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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 19, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/19\hfil Exponential stability of solutions]
{Exponential stability of solutions to nonlinear
time-delay systems of neutral type}

\author[G. V. Demidenko, I. I. Matveeva \hfil EJDE-2016/19\hfilneg]
{Gennadii V. Demidenko, Inessa I. Matveeva}

\address{Gennadii V. Demidenko \newline
Laboratory of Differential and Difference Equations,
Sobolev Institute of Mathematics,
4, Acad. Koptyug avenue, Novosibirsk 630090, Russia.\newline
Department of Mechanics and Mathematics,
Novosibirsk State University,
2, Pirogov street, Novosibirsk 630090, Russia}
\email{demidenk@math.nsc.ru}

\address{Inessa I. Matveeva \newline
Laboratory of Differential and Difference Equations,
Sobolev Institute of Mathematics,
4, Acad. Koptyug avenue, Novosibirsk 630090, Russia.\newline
Department of Mechanics and Mathematics,
Novosibirsk State University,
2, Pirogov street, Novosibirsk 630090, Russia}
\email{matveeva@math.nsc.ru}

\thanks{Submitted October 6, 2015. Published January 11, 2016.}
\subjclass[2010]{34K20}
\keywords{Time-delay systems; neutral equation; exponential stability;
\hfill\break\indent attraction sets}

\begin{abstract}
 We consider a nonlinear time-delay system of neutral equations
 with constant coefficients in the linear terms
 $$
 \frac{d}{dt}\big(y(t) + D y(t-\tau)\big)
 = A y(t) + B y(t-\tau) + F(t, y(t), y(t-\tau)),
 $$
 where
 $$
 \|F(t,u,v)\| \le q_1\|u\|^{1+\omega_1} + q_2\|v\|^{1+\omega_2},
 \quad q_1, q_2,  \omega_1, \omega_2 > 0.
 $$
 We obtain estimates characterizing the exponential decay of solutions
 at infinity and estimates for attraction sets of the zero solution.
\end{abstract}

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


\section{Introduction}


There is large number of works devoted to the study of delay differential
equations (see for instance the books \cite{AgBeBrDo, AzMaRa, ElNo, Gil, GuKhaChe,
GyLa, Ha, Kha, KhuSha, KoMy, Ko, MiNi, WuHeShe} and the bibliography therein).
The question of asymptotic stability of solutions is very important
from the theoretical and practical viewpoints because delay differential
equations arise in many applied problems when describing the processes
whose speeds are defined by present and previous states
(see for example \cite{Er, Ku, MacDo} and the bibliography therein).

This article presents a continuation of our works on stability of solutions
to delay differential equations \cite{DeMa05, DeMa06, DeMa07, De09, DeKoSk,
DeVoSk, DeMa14-1, DeMa14-2, DeMa15-1, MaShche, Ma}.
We consider the system of nonlinear delay differential equations
\begin{equation}
\frac{d}{dt}\big(y(t) + D y(t-\tau)\big)
= A y(t) + B y(t-\tau) + F(t, y(t), y(t-\tau)), \quad t > 0, \label{e1.1}
\end{equation}
where $A$, $B$, $D$ are constant $(n\times n)$ matrices,
$\tau > 0$ is the time delay, and $F$
is a continuous vector function mapping
$[0, \infty) \times {\mathbb C}^n \times {\mathbb C}^n$
into ${\mathbb C}^n$.
We assume that $F(t,u,v)$ satisfies the Lipschitz condition with respect to
$u$ on every compact
$G \subset [0, \infty) \times {\mathbb C}^n \times {\mathbb C}^n$
and the inequality
\begin{equation}
\|F(t,u,v)\| \le q_1\|u\|^{1+\omega_1} + q_2\|v\|^{1+\omega_2},
\quad t \ge 0,\; u,  v \in {\mathbb C}^n,
\label{e1.2}
\end{equation}
for some constants $q_1$, $q_2$, $\omega_1$, $\omega_2 > 0$.
Here and hereafter we use the following dot product and vector norm
$$
\langle x, z \rangle = \sum^n_{j=1}x_j \bar z_j,
\quad
\|x\| =  \sqrt{\langle x, x \rangle}.
$$

Our aim is to study the exponential stability of the zero solution; namely,
to obtain estimates characterizing the decay rate of solutions at infinity
and estimates for attraction sets of the zero solution.
To establish conditions of stability, researchers often use
various Lyapunov or Lyapunov--Krasovskii functionals. At present, there is
large number of works in this direction;
for example, see the bibliographies in the survey \cite{An} and in
the book \cite{WuHeShe}
devoted wholly to obtaining conditions of stability by the use of Lyapunov-Krasovskii
functionals. However, not every Lyapunov-Krasovskii functional makes it possible to obtain
estimates characterizing exponential decay of solutions at infinity.
In recent years, the study in this direction has developed rapidly.
For constant coefficients, there are a lot of works for
linear delay differential equations including equations of neutral
type (for example, see \cite{Gil,Kha} and the bibliography therein).

The case of nonlinear equations is of special interest and is more complicated
in comparison with the case of linear equations.
Along with estimates of exponential decay of solutions,
a very important question is
deriving estimates of attraction sets for nonlinear equations.
The natural problem is to obtain such estimates
by means of the Lyapunov--Krasovskii functionals
used for exponential stability analysis of equations defined by the linear part.
To the best of our knowledge, the first constructive estimates of
attraction sets for the system
\begin{equation}
\frac{d}{dt}y(t) = A y(t) + B y(t-\tau) + F(t, y(t), y(t-\tau)),
\label{e1.3}
\end{equation}
using Lyapunov--Krasovskii functionals associated with
the exponentially stable linear system
\begin{equation}
\frac{d}{dt}y(t) = A y(t) + B y(t-\tau),
\label{e1.4}
\end{equation}
were obtained in \cite{DeMa05, DeMa06, DeMa07, MeAgNi}.

To study asymptotic stability of solutions to \eqref{e1.4}
the authors in \cite{DeMa05} proposed to use the Lyapunov--Krasovskii
functional
\begin{equation}
\langle H y(t),y(t) \rangle
+ \int^t_{t-\tau} \langle K(t-s)y(s), y(s) \rangle \, ds,
\label{e1.5}
\end{equation}
where the matrices
$H$
and
$K(s)$
satisfy
\begin{equation}
H = H^* > 0, \quad
K(s) = K^*(s) \in C^1[0,\tau], \quad
K(s) > 0, \quad
\frac{d}{ds}K(s) < 0, 
\label{e1.6}
\end{equation}
for $s \in [0,\tau]$.
Here $H > 0$ means that $H$ is positive definite.
Using \eqref{e1.5}, we obtained estimates of exponential decay
of solutions to linear systems of the form \eqref{e1.4}.
In \cite{DeMa05, DeMa06} the authors considered nonlinear systems
of delay differential
equations of the form \eqref{e1.3} with
$F(t,u,v)$ satisfying \eqref{e1.2}.
Conditions of asymptotic stability of the zero solution
were obtained, estimates characterizing the decay rate at infinity were established,
and estimates of attraction sets of the zero solution were derived.
Using a generalization of the functional in \eqref{e1.5},
analogous results were obtained for linear and
nonlinear systems of delay differential equations with periodic coefficients in the
linear terms \cite{DeMa05, DeMa06, DeMa07, DeMa14-2, MaShche, Ma}.

To study exponential stability of solutions to the system of linear differential
equations of neutral type
\begin{equation}
\frac{d}{dt}(y(t) + Dy(t-\tau)) = A y(t) + B y(t-\tau) \label{e1.7}
\end{equation}
the first author in \cite{De09} introduced the Lyapunov-Krasovskii functional
\begin{equation}
\begin{aligned}
V(\varphi)
&= \langle H(\varphi(0) + D\varphi(- \tau)), (\varphi(0)
+ D\varphi(- \tau)) \rangle \\
&\quad + \int^0_{-\tau} \langle K(-s)\varphi(s), \varphi(s) \rangle \, ds,
\quad
\varphi(s) \in C[-\tau, 0],
\end{aligned} \label{e1.8}
\end{equation}
where the matrices $H$ and $K(s)$ satisfy \eqref{e1.6}.
In particular, the following result was obtained.


\begin{theorem} \label{thm1}
Suppose that there exist matrices $H$ and $K(s)$ satisfying \eqref{e1.6}
and the matrix
\begin{equation}
C = -\begin{pmatrix}
HA+A^*H+K(0) & HB+A^*HD
\\
B^*H+D^*HA & D^*HB+B^*HD-K(\tau)
\end{pmatrix}
\label{e1.9}
\end{equation}
is positive definite.
Then the zero solution to \eqref{e1.7} is exponentially stable.
\end{theorem}

Using the functional \eqref{e1.8}, the study of exponential stability 
of solutions to time-delay
systems of the form \eqref{e1.1} was conducted in  
\cite{De09, DeKoSk, DeVoSk, DeMa14-1, DeMa15-1, Sk}.
Note that in \cite{De09, DeKoSk, Sk} the estimates of exponential decay 
of solutions to \eqref{e1.1}
were obtained in the case $\|D\| < 1$
(here and hereafter we use the spectral norm of matrices).
In \cite{DeVoSk}, for the linear case
($F(t,u,v) \equiv 0$),
analogous estimates were established when the spectrum of the matrix
$D$ belongs to the unit disk
$\{\lambda \in {\mathbb C}: |\lambda| < 1\}$.
However, in the case $\|D\| < 1$,
the estimates are weaker in comparison with the estimates obtained in \cite{De09}.
More precise exponential estimates for the linear systems were obtained
in \cite{DeMa14-1}.
If the spectrum of the matrix $D$
belongs to the unit disk, the authors in \cite{DeMa15-1}
investigated the time-delay system \eqref{e1.1} with
$F(t,u,v)$ satisfying \eqref{e1.2} with
$\omega_1=\omega_2=0$.

In this article we study a more complicated case; namely, we consider
the nonlinear time-delay system \eqref{e1.1} if
$\omega_1$, $\omega_2 > 0$.
Supposing that the spectrum of $D$
belongs to the unit disk, we establish estimates characterizing
exponential decay of solutions at infinity and estimates for attraction
sets of the zero solution. The main results are formulated in 
Theorems \ref{thm3}--\ref{thm5}
and their proofs are given in the next section.
It should be noted that some sufficient conditions for
exponential stability of the zero solution to \eqref{e1.1} are 
established in \cite[Theorem~7.5]{Gil} in the case $\|D\| < 1$.


\section{Main results}

Suppose that the conditions of Theorem \ref{thm1} are satisfied.
Using the matrices
$H$ and $K(s)$, introduce the following notation
\begin{gather}
S =\begin{pmatrix}
S_{11} & S_{12} \\
S^*_{12} & S_{22}
\end{pmatrix},  \label{e2.1}\\
S_{11} = - HA - A^*H - K(0), \quad
S_{12} = HAD + K(0)D - HB, \nonumber \\
S_{22} =  K(\tau) - D^*K(0)D, \nonumber \\
R = S_{11} - S_{12}S^{-1}_{22}S^*_{12}, \quad
P = S_{22}. \label{e2.2}
\end{gather}
It is not hard to verify that the matrix $C$
in \eqref{e1.9} is positive definite if and only if the matrix $S$
is positive definite (see the proof of Theorem \ref{thm2} for details).
Obviously, $R$ and $P$
are positive definite if and only if the matrix $S$ is positive definite.
Denote by $r_{\rm min}>0$ and $p_{\rm min}>0$
the minimal eigenvalues of $R$ and $P$, respectively.
Let $\kappa > 0$ be the maximal number such that
\begin{equation}
\frac{d}{ds}K(s) + \kappa K(s) \le 0,\quad
s \in [0,\tau].
\label{e2.3}
\end{equation}
We consider the initial value problem for \eqref{e1.1},
\begin{equation}
\begin{gathered}
{\frac{d}{dt}}(y(t) + Dy(t-\tau)) = Ay(t) + By(t-\tau) + F(t,y(t), y(t-\tau)),
\quad t > 0, \\
y(t)=\varphi(t), \quad t \in [-\tau,0], \\
y(+0) = \varphi(0),
\end{gathered}
\label{e2.4}
\end{equation}
where $\varphi(t) \in C^{1}[-\tau,0]$ is a given vector function.
Let $y(t)$ be a noncontinuable solution to the initial value problem \eqref{e2.4},
defined for $t \in [0, t')$.
Using the matrices $H$ and $K(s)$
indicated in Theorem \ref{thm1}, we consider the Lyapunov-Krasovskii functional \eqref{e1.8}.
Introducing the conventional notation
$$
y_t:  \theta \to y(t+\theta), \quad
\theta \in [-\tau,0],
$$
we have
\begin{equation}
\begin{aligned}
V(y_t) &= \langle H(y_t(0) + Dy_t(- \tau)), (y_t(0) + Dy_t(- \tau)) \rangle
+ \int^0_{-\tau} \langle K(-\theta)y_t(\theta), y_t(\theta) \rangle \, d\theta \\
&=\langle H(y(t) + Dy(t- \tau)), (y(t) + Dy(t- \tau)) \rangle\\
&\quad + \int^t_{t-\tau} \langle K(t-s)y(s), y(s) \rangle \, ds.
\end{aligned}\label{e2.5}
\end{equation}

\begin{theorem} \label{thm2}
 Let the conditions of Theorem \ref{thm1} be satisfied.
Then 
\begin{equation}
\begin{aligned}
\frac{d}{dt}V(y_t)
&\le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- (r_{\rm min} - \delta_1(\|y(t-\tau)\|)) \|y(t)+Dy(t-\tau)\|^2 \\
&\quad - (p_{\rm min} - \delta_2(\|y(t-\tau)\|)) \|z(t)\|^2
- \kappa\int^t_{t-\tau} \langle K(t-s)y(s),y(s) \rangle\,ds,
\end{aligned}\label{e2.6}
\end{equation}
for $t \in [0, t')$, where
\begin{gather}
\varepsilon_0 = \frac{2q_1\|H\|(1+\varepsilon_1)^{\omega_1}}
{h^{1+\omega_1/2}_{\rm min}}, \label{e2.7} \\
\delta_1(s) = \Big(\|S_{22}^{-1} S^*_{12}\| + \frac{1}{4\varepsilon_2} \Big)
\delta_0(s), \quad
\delta_2(s) = \varepsilon_2 \delta_0(s), \quad
s \ge 0, \label{e2.8} \\
\delta_0(s) =
2\|H\| \Big[q_1\Big(\frac{1+\varepsilon_1}{\varepsilon_1}\Big)^{\omega_1}
\|D\|^{1+\omega_1} s^{\omega_1} + q_2 s^{\omega_2}\Big], \nonumber \\
\varepsilon_1 > 0, \quad \varepsilon_2 > 0, \nonumber \\
z(t) = S_{22}^{-1} S^*_{12}(y(t)+Dy(t-\tau)) + y(t-\tau),
\label{e2.9}
\end{gather}
and $h_{\rm min} > 0$ is the minimal eigenvalue of the matrix $H$.
\end{theorem}


\begin{proof}
We use the proof scheme in \cite{DeMa05}. Obviously, the time derivative 
of the functional $V(y_t)$ is
\begin{align*}
\frac{d}{dt}V(y_t)
& \equiv \langle H (Ay(t)+By(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&\quad + \langle H (y(t)+Dy(t-\tau)), (Ay(t)+By(t-\tau)) \rangle \\
&\quad + \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle \\
&\quad + \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\quad + \langle K(0)y(t),y(t) \rangle - \langle K(\tau)y(t-\tau),y(t-\tau) \rangle \\
&\quad + \int_{t-\tau}^{t} \langle \frac{d}{dt}K(t-s)y(s),y(s) \rangle ds.
\end{align*}
Using the matrix  $C$ defined in \eqref{e1.9},
we obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt} V(y_t)
& \equiv -\big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}, \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\big\rangle \\
&\quad + \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle \\
&\quad + \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\quad + \int^t_{t-\tau} \langle \frac{d}{dt}K(t-s)y(s),y(s) \rangle ds.
\end{aligned} \label{e2.10}
\end{equation}
We consider the first summand in the right-hand side of \eqref{e2.10}.
Obviously,
$$
\begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix} =
\begin{pmatrix}
I & -D
\\
0 & I
\end{pmatrix}
\begin{pmatrix} y(t) + D y(t-\tau)\\ y(t-\tau)\end{pmatrix}.
$$
Then
$$
\big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\big\rangle
\equiv \big\langle S
\begin{pmatrix} y(t) + Dy(t-\tau)\\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t) + Dy(t-\tau)\\ y(t-\tau)\end{pmatrix}\big\rangle,
$$
where
$$
S =\begin{pmatrix}
I & 0
\\
-D^* & I
\end{pmatrix}
C
\begin{pmatrix}
I & -D
\\
0 & I
\end{pmatrix}.
$$
Taking into account \eqref{e1.9}, the matrix $S$ has the form \eqref{e2.1}.
By the conditions of Theorem \ref{thm1}, the matrix $C$ is positive definite.
Clearly, $S$ is positive definite if and only if
$C$ is positive definite. Using the representation
$$
S =\begin{pmatrix}
I & S_{12} S_{22}^{-1}
\\
0 & I
\end{pmatrix}
\begin{pmatrix}
S_{11} - S_{12} S_{22}^{-1} S^*_{12} & 0
\\
0 & S_{22}
\end{pmatrix}
\begin{pmatrix}
I & 0
\\
S_{22}^{-1} S^*_{12} & I
\end{pmatrix},
$$
we have
\begin{align*}
&\big\langle S \begin{pmatrix} y(t)+Dy(t-\tau) \\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t)+Dy(t-\tau) \\ y(t-\tau)\end{pmatrix}
\big\rangle \\
&= \langle R (y(t)+Dy(t-\tau)), (y(t)+Dy(t-\tau)) \rangle
+ \langle P z(t), z(t) \rangle,
\end{align*}
where $R$, $P$ and $z(t)$
are defined by \eqref{e2.2} and \eqref{e2.9}, respectively.
Obviously, the matrix $S$ is positive definite if and only if the matrices
$R$ and $P$ are positive definite.
Consequently, we derive
\begin{equation}
\big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix},
 \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\big\rangle
\ge r_{\rm min} \|y(t)+Dy(t-\tau)\|^2 + p_{\rm min} \|z(t)\|^2,
\label{e2.11}
\end{equation}
where $r_{\rm min}>0$ and $p_{\rm min}>0$
are the minimal eigenvalues of $R$ and $P$, respectively.

We consider the second and the third summands in the right-hand side of \eqref{e2.10}.
In view of \eqref{e1.2} we have
\begin{align*}
&\langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&+ \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\le 2\|H\|\bigl(q_1\|y(t)\|^{1+\omega_1} + q_2 \|y(t-\tau)\|^{1+\omega_2}\bigr)
\|y(t)+Dy(t-\tau)\| \\
&\le 2\|H\|\bigl(q_1 (\|y(t)+Dy(t-\tau)\| + \|D\|\|y(t-\tau)\|)^{1+\omega_1}
+ q_2 \|y(t-\tau)\|^{1+\omega_2}\bigr) \\
&\quad\times \|y(t)+Dy(t-\tau)\|.
\end{align*}
It is not hard to show that 
$$
(a + b)^{1+\omega} \le (1+\varepsilon_1)^\omega a^{1+\omega}
+ \big(\frac{1+\varepsilon_1}{\varepsilon_1}\big)^\omega b^{1+\omega},
\quad
a,  b \ge 0, \;  \varepsilon_1 > 0.
$$
Hence,
\begin{align*}
&(\|y(t)+Dy(t-\tau)\| + \|D\|\|y(t-\tau)\|)^{1+\omega_1} \\
& \le (1+\varepsilon_1)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{1+\omega_1}
+ \big(\frac{1+\varepsilon_1}{\varepsilon_1}\big)^{\omega_1}
\|D\|^{1+\omega_1}\|y(t-\tau)\|^{1+\omega_1}.
\end{align*}
For example, choosing $\varepsilon_1 = \|D\|$, we have
\begin{align*}
&(\|y(t)+Dy(t-\tau)\| + \|D\|\|y(t-\tau)\|)^{1+\omega_1} \\
& \le (1+\|D\|)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{1+\omega_1}
+ (1+\|D\|)^{\omega_1} \|D\| \|y(t-\tau)\|^{1+\omega_1}.
\end{align*}
Consequently,
\begin{align*}
&\langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&+ \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle\\
&\le 2q_1\|H\|(1+\varepsilon_1)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{2+\omega_1} \\
&\quad + 2\|H\|\Big[q_1\Big(\frac{1+\varepsilon_1}{\varepsilon_1}\Big)^{\omega_1}
\|D\|^{1+\omega_1} \|y(t-\tau)\|^{\omega_1}
+ q_2 \|y(t-\tau)\|^{\omega_2}\Big] \\
&\quad \times \|y(t-\tau)\|\|y(t)+Dy(t-\tau)\|.
\end{align*}
By the definition of $z(t)$,
$$
\|y(t-\tau)\| \le \|S_{22}^{-1} S^*_{12}\| \|y(t)+Dy(t-\tau)\| + \|z(t)\|.
$$
Hence,
\begin{align*}
&\|y(t-\tau)\|\|y(t)+Dy(t-\tau)\|\\
&\le \|S_{22}^{-1} S^*_{12}\| \|y(t)+Dy(t-\tau)\|^2 + \|z(t)\|\|y(t)+Dy(t-\tau)\| \\
&\le \Big(\|S_{22}^{-1} S^*_{12}\| + \frac{1}{4\varepsilon_2} \Big)
 \|y(t)+Dy(t-\tau)\|^2 + \varepsilon_2 \|z(t)\|^2,
\quad \varepsilon_2 > 0.
\end{align*}
Then we obtain
\begin{equation}
\begin{aligned}
&\langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&+ \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\le 2q_1\|H\|(1+\varepsilon_1)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{2+\omega_1} \\
&\quad + \delta_1(\|y(t-\tau)\|) \|y(t)+Dy(t-\tau)\|^2
  + \delta_2(\|y(t-\tau)\|) \|z(t)\|^2,
\end{aligned} \label{e2.12}
\end{equation}
where
$\delta_1(s)$, $\delta_2(s)$
are defined by \eqref{e2.8}.

By \eqref{e2.11} and \eqref{e2.12}, we have
\begin{align*}
&-\big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}, 
\begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\big\rangle \\
&+ \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle \\
&+ \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
& \le 2q_1\|H\|(1+\varepsilon_1)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{2+\omega_1} \\
&\quad - (r_{\rm min} - \delta_1(\|y(t-\tau)\|)) \|y(t)+Dy(t-\tau)\|^2
- (p_{\rm min} - \delta_2(\|y(t-\tau)\|)) \|z(t)\|^2.
\end{align*}
It follows from \eqref{e2.10} that
\begin{align*}
\frac{d}{dt}V(y_t)
&\le 2q_1\|H\|(1+\varepsilon_1)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{2+\omega_1} \\
&\quad - (r_{\rm min} - \delta_1(\|y(t-\tau)\|)) \|y(t)+Dy(t-\tau)\|^2\\
&\quad - (p_{\rm min} - \delta_2(\|y(t-\tau)\|)) \|z(t)\|^2 \\
&\quad + \int^t_{t-\tau} \big\langle \frac{d}{dt}K(t-s)y(s),y(s) \big\rangle ds.
\end{align*}
By \eqref{e2.3}, we obtain
\begin{align*}
\frac{d}{dt}V(y_t)
&\le 2q_1\|H\|(1+\varepsilon_1)^{\omega_1} \|y(t)+Dy(t-\tau)\|^{2+\omega_1} \\
&\quad - (r_{\rm min} - \delta_1(\|y(t-\tau)\|)) \|y(t)+Dy(t-\tau)\|^2\\
&\quad - (p_{\rm min} - \delta_2(\|y(t-\tau)\|)) \|z(t)\|^2
- \kappa\int^t_{t-\tau} \langle K(t-s)y(s),y(s) \rangle\,ds.
\end{align*}
Using the matrix $H$, we have
\begin{equation}
\begin{aligned}
&\frac{1}{\|H\|}
\big\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \big\rangle \\
&\le \|y(t) + D y(t-\tau)\|^2 \\
&\le \frac{1}{h_{\rm min}}
\big\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \big\rangle,
\end{aligned}\label{e2.13}
\end{equation}
where $h_{\rm min}>0$ is the minimal eigenvalue of $H$.
Hence,
\begin{align*}
\frac{d}{dt}V(y_t)
&\le \frac{2q_1\|H\|(1+\varepsilon_1)^{\omega_1}}{h^{1+\omega_1/2}_{\rm min}}
\big\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \big\rangle^{1+\omega_1/2}\\
&\quad - (r_{\rm min} - \delta_1(\|y(t-\tau)\|)) \|y(t)+Dy(t-\tau)\|^2 \\
&\quad - (p_{\rm min} - \delta_2(\|y(t-\tau)\|)) \|z(t)\|^2
- \kappa\int^t_{t-\tau} \langle K(t-s)y(s),y(s) \rangle\,ds.
\end{align*}
By the definition of $V(y_t)$, we obtain
\begin{align*}
\frac{d}{dt}V(y_t)
&\le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- (r_{\rm min} - \delta_1(\|y(t-\tau)\|)) \|y(t)+Dy(t-\tau)\|^2\\
&\quad - (p_{\rm min} - \delta_2(\|y(t-\tau)\|)) \|z(t)\|^2
- \kappa\int^t_{t-\tau} \langle K(t-s)y(s),y(s) \rangle\,ds,
\end{align*}
where $\varepsilon_0$ is defined by \eqref{e2.7}.
The proof is complete.
\end{proof}

The parameters $\varepsilon_1$, $\varepsilon_2 > 0$ allow us to control
$\varepsilon_0$, $\delta_1(s)$, $\delta_2(s)$
in \eqref{e2.6}.


Let $\sigma > 0$ be a number such that
$$
\delta_0(\sigma) < \min\Big\{
\frac{r_{\rm min}}{\big(\|S_{22}^{-1} S^*_{12}\| + \frac{1}{4\varepsilon_2} \big)},
\frac{p_{\rm min}}{\varepsilon_2} \Big\}.
$$
Then
$r_{\rm min} - \delta_1(\sigma) > 0$ and $p_{\rm min} - \delta_2(\sigma) > 0$.
We introduce the  notation
\begin{equation}
\gamma = \min\{r_{\rm min} - \delta_1(\sigma), \kappa\|H\|\} > 0,
\quad
\beta = \frac{\gamma}{2\|H\|}.
\label{e2.14}
\end{equation}
Since the spectrum of the matrix $D$ belongs to the unit disk
$\{\lambda \in {\mathbb C}: |\lambda| < 1\}$,
it follows that $\|D^j\| \to 0$ as $j \to \infty$.
Let $l > 0$ be the minimal integer such that
$\|D^l\| < 1$. We distinguish three cases
$$
\|D^l\| < e^{-l\beta \tau},
\quad
\|D^l\| = e^{-l\beta \tau},
\quad
e^{-l\beta \tau} < \|D^l\| < 1.
$$
We describe in every case an attraction set for the initial function
$\varphi(t)$ and show that the solution to the initial value problem \eqref{e2.4}
with this function is defined for $t > 0$.
We establish estimates characterizing exponentially decay of this
 solution at infinity.

\begin{theorem} \label{thm3}
Let the conditions of Theorem \ref{thm1} be satisfied and
\begin{equation}
\|D^l\| < e^{-l\beta \tau}. \label{e2.15}
\end{equation}
Suppose that $\varphi(t) \in \mathcal{E}_1$, where
\begin{gather}
\begin{aligned}
\mathcal{E}_1
&= \Bigl\{\varphi(s) \in C^1[-\tau, 0]:
\Phi < \sigma, \;
V(\varphi) < \Big( \frac{\gamma}{\varepsilon_0 \|H\|} \Big)^{2/\omega_1},\\
&\quad \alpha \bigl(1-\|D^l\|e^{l\beta \tau}\bigr)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \{\|D\|, \dots, \|D^l\|\} \Phi < \sigma
\Bigr\},
\end{aligned}\label{e2.16}\\
\alpha = \big[1 - \frac{\varepsilon_0 \|H\|}{\gamma} V^{\omega_1/2}(\varphi)
\big]^{-1/\omega_1} \sqrt{\frac{V(\varphi)}{h_{\rm min}}},
\quad
\Phi = \max_{s \in [-\tau,0]}\|\varphi(s)\|.
\label{e2.17}
\end{gather}
Then the solution to the initial value problem \eqref{e2.4}
is defined for $t > 0$ and
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le \Bigl[ \alpha \bigl(1-\|D^l\|e^{l\beta \tau}\bigr)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\quad + \max \left\{\|D\|e^{\beta\tau}, \dots, \|D^l\| e^{l\beta\tau}\right\} \Phi
\Bigr] e^{-\beta t},
\quad t > 0.
\end{aligned} \label{e2.18}
\end{equation}
\end{theorem}

The proof of the above theorem is based on the next two lemmas.



\begin{lemma} \label{lem1}
Let
$$
\|D^l\| < e^{-l\beta \tau}.
$$
Then
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} + \|D^{k+1}\|\Phi \\
& \le \Bigl[
\alpha \bigl(1-\|D^l\|e^{l\beta \tau}\bigr)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\quad + \max \left\{\|D\|e^{\beta\tau}, \dots, \|D^l\| e^{l\beta\tau}\right\} \Phi
\Bigr] e^{-\beta t}
\end{aligned} \label{e2.19}
\end{equation}
for
$t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$,
where $\alpha$, $\beta$, and $\Phi$
are defined in \eqref{e2.14}, \eqref{e2.17}.
\end{lemma}

\begin{proof} 
Obviously,
\begin{align*}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} + \|D^{k+1}\|\Phi \\
& \le
\Big[\alpha \sum^k_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^{k+1}\| e^{(k+1)\beta\tau} \Phi \Big] e^{-\beta t},
\quad
t \in [k\tau, (k+1)\tau).
\end{align*}
In view of the condition on
$\|D^l\|$,
we obtain the estimate
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} + \|D^{k+1}\|\Phi \\
&\le \Bigl[\alpha \sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \left\{\|D\|e^{\beta\tau}, \dots, \|D^l\| e^{l\beta\tau}\right\} \Phi
\Bigl] e^{-\beta t}.
\end{aligned} \label{e2.20}
\end{equation}
We consider the series
$ \sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}$.
Obviously,
\begin{align*}
&\sum^\infty_{j=0} \|D^j\| e^{j\beta\tau} \\
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{2l-1}_{j=l} \|D^j\| e^{j\beta\tau}
+ \sum^{3l-1}_{j=2l} \|D^j\| e^{j\beta\tau} + \dots \\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+\|D^l\| e^{l\beta\tau} \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \left(\|D^l\| e^{l\beta\tau}\right)^2
 \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} + \dots \\
& = \left(1 + \|D^l\| e^{l\beta\tau} + \left(\|D^l\| e^{l\beta\tau}\right)^2
+ \dots \right)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
By \eqref{e2.15}, we have
$$
\sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}
\le \left(1 - \|D^l\| e^{l\beta\tau} \right)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Using this inequality, we derive \eqref{e2.19} from \eqref{e2.20}.
The proof is complete.
\end{proof}

\begin{lemma} \label{lem2}
Let the conditions of Theorem \ref{thm3} be satisfied.
Then the solution to the initial value problem \eqref{e2.4} is defined for
$t > 0$;
moreover,  on each segment 
$t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$, it satisfies
the  estimate
\begin{equation}
\|y(t)\| \le \alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)}
+ \|D^{k+1}\|\Phi,
\label{e2.21}
\end{equation}
where $\alpha$, $\beta$, and $\Phi$
are defined in \eqref{e2.14}, \eqref{e2.17}.
\end{lemma}

\begin{proof}
Let $t \in [0, \tau)$.
If the initial function
$\varphi(t)$ belongs to the attraction set
$\mathcal{E}_1$ defined by \eqref{e2.16}, then
$$
\max_{s \in [-\tau,0]}\|\varphi(s)\| < \sigma\,.
$$
Consequently,
\begin{gather*}
r_{\rm min} - \delta_1(\|y(t-\tau)\|) = r_{\rm min} - \delta_1(\|\varphi(t-\tau)\|) >
r_{\rm min} - \delta_1(\sigma) > 0, \\
p_{\rm min} - \delta_2(\|y(t-\tau)\|) = p_{\rm min} - \delta_2(\|\varphi(t-\tau)\|) >
p_{\rm min} - \delta_2(\sigma) > 0.
\end{gather*}
By Theorem \ref{thm2},
\begin{align*}
\frac{d}{dt}V(y_t)
&\le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- (r_{\rm min} - \delta_1(\sigma)) \|y(t)+Dy(t-\tau)\|^2\\
&\quad - \kappa\int^t_{t-\tau}
\langle K(t-s)y(s),y(s) \rangle\,ds
\end{align*}
for $t \in [0, t_1)$, where $t_1 = \min\{\tau, t'\}$.
Using \eqref{e2.13}, we have
\begin{align*}
&\frac{d}{dt} V(y_t) \\
& \le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- \frac{r_{\rm min} - \delta_1(\sigma)}{\|H\|}
\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \rangle \\
&\quad - \kappa \int^t_{t-\tau}
\langle K(t-s)y(s),y(s) \rangle ds.
\end{align*}
Taking into account \eqref{e2.5}, we obtain
$$
\frac{d}{dt} V(y_t) \le \varepsilon_0 V^{1+\omega_1/2}(y_t) 
- \frac{\gamma}{\|H\|} V(y_t),
$$
where $\gamma$ is defined in \eqref{e2.14}.
If $\varphi(t) \in \mathcal{E}_1$ then
$$
\frac{\varepsilon_0\|H\|}{\gamma}V^{\omega_1/2}(\varphi) < 1.
$$
By a Gronwall-like inequality (for example, see \cite{Har}), 
we have the estimate
$$
V(y_t) \le \big[1 - \frac{\varepsilon_0\|H\|}{\gamma}V^{\omega_1/2}(\varphi)
\big]^{-2/\omega_1} V(\varphi)\exp\big(-\frac{\gamma t}{\|H\|}\big).
$$
Using the definition of the functional \eqref{e2.5}, we obtain
$$
\|y(t) + Dy(t-\tau)\|
\le \sqrt{\frac{V(y_t)}{h_{\rm min}}}.
$$
Consequently,
$$
\|y(t) + Dy(t-\tau)\|
\le
\Big[1 - \frac{\varepsilon_0\|H\|}{\gamma}V^{\omega_1/2}(\varphi)\Big]^{-1/\omega_1}
\sqrt{\frac{V(\varphi)}{h_{\rm min}}} \exp\Big(-\frac{\gamma t}{2 \|H\|}\Big).
$$
Hence,
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le \|y(t) + Dy(t-\tau)\| + \|Dy(t-\tau)\| \\
&\le \alpha e^{-\beta t} + \|D\| \,\|\varphi(t-\tau)\|
\le \alpha e^{-\beta t} + \|D\| \Phi,
\quad t \in [0, t_1),
\end{aligned} \label{e2.22}
\end{equation}
where $\alpha$, $\beta$, and $\Phi$
are defined in \eqref{e2.14} and \eqref{e2.17}.
The function in the right-hand side of \eqref{e2.22} is continuous
and bounded for $t > 0$.
Then $t_1 = \tau$  and the noncontinuable solution
$y(t)$ to \eqref{e2.4} is defined for $t \in [0, \tau]$.
Consequently, $t' > \tau$.
It follows from \eqref{e2.22} that $y(t)$
satisfies \eqref{e2.21} for $k=0$.
Obviously, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_1$ then
$$
\|y(t)\| < \sigma, \quad t \in [0, \tau].
$$
Let $t \in [\tau, 2\tau)$. Consequently,
\begin{gather*}
r_{\rm min} - \delta_1(\|y(t-\tau)\|) >
r_{\rm min} - \delta_1(\sigma) > 0, \\
p_{\rm min} - \delta_2(\|y(t-\tau)\|) >
p_{\rm min} - \delta_2(\sigma) > 0.
\end{gather*}
By Theorem \ref{thm2},
\begin{align*}
\frac{d}{dt}V(y_t)
&\le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- (r_{\rm min} - \delta_1(\sigma)) \|y(t)+Dy(t-\tau)\|^2\\
&\quad - \kappa\int^t_{t-\tau}
\langle K(t-s)y(s),y(s) \rangle\,ds
\end{align*}
for $t \in [0, t_2)$, where
$t_2 = \min\{2\tau, t'\}$.
Repeating the same reasoning as above, we have
$$
\|y(t) + Dy(t-\tau)\| \le \alpha e^{-\beta t},
\quad t \in [0, t_2),
$$
where
$\alpha$ and $\beta$
are defined in \eqref{e2.17} and \eqref{e2.14}, respectively.
Hence,
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le \|y(t) + Dy(t-\tau)\| + \|Dy(t-\tau)\| \\
&\le \alpha e^{-\beta t} + \|Dy(t-\tau) - D^2y(t-2\tau)\| + \|D^2y(t-2\tau)\| \\
& \le \alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi,
\quad
t \in [\tau, t_2).
\end{aligned}
\label{e2.23}
\end{equation}
The function in the right-hand side of \eqref{e2.23} is continuous
and bounded for $t > 0$.
Then $t_2 = 2\tau$ and the noncontinuable solution
$y(t)$ to \eqref{e2.4} is defined for $t \in [0, 2\tau]$.
Consequently, $t' > 2\tau$.
It follows from \eqref{e2.23} that $y(t)$
satisfies \eqref{e2.21} for $k=1$.
By Lemma \ref{lem1},
\begin{align*}
&\alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi\\
&\le \Bigl[
\alpha \bigl(1-\|D^l\|e^{l\beta \tau}\bigr)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
\Bigr] e^{-\beta t}
+ \|D^2\| \Phi,
\quad
t \in [\tau, 2\tau].
\end{align*}
Consequently, if the initial function
$\varphi(t)$
belongs to the attraction set
$\mathcal{E}_1$,
then
$$
\|y(t)\| < \sigma, \quad
t \in [0, 2\tau].
$$

Repeating the same reasoning, we obtain that the solution to \eqref{e2.4}
is defined for $t > 0$,
and it satisfies \eqref{e2.21} on each segment
$t \in [k\tau, (k+1)\tau)$, $k \in {\mathbb N}$.
By Lemma \ref{lem1} and the condition
$\|D^l\| < 1$, we have
$$
\|y(t)\|
\le
\Bigl[
\alpha \bigl(1-\|D^l\|e^{l\beta \tau}\bigr)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
\Bigr] e^{-\beta t}
+ \max\{\|D\|, \dots, \|D^l\|\} \Phi, \quad
t > 0.
$$
Consequently, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_1$, then
$$
\|y(t)\| < \sigma, \quad
t > 0.
$$
The proof is complete.
\end{proof}


By Lemmas \ref{lem1} and \ref{lem2}, we obtain that the solution to \eqref{e2.4} 
satisfies \eqref{e2.18}. Therefore,  Theorem \ref{thm3} is proved.


\begin{theorem} \label{thm4}
Let the conditions of Theorem \ref{thm1} be satisfied and
\begin{equation}
\|D^l\| = e^{-l\beta \tau}. \label{e2.24}
\end{equation}
Suppose that $\varphi(t) \in \mathcal{E}_2$, where
\begin{equation}
\begin{aligned}
\mathcal{E}_2 &= \Bigl\{\varphi(s) \in C^1[-\tau, 0]:
\Phi < \sigma, \;
V(\varphi) < \big( \frac{\gamma}{\varepsilon_0 \|H\|} \big)^{2/\omega_1},\\
&\quad \frac{\alpha}{l\tau\beta} e^{l\tau\beta-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \{\|D\|, \dots, \|D^l\|\} \Phi < \sigma
\Bigr\}.
\end{aligned} \label{e2.25}
\end{equation}
Then the solution to the initial value problem \eqref{e2.4}
is defined for $t > 0$ and the estimate holds
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le \Bigl[ \alpha \big(1+\frac{t}{l\tau}\big)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\\
&\quad + \max \left\{1, \|D\|e^{\beta\tau}, \dots, \|D^{l-1}\| e^{(l-1)\beta\tau}
\right\} \Phi \Bigr] e^{-\beta t},
\quad t > 0,
\end{aligned} \label{e2.26}
\end{equation}
where $\alpha$, $\beta$, and $\Phi$
are defined in \eqref{e2.14}, \eqref{e2.17}.
\end{theorem}

The proof of the above theorem is based on the next two lemmas.

\begin{lemma} \label{lem3}
 Let
$$
\|D^l\| = e^{-l\beta \tau}.
$$
Then
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} + \|D^{k+1}\|\Phi \\
& \le \Bigl[ \alpha \big(1+\frac{t}{l\tau}\big)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\\
&\quad + \max \left\{1, \|D\|e^{\beta\tau}, \dots, \|D^{l-1}\| e^{(l-1)\beta\tau}\right\}
\Phi \Bigr] e^{-\beta t}
\end{aligned} \label{e2.27}
\end{equation}
for $t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$.
where $\alpha$, $\beta$, and $\Phi$ are defined in \eqref{e2.14}, \eqref{e2.17}.
\end{lemma}

\begin{proof} 
Obviously,
\begin{align*}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} + \|D^{k+1}\|\Phi \\
& \le \Big[\alpha \sum^k_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^{k+1}\| e^{(k+1)\beta\tau} \Phi \Big]e^{-\beta t},
\quad
t \in [k\tau, (k+1)\tau).
\end{align*}
In view of the condition \eqref{e2.24} on $\|D^l\|$,
we obtain the estimate
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)}+ \|D^{k+1}\|\Phi\\
&\le \Big[
\alpha \sum^k_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \left\{1, \|D\|e^{\beta\tau}, \dots,
\|D^{l-1}\| e^{(l-1)\beta\tau}\right\} \Phi \Big]
e^{-\beta t}.
\end{aligned} \label{e2.28}
\end{equation}
If $k \le l-1$ then \eqref{e2.27} follows from~\eqref{e2.28}.


Let $l \le k \le 2l-1$; i.e., $ 1 \le \frac{t}{l\tau} < 2$.
We consider the sum $ \sum^k_{j=0} \|D^j\| e^{j\beta\tau}$.
Clearly,
\begin{align*}
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} + \sum^k_{j=l} \|D^j\| e^{j\beta\tau}\\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^l\| e^{l\beta\tau} \sum^{k-l}_{j=0} \|D^j\| e^{j\beta\tau} \\
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{k-l}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Then we have
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \frac{t}{l \tau}\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Using this inequality,
\eqref{e2.27} follows from \eqref{e2.28}.

Let $ml \le k \le (m+1)l-1$, $m = 2, 3, \dots$;
i.e., $ m \le \frac{t}{l\tau} < m+1$.
We consider the sum $ \sum^k_{j=0} \|D^j\| e^{j\beta\tau}$.
It is not difficult to see that
\begin{align*}
&\sum^k_{j=0} \|D^j\| e^{j\beta\tau}\\
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{2l-1}_{j=l} \|D^j\| e^{j\beta\tau}
+ \dots + \sum^{k}_{j=ml} \|D^j\| e^{j\beta\tau} \\
& \le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^l\| e^{l\beta\tau} \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \dots
+ \|D^{ml}\| e^{ml\beta\tau} \sum^{k-ml}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \dots
+ \sum^{k-ml}_{j=0} \|D^j\| e^{j\beta\tau}\\
&\le (1+m)\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Consequently,
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le
\big(1 + \frac{t}{l \tau} \big)\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
In view of this estimate, \eqref{e2.27} follows from \eqref{e2.28}.
Owing to the arbitrariness of $m$,
the proof is complete.
\end{proof}

\begin{lemma} \label{lem4}
Let the conditions of Theorem \ref{thm4} be satisfied.
Then the solution to the initial value problem \eqref{e2.4} is defined for
$t > 0$; moreover, it satisfies \eqref{e2.21} on each segment
$t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$.
\end{lemma}

\begin{proof}
Recall that the noncontinuable solution to \eqref{e2.4} is defined for
$t \in [0, t')$.
Let $t \in [0, \tau)$.
Repeating the same reasoning as in the proof of Lemma \ref{lem2}, we obtain that
$y(t)$ satisfies \eqref{e2.22} and is defined for
$t \in [0, \tau]$. Consequently, $t' > \tau$.
It follows from \eqref{e2.22} that
$y(t)$ satisfies \eqref{e2.21} for $k=0$.
Obviously, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_2$
defined by \eqref{e2.25}, then
$$
\|y(t)\| < \sigma, \quad t \in [0, \tau].
$$
Let $t \in [\tau, 2\tau)$. Consequently,
\begin{gather*}
r_{\rm min} - \delta_1(\|y(t-\tau)\|) >
r_{\rm min} - \delta_1(\sigma) > 0, \\
p_{\rm min} - \delta_2(\|y(t-\tau)\|) >
p_{\rm min} - \delta_2(\sigma) > 0.
\end{gather*}
By Theorem \ref{thm2},
\begin{align*}
\frac{d}{dt}V(y_t)
&\le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- (r_{\rm min} - \delta_1(\sigma)) \|y(t)+Dy(t-\tau)\|^2 \\
&\quad - \kappa\int^t_{t-\tau}
\langle K(t-s)y(s),y(s) \rangle\,ds
\end{align*}
for $t \in [0, t_2)$, where $t_2 = \min\{2\tau, t'\}$.
By a similar way as in the proof of Lemma \ref{lem2}, we have
$$
\|y(t) + Dy(t-\tau)\| \le \alpha e^{-\beta t}, \quad
t \in [0, t_2),
$$
where $\alpha$ and $\beta$ are defined in \eqref{e2.17} and \eqref{e2.14}, 
respectively. Hence,
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le \|y(t) + Dy(t-\tau)\| + \|Dy(t-\tau)\|\\
&\le \alpha e^{-\beta t} + \|Dy(t-\tau) - D^2y(t-2\tau)\| + \|D^2y(t-2\tau)\|\\
& \le \alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi,
\quad
t \in [\tau, t_2).
\end{aligned}\label{e2.29}
\end{equation}
The function in the right-hand side of \eqref{e2.29} is continuous and bounded for
$t > 0$.
Then $t_2 = 2\tau$ and the noncontinuable solution $y(t)$
to \eqref{e2.4} is defined for $t \in [0, 2\tau]$. Consequently,
$t' > 2\tau$. It follows from \eqref{e2.29} that $y(t)$
satisfies \eqref{e2.21} for $k=1$.
By Lemma \ref{lem3},
\begin{equation}
\begin{aligned}
&\alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi \\
& \le \Bigl[\alpha \left(1+\frac{t}{l\tau}\right)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \Bigr] e^{-\beta t} + \|D^2\| \Phi,
\quad t \in [\tau, 2\tau].
\end{aligned} \label{e2.30}
\end{equation}
We consider the function
$$
f(t) = \big(1+\frac{t}{l\tau}\big)e^{-\beta t}, \quad t \ge 0.
$$
It is not difficult to show that
$$
f(t) \le \begin{cases}
\frac{1}{l\tau\beta} e^{l\tau\beta - 1}, \quad & l\tau\beta \le 1,
\\
1, & l\tau\beta \ge 1.
\end{cases}
$$
Obviously, $ \frac{1}{l\tau\beta} e^{l\tau\beta - 1} \ge 1$
for $\tau$, $\beta > 0$.
Then
$$
f(t) \le \frac{1}{l\tau\beta} e^{l\tau\beta - 1},
\quad t \ge 0,
$$
for any $\tau$, $\beta > 0$.
Taking into account the last inequality, from \eqref{e2.30} we have
\begin{align*}
&\alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi \\
&\le \frac{\alpha}{l\tau\beta} e^{l\tau\beta - 1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^2\| \Phi, \quad
t \in [\tau, 2\tau].
\end{align*}
Consequently, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_2$ then
$$
\|y(t)\| < \sigma, \quad t \in [0, 2\tau].
$$

Repeating the same reasoning, we obtain that the solution to \eqref{e2.4}
is defined for $t > 0$, and it satisfies \eqref{e2.21} on each segment
$t \in [k\tau, (k+1)\tau)$, $k \in {\mathbb N}$.
By Lemma \ref{lem3} and the condition $\|D^l\| < 1$, we have
\begin{align*}
\|y(t)\| 
&\le \Bigl[\alpha \big(1+\frac{t}{l\tau}\big)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\Bigr] e^{-\beta t}
+ \max\{\|D\|, \dots, \|D^l\|\} \Phi \\
&\le \frac{\alpha}{l\tau\beta} e^{l\tau\beta - 1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \max\{\|D\|, \dots, \|D^l\|\} \Phi, \quad t > 0.
\end{align*}
Consequently, if the initial function $\varphi(t)$
belongs to the attraction set
$\mathcal{E}_2$
then
$$
\|y(t)\| < \sigma,
\quad
t > 0.
$$
The proof is complete.
\end{proof}

By Lemmas \ref{lem3} and \ref{lem4}, we obtain that the solution to \eqref{e2.4} 
satisfies \eqref{e2.26}.
Therefore, Theorem \ref{thm4} is proved.

\begin{theorem} \label{thm5}
Let the conditions of Theorem \ref{thm1} be satisfied and
\begin{equation}
e^{-l\beta \tau} < \|D^l\| < 1. \label{e2.31}
\end{equation}
Suppose that $\varphi(t) \in \mathcal{E}_3$, where
\begin{equation}
\begin{aligned}
\mathcal{E}_3 &= \Bigl\{\varphi(s) \in C^1[-\tau, 0]:
\Phi < \sigma, \;
V(\varphi) < \big( \frac{\gamma}{\varepsilon_0 \|H\|} \big)^{2/\omega_1}, \\
&\alpha \Big(1 - \big(\|D^l\|e^{l\beta\tau}\big)^{-1}\Big)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \{\|D\|, \dots, \|D^l\|\} \Phi < \sigma
\Bigr\}.
\end{aligned} \label{e2.32}
\end{equation}
Then the solution to the initial value problem \eqref{e2.4} is defined for
$t > 0$ and
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le  \Bigr[ \alpha
\Big(1 - \big(\|D^l\|e^{l\beta\tau}\big)^{-1}\Big)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\quad + \|D^l\|^{\frac{1}{l} - 1}\max \left\{1, \|D\|, \dots, \|D^{l-1}\|
 \right\} \Phi \Bigr]
\exp\Big( \frac{t}{l\tau} \ln\|D^l\| \Big),
\end{aligned}\label{e2.33}
\end{equation}
for $t > 0$, where $\alpha$, $\beta$, and $\Phi$ are defined in 
\eqref{e2.14}, \eqref{e2.17}.
\end{theorem}

The proof of the above theorem is based on the next two lemmas.


\begin{lemma} \label{lem5}
Let
$$
e^{-l\beta \tau} < \|D^l\| < 1.
$$
Then
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)}
+ \|D^{k+1}\|\Phi \\
&\le \Bigr[\alpha
\Big(1 - \big(\|D^l\|e^{l\beta\tau}\big)^{-1}\Big)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\quad + \|D^l\|^{\frac{1}{l} - 1}\max \left\{1, \|D\|, \dots, \|D^{l-1}\| \right\} \Phi
\Bigr]
\exp\Big( \frac{t}{l\tau} \ln\|D^l\| \Big)
\end{aligned} \label{e2.34}
\end{equation}
for $t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$.
where $\alpha$, $\beta$, and $\Phi$ are defined in \eqref{e2.14}, \eqref{e2.17}.
\end{lemma}

\begin{proof}
First we consider the first summand in the left-hand side of \eqref{e2.34}.
For $k\le l-1$ we obviously have
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Let
$ml \le k \le (m+1)l-1$,
$m = 1, 2, 3, \dots$.
Clearly,
\begin{align*}
&\sum^k_{j=0} \|D^j\| e^{j\beta\tau}\\
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{2l-1}_{j=l} \|D^j\| e^{j\beta\tau} + \dots
+ \sum^{k}_{j=ml} \|D^j\| e^{j\beta\tau} \\
& \le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^l\| e^{l\beta\tau} \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} + \dots
+ \|D^{ml}\| e^{ml\beta\tau} \sum^{k-ml}_{j=0} \|D^j\| e^{j\beta\tau} \\
& \le \left[1 + \|D^l\|e^{l\beta\tau} + \dots + \|D^l\|^m e^{ml\beta\tau}\right]
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Consequently,
\begin{align*}
&\sum^k_{j=0} \|D^j\| e^{j\beta\tau}\\
&\le \|D^l\|^m e^{ml\beta\tau}
\Big[1 + \left(\|D^l\|e^{l\beta\tau}\right)^{-1} + \dots +
\left(\|D^l\| e^{l\beta\tau}\right)^{-m}\Big]
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\le \|D^l\|^m e^{ml\beta\tau}
\Big[1 + \left(\|D^l\|e^{l\beta\tau}\right)^{-1} + \dots +
\left(\|D^l\| e^{l\beta\tau}\right)^{-m} + \dots\Big]
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Since $\|D^l\|e^{l\beta\tau}>1$ owing to \eqref{e2.31}, we have
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le \|D^l\|^m e^{ml\beta\tau}
\left[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\right]^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Taking into account that $m l \tau \le t < (m+1) l \tau$,
we obtain
\begin{align*}
\sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)}
&\le \|D^l\|^m e^{-\beta (t-ml\tau)}
\Big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\Big]^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\le \|D^l\|^{\frac{t}{l\tau}}
\Big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\Big]^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
As a result, we derive the estimate for the first summand in
\eqref{e2.34} for every $k$,
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} \\
&\le \alpha \Big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\Big]^{-1}
\Big(\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\Big)
\exp\big(\frac{t}{l \tau} \ln \|D^l\| \big).
\end{aligned} \label{e2.35}
\end{equation}
We now we consider the second summand in the left-hand side of \eqref{e2.34}.
Obviously, for $0 \le k \le l-2$ we have
$$
\|D^{k+1}\| \le \max \big\{\|D\|, \dots, \|D^{l-1}\| \big\}.
$$
Let $ml-1 \le k \le (m+1)l-2$, $m = 1, 2, \dots$. Hence,
$$
\|D^{k+1}\| \le \|D^l\|^m \|D^{k+1-ml}\|
\le \|D^l\|^m \max \left\{1, \|D\|, \dots, \|D^{l-1}\| \right\}.
$$
Since $\|D^l\|<1$ and $t < ((m+1)l-1)\tau$, it follows that
$$
\|D^l\|^m \le \|D^l\|^{\frac{t-(l-1)\tau}{l\tau}}
= \|D^l\|^{\frac{1}{l}-1}\exp\Big(\frac{t}{l \tau} \ln \|D^l\| \Big).
$$
Owing to arbitrariness of $m$, we infer that
$$
\|D^{k+1}\|
\le \|D^l\|^{\frac{1}{l}-1}
\max \left\{1, \|D\|, \dots, \|D^{l-1}\| \right\}
\exp\big(\frac{t}{l \tau} \ln \|D^l\| \big)
$$
for every $k$.
Taking into account the estimate~\eqref{e2.35}, we derive~\eqref{e2.34}.
The proof is complete.
\end{proof}

\begin{lemma} \label{lem6}
Let the conditions of Theorem \ref{thm5} be satisfied.
Then the solution to the initial value problem \eqref{e2.4} is defined for
$t > 0$; moreover, it satisfies \eqref{e2.21} on each segment
$t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$.
\end{lemma}

\begin{proof}
 Recall that the noncontinuable solution to \eqref{e2.4}
is defined for $t \in [0, t')$.
Let $t \in [0, \tau)$.
Repeating the same reasoning as in the proof of Lemma \ref{lem2}, we obtain that
$y(t)$ satisfies \eqref{e2.22} and is defined for $t \in [0, \tau]$.
Consequently, $t' > \tau$.
It follows from \eqref{e2.22} that $y(t)$
satisfies \eqref{e2.21} for $k=0$.
Obviously, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_3$
defined by \eqref{e2.32}, then
$$
\|y(t)\| < \sigma, \quad
t \in [0, \tau].
$$
Let $t \in [\tau, 2\tau)$. Consequently,
\begin{gather*}
r_{\rm min} - \delta_1(\|y(t-\tau)\|) >
r_{\rm min} - \delta_1(\sigma) > 0, \\
p_{\rm min} - \delta_2(\|y(t-\tau)\|) >
p_{\rm min} - \delta_2(\sigma) > 0.
\end{gather*}
By Theorem \ref{thm2},
\begin{align*}
\frac{d}{dt}V(y_t)
&\le \varepsilon_0 V^{1+\omega_1/2}(y_t)
- (r_{\rm min} - \delta_1(\sigma)) \|y(t)+Dy(t-\tau)\|^2\\
&- \kappa\int^t_{t-\tau}
\langle K(t-s)y(s),y(s) \rangle\,ds
\end{align*}
for $t \in [0, t_2)$, where $t_2 = \min\{2\tau, t'\}$.
Repeating the same reasoning as above, we have
$$
\|y(t) + Dy(t-\tau)\| \le \alpha e^{-\beta t}, \quad
t \in [0, t_2),
$$
where
$\alpha$ and $\beta$
are defined in \eqref{e2.17} and \eqref{e2.14}, respectively.
Hence,
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le \|y(t) + Dy(t-\tau)\| + \|Dy(t-\tau)\|\\
&\le \alpha e^{-\beta t} + \|Dy(t-\tau) - D^2y(t-2\tau)\| + \|D^2y(t-2\tau)\|\\
&\le \alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi,
\quad
t \in [\tau, t_2).
\end{aligned}\label{e2.36}
\end{equation}
The function in the right-hand side of \eqref{e2.36} is continuous and bounded for
$t > 0$.
Then $t_2 = 2\tau$ and the noncontinuable solution
$y(t)$ to \eqref{e2.4} is defined for $t \in [0, 2\tau]$.
Consequently, $t' > 2\tau$.
It follows from \eqref{e2.36} that $y(t)$
satisfies \eqref{e2.21} for $k=1$.
By Lemma \ref{lem5},
\begin{align*}
&\alpha e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)} + \|D^2\| \Phi \\
&\le \alpha
\Big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\Big]^{-1}
\Big(\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\Big)
\exp\Big(\frac{t}{l \tau} \ln \|D^l\| \Big)
+ \|D^2\| \Phi, 
\end{align*}
for $t \in [\tau, 2\tau]$.
Consequently, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_3$, then
$$
\|y(t)\| < \sigma, \quad
t \in [0, 2\tau].
$$

Repeating the same reasoning, we obtain that the solution to \eqref{e2.4}
is defined for $t > 0$, and it satisfies \eqref{e2.21} on each segment
$t \in [k\tau, (k+1)\tau)$, $k \in {\mathbb N}$.
By Lemma \ref{lem5} and the condition $\|D^l\| < 1$,
we have
\begin{align*}
\|y(t)\|
&\le \alpha
\Big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\Big]^{-1}
\Big(\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\Big)
\exp\big(\frac{t}{l \tau} \ln \|D^l\| \big) \\
&\quad + \max\{\|D\|, \dots, \|D^l\|\} \Phi, \quad
t > 0.
\end{align*}
Consequently, if the initial function $\varphi(t)$
belongs to the attraction set $\mathcal{E}_3$ then
$$
\|y(t)\| < \sigma, \quad t > 0.
$$
The proof is complete.
\end{proof}

By Lemmas \ref{lem5} and \ref{lem6}, we obtain that the solution
 to \eqref{e2.4} satisfies  \eqref{e2.33}.
Therefore, Theorem \ref{thm5} is proved.



\subsection*{Conclusion}
In this article, we investigated the nonlinear time-delay system \eqref{e1.1}
of neutral type with constant coefficients in the linear terms.
Supposing that the spectrum of $D$
belongs to the unit disk, we indicated sufficient conditions under which
the zero solution to \eqref{e1.1} is exponentially stable.
Depending on the norms of the powers of $D$,
we established the constructive estimates for solutions to \eqref{e1.1} and
attraction sets of the zero solution to \eqref{e1.1} (see Theorems
\ref{thm3}, \ref{thm4}, \ref{thm5}).
All the values characterizing the exponential decay rate of the solutions
at infinity and the attraction sets are written out in explicit form.


\subsection*{Acknowledgments}
The authors are grateful to the anonymous referee for the helpful comments and
suggestions.
This research was supported by the Russian Foundation for Basic Research
(project no. 16-01-00592).

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\end{document}