^{t}z(u)du\Big| dV(\eta
)(s) \\
&\leq \int_{-\sigma }^0e^{\lambda_0s}\left| z(t+s)\right| dV(\zeta
)(s)+\left| \lambda_0\right| \int_{-\sigma }^0e^{\lambda_0s}\Big[ %
\int_{t+s}^{t}\left| z(u)\right| du\Big] dV(\zeta )(s) \\
&\quad+\int_{-\tau }^0e^{\lambda_0s}\Big[ \int_{t+s}^{t}\left| z(u)\right| du%
\Big] dV(\eta )(s) \\
&\leq \Big[ \int_{-\sigma }^0e^{\lambda_0s}dV(\zeta )(s)+\left|
\lambda_0\right| \int_{-\sigma }^0e^{\lambda_0s}(-s)dV(\zeta )(s) \\
&\quad + \int_{-\tau }^0e^{\lambda_0s}(-s)dV(\eta )(s)\Big] M(\lambda_0;\phi
) \\
&= \Big\{ \int_{-\sigma }^0\left[ 1+\left| \lambda_0\right| (-s)\right]
e^{\lambda_0s}dV(\zeta )(s)+\int_{-\tau }^0e^{\lambda_0s}(-s)dV(\eta )(s)%
\Big\} M(\lambda_0;\phi )\,.
\end{align*}
Consequently, by the definition of $\mu (\lambda_0)$, we have
\begin{equation}
\left| z(t)\right| \leq \mu (\lambda_0)M(\lambda_0;\phi )\quad \text{for
every }t\geq 0. \label{3.11}
\end{equation}
Using \eqref{3.5} and taking into account the definition of $\mu (\lambda_0)$
as well as \eqref{3.9} and \eqref{3.11}, one can show, by an easy induction,
that $z$ satisfies
\begin{equation}
\left| z(t)\right| \leq \left[ \mu (\lambda_0)\right] ^{\nu }M(\lambda
_0;\phi )\quad\text{for all }t\geq \nu r-r\quad (\nu =0,1,2,\dots).
\label{3.12}
\end{equation}
Because of \eqref{3.1}, we have $\lim_{\nu \rightarrow \infty }\left[\mu
(\lambda_0)\right] ^{\nu }=0$. Thus, from \eqref{3.12} it follows that $%
\lim_{t\rightarrow \infty }z(t)=0$, i.e. \eqref{3.7} holds. The proof of
Theorem \ref{thm1} is complete.
\section{Proof or Theorem \ref{thm2}}
We first notice that, as in the proof of Theorem \ref{thm1}, we have $0<\mu
(\lambda_0)<1$, $\left| \gamma (\lambda_0)\right| \leq \mu (\lambda _0)$ and
$1+\gamma (\lambda_0)>0$. It follows immediately that $N(\lambda_0)>1$.
Consider an arbitrary function $\phi $ in $C([-r,0],\mathbb{R})$ and let $x$
be the solution of \eqref{E}-\eqref{C}. Let $y$ and $z$ be defined as in the
proof of Theorem \ref{thm1}, i.e.
\begin{equation*}
y(t)=e^{-\lambda_0t}x(t)\quad\text{for }t\geq -r, \quad\text{and}\quad
z(t)=y(t)-\frac{L(\lambda_0;\phi )}{1+\gamma (\lambda_0)}\quad \text{for }%
t\geq -r,
\end{equation*}
where $L(\lambda_0;\phi )$ is defined as in Theorem \ref{thm1}. Moreover,
let $M(\lambda_0;\phi )$ be defined as in the proof of Theorem \ref{thm1},
i.e.
\begin{equation*}
M(\lambda_0;\phi )=\underset{t\in [-r,0]}{\max }\Big| e^{-\lambda _0t}\phi
(t)-\frac{L(\lambda_0;\phi )}{1+\gamma (\lambda_0)}\Big| .
\end{equation*}
Then, as in the proof of Theorem \ref{thm1}, we can show that $z$ satisfies %
\eqref{3.11}, namely
\begin{equation*}
\left| z(t)\right| \leq \mu (\lambda_0)M(\lambda_0;\phi )\quad\text{for
every } t\geq 0.
\end{equation*}
By the definition of $z$, from the last inequality it follows that
\begin{equation}
\left| y(t)\right| \leq \frac{\left| L(\lambda_0;\phi )\right| }{1+\gamma
(\lambda_0)}+\mu (\lambda_0)M(\lambda_0;\phi )\quad\text{for }t\geq 0.
\label{4.1}
\end{equation}
On the other hand, from the definition of $M(\lambda_0;\phi )$ we get
\begin{equation*}
M(\lambda_0;\phi )\leq \left\| \phi \right\| \max \{1,e^{\lambda_0r}\}+\frac{%
\left| L(\lambda_0;\phi )\right| }{1+\gamma (\lambda_0)}.
\end{equation*}
So, \eqref{4.1} gives
\begin{equation}
\left| y(t)\right| \leq \frac{1+\mu (\lambda_0)}{1+\gamma (\lambda_0)}
\left| L(\lambda_0;\phi )\right| +\left\| \phi \right\| \mu (\lambda _0)\max
\{1,e^{\lambda_0r}\},\quad\text{}t\geq 0\text{.} \label{4.2}
\end{equation}
Furthermore, by the definition of $L(\lambda_0;\phi )$, we obtain
\begin{align*}
\left| L(\lambda_0;\phi )\right| &\leq | \phi (0)| +\Big|\int_{-\sigma }^0%
\Big[ \phi (s) -\lambda_0e^{\lambda_0s}\int_{s}^0e^{-\lambda_0u}\phi (u)du%
\Big] d \zeta (s)\Big| \\
&\quad +\Big| \int_{-\tau }^0e^{\lambda_0s}\Big[ \int_{s}^0e^{-\lambda_0u}%
\phi (u)du\Big] d\eta (s)\Big| \\
&= | \phi (0)| +\Big| \int_{-\sigma }^0\Big[ e^{-\lambda _0s}\phi
(s)-\lambda_0\int_{s}^0e^{-\lambda_0u}\phi (u)du\Big] e^{\lambda_0s}d\zeta
(s)\Big| \\
&\quad+\Big| \int_{-\tau }^0\Big[ \int_{s}^0e^{-\lambda_0u}\phi (u)du\Big] %
e^{\lambda_0s}d\eta (s)\Big| \\
&\leq | \phi (0)| +\int_{-\sigma }^0\Big| e^{-\lambda _0s}\phi
(s)-\lambda_0\int_{s}^0e^{-\lambda_0u}\phi (u)du\Big| e^{\lambda_0s}dV(\zeta
)(s) \\
&\quad +\int_{-\tau }^0\Big| \int_{s}^0e^{-\lambda _0u}\phi (u)du\Big| %
e^{\lambda_0s}dV(\eta )(s) \\
&\leq | \phi (0)| +\int_{-\sigma }^0\Big[ e^{-\lambda _0s}| \phi (s)|
+\left| \lambda_0\right| \int_{s}^0e^{-\lambda_0u}\left| \phi (u)\right| du%
\Big] e^{\lambda _0s}dV(\zeta )(s) \\
&\quad +\int_{-\tau }^0\Big[ \int_{s}^0e^{-\lambda _0u}\left| \phi
(u)\right| du\Big] e^{\lambda_0s}dV(\eta )(s)\,.
\end{align*}
Consequently
\begin{equation} \label{4.3}
\begin{aligned} \left| L(\lambda_0;\phi )\right| &\leq \| \phi\| \Big[
1+\int_{-\sigma }^0\Big( e^{-\lambda _0s}+\left| \lambda_0\right|
\int_{s}^0e^{-\lambda_0u}du\Big) e^{\lambda_0s}dV(\zeta )(s)\\ &\quad
+\int_{-\tau }^0\Big(\int_{s}^0e^{-\lambda_0u}du\Big) e^{\lambda_0s}dV(\eta
)(s)\Big]. \end{aligned}
\end{equation}
We have previously used the elementary inequality $e^{-\lambda_0t}\leq \max
\{1,e^{\lambda_0r}\}$ for each $t\in [-r,0]$. Therefore,
\begin{gather*}
e^{-\lambda_0s}\leq \max \{1,e^{\lambda_0r}\}\quad\text{for }s\in [-\sigma
,0], \\
\int_{s}^0e^{-\lambda_0u}du\leq (-s)\max \{1,e^{\lambda_0r}\} \quad\text{for
}s\in [-\sigma ,0], \\
\int_{s}^0e^{-\lambda_0u}du\leq (-s)\max \{1,e^{\lambda_0r}\} \quad\text{for
}s\in [-\tau ,0].
\end{gather*}
Thus, \eqref{4.3} leads to
\begin{align*}
\left| L(\lambda_0;\phi )\right| &\leq \left\| \phi \right\| \Big\{1+\Big( %
\int_{-\sigma }^0\left[ 1+\left| \lambda_0\right| (-s)\right]
e^{\lambda_0s}dV(\zeta )(s) \\
&\quad+\int_{-\tau}^0(-s)e^{\lambda_0s}dV(\eta )(s)\Big) \max
\{1,e^{\lambda_0r}\}\Big\} ,
\end{align*}
which, in view of the definition of $\mu (\lambda_0)$, can be written as
\begin{equation*}
\left| L(\lambda_0;\phi )\right| \leq \left\| \phi \right\| \left[ 1+\mu
(\lambda_0)\max \{1,e^{\lambda_0r}\}\right] .
\end{equation*}
Hence, for $t\geq 0$, \eqref{4.2} gives
\begin{align*}
\left| y(t)\right| &\leq \Big\{ \frac{1+\mu (\lambda_0)}{1+\gamma (\lambda_0)%
}\left[ 1+\mu (\lambda_0)\max \{1,e^{\lambda_0r}\}\right] +\mu
(\lambda_0)\max \{1,e^{\lambda_0r}\}\Big\} \left\| \phi \right\| \\
&= \Big\{ \frac{1+\mu (\lambda_0)}{1+\gamma (\lambda_0)} +\Big[ 1+\frac{%
1+\mu (\lambda_0)}{1+\gamma (\lambda_0)}\Big] \mu (\lambda_0)\max
\{1,e^{\lambda_0r}\}\Big\} \| \phi\|
\end{align*}
and so, because of the definition of $N(\lambda_0)$, we have
\begin{equation*}
\left| y(t)\right| \leq N(\lambda_0)\left\| \phi \right\| \quad\text{for
every }t\geq 0.
\end{equation*}
Finally, in view of the definition of $y$, we obtain
\begin{equation}
\left| x(t)\right| \leq N(\lambda_0)\left\| \phi \right\| e^{\lambda
_0t}\quad\text{for all }t\geq 0. \label{4.4}
\end{equation}
This completes the proof of the first part of the theorem. It remains to
show the stability criterion contained in the theorem.
Let us suppose that $\lambda_0\leq 0$. Let $\phi \in C([-r,0],\mathbb{R})$
be an arbitrary initial function and let $x$ be the solution of \eqref{E}-%
\eqref{C}. Then \eqref{4.4} holds and hence
\begin{equation*}
\left| x(t)\right| \leq N(\lambda_0)\left\| \phi \right\| \quad \text{for
every }t\geq 0.
\end{equation*}
Since $N(\lambda_0)>1$, it follows that $\left| x(t)\right| \leq
N(\lambda_0)\left\| \phi \right\|$ for all $t\geq -r$. Using this
inequality, we can immediately verify that the trivial solution of \eqref{E}
is stable (at 0). Moreover, if $\lambda_0<0$, then \eqref{4.4} guarantees
that
\begin{equation*}
\lim_{t\rightarrow \infty } x(t)=0.
\end{equation*}
Thus, for $\lambda_0<0$\ the trivial solution of \eqref{E} is asymptotically
stable (at 0). Because of the autonomous character of \eqref{E}, the trivial
solution of \eqref{E} is uniformly stable if $\lambda_0=0$ \ and it is
uniformly asymptotically stable if $\lambda_0<0$.
Finally, we assume that $\lambda_0>0$ and we will show that the trivial
solution of \eqref{E} is unstable. Suppose, for the sake of contradiction,
that the trivial solution of \eqref{E} is stable (at 0). Then we can choose
a number $\delta >0$ such that, for each $\phi \in C([-r,0],\mathbb{R})$
with $\left\| \phi \right\| <\delta $, the solution $x$ of \eqref{E}-%
\eqref{C} satisfies
\begin{equation}
\left| x(t)\right| <1\quad \text{for all }t\geq -r. \label{4.5}
\end{equation}
Set
\begin{equation*}
\phi_0(t)=e^{\lambda_0t}\quad \text{for }t\in [-r,0].
\end{equation*}
We see that $\phi_0\in C([-r,0],\mathbb{R})$ and, as in Section 2, we can
verify that
\begin{equation}
L(\lambda_0;\phi_0)=1+\gamma (\lambda_0)>0, \label{4.6}
\end{equation}
where $\gamma (\lambda_0)$ and, for any $\phi \in C([-r,0],\mathbb{R})$, $%
L(\lambda_0;\phi )$ are defined as in Theorem \ref{thm1}. Next, we consider
a number $\delta_0$ with $0<\delta_0<\delta $ and we put
\begin{equation*}
\phi =\frac{\delta_0}{\left\| \phi_0\right\| }\phi_0.
\end{equation*}
Clearly, $\phi $ belongs to $C([-r,0],\mathbb{R})$ and $\left\| \phi\right\|
=\delta_0<\delta $. Hence, for this initial function, the solution $x$ of %
\eqref{E}-\eqref{C} satisfies \eqref{4.5}. On the other hand, by applying
Theorem \ref{thm1} and taking into account \eqref{4.6} as well as the
linearity of the operator $L(\lambda_0;\cdot )$, we obtain
\begin{equation*}
\lim_{t\rightarrow \infty }\left[ e^{-\lambda_0t}x(t)\right] =\frac{%
L(\lambda_0;\phi )}{1+\gamma (\lambda_0)} =\frac{\left( \delta_0/\left\|
\phi_0\right\| \right) L(\lambda_0;\phi_0)}{1+\gamma(\lambda_0)} =\frac{%
\delta_0}{\left\| \phi_0\right\| }>0.
\end{equation*}
But, since $\lambda_0>0$, from \eqref{4.5} it follows that
\begin{equation*}
\lim_{t\rightarrow \infty } \left[ e^{-\lambda_0t}x(t)\right] =0.
\end{equation*}
We have thus arrived at a contradiction. The proof of Theorem \ref{thm2} is
now complete.
\section{Application of the main results to the special case of non-neutral
equations}
In this section, we will concentrate on the (non-neutral) delay differential
equation \eqref{E0} and we shall apply our main results to this equation.
For the delay differential equation \eqref{E0}, the following results hold.
\begin{theorem} \label{thmI}
Let $\lambda_0$ be a real root of the characteristic equation \eqref{*0}
with the property \eqref{pl0}. Then, for any
$\phi \in C([-\tau ,0],\mathbb{R})$, the solution $x$ of \eqref{E0}-\eqref{C0}
satisfies
\[
\lim_{t\rightarrow \infty } \left[ e^{-\lambda_0t}x(t)\right]
=\frac{\ell (\lambda_0;\phi )}{1+\int_{-\tau }^0(-s)e^{\lambda_0s}d\eta (s)},
\]
where
\[
\ell (\lambda_0;\phi )=\phi (0)+\int_{-\tau }^0e^{\lambda_0s}
\Big[\int_{s}^0e^{-\lambda_0u}\phi (u)du\Big] d\eta (s).
\]
\end{theorem}
Note that Property \eqref{pl0} guarantees that $1+\int_{-\tau}^0(-s)e^{%
\lambda_0s}d\eta (s)>0$.
\begin{corollary} \label{coroI}
Assume that
\begin{equation} \label{Q0}
\eta (-\tau )=\eta (0)\quad\text{and} \quad
\int_{-\tau}^0(-s)dV(\eta )(s)<1.
\end{equation}
Then, for any $\phi \in C([-\tau ,0],\mathbb{R})$, the
solution $x$ of \eqref{E0}-\eqref{C0} satisfies
\[
\lim_{t\rightarrow \infty } x(t)
=\frac{\phi (0)+\int_{-\tau }^0\big[ \int_{s}^0\phi (u)du\big] d\eta (s)}
{1+\int_{-\tau }^0(-s)d\eta (s)}.
\]
\end{corollary}
Note that the second assumption of \eqref{Q0} ensures that $1+\int_{-\tau
}^0(-s)d\eta (s)>0$.
\begin{corollary} \label{coroII}
Let $\lambda_0$ be a real root of the characteristic equation
\eqref{*0} with the property \eqref{pl0}. Then, for any
$\phi \in C([-\tau ,0],\mathbb{R})$, the solution $x$ of
\eqref{E0}-\eqref{C0} will be nonoscillatory, except possibly if
$\phi $ satisfies $\ell (\lambda_0;\phi )=0$, where
$\ell (\lambda_0;\phi )$ is defined as in Theorem \ref{thmI}.
\end{corollary}
As a complement to Corollary \ref{coroII}, we have: \textit{Let $\lambda_0$ be a
real root of the characteristic equation \eqref{*0} with the property %
\eqref{pl0}. Moreover, for any $\phi \in C([-\tau ,0],\mathbb{R})$, let $%
\ell (\lambda_0;\phi)$ be defined as in Theorem \ref{thmI}. Then the set of
all functions $\phi \in C([-\tau ,0],\mathbb{R})$ which satisfy $\ell
(\lambda_0;\phi )=0$ is a nowhere dense subset of the Banach space $C([-\tau
,0],\mathbb{R})$ (with the sup-norm).}
\begin{theorem} \label{thmII}
Let $\lambda_0$ be a real root of the characteristic equation
\eqref{*0} with the property \eqref{pl0}.
Then, for any $\phi \in C([-\tau ,0],\mathbb{R})$, the
solution $x$ of \eqref{E0}-\eqref{C0} satisfies
\[
\left| x(t)\right| \leq n(\lambda_0)\| \phi \| e^{\lambda
_0t}\text{ \ \textit{for all}\ }t\geq 0,
\]
where
\begin{align*}
n(\lambda_0)
&= \frac{1+\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s)}
{1+\int_{-\tau }^0(-s)e^{\lambda_0s}d\eta (s)}\\
&\quad+\Big[ 1+\frac{1+\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s)}
{1+\int_{-\tau}^0(-s)e^{\lambda_0s}d\eta (s)}\Big]
\Big[ \int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s)\Big]
\max \{1,e^{\lambda_0\tau }\}
\end{align*}
with the constant $n(\lambda_0)$ being greater than 1.
Moreover, the trivial solution of \eqref{E0} is uniformly
stable if $\lambda_0=0$, uniformly asymptotically stable if
$\lambda_0<0$, and unstable if $\lambda_0>0$.
\end{theorem}
We observe that, concerning the uniform stability, the corresponding result
in Theorem \ref{thmII} can be equivalently stated as: \textit{The trivial
solution of \eqref{E0} is uniformly stable if Condition \eqref{Q0} holds.}
\section{Sufficient conditions for the characteristic equation to have a
real root with the property required}
In this section, we give some conditions, under which the characteristic
equation \eqref{*} (and, in particular, the characteristic equation %
\eqref{*0}) has a real root $\lambda_0$ with the property \eqref{Pl0} (and,
in particular, with the property \eqref{pl0}).
\begin{lemma} \label{lm1}
Assume that
\begin{gather}
\int_{-\sigma }^0e^{-s/r}d\zeta (s)+r\int_{-\tau }^0e^{-s/r}d\eta (s)>-1,
\label{H1}\\
-\int_{-\sigma }^0e^{s/r}d\zeta (s)+r\int_{-\tau }^0e^{s/r}d\eta (s)<1,
\label{H2}\\
\int_{-\sigma }^0\left[ 1+(-s)/r\right] e^{-s/r}dV(\zeta )(s)+\int_{-\tau
}^0(-s)e^{-s/r}dV(\eta )(s)\leq 1. \label{H3}
\end{gather}
Then, in the interval $(-1/r,1/r)$, the characteristic
equation \eqref{*} has a unique root $\lambda_0$, and this
root satisfies the property \eqref{Pl0}.
\end{lemma}
\begin{proof} Define
\[
F(\lambda )=\lambda \Big[ 1+\int_{-\sigma }^0e^{\lambda s}d\zeta (s)%
\Big] -\int_{-\tau }^0e^{\lambda s}d\eta (s)\text{\ \ \ for\ }\lambda
\in [-1/r,1/r].
\]
We have
\begin{align*}
F(-1/r) &= -\frac{1}{r}\Big[ 1+\int_{-\sigma }^0e^{-s/r}d\zeta (s)\Big]
-\int_{-\tau }^0e^{-s/r}d\eta (s) \\
&= -\frac{1}{r}\Big[ 1+\int_{-\sigma }^0e^{-s/r}d\zeta (s)
+r\int_{-\tau}^0e^{-s/r}d\eta (s)\Big]
\end{align*}
and so, by \eqref{H1}, we get $F(-1/r)<0$.
Moreover,
\begin{align*}
F(1/r) &= \frac{1}{r}\Big[ 1+\int_{-\sigma }^0e^{s/r}d\zeta (s)\Big]
-\int_{-\tau }^0e^{s/r}d\eta (s) \\
&= -\frac{1}{r}\Big[ -1-\int_{-\sigma }^0e^{s/r}d\zeta (s)+r\int_{-\tau
}^0e^{s/r}d\eta (s)\Big]
\end{align*}
and hence from \eqref{H2} it follows that $F(1/r)>0$.
Furthermore, by taking into account \eqref{H3}, for $\lambda \in (-1/r,1/r)$,
we obtain
\begin{align*}
F'(\lambda )
&= 1+\int_{-\sigma }^0\left[ 1-\lambda (-s)\right]
e^{\lambda s}d\zeta (s)+\int_{-\tau }^0(-s)e^{\lambda s}d\eta (s) \\
&\geq 1-\left| \int_{-\sigma }^0\left[ 1-\lambda (-s)\right] e^{\lambda
s}d\zeta (s)\right| -\Big| \int_{-\tau }^0(-s)e^{\lambda s}d\eta(s)\Big| \\
&\geq 1-\int_{-\sigma }^0\left| 1-\lambda (-s)\right| e^{\lambda
s}dV(\zeta )(s)-\int_{-\tau }^0(-s)e^{\lambda s}dV(\eta )(s) \\
&\geq 1-\int_{-\sigma }^0\left[ 1+\left| \lambda \right| (-s)\right]
e^{\lambda s}dV(\zeta )(s)-\int_{-\tau }^0(-s)e^{\lambda s}dV(\eta )(s) \\
&> 1-\int_{-\sigma }^0\left[ 1+(-s)/r\right] e^{-s/r}dV(\zeta
)(s)-\int_{-\tau }^0(-s)e^{-s/r}dV(\eta )(s) \\
&\geq 0\,.
\end{align*}
Therefore, $F$ is strictly increasing on the interval $(-1/r,1/r)$. So,
in the interval $(-1/r,1/r)$, the equation $F(\lambda )=0$ (which coincides
with \eqref{*}) has a unique root $\lambda_0$. This root satisfies
\eqref{Pl0}. Indeed, by using again \eqref{H3}, we have
\begin{align*}
&\int_{-\sigma }^0\left[ 1+\left| \lambda_0\right| (-s)\right]
e^{\lambda_0s}dV(\zeta )(s)+\int_{-\tau }^0(-s)
e^{\lambda_0s}dV(\eta)(s)\\
&<\int_{-\sigma }^0\left[ 1+(-s)/r\right] e^{-s/r}dV(\zeta )(s)+\int_{-\tau
}^0(-s)e^{-s/r}dV(\eta )(s)\leq 1.
\end{align*}
This completes the proof.
\end{proof}
Now, we will confine our attention to the special case of the (non-neutral)
delay differential equation \eqref{E0}, for which the characteristic equation
is \eqref{*0}. In this case, Conditions \eqref{H1}, \eqref{H2}, \eqref{H3} take the form
\begin{gather}
\tau \int_{-\tau }^0e^{-s/\tau }d\eta (s)>-1, \label{H10}\\
\tau \int_{-\tau }^0e^{s/\tau }d\eta (s)<1, \label{H20}\\
\int_{-\tau }^0(-s)e^{-s/\tau }dV(\eta )(s)\leq 1\,. \label{H30}
\end{gather}
Lemma \ref{lm1} can be applied to the case of the characteristic equation
\eqref{*0} with the assumptions \eqref{H10}--\eqref{H30} instead of
\eqref{H1}--\eqref{H3}. However, we have the following result which is slightly
better.
\begin{lemma} \label{lm2}
Let \eqref{H10} and \eqref{H30} be satisfied. Then, in the interval
$(-1/\tau ,\infty )$, the characteristic equation \eqref{*0} has a unique
root $\lambda_0$; this root has the property \eqref{pl0}
and, provided that \eqref{H20} holds, the root $\lambda_0$ is less than
$1/\tau$.
\end{lemma}
\begin{proof} Set
\[
F_0(\lambda )=\lambda -\int_{-\tau }^0e^{\lambda s}d\eta (s)\text{\ \ \
for\ }\lambda \geq -1/\tau .
\]
From \eqref{H10}, it follows immediately that $F_0(-1/\tau )<0$.
Next, for every $\lambda \geq -1/\tau $, we obtain
\[
F_0(\lambda )\geq \lambda -\Big| \int_{-\tau }^0e^{\lambda s}d\eta(s)\Big|
\geq \lambda -\int_{-\tau }^0e^{\lambda s}dV(\eta )(s)
\geq \lambda -\int_{-\tau }^0e^{-s/\tau }dV(\eta )(s)
\]
and consequently $F_0(\infty )=\infty$.
Moreover, for $\lambda >-1/\tau $, we have
\begin{align*}
F_0'(\lambda )
&= 1+\int_{-\tau }^0(-s)e^{\lambda s}d\eta
(s)\geq 1-\Big| \int_{-\tau }^0(-s)e^{\lambda s}d\eta (s)\Big| \\
&\geq 1-\int_{-\tau }^0(-s)e^{\lambda s}dV(\eta )(s)>1
-\int_{-\tau}^0(-s)e^{-s/\tau }dV(\eta )(s)
\end{align*}
and so, by \eqref{H30}, it follows that $F_0$ is strictly increasing
on $(-1/\tau ,\infty )$. Hence, in the interval $(-1/\tau ,\infty )$, there
exists a unique root $\lambda_0$ of the equation $F_0(\lambda )=0$ (or,
equivalently, of \eqref{*0}). By using again \eqref{H30}, we get
\[
\int_{-\tau }^0(-s)e^{\lambda_0s}dV(\eta )(s)<\int_{-\tau
}^0(-s)e^{-s/\tau }dV(\eta )(s)\leq 1\,.
\]
Consequently the root $\lambda_0$ satisfies \eqref{pl0}.
Now assume that \eqref{H20} is also satisfied. This
assumption implies that $F_0(1/\tau )>0$. Thus, we can immediately conclude
that the root $\lambda_0$ is always less than $1/\tau $. The proof is
now complete.
\end{proof}
We remark that \textit{Conditions \eqref{H10}--\eqref{H30} are satisfied if the
following stronger condition holds:}
\begin{equation}
\tau \int_{-\tau }^0e^{-s/\tau }dV(\eta )(s)<1. \label{H0}
\end{equation}
In fact, we have
\begin{equation*}
\tau \int_{-\tau }^0e^{-s/\tau }d\eta (s)\geq -\tau \Big| \int_{-\tau
}^0e^{-s/\tau }d\eta (s)\Big| \geq -\tau \int_{-\tau }^0e^{-s/\tau }dV(\eta
)(s),
\end{equation*}
\begin{align*}
\tau \int_{-\tau }^0e^{s/\tau }d\eta (s) &\leq \tau \Big| \int_{-\tau
}^0e^{s/\tau }d\eta (s)\Big| \leq \tau \int_{-\tau }^0e^{s/\tau }dV(\eta )(s)
\\
&\leq \tau \int_{-\tau }^0dV(\eta )(s)\leq \tau \int_{-\tau }^0e^{-s/\tau
}dV(\eta )(s)
\end{align*}
and
\begin{equation*}
\int_{-\tau }^0(-s)e^{-s/\tau }dV(\eta )(s)\leq \tau \int_{-\tau
}^0e^{-s/\tau }dV(\eta )(s)
\end{equation*}
and so our assertion is true. Furthermore, since
\begin{equation*}
\tau \int_{-\tau }^0e^{-s/\tau }dV(\eta )(s)\leq \tau e\int_{-\tau
}^0dV(\eta )(s)=\tau eV(\eta )(0),
\end{equation*}
we conclude that \textit{Condition \eqref{H0} holds if}
\begin{equation}
\tau eV(\eta )(0)<1. \label{tH0}
\end{equation}
Note that $V(\eta )(0)$ is the total variation of $\eta $ on the interval $%
[-\tau ,0]$. Condition \eqref{H0} and, in particular, Condition \eqref{tH0}
were used by Driver \cite{d2}.
Note that it is an interesting question to find other conditions on $\sigma $
and $\tau $ and on the integrators $\zeta $ and $\eta $, which are
sufficient for the characteristic equation \eqref{*} to have a real root $%
\lambda_0$ with the property \eqref{Pl0}. This problem remains interesting
still in the special case of the characteristic equation \eqref{*0}.
Before closing this section and the paper, we will use Lemma \ref{lm1} (and,
in particular, Lemma \ref{lm2}) to find some explicit conditions in terms of
$\sigma$, $\tau $ and $\zeta $, $\eta $ (and, in particular, in terms of $%
\tau $ and $\eta $), under which the trivial solution of \eqref{E} (and, in
particular, of \eqref{E0}) is uniformly asymptotically stable or unstable.
Note that analogous conditions for the uniform stability of the trivial
solution of \eqref{E} (and, in particular, of \eqref{E0}) have already been
given in previous sections.
Let us assume that \eqref{H1}--\eqref{H3} hold. Then Lemma \ref{lm1}
guarantees that, in the interval $(-1/r,1/r)$, the characteristic equation %
\eqref{*} has a unique root $\lambda_0$; this root satisfies the property %
\eqref{Pl0}. Let $F$ be defined as in the proof of Lemma \ref{lm1}. For this
function, as in the proof of Lemma \ref{lm1}, we have
\begin{equation*}
F(-1/r)<0\quad \text{and}\quad F(1/r)>0.
\end{equation*}
Clearly, $\lambda_0$ is negative if $F(0)>0$, and $\lambda_0$ is positive if
$F(0)<0$. On the other hand,
\begin{equation*}
F(0)=-\int_{-\tau }^0d\eta (s)=-[\eta (0)-\eta (-\tau )].
\end{equation*}
So, $\lambda_0<0$ if $\eta (0)<\eta (-\tau )$, and $\lambda_0>0$\ if $\eta
(0)>\eta (-\tau )$. Hence, from the stability criterion contained in Theorem
\ref{thm2} we can obtain the following result.
\begin{corollary} \label{coro3}
Assume that \eqref{H1}--\eqref{H3} are satisfied. Then the trivial solution of
\eqref{E} is uniformly asymptotically stable if $\eta (0)<\eta (-\tau )$
and it is unstable if $\eta (0)>\eta (-\tau )$.
\end{corollary}
By an analogous way, we can use Lemma \ref{lm2} and the stability criterion
contained in Theorem \ref{thmII} to derive the following result.
\begin{corollary} \label{coro4}
Assume that \eqref{H10} and \eqref{H30} are satisfied. Then the trivial solution of
\eqref{E0} is uniformly asymptotically stable if $\eta (0)<\eta (-\tau )$
and it is unstable if $\eta (0)>\eta (-\tau )$.
\end{corollary}
\begin{thebibliography}{99}
\bibitem{a1} O. Arino and M. Pituk, More on linear differential systems
with small delays, \textit{J. Differential Equations} \textbf{170} (2001),
381-407.
\bibitem{d1} O. Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H.-O.
Walther, \textit{Delay Equations: Functional-, Complex-, and Nonlinear
Analysis}, Springer-Verlag, New York, 1995.
\bibitem{d2} R. D. Driver, Some harmless delays, \textit{Delay and
Functional Differential Equations and Their Applications,} Academic Press,
New York, 1972, pp. 103-119.
\bibitem{d3} R. D. Driver, Linear differential systems with small delays,
\textit{J. Differential Equations} \textbf{21} (1976), 148-166.
\bibitem{d4} R. D. Driver,\textit{\ Ordinary and Delay Differential
Equations}, Springer-Verlag, New York, 1977.
\bibitem{d5} R. D. Driver, D. W. Sasser and M. L. Slater, The equation $%
x^{\prime}(t)=ax(t)+bx(t-\tau )$ with ``small'' delay, \textit{Amer. Math.
Monthly} \textbf{80 }(1973), 990-995.
\bibitem{f1} M. V. S. Frasson and S. M. Verduyn Lunel, Large time behaviour
of linear functional differential equations, \textit{Integral Equations
Operator Theory} \textbf{47} (2003), 91-121.
\bibitem{g1} J. R. Graef and C. Qian, Asymptotic behavior of forced delay
equations with periodic coefficients, \textit{Commun. Appl. Anal.} \textbf{2}
(1998), 551-564.
\bibitem{g2} I. Gy\"{o}ri, Invariant cones of positive initial functions
for delay differential equations, \textit{Appl. Anal. }\textbf{35} (1990),
21-41.
\bibitem{h1} J. K. Hale, \textit{Theory of Functional Differential Equations%
}, Springer-Verlag, New York, 1977.
\bibitem{h2} J. K. Hale and S. M. Verduyn Lunel, \textit{Introduction to
Functional Differential Equations}, Springer-Verlag, New York, 1993.
\bibitem{k1} I.-G. E. Kordonis, N. T. Niyianni and Ch. G. Philos, On the
behavior of the solutions of scalar first order linear autonomous neutral
delay differential equations, \textit{Arch. Math. (Basel)} \textbf{71}
(1998), 454-464.
\bibitem{p1} Ch. G. Philos, Asymptotic behaviour, nonoscillation and
stability in periodic first-order linear delay differential equations,
\textit{Proc. Roy. Soc. Edinburgh Sect. A} \textbf{128} (1998), 1371-1387.
\bibitem{p2} Ch. G. Philos and I. K. Purnaras, Periodic first order linear
neutral delay differential equations, \textit{Appl. Math. Comput.}
\textbf{117} (2001), 203-222.
\end{thebibliography}
\end{document}