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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 110, pp. 1--25.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/110\hfil Periodic linear neutral delay equations]
{On periodic linear neutral delay \\ differential and difference equations}

\author[Ch. G. Philos, I. K. Purnaras\hfil EJDE-2006/110\hfilneg]
{Christos G. Philos,  Ioannis K. Purnaras}  % in alphabetical order

\address{Christos G. Philos \newline
Department of Mathematics, University of Ioannina, P. O. Box 1186,
451 10 Ioannina, Greece}
\email{cphilos@cc.uoi.gr}

\address{Ioannis K. Purnaras \newline
Department of Mathematics, University of Ioannina, P. O. Box 1186,
451 10 Ioannina, Greece}
\email{ipurnara@cc.uoi.gr}

\date{}
\thanks{Submitted April 13, 2006. Published September 11, 2006.}
\subjclass[2000]{34K25, 34K40, 39A11}
\keywords{Neutral delay differential equation;
neutral delay difference equation; \hfill\break\indent
periodic coefficients;
characteristic equation; behavior of solutions}

\begin{abstract}
 This article  concerns  the behavior of the solutions to periodic
 linear neutral delay differential equations as well as to periodic linear
 neutral delay difference equations. Some new results are obtained via two
 appropriate distinct roots of the corresponding (so called) characteristic
 equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}
\allowdisplaybreaks

\section{Introduction}

Motivated by the old but very interesting asymptotic and stability results
for delay differential equations due to Driver \cite{d4,d5}
 and to Driver, Sasser and Slater \cite{d7}, a number of articles has been published during the last few
years, which are concerned with the asymptotic behavior (and, more general,
the behavior) and the stability for delay differential equations, neutral
delay differential equations and (neutral or non-neutral)
integrodifferential equations with unbounded delay as well as for delay
difference equations (with discrete or continuous variable), neutral delay
difference equations and (neutral or non-neutral) Volterra difference
equations with infinite delay. See \cite{a1,d2,d3,f1,g1,k1,k2,k3,k4,k5},
\cite{p1}--\cite{p13}, \cite{p17};
for some related results, see \cite{b1,d6,g2,g3,n1,p15,p16}.

Recently, the authors \cite{p9} obtained some results concerning the behavior of
the solutions to autonomous linear delay differential equations as well as
to autonomous linear neutral delay differential equations; these results are
essentially motivated by a result due to Driver \cite[Theorem 2]{d4}.
In \cite{p10}, the
authors continued the work in \cite{p9} to a wide class of autonomous linear
neutral delay differential equations (and, especially, delay differential
equations) with infinitely many distributed type delays. The authors' paper
[28] contains the discrete analogues of the results given in \cite{p9} for
autonomous linear delay difference equations as well as for autonomous
linear neutral delay difference equations; a result of the same type for
autonomous linear delay difference equations with continuous variable is
also contained in \cite{p7}. The study in \cite{p7,p9,p10} was continued by the
authors in \cite{p12} to linear neutral integrodifferential equations with
unbounded delay, and, especially, to linear (non-neutral)
integrodifferential equations with unbounded delay; the discrete analogues
of the results in \cite{p12} for linear neutral (and, especially, non-neutral)
Volterra difference equations with infinite delay have been presented by the
authors in [34]. It must be noted that the results in
\cite{p7,p9,p10,p12,p13}
are obtained via two distinct roots of the corresponding characteristic
equation.

A further continuation of the study in \cite{p7,p9,p10,p12,p13} was, very
recently, presented by the authors in \cite{p14}; the paper \cite{p14}
contains some results on the behavior of the solutions to
\textit{periodic} linear delay
differential equations as well as to \textit{periodic} linear delay
difference equations, which are derived by the use of two (appropriate)
distinct roots of the corresponding (so called) characteristic equation.

It is the subject of this paper to present a study analogous to the one in
\cite{p14} for the behavior of the solutions to \textit{periodic} linear
\textit{neutral} delay differential equations as well as to \textit{periodic} linear
\textit{neutral} delay difference equations. Section 2 is devoted to the
case of differential equations, and Section 3 is concerned with the case of
difference equations. It is remarkable that an application of the main
results of the present work to the special case of (non-neutral) periodic
linear delay differential equations or to the special case of (non-neutral)
periodic linear delay difference equations leads to the main results of the
previous authors' paper \cite{p14}, \textit{under some additional hypotheses};
but, these (additional) hypotheses are not needed for the main results in
\cite{p14} to hold. So, although the differential and difference equations treated
in \cite{p14} can be considered as special cases of the ones studied in this
paper, the main results in \cite{p14} cannot be obtained as corollaries of those
given here.

Finally, we note that some considerable difficulty arises in the attempt to
extend the results of this paper to the more general case of periodic linear
neutral delay differential equations with several delays (such as those
studied by the authors in \cite{p2}) as well as of periodic linear neutral delay
difference equations with several delays (such as the ones treated by the
authors in \cite{p8}).

\section{On the behavior of the solutions to periodic linear neutral delay
differential equations}

This section is concerned with the behavior of the solutions of the linear
neutral delay differential equation
\begin{equation}
[ x(t)+cx(t-\sigma )] '=a(t)x(t)+b(t)x(t-\tau ),
\label{e2.1}
\end{equation}
where  $c$ \textit{is a real number, $a$ and $b$
are continuous real-valued functions on the interval
$[0,\infty )$, and $\sigma $ and
$\tau $ are positive real numbers.
The function $b$ is assumed to be not identically zero on
$[0,\infty )$.} Moreover, \textit{it will be supposed that the
coefficients $a$ and $b$ are periodic functions with a
common period  $T>0$ and that there exist positive integers $\ell $
 and $m$ such that}
\begin{equation*}
\sigma =\ell T\quad \textit{and}\quad  \tau =mT.
\end{equation*}

Consider the positive real number
\begin{equation*}
r=\max \{\sigma ,\tau \}.
\end{equation*}

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{e2.1} if the function $x(t)+cx(t-\sigma )$ is
continuously differentiable for $t\geq 0$ and $x$ satisfies \eqref{e2.1} for all
$t\geq 0$.

Together with the neutral delay differential equation \eqref{e2.1}, it is customary
to specify an \textit{initial condition} of the form
\begin{equation}
x(t)=\phi (t)\quad \text{for }-r\leq t\leq 0,  \label{e2.2}
\end{equation}
where \textit{the initial function $\phi $ is a given continuous
real-valued function on the interval}  $[-r,0]$.

Equations \eqref{e2.1} and \eqref{e2.2} constitute an
\textit{initial value problem} (IVP, for short). It is well-known
(see, for example, Diekmann \textit{et al.} \cite{d1},
Hale \cite{h1}, or Hale and Verduyn Lunel \cite{h2}) that there
exists a unique solution $x$ of the neutral delay differential
equation \eqref{e2.1} which
satisfies the initial condition \eqref{e2.2}; this unique solution $x$ will be
called the \textit{solution} of the initial value problem \eqref{e2.1}, \eqref{e2.2} or,
more briefly, the \textit{solution} of the IVP \eqref{e2.1}, \eqref{e2.2}.

Throughout this section, we shall use the notation
\begin{equation*}
A=\frac{1}{T}\int_{0}^{T}a(t)dt\quad \text{and}\quad
B=\frac{1}{T}\int_{0}^{T}b(t)dt.
\end{equation*}
We note that $A$ and $B$ are real constants. Also, it must be noted that
$B\neq 0$ in the case where the coefficient $b$ is assumed to be of one sign
on the interval $[0,\infty )$.

Along with the neutral delay differential equation \eqref{e2.1}, one
associates the equation
\begin{equation}
\lambda (1+ce^{-\lambda \sigma })=A+Be^{-\lambda \tau },  \label{e2.3}
\end{equation}
which will be called the \textit{characteristic equation} of \eqref{e2.1}
(see the authors' paper \cite{p2}).

In the sequel, by $\widetilde{a}$ and $\widetilde{b}$ we shall denote the $T$-periodic extensions of the coefficients $a$ and $b$, respectively, on the
interval $[-r,\infty )$. Moreover, for any real number $\lambda $, by $
f_{\lambda }$ we will denote the continuous real-valued function defined on
the interval $[-r,\infty )$ as follows
\begin{equation*}
f_{\lambda }(t)=\widetilde{a}(t)+\widetilde{b}(t)e^{-\lambda \tau }\quad\text{for }t\geq -r.
\end{equation*}

Theorem \ref{thm2.1} below has been proved by the authors in \cite{p2} for more general
periodic linear neutral delay differential equations with several delays.
This theorem is closely related to the main result (Theorem \ref{thm2.4} below) of
this section and constitutes a fundamental asymptotic result for the
solutions of the neutral delay differential equation \eqref{e2.1}. In order to
state Theorem \ref{thm2.1}, we introduce the notation
\begin{equation*}
\widehat{B}=\frac{1}{T}\int_{0}^{T}| b(t)| dt.
\end{equation*}
Clearly, $\widehat{B}$ is a positive constant. It is obvious that
$|B| \leq \widehat{B}$. Moreover, we note that
$| B| =\widehat{B}$ in the case where the coefficient $b$
is assumed to be of constant sign on the interval $[0,\infty )$.

\begin{theorem} \label{thm2.1}
Let $\lambda _{0}$ be a real root
of the characteristic equation \eqref{e2.3}, and set
\begin{equation}
\rho _{\lambda _{0}}=1+ce^{-\lambda _{0}\sigma }  \label{e2.4}
\end{equation}
and
\begin{equation*}
\widehat{F}_{\lambda _{0}}=\frac{1}{T}\int_{0}^{T}| f_{\lambda
_{0}}(t)| dt.
\end{equation*}
Assume that the root $\lambda _{0}$ has the
property
\begin{equation}
| c| (| \rho _{\lambda _{0}}| +
\widehat{F}_{\lambda _{0}}\sigma )e^{-\lambda _{0}\sigma }+| \rho
_{\lambda _{0}}| \widehat{B}\tau e^{-\lambda _{0}\tau }<|
\rho _{\lambda _{0}}| .  \label{e2.5}
\end{equation}
Define
\begin{equation}
\gamma _{\lambda _{0}}=c(1-\lambda _{0}\sigma )e^{-\lambda _{0}\sigma
}+B\tau e^{-\lambda _{0}\tau }.  \label{e2.6}
\end{equation}
(Note that Property \eqref{e2.5} guarantees that
$\rho _{\lambda_{0}}>0$ and  $1+\gamma _{\lambda _{0}}>0$.)


Then the solution $x$ of the IVP \eqref{e2.1}, \eqref{e2.2}
satisfies
\begin{equation}
\lim_{t\to \infty }\left\{ x(t)\exp \left[ -\frac{1}{\rho
_{\lambda _{0}}}\int_{0}^{t}f_{\lambda _{0}}(u)du\right] \right\}
=\frac{L_{\lambda _{0}}(\phi )}{1+\gamma _{\lambda _{0}}},  \label{e2.7}
\end{equation}
where
\begin{equation} \label{e2.8}
\begin{aligned}
L_{\lambda _{0}}(\phi )
&=\phi (0)+c\Big\{ \phi (-\sigma )\\
&\quad -\frac{1}{\rho_{\lambda _{0}}}e^{-\lambda _{0}\sigma }
\int_{-\sigma }^{0}f_{\lambda_{0}}(s)\phi (s)\exp
\left[ -\frac{1}{\rho _{\lambda _{0}}}
\int_{0}^{s}f_{\lambda _{0}}(u)du\right] ds\Big\}   \\
&\quad +e^{-\lambda _{0}\tau }\int_{-\tau }^{0}\widetilde{b}(s)\phi (s)
\exp \left[ -\frac{1}{\rho _{\lambda _{0}}}\int_{0}^{s}
f_{\lambda _{0}}(u)du\right] ds.
\end{aligned}
\end{equation}
\end{theorem}

In the main result (Theorem \ref{thm2.4} below) of this section, \textit{it is
supposed that $c\leq 0$ and that $b$ is nonpositive on
the interval} $[0,\infty )$. The hypothesis that $b$ is nonpositive on $
[0,\infty )$ together with the assumptions that $b$ is not identically zero
on $[0,\infty )$ and that $b$ is a $T$-periodic function imply that
 \textit{the constant $B$ is always negative.} Furthermore, we notice
that Theorem \ref{thm2.4} is obtained by the use of \textit{two real roots
$\lambda_{0}$ and $\lambda _{1}$, $\lambda _{0}\neq \lambda _{1}$,
 of the characteristic equation \eqref{e2.3}; for the root
$\lambda_{0} $ it is assumed that $\rho _{\lambda _{0}}\neq 0$ and}
\begin{equation}
\rho _{\lambda _{0}}\left[ a(t)+b(t)e^{-\lambda _{0}\tau }\right] \leq 0
\quad\text{\textit{for} }t\geq 0,  \label{e2.9}
\end{equation}
\textit{where $\rho _{\lambda _{0}}$ is defined by} \eqref{e2.4}. Under
these assumptions for the real root $\lambda _{0}$, we obviously have
\begin{equation*}
\frac{1}{\rho _{\lambda _{0}}}\left[ a(t)+b(t)e^{-\lambda _{0}\tau }\right]
\leq 0\quad \text{for }t\geq 0,
\end{equation*}
which gives
\begin{equation*}
\frac{1}{\rho _{\lambda _{0}}}\left\{ \left[ \frac{1}{T}\int_{0}^{T}a(t)dt
\right] +\left[ \frac{1}{T}\int_{0}^{T}b(t)dt\right] e^{-\lambda _{0}\tau
}\right\} \leq 0,
\end{equation*}
i.e.,
\begin{equation*}
\frac{1}{\rho _{\lambda _{0}}}( A+Be^{-\lambda _{0}\tau }) \leq 0.
\end{equation*}
So, by \eqref{e2.4} and the fact that $\lambda _{0}$ is a root of \eqref{e2.3}, it
follows immediately that \textit{the root $\lambda _{0}$ is always
nonpositive.}

After the above observations, we give a lemma (Lemma \ref{lem2.2} below)
concerning the real roots of the characteristic equation \eqref{e2.3}.
This lemma is a special
case of a more general lemma due to the authors \cite{p9}.

\begin{lemma} \label{lem2.2}
Suppose that $c\leq 0$ and $B<0$.

(I) Let $\lambda _{0}$ be a nonpositive real root of the
characteristic equation \eqref{e2.3}, and let $\gamma _{\lambda _{0}}$
be defined by \eqref{e2.6}. Then
\begin{equation*}
1+\gamma _{\lambda _{0}}>0
\end{equation*}
if \eqref{e2.3} has another real root less than $\lambda _{0}$, and
\begin{equation*}
1+\gamma _{\lambda _{0}}<0
\end{equation*}
if \eqref{e2.3} has another nonpositive real root greater than
$\lambda _{0}$.

(II)  If $A=0$, then $\lambda =0$ is not a root
of the characteristic equation \eqref{e2.3}.

(III) Assume that $A=0$ and that $c\geq -1$.
Then the characteristic equation \eqref{e2.3} has no positive real roots.

(IV)  Assume that $A+B\leq 0$ and $c+B\tau \geq -1$.
Then the characteristic equation \eqref{e2.3} has no positive real
roots.

(V) Assume that $A+B\leq 0$, $Ar<1$, and
\begin{equation*}
(1-Ar)ce^{-( A-\frac{1}{r}) \sigma }+Bre^{-( A-\frac{1}{r}) \tau }>-1.
\end{equation*}
Then: (i) $\lambda =A-\frac{1}{r}$ is not a root of the
characteristic equation \eqref{e2.3}.
 (ii) In the interval $( A-\frac{1}{r},0] $, \eqref{e2.3} has a unique root.
 (iii) In the interval $( -\infty ,A-\frac{1}{r}) $, \eqref{e2.3} has a
unique root.
\end{lemma}

As we have previously noted, in Theorem \ref{thm2.4} (the main result of this
section) we use two suitable distinct real roots of the characteristic
equation \eqref{e2.3}. We will give here a lemma (Lemma \ref{lem2.3} below),
 which plays a
crucial role in proving Theorem \ref{thm2.4}, although this lemma is rather
technical.

\begin{lemma} \label{lem2.3}  Let $\lambda _{0}$ and
$\lambda_{1} $, $\lambda _{0}\neq \lambda _{1}$, be two real roots
of the characteristic equation \eqref{e2.3} with
$\rho _{\lambda_{0}}\neq 0 $ and $\rho _{\lambda _{1}}\neq 0$, where
$\rho _{\lambda _{0}}$ is defined by \eqref{e2.4} and
$\rho_{\lambda _{1}}$ is defined in an analogous way, i.e.,
\begin{equation}
\rho _{\lambda _{1}}=1+ce^{-\lambda _{1}\sigma }.  \label{e2.10}
\end{equation}
Then, for each $t\geq 0$, we have
\begin{equation} \label{e2.11}
\begin{aligned}
1&=-ce^{-\lambda _{1}\sigma }+ce^{-\lambda _{0}\sigma }\frac{1}{\rho
_{\lambda _{0}}}\int_{t-\sigma }^{t}f_{\lambda _{0}}(s)\exp
\left\{\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds
\\
&\quad -e^{-\lambda _{0}\tau }\int_{t-\tau }^{t}\widetilde{b}(s)\exp
 \left\{\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof} First of all, we will establish some equalities needed
below. It is obvious that
\begin{equation}
f_{\lambda _{0}}(t)-f_{\lambda _{1}}(t)=\widetilde{b}(t)\left( e^{-\lambda
_{0}\tau }-e^{-\lambda _{1}\tau }\right) \quad \text{for }t\geq -r.
\label{e2.12}
\end{equation}
Also, in view of \eqref{e2.4} and \eqref{e2.10}, we have
\begin{equation}
\rho _{\lambda _{0}}-\rho _{\lambda _{1}}=c\left( e^{-\lambda _{0}\sigma
}-e^{-\lambda _{1}\sigma }\right) .  \label{e2.13}
\end{equation}
Furthermore, the $T$-periodicity of the coefficients $a$ and $b$ implies
that the functions $f_{\lambda _{0}}$ and $f_{\lambda _{1}}$ are also
$T$-periodic. So, by taking into account the fact that $\tau =mT$, we obtain
for $t\geq 0$,
\begin{align*}
\int_{t-\tau }^{t}f_{\lambda _{0}}(u)du
&=\int_{0}^{\tau }f_{\lambda_{0}}(u)du\\
&=\left[ \frac{1}{\tau }\int_{0}^{\tau }f_{\lambda _{0}}(u)du\right]\tau \\
&=\left[ \frac{1}{T}\int_{0}^{T}f_{\lambda _{0}}(u)du\right] \tau \\
&=\left\{ \left[ \frac{1}{T}\int_{0}^{T}a(u)du\right]
  +\left[ \frac{1}{T}\int_{0}^{T}b(u)du\right] e^{-\lambda _{0}\tau }\right\} \tau \\
&=( A+Be^{-\lambda _{0}\tau }) \tau .
\end{align*}
Thus, because of \eqref{e2.4} and the fact that $\lambda _{0}$ is a
root of \eqref{e2.3}, it follows that
\begin{equation}
\frac{1}{\rho _{\lambda _{0}}}\int_{t-\tau }^{t}f_{\lambda
_{0}}(u)du=\lambda _{0}\tau \quad\text{for every }t\geq 0.  \label{e2.14}
\end{equation}
In a similar way, by using \eqref{e2.10} and the fact that $\lambda _{1}$
 is also a root of \eqref{e2.3}, we can see that
\begin{equation}
\frac{1}{\rho _{\lambda _{1}}}\int_{t-\tau }^{t}f_{\lambda
_{1}}(u)du=\lambda _{1}\tau \quad\text{for every }t\geq 0.  \label{e2.15}
\end{equation}
Moreover, by taking again into account the fact that the function $
f_{\lambda _{0}}$ is $T$-periodic, for any positive integer $\nu $, we get
\begin{align*}
\int_{0}^{\nu T}f_{\lambda _{0}}(u)du
&=\left[ \frac{1}{\nu T}\int_{0}^{\nu
T}f_{\lambda _{0}}(u)du\right] (\nu T)\\
&=\left[ \frac{1}{T}\int_{0}^{T}f_{
\lambda _{0}}(u)du\right] (\nu T) \\
&=( A+Be^{-\lambda _{0}\tau }) (\nu T).
\end{align*}
Hence, as $\rho _{\lambda _{0}}$ is given by \eqref{e2.4} and
$\lambda _{0}$ is a
root of \eqref{e2.3}, we have
\begin{equation}
\frac{1}{\rho _{\lambda _{0}}}\int_{0}^{\nu T}f_{\lambda _{0}}(u)du=\nu
\left( \lambda _{0}T\right) \quad (\nu =1,2,\dots ).  \label{e2.16}
\end{equation}
In a similar manner, by taking into account the fact that $f_{\lambda _{1}}$
 is a $T$-periodic function and using \eqref{e2.10} and the fact that
 $\lambda_{1} $ is also a root of \eqref{e2.3}, one can verify that
\begin{equation}
\frac{1}{\rho _{\lambda _{1}}}\int_{0}^{\nu T}f_{\lambda _{1}}(u)du=\nu
\left( \lambda _{1}T\right) \quad (\nu =1,2,\dots ).  \label{e2.17}
\end{equation}

Now, let us fix a point $t\geq 0$. We will show that, for this fixed point $
t $, equality \eqref{e2.11} holds true.

By using \eqref{e2.12} and \eqref{e2.13}, we obtain
\begin{align*}
&\int_{t-\tau }^{t}\widetilde{b}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\}  ds \\
&= \frac{1}{e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }}\int_{t-\tau }^{t}
\widetilde{b}(s)\left( e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }\right)
\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda
_{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds
\\
&= \frac{1}{e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }}\int_{t-\tau }^{t}
[ f_{\lambda _{0}}(s)-f_{\lambda _{1}}(s)] \exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\frac{1}{e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }}\int_{t-\tau
}^{t}\left\{ \left( \rho _{\lambda _{0}}-\rho _{\lambda _{1}}\right) \frac{
f_{\lambda _{0}}(s)}{\rho _{\lambda _{0}}}+\rho _{\lambda _{1}}\left[ \frac{
f_{\lambda _{0}}(s)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(s)}{\rho
_{\lambda _{1}}}\right] \right\} \times 
\\
&\quad \times \exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} ds 
\\
&= c\frac{e^{-\lambda _{0}\sigma }-e^{-\lambda _{1}\sigma }}{e^{-\lambda
_{0}\tau }-e^{-\lambda _{1}\tau }}\cdot \frac{1}{\rho _{\lambda _{0}}}
\int_{t-\tau }^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds 
\\
&\quad +\frac{1}{e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }}\rho _{\lambda
_{1}}\int_{t-\tau }^{t}\left[ \frac{f_{\lambda _{0}}(s)}{\rho _{\lambda _{0}}
}-\frac{f_{\lambda _{1}}(s)}{\rho _{\lambda _{1}}}\right] \exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds.
\end{align*}
But, by \eqref{e2.14} and \eqref{e2.15}, we have
\begin{align*}
&\int_{t-\tau }^{t}\left[ \frac{f_{\lambda _{0}}(s)}{\rho _{\lambda _{0}}}-
\frac{f_{\lambda _{1}}(s)}{\rho _{\lambda _{1}}}\right] \exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds \\
&=-\left( \exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} \right) _{s=t-\tau }^{s=t}
\\
&=-1+\exp \left\{ \int_{t-\tau }^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} \\
&=-1+\exp \left[ \frac{1}{\rho _{\lambda _{0}}}\int_{t-\tau }^{t}f_{\lambda
_{0}}(u)du-\frac{1}{\rho _{\lambda _{1}}}\int_{t-\tau }^{t}f_{\lambda
_{1}}(u)du\right] \\
&=-1+e^{\lambda _{0}\tau -\lambda _{1}\tau }=-1+e^{\left( \lambda
_{0}-\lambda _{1}\right) \tau }.
\end{align*}
So, it holds
\begin{equation} \label{e2.18}
\begin{aligned}
&\int_{t-\tau }^{t}\widetilde{b}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds   \\
&=c\frac{e^{-\lambda _{0}\sigma }-e^{-\lambda _{1}\sigma }}{e^{-\lambda
_{0}\tau }-e^{-\lambda _{1}\tau }}\cdot \frac{1}{\rho _{\lambda _{0}}}
\int_{t-\tau }^{t}f_{\lambda _{0}}(s)\\
&\quad\times \exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds
-\frac{1}{e^{-\lambda _{0}\tau }}\rho _{\lambda _{1}}.
\end{aligned}
\end{equation}

Furthermore, in view of \eqref{e2.10}, equality \eqref{e2.11} becomes
\begin{align*}
\rho _{\lambda _{1}}
&=ce^{-\lambda _{0}\sigma }\frac{1}{\rho _{\lambda _{0}}}
\int_{t-\sigma }^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds \\
&\quad -e^{-\lambda _{0}\tau }\int_{t-\tau }^{t}\widetilde{b}(s)\exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds.
\end{align*}
Hence, by using \eqref{e2.18}, we see that \eqref{e2.11} can equivalently
 be written as follows
\begin{align*}
0&=ce^{-\lambda _{0}\sigma }\frac{1}{\rho _{\lambda _{0}}}\int_{t-\sigma
}^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda
_{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda
_{1}}}\right] du\right\} ds \\
&\quad -e^{-\lambda _{0}\tau }c\frac{e^{-\lambda _{0}\sigma }-e^{-\lambda
_{1}\sigma }}{e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }}\cdot \frac{1}{
\rho _{\lambda _{0}}}\int_{t-\tau }^{t}f_{\lambda _{0}}(s)\exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds.
\end{align*}
It is clear that this equality holds if
\begin{align*}
&\frac{e^{-\lambda _{0}\sigma }}{e^{-\lambda _{0}\sigma }-e^{-\lambda
_{1}\sigma }}\int_{t-\sigma }^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}
\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda
_{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds \\
&=\frac{e^{-\lambda _{0}\tau }}{e^{-\lambda _{0}\tau }-e^{-\lambda _{1}\tau }}
\int_{t-\tau }^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds
\end{align*}
or, equivalently,
\begin{equation} \label{e2.19}
\begin{aligned}
&\frac{1}{1-e^{(\lambda _{0}-\lambda _{1})\sigma }}\int_{t-\sigma
}^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda
_{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda
_{1}}}\right] du\right\} ds   \\
&=\frac{1}{1-e^{(\lambda _{0}-\lambda _{1})\tau }}\int_{t-\tau
}^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda
_{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda
_{1}}}\right] du\right\} ds.
\end{aligned}
\end{equation}
So, the proof of the lemma can be accomplished by proving that
\eqref{e2.19} holds.
It suffices to show that
\begin{equation}
\begin{aligned}
&\frac{1}{1-e^{(\lambda _{0}-\lambda _{1})\sigma }}\int_{t-\sigma
}^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda
_{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda
_{1}}}\right] du\right\} ds  \label{e2.20} \\
&=\frac{1}{1-e^{(\lambda _{0}-\lambda _{1})T}}\int_{t-T}^{t}f_{\lambda
_{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} ds
\end{aligned}
\end{equation}
and
\begin{equation}  \label{e2.21}
\begin{aligned}
&\frac{1}{1-e^{(\lambda _{0}-\lambda _{1})\tau }}\int_{t-\tau }^{t}f_{\lambda
_{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} ds  \\
&=\frac{1}{1-e^{(\lambda _{0}-\lambda _{1})T}}\int_{t-T}^{t}f_{\lambda
_{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} ds.
\end{aligned}
\end{equation}

Next, we will establish \eqref{e2.20}. By taking into account the fact that the
functions $f_{\lambda _{0}}$ and $f_{\lambda _{1}}$ are $T$-periodic and
that $\sigma =\ell T$ and using \eqref{e2.16} and \eqref{e2.17}, we obtain
\begin{align*}
&\int_{t-\sigma }^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[
\frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{
\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\sum_{i=1}^{\ell }\int_{t-iT}^{t-(i-1)T}f_{\lambda _{0}}(s)\exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\sum_{i=1}^{\ell }\int_{t-T}^{t}f_{\lambda _{0}}(s-(i-1)T)\exp \left\{
\int_{s-(i-1)T}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-
\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\sum_{i=1}^{\ell }\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \left\{
\int_{s-(i-1)T}^{s}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-
\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right. 
\\
&\quad \left. +\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&= \sum_{i=1}^{\ell }\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \left\{ \left[
\frac{1}{\rho _{\lambda _{0}}}\int_{s-(i-1)T}^{s}f_{\lambda _{0}}(u)du-\frac{
1}{\rho _{\lambda _{1}}}\int_{s-(i-1)T}^{s}f_{\lambda _{1}}(u)du\right]\right. 
 \\
&\quad \left. +\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\sum_{i=1}^{\ell }\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \left\{ \left[
\frac{1}{\rho _{\lambda _{0}}}\int_{0}^{(i-1)T}f_{\lambda _{0}}(u)du-\frac{1
}{\rho _{\lambda _{1}}}\int_{0}^{(i-1)T}f_{\lambda _{1}}(u)du\right]\right.
\\
&\quad \left. +\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\sum_{i=1}^{\ell }\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \Big\{ \left[
(i-1)(\lambda _{0}T)-(i-1)(\lambda _{1}T)\right]
\\
&\quad \left. +\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\left[ \sum_{i=1}^{\ell }e^{(i-1)(\lambda _{0}-\lambda _{1})T}\right]
\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds 
\\
&=\frac{1-e^{\ell (\lambda _{0}-\lambda _{1})T}}{1-e^{(\lambda _{0}-\lambda
_{1})T}}\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \left\{ \int_{s}^{t}\left[
\frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{
\rho _{\lambda _{1}}}\right] du\right\} ds 
\\
&=\frac{1-e^{(\lambda _{0}-\lambda _{1})\sigma }}{1-e^{(\lambda
_{0}-\lambda _{1})T}}\int_{t-T}^{t}f_{\lambda _{0}}(s)\exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds.
\end{align*}
This proves \eqref{e2.20}.

Finally, by taking into account the fact that $f_{\lambda _{0}}$ and $
f_{\lambda _{1}}$ are $T$-periodic functions and that $\tau =mT$ and using
again \eqref{e2.16} and \eqref{e2.17}, we can follow the same procedure to show that
\eqref{e2.21} is also satisfied.

The proof of the lemma is now complete.
\end{proof}

Now, we are in a position to present the main result of this section, i.e.,
the following theorem.

\begin{theorem} \label{thm2.4}
Suppose that $c\leq 0$ and that $b $ is nonpositive on the interval
$[0,\infty )$. Let $\lambda _{0}$ be a real root of the characteristic equation
 \eqref{e2.3} with $\rho _{\lambda _{0}}\neq 0$ and satisfying
 \eqref{e2.9}, where $\rho _{\lambda _{0}}$ is defined by \eqref{e2.4},
 and assume that
\begin{equation*}
1+\gamma _{\lambda _{0}}\neq 0,
\end{equation*}
where $\gamma _{\lambda _{0}}$ is defined by \eqref{e2.6}.
Let also $\lambda _{1}$, $\lambda _{1}\neq \lambda _{0}$, be
another real root of \eqref{e2.3} with $\rho _{\lambda _{1}}\neq 0$,
where $\rho _{\lambda _{1}}$ is defined by \eqref{e2.10}.

Then the solution $x$ of the IVP \eqref{e2.1}, \eqref{e2.2}
satisfies
\begin{align*}
U_{1}(\lambda _{0},\lambda _{1};\phi )
&\leq \left\{ x(t)-\frac{L_{\lambda
_{0}}(\phi )}{1+\gamma _{\lambda _{0}}}\exp \left[ \frac{1}{\rho _{\lambda
_{0}}}\int_{0}^{t}f_{\lambda _{0}}(u)du\right] \right\} \exp \left[ -\frac{1
}{\rho _{\lambda _{1}}}\int_{0}^{t}f_{\lambda _{1}}(u)du\right] \\
&\leq U_{2}(\lambda _{0},\lambda _{1};\phi )\quad \text{for all }
t\geq 0,
\end{align*}
where $L_{\lambda _{0}}(\phi )$ is defined by \eqref{e2.8},
and
\begin{align*}
U_{1}(\lambda _{0},\lambda _{1};\phi )
&=\min_{-r\leq t\leq 0}\left( \left\{ \phi (t)-\frac{L_{\lambda _{0}}(\phi )}{1+\gamma _{\lambda
_{0}}}\exp \left[ \frac{1}{\rho _{\lambda _{0}}}\int_{0}^{t}f_{\lambda
_{0}}(u)du\right] \right\} \times \right. \\
&\quad \left. \times \exp \left[ -\frac{1}{\rho _{\lambda _{1}}}\int_{0}^{t}f_{
\lambda _{1}}(u)du\right] \right) ,
\end{align*}
\begin{align*}
U_{2}(\lambda _{0},\lambda _{1};\phi )
&=\underset{-r\leq t\leq 0}{\max }
\left( \left\{ \phi (t)-\frac{L_{\lambda _{0}}(\phi )}{1+\gamma _{\lambda
_{0}}}\exp \left[ \frac{1}{\rho _{\lambda _{0}}}\int_{0}^{t}f_{\lambda
_{0}}(u)du\right] \right\} \times \right. \\
&\quad \left. \times \exp \left[ -\frac{1}{\rho _{\lambda _{1}}}\int_{0}^{t}f_{
\lambda _{1}}(u)du\right] \right) .
\end{align*}

Note: The constant $B$ is negative and the root $\lambda _{0}$
is necessarily nonpositive; hence, by Part (I) of Lemma \ref{lem2.2}, we always have
 $1+\gamma _{\lambda _{0}}\neq 0$ if $\lambda _{1}$ is also
nonpositive.
\end{theorem}

It is remarkable that the double inequality in the conclusion of the above
theorem can equivalently be written as follows
\begin{align*}
&U_{1}(\lambda _{0},\lambda _{1};\phi )\exp \left\{ \int_{0}^{t}\left[
\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}-\frac{f_{\lambda _{0}}(u)}{
\rho _{\lambda _{0}}}\right] du\right\} \\
&\leq x(t)\exp \left[ -\frac{1}{\rho _{\lambda _{0}}}\int_{0}^{t}f_{\lambda
_{0}}(u)du\right] -\frac{L_{\lambda _{0}}(\phi )}{1+\gamma _{\lambda _{0}}}
\\
&\leq U_{2}(\lambda _{0},\lambda _{1};\phi )\exp \left\{ \int_{0}^{t}\left[
\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}-\frac{f_{\lambda _{0}}(u)}{
\rho _{\lambda _{0}}}\right] du\right\} \quad\text{for }t\geq 0.
\end{align*}
Hence, if the roots $\lambda _{0}$ and $\lambda _{1}$ of the
characteristic equation \eqref{e2.3} are such that
\begin{equation*}
\underset{t\to \infty }{\lim }\int_{0}^{t}\left[ \frac{f_{\lambda
_{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda
_{1}}}\right] du=\infty ,
\end{equation*}
then the solution $x$ of the IVP \eqref{e2.1}, \eqref{e2.2} satisfies
\eqref{e2.7}. Furthermore,
we see that the double inequality in the conclusion of Theorem \ref{thm2.4} is
equivalent to
\begin{align*}
&U_{1}(\lambda _{0},\lambda _{1};\phi )\exp \left[ \frac{1}{\rho _{\lambda
_{1}}}\int_{0}^{t}f_{\lambda _{1}}(u)du\right] +\frac{L_{\lambda _{0}}(\phi )
}{1+\gamma _{\lambda _{0}}}\exp \left[ \frac{1}{\rho _{\lambda _{0}}}
\int_{0}^{t}f_{\lambda _{0}}(u)du\right]\\
& \leq x(t) \\
& \leq U_{2}(\lambda _{0},\lambda _{1};\phi )\exp \left[ \frac{1}{\rho
_{\lambda _{1}}}\int_{0}^{t}f_{\lambda _{1}}(u)du\right] +\frac{L_{\lambda
_{0}}(\phi )}{1+\gamma _{\lambda _{0}}}\exp \left[ \frac{1}{\rho _{\lambda
_{0}}}\int_{0}^{t}f_{\lambda _{0}}(u)du\right]
\end{align*}
for $t\geq 0$.


\begin{proof}[Proof of Theorem \ref{thm2.4}]
Let $x$ be the solution of the IVP  \eqref{e2.1}, \eqref{e2.2},
and consider the function $y$ defined by
\begin{equation*}
y(t)=x(t)\exp \left[ -\frac{1}{\rho _{\lambda _{0}}}\int_{0}^{t}f_{\lambda
_{0}}(u)du\right] \quad\text{for }t\geq -r.
\end{equation*}
Furthermore, let us define
\begin{equation*}
z(t)=y(t)-\frac{L_{\lambda _{0}}(\phi )}{1+\gamma _{\lambda _{0}}}\quad\text{
for }t\geq -r.
\end{equation*}
As it has been shown by the authors \cite{p2} (for more general periodic linear
neutral delay differential equations with several delays), the fact that $x$
satisfies \eqref{e2.1} for $t\geq 0$ is equivalent to the fact that
$z$ satisfies
\begin{equation} \label{e2.22}
\begin{aligned}
&z(t)+ce^{-\lambda _{0}\sigma }z(t-\sigma ) \\
&=ce^{-\lambda _{0}\sigma }\frac{1}{
\rho _{\lambda _{0}}}\int_{t-\sigma }^{t}f_{\lambda _{0}}(s)z(s)ds
-e^{-\lambda _{0}\tau }\int_{t-\tau }^{t}\widetilde{b}(s)z(s)ds\quad
\text{for }t\geq 0.
\end{aligned}
\end{equation}
(Note that in \cite{p2} we have $\rho _{\lambda _{0}}>0$, but it suffices to have
$\rho _{\lambda _{0}}\neq 0$.)


Next, we introduce the function
\begin{equation*}
w(t)=z(t)\exp \left\{ \int_{0}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} \quad\text{for }t\geq -r.
\end{equation*}
By using this function, we can immediately see that \eqref{e2.22} takes the
 equivalent form
\begin{equation}  \label{e2.23}
\begin{aligned}
&w(t)+ce^{-\lambda _{0}\sigma }w(t-\sigma )\exp \left\{ \int_{t-\sigma }^{t}
\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda
_{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\}  
\\
&=ce^{-\lambda _{0}\sigma }\frac{1}{\rho _{\lambda _{0}}}\int_{t-\sigma
}^{t}f_{\lambda _{0}}(s)w(s)\exp \left\{ \int_{s}^{t}
\left[ \frac{f_{\lambda_{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda_{1}}}\right] du\right\} ds  
\\
&\quad -e^{-\lambda _{0}\tau }\int_{t-\tau }^{t}\widetilde{b}(s)w(s)\exp \left\{
\int_{s}^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{
f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds\quad
\text{for }t\geq 0.
\end{aligned}
\end{equation}
In view of the fact that the coefficients $a$ and $b$ are $T$-periodic
functions, it follows that the functions $f_{\lambda _{0}}$ and $f_{\lambda
_{1}}$ are also $T$-periodic. So, by taking into account the fact that $
\sigma =\ell T$ and following the same procedure as in proving \eqref{e2.14}
and \eqref{e2.15} (in the proof of Lemma \ref{lem2.3}), we can show that
\begin{equation*}
\frac{1}{\rho _{\lambda _{0}}}\int_{t-\sigma }^{t}f_{\lambda
_{0}}(u)du=\lambda _{0}\sigma \quad\text{for }t\geq 0
\end{equation*}
and
\begin{equation*}
\frac{1}{\rho _{\lambda _{1}}}\int_{t-\sigma }^{t}f_{\lambda
_{1}}(u)du=\lambda _{1}\sigma \quad\text{for }t\geq 0.
\end{equation*}
Consequently,
\begin{equation*}
\exp \left\{ \int_{t-\sigma }^{t}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} =e^{(\lambda _{0}-\lambda _{1})\sigma }\quad\text{for every }
t\geq 0.
\end{equation*}
By using this fact, we see that \eqref{e2.23} can equivalently be written as
follows
\begin{equation} \label{e2.24}
\begin{aligned}
&w(t)+ce^{-\lambda _{1}\sigma }w(t-\sigma )  
 \\
&=ce^{-\lambda _{0}\sigma }\frac{1}{\rho _{\lambda _{0}}}\int_{t-\sigma
}^{t}f_{\lambda _{0}}(s)w(s)\exp \left\{ \int_{s}^{t}
\left[ \frac{f_{\lambda_{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds  
 \\
&\quad -e^{-\lambda _{0}\tau }\int_{t-\tau }^{t}\widetilde{b}(s)w(s)
\exp \left\{\int_{s}^{t}\left[
\frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}
-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds\quad
\text{for }t\geq 0.
\end{aligned}
\end{equation}
Combining the definitions of the functions $y$, $z$ and $w$, we have
\[
w(t)=\left\{ x(t)-\frac{L_{\lambda _{0}}(\phi )}{1+\gamma _{\lambda _{0}}}
\exp \left[ \frac{1}{\rho _{\lambda _{0}}}\int_{0}^{t}f_{\lambda _{0}}(u)du
\right] \right\} \exp \left[ -\frac{1}{\rho _{\lambda _{1}}}
\int_{0}^{t}f_{\lambda _{1}}(u)du\right]
\]
for $t\geq -r$.
Thus, by taking into account the initial condition \eqref{e2.2} and the
 definitions
of the constants $U_{1}(\lambda _{0},\lambda _{1};\phi )$ and
$U_{2}(\lambda _{0},\lambda _{1};\phi )$, we can immediately conclude that
the double inequality in the conclusion of our theorem can be written in the
equivalent form
\begin{equation*}
\underset{-r\leq s\leq 0}{\min }w(s)\leq w(t)\leq \underset{-r\leq s\leq 0}{
\max }w(s)\quad\text{for all }t\geq 0.
\end{equation*}
What we have to prove is that the last double inequality is valid. We will
confine our attention in establishing that
\begin{equation}
w(t)\geq \underset{-r\leq s\leq 0}{\min }w(s)\quad\text{for every }t
\geq 0.  \label{e2.25}
\end{equation}
In a similar way, one can show that
\begin{equation*}
w(t)\leq \underset{-r\leq s\leq 0}{\max }w(s)\quad\text{for every }t
\geq 0.
\end{equation*}


It remains to prove \eqref{e2.25}. For this purpose, let us consider an arbitrary
real number $K$ with $K<\min_{-r\leq s\leq 0} w(s)$. Then we
obviously have
\begin{equation}
w(t)>K\quad\text{for }-r\leq t\leq 0.  \label{e2.26}
\end{equation}
We claim that
\begin{equation}
w(t)>K\quad \text{for all }t\geq 0.  \label{e2.27}
\end{equation}
Otherwise, in view of \eqref{e2.26}, there exists a point $t_{0}>0$ so that
\begin{equation}
w(t)>K\quad\text{for }-r\leq t<t_{0},\quad \text{and}\quad w(t_{0})=K.
\label{e2.28}
\end{equation}
We notice that it is supposed that $c\leq 0$. Also, we observe that, as
$\rho _{\lambda _{0}}\neq 0$, the hypothesis \eqref{e2.9} can be written
in the form $\frac{1}{\rho _{\lambda _{0}}}f_{\lambda _{0}}(t)\leq 0$
 for $t\geq 0 $. So, since the function $f_{\lambda _{0}}$ is
$T$-periodic and $r$ is a multiple of the period $T$, we always have
\begin{equation*}
\frac{1}{\rho _{\lambda _{0}}}f_{\lambda _{0}}(t)\leq 0\quad
\text{for every }t\geq -r.
\end{equation*}
Furthermore, since the function $b$ is $T$-periodic and $\tau =mT$, the
assumption that $b$ is not identically zero on $[0,\infty )$ means that
$\widetilde{b}$ is not identically zero on the interval
$[t_{0}-\tau ,t_{0})$, while the hypothesis that $b$ is nonpositive
on $[0,\infty )$ means that $\widetilde{b}$ is nonpositive on
$[t_{0}-\tau ,t_{0})$. Hence, the function $\widetilde{b}$ is nonpositive,
but not identically zero, on the interval $
[t_{0}-\tau ,t_{0})$. We also have $\widetilde{b}(t_{0})\leq 0$. Now, by
using \eqref{e2.28} and taking into account the above observations,
from \eqref{e2.24} we obtain
\begin{align*}
K &=w(t_{0})=-ce^{-\lambda _{1}\sigma }w(t_{0}-\sigma ) \\
&\quad +ce^{-\lambda _{0}\sigma }\frac{1}{\rho _{\lambda _{0}}}\int_{t_{0}-\sigma
}^{t_{0}}f_{\lambda _{0}}(s)w(s)\exp \left\{ \int_{s}^{t_{0}}\left[ \frac{
f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho
_{\lambda _{1}}}\right] du\right\} ds \\
&\quad-e^{-\lambda _{0}\tau }\int_{t_{0}-\tau }^{t_{0}}\widetilde{b}(s)w(s)\exp
\left\{ \int_{s}^{t_{0}}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda
_{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds
\\
&> K\left( -ce^{-\lambda _{1}\sigma }+ce^{-\lambda _{0}\sigma }\frac{1}{\rho
_{\lambda _{0}}}\int_{t_{0}-\sigma }^{t_{0}}f_{\lambda _{0}}(s)\exp \left\{
\int_{s}^{t_{0}}\left[ \frac{f_{\lambda _{0}}(u)}{\rho _{\lambda _{0}}}-
\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right] du\right\} ds\right.
\\
&\quad -\left. e^{-\lambda _{0}\tau }\int_{t_{0}-\tau }^{t_{0}}\widetilde{b}
(s)\exp \left\{ \int_{s}^{t_{0}}\left[ \frac{f_{\lambda _{0}}(u)}{\rho
_{\lambda _{0}}}-\frac{f_{\lambda _{1}}(u)}{\rho _{\lambda _{1}}}\right]
du\right\} ds\right) .
\end{align*}
Thus, using \eqref{e2.11} for $t=t_{0}$, we arrive at the contradiction $K>K$.
This contradiction establishes our claim, i.e., \eqref{e2.27} holds. Since
\eqref{e2.27} is satisfied for all real numbers $K$ such that
$K<\min_{-r\leq s\leq 0}w(s)$, it follows that \eqref{e2.25} is always
fulfilled.

The proof of the theorem is complete.
\end{proof}

Before closing this section, we will consider two special cases, namely the
non-neutral case and the autonomous case.

Consider, first, the periodic linear (\textit{non-neutral}) delay
differential equation
\begin{equation}
x'(t)=a(t)x(t)+b(t)x(t-\tau ).  \label{e2.29}
\end{equation}
Equation \eqref{e2.29} can be obtained (as a special case)
from \eqref{e2.1} \textit{by taking $c=0$ and} $\sigma =\tau $.
As it concerns the (non-neutral) delay differential equation \eqref{e2.29},
we have the number $\tau $ in place of $r$. The characteristic equation
of \eqref{e2.29} is
\begin{equation}
\lambda =A+Be^{-\lambda \tau }.  \label{e2.30}
\end{equation}
By applying Theorem \ref{thm2.4} to the (non-neutral) delay differential equation
\eqref{e2.29}, we are led to \cite[Theorem 2.3]{p14} in the previous
authors' paper,
\textit{under the additional hypotheses that the root $\lambda _{0}$
 of the characteristic equation \eqref{e2.30} is such that
$a(t)+b(t)e^{-\lambda _{0}\tau }\leq 0$ for $t\geq 0$} (and so $
\lambda _{0}$ is always nonpositive) \textit{and that $\lambda _{0}$
 satisfies} $1+B\tau e^{-\lambda _{0}\tau }\neq 0$. (Note
that we always have $1+B\tau e^{-\lambda _{0}\tau }\neq 0$ if the other
root $\lambda _{1}$ of \eqref{e2.30} is also nonpositive.)
But, these (additional)
hypotheses are not needed for \cite[Theorem 2.3]{p14} to hold. Hence,
\cite[Theorem 2.3]{p14} cannot be obtained as a corollary from
Theorem \ref{thm2.4}, the main result of this section.

Next, let us consider the \textit{autonomous} linear neutral delay
differential equation
\begin{equation}
\left[ x(t)+cx(t-\sigma )\right] '=ax(t)+bx(t-\tau ),
\label{e2.31}
\end{equation}
where $c$, $a$ \textit{and $b\neq 0$ are real numbers, and
$\sigma $ and $\tau $ are positive real constants.} The
characteristic equation of \eqref{e2.31} is the following one
\begin{equation}
\lambda (1+ce^{-\lambda \sigma })=a+be^{-\lambda \tau }.  \label{e2.32}
\end{equation}
The constant coefficients $a$ and $b$ of the autonomous neutral delay
differential equation \eqref{e2.31} can be considered as $T$-periodic functions,
for any real number $T>0$. We observe that the hypothesis that the root
$\lambda _{0}$ of the characteristic equation \eqref{e2.32}
satisfies \eqref{e2.9} is
equivalent to the hypothesis that $\lambda _{0}$ is nonpositive. After
these observations, it is not difficult to apply Theorem \ref{thm2.4} to the special
case of the autonomous linear neutral delay differential equation \eqref{e2.31}.
The result obtained by such an application is a special case of a more
general result given by the authors in \cite{p9}
 (for autonomous linear neutral
delay differential equations \textit{with several delays}), \textit{under
the assumption that there exist $T>0$ and positive integers
$\ell $ and $m$ with $\sigma =\ell T$ and $\tau =mT$, and the hypothesis
that the roots $\lambda _{0}$ and $\lambda _{1}$ of the characteristic equation
 \eqref{e2.32} satisfy $1+ce^{-\lambda _{0}\sigma }\neq 0$ and}
$1+ce^{-\lambda _{1}\sigma }\neq 0$. Note that these restrictions are not necessary for the
special case of the more general result in \cite{p9} to be valid. Such
restrictions are not imposed in the corresponding result in \cite{p9}
 concerning
more general autonomous linear neutral delay differential equations.



\section{On the behavior of the solutions to periodic linear neutral delay
difference equations}

This section is devoted to the study of the behavior of the solutions of the
linear neutral delay difference equation
\begin{equation}
\Delta \left( x_{n}+cx_{n-\sigma }\right) =a(n)x_{n}+b(n)x_{n-\tau },
\label{e3.1}
\end{equation}
where  $c$ \textit{is a real number,
 $(a(n)) _{n\geq 0} $ and $( b(n)) _{n\geq 0}$
are sequences of real numbers, and $\sigma $ and $\tau $ are positive
integers. It is supposed that the sequence $( b(n)) _{n\geq 0}$
is not identically zero.} Moreover, \textit{it will be assumed
that the coefficients $(a(n)) _{n\geq 0}$ and $( b(n)) _{n\geq 0}$
are periodic sequences with a common period
$T$ (where $T$ is a positive integer) and that
there exist positive integers $\ell $ and $m$ such that}

\begin{equation*}
\sigma =\ell T\quad \textit{and}\quad  \tau =mT.
\end{equation*}

Let us consider the positive integer $r$ defined by
\begin{equation*}
r=\max \{\sigma ,\tau \}.
\end{equation*}

A \textit{solution} of the neutral delay difference equation \eqref{e3.1} is a
sequence of real numbers $(x_{n})_{n\geq -r}$, which satisfies \eqref{e3.1}
for all $n\geq 0$.

With the neutral delay difference equation \eqref{e3.1}, one associates
an \textit{initial condition} of the form
\begin{equation}
x_{n}=\phi _{n}\quad\text{for }n=-r,\dots ,0,  \label{e3.2}
\end{equation}
where \textit{the initial values $\phi _{n}$ $(n=-r,\dots ,0)$ are
given real numbers.} For convenience, we will use the notation
$\phi =(\phi_{n})_{n=-r}^{0}$.

Equations \eqref{e3.1} and \eqref{e3.2} constitute an
\textit{initial value problem }(IVP, for short). It is clear that there
exists exactly one solution $(x_{n}) _{n\geq -r}$ of the neutral delay
difference equation \eqref{e3.1}
which satisfies the initial condition \eqref{e3.2}; we shall call this unique
solution $(x_n) _{n\geq -r}$ the \textit{solution} of the
initial value problem \eqref{e3.1}, \eqref{e3.2} or, more briefly,
the \textit{solution} of the IVP \eqref{e3.1}, \eqref{e3.2}.

With the neutral delay difference equation \eqref{e3.1}, we associate
the equation
\begin{equation}
\left[ \lambda (1+c\lambda ^{-\sigma })\right] ^{T}=\prod_{k=0}^{T-1}\left[
1+c\lambda ^{-\sigma }+a(k)+b(k)\lambda ^{-\tau }\right] \text{;}  \label{e3.3}
\end{equation}
this equation will be called the \textit{characteristic equation }
of \eqref{e3.1} (see the authors' paper \cite{p8}).

Now, we shall introduce certain notation, which will be used throughout
this section without any further mention.

By  $( \widetilde{a}(n)) _{n\geq -r}$  and
  $(\widetilde{b}(n))_{n\geq -r}$  we will denote the $T$-periodic extensions of the coefficients   $(a(n))_{n\geq 0}$
and  $( b(n)) _{n\geq 0}$, respectively. (Clearly, $r$ is
 a multiple of the period $T$.)

We consider positive roots $\lambda $ of the characteristic equation \eqref{e3.3}
with the following property:
\begin{equation}
1+c\lambda ^{-\sigma }\neq 0.  \label{ep0lamb}
\end{equation}
We immediately observe that a positive root $\lambda $ of \eqref{e3.3} with the
property \eqref{ep0lamb} satisfies
\begin{equation*}
\lambda ^{T}=\prod_{k=0}^{T-1}\left\{ 1+\frac{1}{1+c\lambda ^{-\sigma }}
\left[ a(k)+b(k)\lambda ^{-\tau }\right] \right\} .
\end{equation*}

Furthermore, for any positive root $\lambda $ of the characteristic equation
\eqref{e3.3} with the property \eqref{ep0lamb}, by
$\left(h_{\lambda }(n)\right) _{n\geq -r}$ we shall denote the sequence of real
numbers defined as follows
\begin{equation*}
h_{\lambda }(n)=1+\frac{1}{1+c\lambda ^{-\sigma }}[\widetilde{a}(n)+
\widetilde{b}(n)\lambda ^{-\tau }]\quad\text{for }n\geq -r.
\end{equation*}
Since the sequences $(a(n)) _{n\geq 0}$  and
$( b(n)) _{n\geq 0}$  are $T$-periodic, it follows
immediately that, for each positive root $\lambda $ of \eqref{e3.3} with the
property \eqref{ep0lamb}, the sequence
$\left( h_{\lambda }(n)\right)_{n\geq -r}$ is also $T$-periodic.

By the use of the above notation, we have
\begin{equation*}
\lambda ^{T}=\prod_{k=0}^{T-1}h_{\lambda }(k)
\end{equation*}
for each positive root $\lambda $ of the characteristic equation \eqref{e3.3}
with the property \eqref{ep0lamb}. This fact will be used quite frequently
in the sequel without any specific mention.

We will make use of positive roots $\lambda $ of the characteristic equation
\eqref{e3.3} with the property \eqref{ep0lamb} and the additional property:
\textit{If $T>1$, then}
\begin{equation} \label{eplamb}
h_{\lambda }(k)\equiv 1+\frac{1}{1+c\lambda ^{-\sigma }}[
a(k)+b(k)\lambda ^{-\tau }] >0\quad(k=1,\dots ,T-1).
\end{equation}
(It must be noted that \eqref{eplamb} holds by itself when $T=1$.)

The following simple result will be kept in mind in what follows:

\textit{If $\lambda $ is a positive root of the characteristic
equation \eqref{e3.3} with the properties \eqref{ep0lamb} and \eqref{eplamb},
then}
\begin{equation*}
h_{\lambda }(n)>0\quad \textit{for all } n\geq -r.
\end{equation*}

This result has been established by the authors in \cite{p8}
 for the case of more
general periodic linear neutral delay difference equations with several
delays. Note that in \cite{p8} we have used positive roots $\lambda $ of the
characteristic equation \eqref{e3.3} with the property
\begin{equation}
1+c\lambda ^{-\sigma }>0  \label{etildep0lamb}
\end{equation}
in place of the property \eqref{ep0lamb} considered here. But, the
above result remains valid with \eqref{ep0lamb} instead of
\eqref{etildep0lamb}.

Let us introduce another notation. If $\lambda $ is a positive root of the
characteristic equation \eqref{e3.3} with the properties \eqref{ep0lamb} and
\eqref{eplamb}, then $(H_{\lambda }(n))_{n\geq -r}$ will stand for the
sequence of positive real numbers defined by
\begin{equation*}
H_{\lambda }(n)=\begin{cases}
\prod_{k=0}^{n-1}h_{\lambda }(k)& \text{for }n\geq 0 \\
\big[ \prod_{k=n}^{-1}h_{\lambda }(k)\big] ^{-1} &
\text{for }n=-r,\dots ,0.
\end{cases}
\end{equation*}
Note that, here and in the sequel, we use the usual convention
$\prod_{k=0}^{-1}=1$.

A basic asymptotic criterion for the solutions of the neutral delay
difference equation \eqref{e3.1} is the following theorem, which is closely related
to the main result (Theorem \ref{thm3.3} below) of this section.

\begin{theorem} \label{thm3.1}
 Let $\lambda _{0}$ be a positive
root of the characteristic equation \eqref{e3.3} with the properties
\eqref{ep0lamb} and \eqref{eplamb}
with $\lambda_0$ instead of $\lambda$,
and the following property:
\begin{equation}
| c| \lambda _{0}^{-\sigma }\left[ 1+\sum_{s=0}^{\sigma
-1}\left| 1-\frac{1}{h_{\lambda _{0}}(s)}\right| \right] +\lambda
_{0}^{-\tau }\sum_{s=0}^{\tau -1}\frac{1}{h_{\lambda _{0}}(s)}|
b(s)| <1.  \label{e3.4}
\end{equation}
Set
\begin{equation}
\beta _{\lambda _{0}}=c\lambda _{0}^{-\sigma }\left\{ 1-\sum_{s=0}^{\sigma
-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \right\} +\lambda
_{0}^{-\tau }\sum_{s=0}^{\tau -1}\frac{1}{h_{\lambda _{0}}(s)}b(s).
\label{e3.5}
\end{equation}
(Note that Property \eqref{e3.4} guarantees that
$1+\beta_{\lambda _{0}}>0$.)

Then the solution $(x_{n})_{n\geq -r}$ of the IVP \eqref{e3.1},
\eqref{e3.2} satisfies
\begin{equation}
\lim_{n\to \infty } \frac{x_{n}}{H_{\lambda _{0}}(n)}=
\frac{M_{\lambda _{0}}(\phi )}{1+\beta _{\lambda _{0}}},  \label{e3.6}
\end{equation}
where
\begin{equation} \label{e3.7}
\begin{aligned}
M_{\lambda _{0}}(\phi )
&=\phi _{0}+c\left\{ \phi _{-\sigma }-\lambda
_{0}^{-\sigma }\sum_{s=-\sigma }^{-1}
\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] 
 \left[ \prod_{k=s}^{-1}h_{\lambda _{0}}(k)\right] \phi _{s}\right\}
\\
&\quad +\lambda _{0}^{-\tau }\sum_{s=-\tau }^{-1}\left[ \prod_{k=s+1}^{-1}h_{
\lambda _{0}}(k)\right] \widetilde{b}(s)\phi _{s}.
\end{aligned}
\end{equation}
\end{theorem}

The above theorem has been established in the authors' paper \cite{p8}
 for more general periodic linear neutral delay difference equations
with several delays. Note that in \cite{p8} we have assumed that the
positive root
$\lambda_{0}$ of the characteristic equation \eqref{e3.3} has the property
\eqref{etildep0lamb} with $\lambda_0$ instead of $\lambda$, while in Theorem \ref{thm3.1} it is supposed that
$\lambda _{0}$ has the property \eqref{ep0lamb} with $\lambda _{0}$ instead
of $\lambda$.  But, if $\lambda _{0}$ has the properties
\eqref{ep0lamb} 
and \eqref{eplamb} with $\lambda_0$ instead of $\lambda$,
 then \eqref{e3.4} makes sense and it implies that
$|c|\lambda _{0}^{-\sigma }<1$, which gives $1+c\lambda _{0}^{-\sigma }>0$,
i.e., $\lambda _{0}$ has always the property
\eqref{etildep0lamb} with $\lambda_0$ instead of $\lambda$.

The main result of this section, i.e., Theorem \ref{thm3.3} below, is derived via two
suitable distinct positive roots of the characteristic equation \eqref{e3.3}.
Before stating and proving Theorem \ref{thm3.3}, we give a lemma
(Lemma \ref{lem3.2} below),
which is rather technical, but it is crucial in proving Theorem \ref{thm3.3}.

\begin{lemma} \label{lem3.2}
Let $\lambda _{0}$ and $\lambda_{1} $, $\lambda _{0}\neq \lambda _{1}$,
be two positive roots of the characteristic equation
\eqref{e3.3} with the properties
\eqref{ep0lamb} and
\eqref{eplamb} with $\lambda_0$ instead of $\lambda$
as well as \eqref{ep0lamb} and
\eqref{eplamb} with $\lambda_1$ instead of $\lambda$, respectively.
Then, for each $n\geq 0$, we have
\begin{equation} \label{e3.8}
\begin{aligned}
1&=-c\lambda _{1}^{-\sigma }+c\lambda _{0}^{-\sigma }\sum_{s=n-\sigma }^{n-1}
\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{
h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}   \\
&\quad -\lambda _{0}^{-\tau }\sum_{s=n-\tau }^{n-1}\frac{1}{h_{\lambda _{0}}(s)}
\widetilde{b}(s)\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)}.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof} Set
$\mu _{\lambda _{0}}=1+c\lambda _{0}^{-\sigma }$
and $\mu _{\lambda _{1}}=1+c\lambda _{1}^{-\sigma }$.
We immediately see that
\begin{equation}
\mu _{\lambda _{0}}\left[ h_{\lambda _{0}}(n)-1\right] -\mu _{\lambda _{1}}
\left[ h_{\lambda _{1}}(n)-1\right] =\widetilde{b}(n)\left( \lambda
_{0}^{-\tau }-\lambda _{1}^{-\tau }\right) \quad\text{for }n\geq -r.
\label{e3.9}
\end{equation}
Also, it is obvious that
\begin{equation}
\mu _{\lambda _{0}}-\mu _{\lambda _{1}}=c\left( \lambda _{0}^{-\sigma
}-\lambda _{1}^{-\sigma }\right) .  \label{e3.10}
\end{equation}
Furthermore, we note that the sequences $(h_{\lambda _{0}}(n))_{n\geq -r}$
and $(h_{\lambda _{1}}(n))_{n\geq -r}$ are $T$-periodic. Thus, by taking
into account the fact that $\tau =mT$, we obtain for $n\geq 0$
\begin{equation*}
\prod_{k=n-\tau }^{n-1}h_{\lambda _{0}}(k)=\prod_{k=0}^{\tau -1}h_{\lambda
_{0}}(k)=\left[ \prod_{k=0}^{T-1}h_{\lambda _{0}}(k)\right] ^{m}
=\left(\lambda _{0}^{T}\right) ^{m}=\lambda _{0}^{mT}=\lambda _{0}^{\tau }.
\end{equation*}
That is,
\begin{equation}
\prod_{k=n-\tau }^{n-1}h_{\lambda _{0}}(k)=\lambda _{0}^{\tau }\quad
\text{for every }n\geq 0.  \label{e3.11}
\end{equation}
Similarly, we have
\begin{equation}
\prod_{k=n-\tau }^{n-1}h_{\lambda _{1}}(k)=\lambda _{1}^{\tau }\quad\text{
for every }n\geq 0.  \label{e3.12}
\end{equation}
Moreover, for any positive integer $\nu $, we get
\begin{equation*}
\prod_{k=0}^{\nu T-1}h_{\lambda _{0}}(k)=\left[ \prod_{k=0}^{T-1}h_{\lambda
_{0}}(k)\right] ^{\nu }=\left( \lambda _{0}^{T}\right) ^{\nu }=\lambda
_{0}^{\nu T}.
\end{equation*}
Consequently,
\begin{equation}
\prod_{k=0}^{\nu T-1}h_{\lambda _{0}}(k)=\lambda _{0}^{\nu T}\quad
(\nu =1,2,\dots ).  \label{e3.13}
\end{equation}
Analogously, we find
\begin{equation}
\prod_{k=0}^{\nu T-1}h_{\lambda _{1}}(k)=\lambda _{1}^{\nu T}\quad
(\nu =1,2,\dots ).  \label{e3.14}
\end{equation}

Now, let us consider an arbitrary but fixed integer $n\geq 0$. We shall show
that, for this fixed integer $n$, equality \eqref{e3.8} is fulfilled.

We take into account \eqref{e3.9} and \eqref{e3.10} to obtain
\begin{align*}
&\sum_{s=n-\tau }^{n-1}\frac{1}{h_{\lambda _{0}}(s)}\widetilde{b}
(s)\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&= \frac{1}{\lambda _{0}^{-\tau }-\lambda _{1}^{-\tau }}\sum_{s=n-\tau
}^{n-1}\frac{1}{h_{\lambda _{0}}(s)}\widetilde{b}(s)\left( \lambda
_{0}^{-\tau }-\lambda _{1}^{-\tau }\right) \prod_{k=s}^{n-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)} \\
&= \frac{1}{\lambda _{0}^{-\tau }-\lambda _{1}^{-\tau }}\sum_{s=n-\tau
}^{n-1}\frac{1}{h_{\lambda _{0}}(s)}\Big\{ \mu _{\lambda _{0}}\left[
h_{\lambda _{0}}(s)-1\right] -\mu _{\lambda _{1}}\left[ h_{\lambda _{1}}(s)-1
\right] \Big\} \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)} \\
&= \frac{1}{\lambda _{0}^{-\tau }-\lambda _{1}^{-\tau }}\sum_{s=n-\tau
}^{n-1}\frac{1}{h_{\lambda _{0}}(s)}\Big\{ (\mu _{\lambda _{0}}-\mu
_{\lambda _{1}})\left[ h_{\lambda _{0}}(s)-1\right] +\mu _{\lambda
_{1}}[h_{\lambda _{0}}(s)-h_{\lambda _{1}}(s)]\Big\} \times \\
&\quad \times \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&= c\frac{\lambda _{0}^{-\sigma }-\lambda _{1}^{-\sigma }}{\lambda
_{0}^{-\tau }-\lambda _{1}^{-\tau }}\sum_{s=n-\tau }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)} \\
&\quad -\frac{1}{\lambda _{0}^{-\tau }-\lambda _{1}^{-\tau }}\mu _{\lambda
_{1}}\sum_{s=n-\tau }^{n-1}\left[ \frac{h_{\lambda _{1}}(s)}{h_{\lambda
_{0}}(s)}-1\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)}.
\end{align*}
But, using \eqref{e3.11} and \eqref{e3.12}, we have
\begin{align*}
\sum_{s=n-\tau }^{n-1}\left[ \frac{h_{\lambda _{1}}(s)}{h_{\lambda _{0}}(s)}
-1\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
&=\sum_{s=n-\tau }^{n-1}\left[ \frac{1}{\frac{h_{\lambda _{0}}(s)}{
h_{\lambda _{1}}(s)}}\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)}-\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
\right] \\
&=\sum_{s=n-\tau }^{n-1}\left[ \prod_{k=s+1}^{n-1}\frac{h_{\lambda _{0}}(k)
}{h_{\lambda _{1}}(k)}-\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)}\right] \\
&=\sum_{s=n-\tau }^{n-1}\Delta \left[ \prod_{k=s}^{n-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)}\right] \\
&=\left[ \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
\right] _{s=(n-1)+1}-\left[ \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)}\right] _{s=n-\tau } \\
&= \prod_{k=n}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
-\prod_{k=n-\tau }^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&= 1-\frac{\prod_{k=n-\tau }^{n-1}h_{\lambda _{0}}(k)}{\prod_{k=n-\tau
}^{n-1}h_{\lambda _{1}}(k)}=1-\frac{\lambda _{0}^{\tau }}{\lambda _{1}^{\tau
}}=1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\tau }.
\end{align*}
Note that we have used the usual convention that
$\prod_{n}^{n-1}=1$.
Hence, it holds
\begin{equation} \label{e3.15}
\begin{aligned}
&\sum_{s=n-\tau }^{n-1}\frac{1}{h_{\lambda _{0}}(s)}\widetilde{b}
(s)\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
 \\
&=c\frac{\lambda _{0}^{-\sigma }-\lambda _{1}^{-\sigma }}{\lambda _{0}^{-\tau
}-\lambda _{1}^{-\tau }}\sum_{s=n-\tau }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)}-\frac{1}{\lambda _{0}^{-\tau }}\mu _{\lambda _{1}}.  
\end{aligned}
\end{equation}

Next, we observe that \eqref{e3.8} can equivalently be written as follows
\begin{equation*}
\mu _{\lambda _{1}}=c\lambda _{0}^{-\sigma }\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)}-\lambda _{0}^{-\tau }\sum_{s=n-\tau }^{n-1}
\frac{1}{h_{\lambda _{0}}(s)}\widetilde{b}(s)\prod_{k=s}^{n-1}\frac{
h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}.
\end{equation*}
So, in view of \eqref{e3.15}, equality \eqref{e3.8} is equivalent to
\begin{align*}
0&=c\lambda _{0}^{-\sigma }\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)} \\
&\quad -\lambda _{0}^{-\tau }c\frac{\lambda _{0}^{-\sigma }-\lambda _{1}^{-\sigma }
}{\lambda _{0}^{-\tau }-\lambda _{1}^{-\tau }}\sum_{s=n-\tau }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)}.
\end{align*}
The above  equality is satisfied if
\begin{align*}
&\frac{\lambda _{0}^{-\sigma }}{\lambda _{0}^{-\sigma }-\lambda _{1}^{-\sigma
}}\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\frac{\lambda _{0}^{-\tau }}{\lambda _{0}^{-\tau }-\lambda _{1}^{-\tau }}
\sum_{s=n-\tau }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)},
\end{align*}
i.e., if the following equality holds,
\begin{equation}  \label{e3.16}
\begin{aligned}
&\frac{1}{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\sigma }}
\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}\\
&=\frac{1}{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\tau }}
\sum_{s=n-\tau }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}.
\end{aligned}
\end{equation}
In the rest of the proof, we will establish \eqref{e3.16}. To this end, it is
sufficient to prove that
\begin{equation} \label{e3.17}
\begin{aligned}
&\frac{1}{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\sigma }}
\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
\\
&=\frac{1}{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{T}}
\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.18}
\begin{aligned}
&\frac{1}{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\tau }}
\sum_{s=n-\tau }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
\\
&=\frac{1}{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{T}}
\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}.
\end{aligned}
\end{equation}

To prove \eqref{e3.17} and \eqref{e3.18}, we shall take into account the fact
that the sequences $(h_{\lambda _{0}}(n))_{n\geq -r}$ and
$(h_{\lambda_{1}}(n))_{n\geq -r}$  are $T$-periodic and that
$\sigma =\ell T$ and $\tau =mT$ and we will use equalities \eqref{e3.13}
and \eqref{e3.14}. We confine ourselves
to showing \eqref{e3.17}. Equality \eqref{e3.18} can be established by
an analogous procedure. We obtain
\begin{align*}
&\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\sum_{i=1}^{\ell }\sum_{s=n-iT}^{n-1-(i-1)T}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)} \\
&=\sum_{i=1}^{\ell }\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda
_{0}}(s-(i-1)T)}\right] \prod_{k=s-(i-1)T}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)} \\
&=\sum_{i=1}^{\ell }\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
 \left[ \prod_{k=s-(i-1)T}^{s-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda
_{1}}(k)}\right] \left[ \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)}\right] \\
&=\sum_{i=1}^{\ell }\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \frac{\prod_{k=s-(i-1)T}^{s-1}h_{\lambda _{0}}(k)}{
\prod_{k=s-(i-1)T}^{s-1}h_{\lambda _{1}}(k)}\prod_{k=s}^{n-1}\frac{
h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\sum_{i=1}^{\ell }\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \frac{\prod_{k=0}^{(i-1)T-1}h_{\lambda _{0}}(k)}{
\prod_{k=0}^{(i-1)T-1}h_{\lambda _{1}}(k)}\prod_{k=s}^{n-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\sum_{i=1}^{\ell }\sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \frac{\lambda _{0}^{(i-1)T}}{\lambda _{1}^{(i-1)T}}\prod_{k=s}^{n-1}
\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\left[ \sum_{i=1}^{\ell }\left( \frac{\lambda _{0}}{\lambda _{1}}\right)
^{(i-1)T}\right] \sum_{s=n-T}^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\frac{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\ell T}}{
1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{T}}\sum_{s=n-T}^{n-1}
\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{
h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)} \\
&=\frac{1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\sigma }}{
1-\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{T}}\sum_{s=n-T}^{n-1}
\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n-1}\frac{
h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}.
\end{align*}
So, we have proved that \eqref{e3.17} holds.


The proof of our lemma is complete.
\end{proof}

Now, we proceed to establish the main result of this section, i.e.,
Theorem \ref{thm3.3} below.

\begin{theorem} \label{thm3.3}
Suppose that $c\leq 0$ and that $( b(n)) _{n\geq 0}$ is nonpositive.
Let $\lambda _{0}$ be a positive root of the characteristic equation
\eqref{e3.3} with the properties \eqref{ep0lamb} and
\eqref{eplamb} with $\lambda_0$ instead of $\lambda$ and such that
\begin{equation}
\left( 1+c\lambda _{0}^{-\sigma }\right) \left[ a(n)+b(n)\lambda _{0}^{-\tau
}\right] \leq 0\quad\text{for }n\geq 0,  \label{e3.19}
\end{equation}
and assume that
$1+\beta _{\lambda _{0}}\neq 0$,
where $\beta _{\lambda _{0}}$ is defined by \eqref{e3.5}.
Let also $\lambda _{1}$, $\lambda _{1}\neq \lambda _{0}$, be
another positive root of \eqref{e3.3} with the properties
\eqref{ep0lamb} and
\eqref{eplamb} with $\lambda_1$ instead of $\lambda$.

Then the solution $(x_{n})_{n\geq -r}$ of the IVP \eqref{e3.1},
\eqref{e3.2} satisfies
\begin{equation*}
V_{1}(\lambda _{0},\lambda _{1};\phi )\leq \frac{1}{H_{\lambda _{1}}(n)}
\left[ x_{n}-\frac{M_{\lambda _{0}}(\phi )}{1+\beta _{\lambda _{0}}}
H_{\lambda _{0}}(n)\right] \leq V_{2}(\lambda _{0},\lambda _{1};\phi )
\quad\text{for all }n\geq 0,
\end{equation*}
where $M_{\lambda _{0}}(\phi )$ is defined by \eqref{e3.7},
 and
\begin{gather*}
V_{1}(\lambda _{0},\lambda _{1};\phi )=\min_{n=-r,\dots ,0}\left\{
\frac{1}{H_{\lambda _{1}}(n)}\left[ \phi _{n}-\frac{M_{\lambda _{0}}(\phi )}{
1+\beta _{\lambda _{0}}}H_{\lambda _{0}}(n)\right] \right\} ,
\\
V_{2}(\lambda _{0},\lambda _{1};\phi )=\max_{n=-r,\dots ,0}\left\{
\frac{1}{H_{\lambda _{1}}(n)}\left[ \phi _{n}-\frac{M_{\lambda _{0}}(\phi )}{
1+\beta _{\lambda _{0}}}H_{\lambda _{0}}(n)\right] \right\} .
\end{gather*}
\end{theorem}

\textit{Let $\lambda _{0}$ be a positive root of the
characteristic equation \eqref{e3.3} with the properties
\eqref{ep0lamb} and \eqref{eplamb} with $\lambda_0$ instead of $\lambda$,
and such that \eqref{e3.19} holds.}
Then we immediately see that \eqref{e3.19} can equivalently
be written as follows
\begin{equation*}
\frac{1}{1+c\lambda _{0}^{-\sigma }}[\widetilde{a}(n)+\widetilde{b}
(n)\lambda _{0}^{-\tau }]\leq 0\quad\text{for }n\geq -r.
\end{equation*}
Hence, \textit{inequality \eqref{e3.19} is equivalent to}
\begin{equation}
h_{\lambda _{0}}(n)\leq 1\quad\text{\textit{for} }n\geq -r.  \label{e3.20}
\end{equation}
Furthermore, it follows easily that \textit{the root $\lambda _{0}$
 is always less than or equal to} $1$.

It must be noted that the double inequality in the conclusion of
Theorem \ref{thm3.3} is equivalent to
\begin{equation*}
V_{1}(\lambda _{0},\lambda _{1};\phi )\frac{H_{\lambda _{1}}(n)}{H_{\lambda
_{0}}(n)}\leq \frac{x_{n}}{H_{\lambda _{0}}(n)}-\frac{M_{\lambda _{0}}(\phi )
}{1+\beta _{\lambda _{0}}}\leq V_{2}(\lambda _{0},\lambda _{1};\phi )\frac{
H_{\lambda _{1}}(n)}{H_{\lambda _{0}}(n)}\text{     for }n\geq 0.
\end{equation*}
Thus, \eqref{e3.6} is satisfied if the roots $\lambda _{0}$ and
$\lambda _{1}$ of \eqref{e3.3} are such that
\begin{equation*}
\underset{n\to \infty }{\lim }\frac{H_{\lambda _{0}}(n)}{H_{\lambda
_{1}}(n)}=\infty .
\end{equation*}
Moreover, we notice that another equivalent form of the double inequality in
the conclusion of Theorem \ref{thm3.3} is the following one
\begin{align*}
&V_{1}(\lambda _{0},\lambda _{1};\phi )H_{\lambda _{1}}(n)+\frac{M_{\lambda
_{0}}(\phi )}{1+\beta _{\lambda _{0}}}H_{\lambda _{0}}(n)\\
&\leq x_{n} \\
&\leq V_{2}(\lambda _{0},\lambda _{1};\phi )H_{\lambda _{1}}(n)+\frac{
M_{\lambda _{0}}(\phi )}{1+\beta _{\lambda _{0}}}H_{\lambda _{0}}(n)
\quad\text{for }n\geq 0.
\end{align*}


\begin{proof}[Proof of Theorem \ref{thm3.3}]  Consider the solution
$(x_{n})_{n\geq -r}$ of the IVP \eqref{e3.1}, \eqref{e3.2}, and set
\begin{equation*}
y_{n}=\frac{x_{n}}{H_{\lambda _{0}}(n)}\quad\text{for }n\geq -r.
\end{equation*}
Define
\begin{equation*}
z_{n}=y_{n}-\frac{M_{\lambda _{0}}(\phi )}{1+\beta _{\lambda _{0}}}
\quad\text{for }n\geq -r.
\end{equation*}
It has been shown by the authors \cite{p8} (for more general periodic linear
neutral delay difference equations with several delays) that
$(x_{n})_{n\geq -r}$ satisfies \eqref{e3.1} for $n\geq 0$ if and
 only if $(z_{n})_{n\geq -r}$   satisfies
\begin{equation} \label{e3.21}
z_{n}+c\lambda _{0}^{-\sigma }z_{n-\sigma }=c\lambda _{0}^{-\sigma
}\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] z_{s}
-\lambda _{0}^{-\tau }\sum_{s=n-\tau }^{n-1}\frac{1}{h_{\lambda _{0}}(s)}
\widetilde{b}(s)z_{s}
\end{equation}
for $n\geq 0$.
Note that in \cite{p8} it is assumed that $\lambda _{0}$ has the property
\eqref{etildep0lamb} with $\lambda_0$ instead of $\lambda$, but it is sufficient to suppose
that $\lambda _{0}$ has the property \eqref{ep0lamb} with $\lambda_0$ instead of $\lambda$.

Next, we define
\begin{equation*}
w_{n}=\frac{H_{\lambda _{0}}(n)}{H_{\lambda _{1}}(n)}z_{n} \quad
\text{for } n\geq -r.
\end{equation*}
Then we see that \eqref{e3.21} reduces to the equivalent equation
\begin{equation} \label{e3.22}
\begin{aligned}
&w_{n}+c\lambda _{0}^{-\sigma }\frac{H_{\lambda _{1}}(n-\sigma )/H_{\lambda
_{1}}(n)}{H_{\lambda _{0}}(n-\sigma )/H_{\lambda _{0}}(n)}w_{n-\sigma }\\
&=c\lambda _{0}^{-\sigma }\sum_{s=n-\sigma }^{n-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \frac{H_{\lambda _{0}}(n)/H_{\lambda _{0}}(s)}{
H_{\lambda _{1}}(n)/H_{\lambda _{1}}(s)}w_{s}\\
&\quad -\lambda _{0}^{-\tau}\sum_{s=n-\tau }^{n-1}
\frac{1}{h_{\lambda _{0}}(s)}\widetilde{b}(s)
\frac{H_{\lambda _{0}}(n)/H_{\lambda _{0}}(s)}{H_{\lambda _{1}}(n)/H_{\lambda
_{1}}(s)}w_{s}  \quad\text{for }n\geq 0.
\end{aligned}
\end{equation}
By taking into account the fact that the sequence $\left( h_{\lambda
_{0}}(n)\right) _{n\geq -r}$ is $T$-periodic and that $\sigma =\ell T$, we
obtain for $n\geq 0$
\begin{align*}
\frac{H_{\lambda _{0}}(n-\sigma )}{H_{\lambda _{0}}(n)}
&=\begin{cases}
\frac{\prod_{k=0}^{n-\sigma -1}h_{\lambda _{0}}(k)}{\prod_{k=0}^{n-1}h_{
\lambda _{0}}(k)}, &\text{if }n\geq \sigma \\
\frac{\left[ \prod_{k=n-\sigma }^{-1}h_{\lambda _{0}}(k)\right] ^{-1}}{
\prod_{k=0}^{n-1}h_{\lambda _{0}}(k)},&\text{if }0\leq n\leq \sigma
\end{cases} \\
&= \begin{cases}
\frac{1}{\left[ \prod_{k=0}^{n-1}h_{\lambda _{0}}(k)\right] \left[
\prod_{k=0}^{n-\sigma -1}h_{\lambda _{0}}(k)\right] ^{-1}}, &\text{if }
n\geq \sigma \\
\frac{1}{\left[ \prod_{k=n-\sigma }^{-1}h_{\lambda _{0}}(k)\right] \left[
\prod_{k=0}^{n-1}h_{\lambda _{0}}(k)\right] }, &\text{if }0\leq n\leq
\sigma \end{cases}
 \\
&=\left[ \prod_{k=n-\sigma }^{n-1}h_{\lambda _{0}}(k)\right] ^{-1}
 =\left[\prod_{k=0}^{\sigma -1}h_{\lambda _{0}}(k)\right] ^{-1}\\
&=\left\{ \left[ \prod_{k=0}^{T-1}h_{\lambda _{0}}(k)\right] ^{\ell }\right\} ^{-1}
 =\left[ \prod_{k=0}^{T-1}h_{\lambda _{0}}(k)\right] ^{-\ell }\\
&=\left(\lambda _{0}^{T}\right) ^{-\ell }=\lambda _{0}^{-\ell T}=\lambda
_{0}^{-\sigma }.
\end{align*}
That is,
\begin{equation}
\frac{H_{\lambda _{0}}(n-\sigma )}{H_{\lambda _{0}}(n)}=\lambda
_{0}^{-\sigma }\quad\text{for all }n\geq 0.  \label{e3.23}
\end{equation}
In a similar way, by using the fact that the sequence $\left( h_{\lambda
_{1}}(n)\right) _{n\geq -r}$ is $T$-periodic and that $\sigma =\ell T$, we
find
\begin{equation}
\frac{H_{\lambda _{1}}(n-\sigma )}{H_{\lambda _{1}}(n)}=\lambda
_{1}^{-\sigma }\quad\text{for all }n\geq 0.  \label{e3.24}
\end{equation}
Furthermore, for any integers $n$ and $s$ with $n\geq 0$ and
$-r\leq s\leq n-1$, we get
\begin{align*}
\frac{H_{\lambda _{0}}(n)}{H_{\lambda _{0}}(s)}
 &=\begin{cases}
\frac{\prod_{k=0}^{n-1}h_{\lambda _{0}}(k)}{\prod_{k=0}^{s-1}h_{\lambda
_{0}}(k)},& \text{if }0\leq s\leq n-1 \\
\frac{\prod_{k=0}^{n-1}h_{\lambda _{0}}(k)}{\left[ \prod_{k=s}^{-1}h_{
\lambda _{0}}(k)\right] ^{-1}}, &\text{if }-r\leq s\leq 0
\end{cases} \\
&= \begin{cases}
\frac{\prod_{k=0}^{n-1}h_{\lambda _{0}}(k)}{\prod_{k=0}^{s-1}h_{\lambda
_{0}}(k)}, &\text{if }0\leq s\leq n-1 \\
\left[ \prod_{k=s}^{-1}h_{\lambda _{0}}(k)\right]
 \left[\prod_{k=0}^{n-1}h_{\lambda _{0}}(k)\right],
& \text{if }-r\leq s\leq 0 .
\end{cases}
\end{align*}
So, we have
\begin{equation}
\frac{H_{\lambda _{0}}(n)}{H_{\lambda _{0}}(s)}
=\prod_{k=s}^{n-1}h_{\lambda_{0}}(k)\quad\text{for }
n\geq 0\text{ and }-r\leq s\leq n-1.
\label{e3.25}
\end{equation}
Similarly,
\begin{equation}
\frac{H_{\lambda _{1}}(n)}{H_{\lambda _{1}}(s)}
=\prod_{k=s}^{n-1}h_{\lambda_{1}}(k)\quad\text{for }
n\geq 0\text{ and }-r\leq s\leq n-1.
\label{e3.26}
\end{equation}
In view of \eqref{e3.23}, \eqref{e3.24}, \eqref{e3.25} and \eqref{e3.26},
equation \eqref{e3.22} is written in
the following equivalent form
\begin{equation} \label{e3.27}
\begin{aligned}
w_{n}+c\lambda _{1}^{-\sigma }w_{n-\sigma }
&=c\lambda _{0}^{-\sigma}\sum_{s=n-\sigma }^{n-1}
  \left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right]
  \left[\prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}\right]
  w_{s} \\
&\quad -\lambda _{0}^{-\tau }\sum_{s=n-\tau }^{n-1}\frac{1}{h_{\lambda _{0}}(s)}
\widetilde{b}(s)\left[ \prod_{k=s}^{n-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)}\right] w_{s}\quad \text{for }n\geq 0.
\end{aligned}
\end{equation}
By the definitions of $(y_{n})_{n\geq -r}$, $(z_{n})_{n\geq -r}$ and
$(w_{n})_{n\geq -r}$, it follows immediately that
\begin{equation*}
w_{n}=\frac{1}{H_{\lambda _{1}}(n)}\left[ x_{n}-\frac{M_{\lambda _{0}}(\phi )
}{1+\beta _{\lambda _{0}}}H_{\lambda _{0}}(n)\right]\quad \text{for }
n\geq -r.
\end{equation*}
Thus, in view of  initial condition \eqref{e3.2} and because of the
definition of $V_{1}(\lambda _{0},\lambda _{1};\phi )$ and
$V_{2}(\lambda _{0},\lambda_{1};\phi )$, we see that what we have to
prove is that $(w_{n})_{n\geq -r}$ satisfies
\begin{equation*}
\min_{s=-r,\dots ,0} w_{s}\leq w_{n}\leq \max_{s=-r,\dots ,0}
w_{s}\quad\text{for all }n\geq 0.
\end{equation*}
We will restrict ourselves to showing the left hand part of the above double
inequality. The right hand part of this double inequality can be established
by an analogous procedure. So, it remains to prove that
\begin{equation}
w_{n}\geq \min_{s=-r,\dots ,0} w_{s}\quad\text{for every }n\geq 0.  \label{e3.28}
\end{equation}

To prove \eqref{e3.28}, it suffices to show that, for any real number $K$
with $K<\min_{s=-r,\dots ,0}w_{s}$, it holds
\begin{equation}
w_{n}>K\quad \text{for all }n\geq 0.  \label{e3.29}
\end{equation}
Let us consider an arbitrary real number $K$ with
$K<\min_{s=-r,\dots ,0} w_{s}$. Then
\begin{equation}
w_{n}>K\quad\text{for }n=-r,\dots ,0.  \label{e3.30}
\end{equation}
Assume, for the sake of contradiction, that \eqref{e3.29} is not valid. Then,
because of \eqref{e3.30}, there exists an integer $n_{0}>0$ so that
\begin{equation}
w_{n}>K\quad \text{for }n=-r,\dots ,n_{0}-1, \text{ and }w_{n_{0}}\leq
K.  \label{e3.31}
\end{equation}
We notice that we have supposed that $c\leq 0$. Also, it follows
from \eqref{e3.20} that
\begin{equation*}
1-\frac{1}{h_{\lambda _{0}}(n)}\leq 0\quad\text{for all }n\geq -r.
\end{equation*}
Furthermore, as the coefficient sequence $( b(n)) _{n\geq 0}$ is
$T$-periodic and $\tau =mT$, the assumption that
$( b(n))_{n\geq 0}$ is not identically zero means that
$(\widetilde{b}(n))_{n=n_{0}-\tau }^{n_{0}-1}$ is not identically zero,
and the hypothesis that $( b(n)) _{n\geq 0}$ is nonpositive means that
$(\widetilde{b}(n))_{n=n_{0}-\tau }^{n_{0}-1}$ is nonpositive.
Hence, $(\widetilde{b}(n))_{n=n_{0}-\tau }^{n_{0}-1}$ is nonpositive,
 but not identically zero.
Now, by using \eqref{e3.31} and taking into account the above observations,
from \eqref{e3.27} we obtain
\begin{align*}
K &\geq w_{n_{0}}\\
&=-c\lambda _{1}^{-\sigma }w_{n_{0}-\sigma }+c\lambda
_{0}^{-\sigma }\sum_{s=n_{0}-\sigma }^{n_{0}-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \left[ \prod_{k=s}^{n_{0}-1}\frac{h_{\lambda _{0}}(k)}{
h_{\lambda _{1}}(k)}\right] w_{s} \\
&\quad -\lambda _{0}^{-\tau }\sum_{s=n_{0}-\tau }^{n_{0}-1}\frac{1}{h_{\lambda
_{0}}(s)}\widetilde{b}(s)\left[ \prod_{k=s}^{n_{0}-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)}\right] w_{s} \\
&> K\left\{ -c\lambda _{1}^{-\sigma }+c\lambda _{0}^{-\sigma
}\sum_{s=n_{0}-\sigma }^{n_{0}-1}\left[ 1-\frac{1}{h_{\lambda _{0}}(s)}\right] \prod_{k=s}^{n_{0}-1}\frac{h_{\lambda _{0}}(k)}{h_{\lambda _{1}}(k)}
 \right.\\
&\quad  \left. -\lambda _{0}^{-\tau }\sum_{s=n_{0}-\tau }^{n_{0}-1}\frac{1}{
h_{\lambda _{0}}(s)}\widetilde{b}(s)\prod_{k=s}^{n_{0}-1}\frac{h_{\lambda
_{0}}(k)}{h_{\lambda _{1}}(k)}\right\} .
\end{align*}
So, we can use \eqref{e3.8} for $n=n_{0}$ to arrive at the contradiction $K>K$.
This contradiction implies that \eqref{e3.29} holds true. We have thus
proved that \eqref{e3.28} is always satisfied.


The proof of the theorem is complete.
\end{proof}

Before closing this section and ending the paper, let us concentrate on the
following two special cases.


Consider the special case of the periodic linear (\textit{non-neutral})
delay difference equation
\begin{equation}
\Delta x_{n}=a(n)x_{n}+b(n)x_{n-\tau }.  \label{e3.32}
\end{equation}
This equation can be obtained (as a particular case) from
\eqref{e3.1} \textit{by
taking $c=0$ and considering the delay $\sigma $ to be
chosen arbitrarily so that} $\sigma \leq \tau $ (for example, $\sigma $ can
be chosen to be equal to $\tau $). In the case considered, we have the
integer $\tau $ instead of $r$. As it concerns the (non-neutral) delay
difference equation \eqref{e3.32}, the characteristic equation
 \eqref{e3.3} becomes
\begin{equation}
\lambda ^{T}=\prod_{k=0}^{T-1}\left[ 1+a(k)+b(k)\lambda ^{-\tau }
\right] .  \label{e3.33}
\end{equation}
We observe that \eqref{ep0lamb} holds by itself, for any positive
root $\lambda $ of the characteristic equation \eqref{e3.33}.
Moreover, it is not difficult to see that, if $\lambda $ is a
positive root of the characteristic equation \eqref{e3.33}
with the property \eqref{eplamb}, then every
positive root $\lambda ^{\ast }$ of \eqref{e3.33} with
$\lambda ^{\ast }>\lambda $  has also the property
\eqref{eplamb} with $\lambda ^{\ast }$ instead of $\lambda$,
provided that $( b(n)) _{n\geq 0}$ is nonpositive.
An application of Theorem \ref{thm3.3} to
the (non-neutral) delay difference equation \eqref{e3.32}
leads to \cite[Theorem 3.4]{p14} in the authors' paper,
\textit{under the additional assumptions that the
root $\lambda _{0}$ of the characteristic equation \eqref{e3.33}
satisfies $a(n)+b(n)\lambda _{0}^{-\tau }\leq 0$ for
$n\geq 0$} (which implies that $\lambda _{0}$ is always less than
or equal to 1) \textit{and that}
$1+\lambda _{0}^{-\tau }\sum_{s=0}^{\tau -1}
\frac{1}{h_{\lambda _{0}}(s)}b(s)\neq 0$.
These additional assumptions are not
necessary for the validity of \cite[Theorem 3.4]{p14}. Thus,
\cite[Theorem 3.4]{p14} is not a corollary of the main result
of this section, i.e., Theorem \ref{thm3.3}.

Finally, let us consider the special case of the \textit{autonomous} linear
neutral delay difference equation
\begin{equation}
\Delta (x_{n}+cx_{n-\sigma })=ax_{n}+bx_{n-\tau },  \label{e3.34}
\end{equation}
where $c$, $a$ \textit{and $b\neq 0$ are real constants, and
$\sigma $ and $\tau $ are positive integers}. The constant
coefficients $a$ and $b$ of \eqref{e3.34} can be considered as $T$-periodic
sequences of real numbers with $T=1$. The assumption that there exist
positive integers $\ell $ and $m$ such that $\sigma =\ell T$ and $\tau =mT$
 holds by itself. The characteristic equation of \eqref{e3.34} is
\begin{equation}
(\lambda -1)\left( 1+c\lambda ^{-\sigma }\right) =a+b\lambda ^{-\tau }.
 \label{e3.35}
\end{equation}
We can immediately see that the hypothesis that the root $\lambda _{0}$ of
the characteristic equation \eqref{e3.35} is such that \eqref{e3.19}
holds is equivalent to the hypothesis that $\lambda _{0}\leq 1$.
By applying Theorem \ref{thm3.3} to the
special case of the autonomous linear neutral delay difference equation
\eqref{e3.34}, we can easily be led to a particular case of a
more general result obtained by the authors in \cite{p7}
(for autonomous linear neutral delay
difference equations \textit{with several delays}), \textit{under the
restriction that the roots $\lambda _{0}$ and $\lambda _{1}$
of the characteristic equation \eqref{e3.35} are such that
$1+c\lambda _{0}^{-\sigma }\neq 0$ and}
$1+c\lambda _{1}^{-\sigma}\neq 0$. It must be noted that this restriction
can be removed; indeed, the corresponding more general result in \cite{p7}
holds without such a restriction.


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\end{thebibliography}
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