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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 56, pp. 1--14.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/56\hfil On the $\Psi $-stability]
{On the $\Psi $-stability of a nonlinear Volterra integro-differential system}

\author[A. Diamandescu\hfil EJDE-2005/56\hfilneg]
{Aurel Diamandescu}
\address{Aurel Diamandescu \hfill\break 
University of Craiova \\
Department of Applied Mathematics\\
13, ``Al. I. Cuza'' st.\\
200585, Craiova, Romania}
\email{adiamandescu@central.ucv.ro}

\date{}
\thanks{Submitted March 29, 2005. Published May 31, 2005.}
\subjclass[2000]{45M10, 45J05}
\keywords{$\Psi$-stability; $\Psi$-uniform stability}

\begin{abstract}
In this paper we prove sufficient conditions for $\Psi$-stability of the
zero solution of a nonlinear Volterra integro-differential system.
\end{abstract}


\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Akinyele \cite{a1} introduced the notion of $\Psi$-stability of degree $k$
with respect to a function $\Psi \in C(\mathbb{R}_{+},\mathbb{R}_{+})$,
increasing and differentiable on $\mathbb{R}_{+}$ and such that $\Psi
(t)\geq 1$ for $t\geq 0$ and $\lim_{t\to \infty} \Psi (t) =b$, 
$b\in [1,\infty )$. The fact that the function $\Psi $ is bounded 
does not enable a
deeper analysis, of the asymptotic properties of the solutions of a
differential equations, than the notion of stability in sense Lyapunov.

Constantin \cite{c3} introduced the notions of degree of stability and
degree of boundedness of solutions of an ordinary differential equation,
with respect to a continuous positive and nondecreasing function 
$\Psi:\mathbb{R}_{+}\to\mathbb{R}_{+}$. Some criteria for these notions 
are proved
there too.

Morchalo \cite{m2} introduced the notions of $\Psi$-stability, 
$\Psi$-uniform stability, and $\Psi$-asymptotic stability of trivial solution of
the nonlinear system $x'= f(t,x)$. Several new and sufficient
conditions for mentioned types of stability are proved for the linear system
$x'= A(t)x$. Furthermore, sufficient conditions are given for the
uniform Lipschitz stability of the system $x'=f(t,x) + g(t,x)$. In
this paper, the function $\Psi$ is a scalar continuous function.

The purpose of our paper is to prove sufficient conditions for 
$\Psi$-(uniform) stability of trivial solution of the nonlinear Volterra
integro-differential system
\begin{equation}
x'= A(t)x + \int_0^t F(t,s,x(s))\,ds  \label{e1}
\end{equation}
which can be seen as a perturbed system of
\begin{equation}
y'= A(t)y  \label{e2}
\end{equation}

We investigate conditions on the fundamental matrix $Y(t)$ for the linear
system \eqref{e2} and on the function $F(t,s,x)$ under which the trivial
solution of \eqref{e1} or \eqref{e2} is $\Psi$-(uniformly) stable on $%
\mathbb{R}_{+}$. Here, $\Psi $ is a matrix function whose introduction
permits us obtaining a mixed asymptotic behavior for the components of
solutions.

Recent works for stability of solutions of \eqref{e1} have been given by
Mahfoud \cite{m1} who used Lyapunov functionals; Lakshmikantham and Rama
Mohana Rao \cite{l1} who used the comparison method; Hara, Yoneyama and Itoh
\cite{h1} who used ``variation of parameters'' formula; in other words, the
solution of equation \eqref{e1} with the initial function $\varphi $ on 
$[0, t_0]$ - namely $x(t) = \varphi (t)$ for t $\in [0, t_0]$ - is written
\begin{equation*}
x(t;t_0,\varphi ) = Y( t) Y^{-1}( {t}_0) \varphi (t_0) + \int_0^t{Y(t)Y}^{-1}
{(s)}\int_0^{s}{F(s,u,x(u;t}_0 ,\varphi ))\,du\,ds\,;
\end{equation*}
and by Avramescu \cite{a2} who used the method of admissibility of a pair of
subspaces with respect to an operator.

\section{Definitions, notation and hypotheses}

Let $\mathbb{R}^n$ denote the Euclidean $n$-space. For 
$x = (x_{1},x_{2}, x_{3},\dots, x_{n})^T$ in 
$\mathbb{R}^n$, let $\| x\| =\max\{ | x_{1}| , |x_{2}|, \dots , | x_{{n}}| \}$ 
be the norm of $x$. For an $n \times n$
matrix $A = ( a_{ij})$, we define the norm 
$|A|  = \sup_{\| x\| \leq 1}\|Ax\| $.

In the system \eqref{e1} we assume that $A$ is a continuous $n \times n$
matrix on $\mathbb{R}_{+} = [0,\infty )$ and $F : D \times \mathbb{R}^n \to
\mathbb{R}^n$, $D = \{ (t,s)\in \mathbb{R}^2 : 0\leq s\leq t<\infty \}$, is
a continuous $n$-vector such that $F(t,s,0) = 0$ for $(t,s)\in D$.

Let $\Psi _{{i}}:\mathbb{R}_{+}\to (0,\infty )$, $i =1,2\dots n$, be
continuous functions and
\begin{equation*}
\Psi = \mathop{\rm diag} [\Psi _{1},\Psi _{2},\dots \Psi _{n}].
\end{equation*}
Now, we give definitions of various types of $\Psi$-stability.

\subsection*{Definitions}

The trivial solution of \eqref{e1} is said to be $\Psi$-stable on 
$\mathbb{R}_{+}$ if for every $\varepsilon >0$ and every $t_0$ in 
$\mathbb{R}_{+}$,
there exists $\delta =\delta(\varepsilon ,t_0) > 0$ such that any solution 
$x(t)$ of \eqref{e1} which satisfies the inequality 
$\|\Psi (t_0)x(t_0)\| <\delta $, also satisfies the inequality 
$\| \Psi (t)x(t) \| < \varepsilon $ for all $t\geq t_0$.

The trivial solution of \eqref{e1} is said to be $\Psi$-uniformly stable on 
$\mathbb{R}_{+}$ if it is $\Psi$-stable on $\mathbb{R}_{+}$ and the above 
$\delta $ is independent of $t_0$.

\subsection*{Remarks}

1. For $\Psi _{i}=1$, $i=1,2\dots n$, we obtain the notions of classical
stability and uniform-stability.

2. If in the definitions above, we replace $\Psi $ with $\Psi ^{k}$, $k\in
\mathbb{Z}\setminus \{0,1\}$, we obtain stability and uniform-stability of
degree $k$ with respect to a scalar function $\Psi $ \cite{c3}.

\section{$\Psi$-stability of linear systems}

The purpose of this section is to study conditions for $\Psi$-(uniform)
stability of trivial solution of linear systems. These conditions can be
expressed in terms of a fundamental matrix for \eqref{e2}.

\begin{theorem} \label{thm1}
Let $Y(t)$ be a fundamental matrix for \eqref{e2}. Then
\begin{itemize}
\item[(a)] The trivial solution of \eqref{e2} is
$\Psi$-stable on $\mathbb{R}_{+}$ if and only if there exists a
positive constant $K$ such that $|\Psi(t)Y(t)| \leq K$ for all
$t\geq 0$.

\item[(b)] The trivial solution of \eqref{e2} is $\Psi$-uniformly stable on
$\mathbb{R}_{+}$  if and only if there exists a positive constant
$K$ such that $|\Psi(t)Y(t)Y^{-1}(s)\Psi ^{-1}(s)|\leq  K$
 for all $0 \leq  s \leq t < \infty $.
\end{itemize}
\end{theorem}

\begin{proof}
The solution of \eqref{e2} which takes the value $y $ in $\mathbb{R}^n$
at $a \geq  0$ is $y(t) = Y(t)Y^{-1}(a)y $ for $t \geq 0$.

Suppose first that the trivial solution of \eqref{e2} is $\Psi $-stable on
$\mathbb{R}_{+}$. Then,  for $\varepsilon = 1$ and $t_0 =0$,
there exists $\delta > 0$ such that any solution $y(t)$ of \eqref{e2}
which satisfies the inequality $\|\Psi (0)y(0) \| < \delta $,
there exists and satisfies the inequality
\[
\| \Psi (t)Y(t)(\Psi (0)Y(0))^{-1}\Psi (0)y(0)
\| < 1 \quad\text{for } t \geq 0.
\]
Let u $\in \mathbb{R}^n$ be such that $\| u \|\leq 1$.
If we take $y(0) = \frac{\delta }{2}\Psi ^{-1}(0)u$, then we have
$\| \Psi (0)y(0) \|  < \delta $. Hence,
$\|\Psi (t)Y(t)(\Psi (0)Y(0))^{-1}\frac{\delta }{2}u \| < 1$
 for $t \geq 0$. Therefore, $| \Psi (t)Y(t)(\Psi(0)Y(0))^{-1}| \leq  2/\delta $
for $t \geq 0$. Hence, $|\Psi (t)Y(t) |\leq  K$, a constant, for $t \geq  0$.

Suppose next that $| \Psi (t)Y(t) | \leq  K$ for $t \geq  0$. For
$\varepsilon  > 0$ and $t_0 \in \mathbb{R}_{+}$, let
$\delta (\varepsilon ,t_0) =
\varepsilon K^{-1}| (\Psi (t_0)Y(t_0))^{-1}| ^{-1}$.
For $\| \Psi (t_0)y(t_0)\|  < \delta $ and $t \geq  t_0$,
 we have
\[
\| \Psi (t)y(t)\| = \| \Psi (t)Y(t)(\Psi (t_0)Y(t_0)^{-1}\Psi
(t_0)y(t_0)\|  < \varepsilon .
\]
Thus, the trivial solution of \eqref{e2} is $\Psi $-stable on $\mathbb{R}_{+}$.

Part (b) is proved similarly and omit its proof. The proof is
complete.
\end{proof}

\subsection*{Remarks}

1. It is easy to see that if $|\Psi (t)|$ and $|\Psi ^{-1}(t)|$ are bounded
on $\mathbb{R}_{+}$, then the $\Psi$-stability is equivalent with the
classical stability.

\noindent 2. Theorem \ref{thm1} generalizes a similar result for classical
stability \cite{c5}.

\noindent 3. In the same manner as in classical stability, we can speak
about $\Psi$-(uniform) stability of a linear system \eqref{e2}.

\begin{example} \label{ex1} \rm
Consider the linear system \eqref{e2} with
\[
A(t)=\begin{pmatrix}
1 & -1 & 0 \\
1 & 1 & 0 \\
0 & 0 & -2
\end{pmatrix}.
\]
Then
\[
{Y(t)}=\begin{pmatrix}
{e}^t\sin {t} & {e}^t\cos {t} & 0 \\
-{e}^t\cos {t} & {e}^t\sin {t} & 0 \\
0 & 0 & {e}^{-2{t}}
\end{pmatrix}
\]
is a fundamental matrix for the system \eqref{e2}.
Because $Y(t$) is unbounded on $\mathbb{R}_{+}$, it follows that the system
\eqref{e2} is not stable on $\mathbb{R}_{+}$.
Consider
\[
\Psi (t) = \begin{pmatrix}
{e}^{-{ t}} & 0 & 0 \\
0 & {e}^{-{ t}} & 0 \\
0 & 0 & e^{2t}
\end{pmatrix}.
\]
Then, for all $0 \leq  s \leq  t < \infty $, we have
\[
\Psi (t)Y(t)Y^{-1}(s)\Psi ^{-1}(s) =
\begin{pmatrix}
\cos {(t - s)} & -\sin {(t - s)} & 0 \\
\sin {(t - s)} & \cos {(t - s)} & 0 \\
0 & 0 & {1}
\end{pmatrix}.
\]
Thus, the system \eqref{e2} is $\Psi$-uniformly stable on $\mathbb{R}_{+}$.
\end{example}

\subsection*{Remark}

The introduction of the matrix function $\Psi$ permits us obtain a mixed
asymptotic behavior of the components of the solutions.

\begin{theorem} \label{thm2}
Let $Y(t)$ be a fundamental matrix for \eqref{e2}. If there exist a
continuous function $\varphi :\mathbb{R}_{+}\to (0,\infty )$
 and the constants $p\geq  1$ and $M > 0$ which
fulfil one of the following conditions:
\begin{itemize}
\item[(i)]
$\int_0^t\varphi (s)| \Psi (t)Y(t)Y^{-1}(s)\Psi^{-1} (s)| ^{{p}}\,ds \leq  M$,
for all t $\geq 0$

\item[(ii)] $\int_0^t\varphi (s)| Y^{-1}(s)\Psi ^{-1}(s)\Psi (t)Y(t)| ^{{p}}\,ds
\leq  M$, for all $t \geq  0$,
\end{itemize}
then, the system \eqref{e2} is $\Psi$-stable on $\mathbb{R}_{+}$.
\end{theorem}

\begin{proof}
For the case (i), first, we consider $p = 1$.
Let q(t) = $| \Psi (t)Y(t) | ^{-1}$ for $t \geq  0$. From the
identity
\[
\Big( \int_0^t\varphi {(s)q(s)\,ds}\Big) \Psi (t)Y(t)
= \int_0^t\varphi {(s)}\Psi {(t)Y(t)Y}^{-1}{(s)}
\Psi ^{-1}{(s)}\Psi {(s)Y(s)q(s)\,ds,}
\]
it follows that
\begin{align*}
&\Big( \int_0^t\varphi {(s)q(s)\,ds}\Big) | \Psi {(t)Y(t)}| \\
&\leq \int_0^t\varphi {(s)}| \Psi {(t)Y(t)Y}^{-1}
{(s)}\Psi ^{-1}{(s)}| | \Psi {(s)Y(s)}| q(s)\,ds.
\end{align*}
Thus, the scalar function $h(t) = \int_0^t\varphi (s)q(s)\,ds$ satisfies
the inequality
\[
{h(t)q}^{-1}{(t)}\leq {M, for t}\geq 0\,.
\]
We have $h'( {t}) =\varphi (t)q(t) \geq  M^{-1}\varphi (t)h(t)$ for
$t \geq 0$. It follows that
\[
h(t)\geq {h(t}_{1}{)e}^{M^{-1}\int_{t_{1}}^t\varphi (s)\,ds}
, \quad\text{for }   t \geq { t}_{1}> 0
\]
and hence
\[
| \Psi {(t)Y(t)}|  ={ q}^{-1}({t}) \leq
{ Mh}^{-1}(t_{1}{)e}^{-M^{-1}\int_{t_{1}}^t\varphi (s)\,ds},
\quad\text{for } t\geq t_{1}> 0\,.
\]

Because $|\Psi (t)Y(t) |$ is a continuous function on $[0,t_{1}]$,
it follows that there exists a positive constant $K$ such that
$|\Psi (t)Y(t) | \leq  K$ for $t\geq 0$.
Hence, the theorem follows immediately from the Theorem \ref{thm1}.

Next, suppose that $p > 1$.
Let $r(t) = | \Psi (t)Y(t) | ^{-{p}}$ for $t \geq  0$. In the
same manner as above, we have
\[
\Big( \int_0^t\varphi {(s)r(s)\,ds}\Big) | \Psi {(t)Y(t)}|
\leq \int_0^t\varphi {(s)}| \Psi {(t)Y(t)Y}^{-1}
(s)\Psi ^{-1}{(s)}| | \Psi {(s)Y(s)}| r(s)\,ds.
\]
Because $\varphi (s)| \Psi (s)Y(s)| r(s)
= ( \varphi {(s)}) ^{1/p}( \varphi {(s)r(s)}) ^{1/q}$,
where $\frac{1}{{p}} + \frac{1}{{q}} = 1$,
we have
\begin{align*}
&\Big( \int_0^t\varphi {(s)r(s)\,ds}\Big) | \Psi (t)Y(t)|\\
&\leq \int_0^t( \varphi {(s)}) ^{1/p}|
\Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{(s)}| (\varphi {(s)r(s)}) ^{1/q}
\,ds\,.
\end{align*}
Using the H\"{o}lder inequality, we obtain
\begin{align*}
&\Big( \int_0^t\varphi {(s)r(s)\,ds}\Big) | \Psi (t)Y(t)|\\
& \leq \Big(\int_0^t\varphi {(s)}| \Psi {(t)Y(t)Y}^{-1}{(s)}
\Psi ^{-1}{(s)}| ^{{p}}{\,ds}\Big)^{1/p}
\Big( \int_0^t\varphi {(s)r(s)\,ds}\Big) ^{1/q},
\quad t \geq  0;
\end{align*}
or
\[
\Big( \int_0^t\varphi {(s)r(s)\,ds}\Big)
| \Psi {(t)Y(t)}| \leq { M}^{1/p}
\Big( \int_0^t\varphi {(s)r(s)\,ds}\Big) ^{1/q}, \quad
t \geq  0\,.
\]
Thus, the matrix $\Psi $(t)Y(t) satisfies the inequality
\[
| \Psi (t)Y(t)|  \leq  M^{1/p}
\Big( \int_0^t\varphi (s)r(s)\,ds\Big) ^{-1/p},\quad \forall
 t \geq  0\,.
\]
Denoting $Q(t) = \int_0^t\varphi (s)r(s)\,ds$ for $t \geq  0$, we
obtain
\[
| \Psi {(t)Y(t)}|  \leq { M}^{\frac{1}{{p}}
}( {Q(t)}) ^{-1/p},\quad \forall t \geq  0.
\]
Because $Q'(t) = \varphi (t)r(t) = \varphi (t)| \Psi (t)Y(t) | ^{-{p}}
\geq M^{-1}\varphi (t)Q(t)$, we have
\[
{Q(t) }\geq { Q(1)e}^{{M}^{-1}\int_{1}^t\varphi
(s)\,ds},\quad  t \geq 1\,.
\]
It follows that
\[
| \Psi {(t)Y(t)}| \leq { M}^{1/p}( {Q(1)}) ^{-1/p}{e}^{-{p}^{-1}
{M}^{-1}\int_{1}^t\varphi {(s)\,ds}}, \quad t \geq 1.
\]
Because $|\Psi (t)Y(t) |$ is a continuous function on $[0,1]$,
it follows that there exists a positive constant $K$ such that
$|\Psi (t)Y(t) | \leq  K$ for $t \geq  0$.
Hence, the theorem follows immediately from the Theorem \ref{thm1}.

For case (ii), the proof is similar and we omit it.
The proof is complete.
\end{proof}

\subsection*{Remarks}

1. The function $\varphi $ can serve to weaken the required hypotheses on
the fundamental matrix $Y$.

\noindent 2. Theorem \ref{thm2} generalizes a result of Dannan and Elaydi
\cite{d1}.

\noindent 3. In the conditions of the Theorem, the linear system \eqref{e2}
can not be $\Psi$-uniformly stable on $\mathbb{R}_{+}$. This is shown in
\cite[Example 2]{d2}. \smallskip

Finally, we consider various $\Psi$-stability problems connected with the
linear system
\begin{equation}
x'= ( {A(t) + B(t)}) {x}  \label{e3}
\end{equation}
as a perturbed system of \eqref{e2}. We seek conditions under which the $%
\Psi $-(uniform) stability of \eqref{e2} implies the $\Psi$-(uniform)
stability of \eqref{e3}.

\begin{theorem} \label{thm3}
Suppose that $B$ is a continuous $n \times n$ matrix
function for $ t \geq 0$. If the linear system \eqref{e2} is
$\Psi$-uniformly stable on $\mathbb{R}_{+}$ and
\[
\int_0^{\infty }| \Psi {(t)B(t)}\Psi ^{-1}{(t)}|
\,dt < +\infty ,
\]
then the linear system \eqref{e3} is also $\Psi$-uniformly
stable on $\mathbb{R}_{+}$.
\end{theorem}

\begin{proof}
Let $Y(t)$ be a fundamental matrix for the homogeneous system \eqref{e2}.
Because the system \eqref{e2} is $\Psi$-uniformly stable on $\mathbb{R}_{+}$,
there exists a positive constant $K$ such that
\[
| \Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{(s)}|
\leq  K  \quad\text{for }  0 \leq  s \leq  t < +\infty .
\]
The solution of \eqref{e3} with initial condition $x(t_0) = x_0$ is unique
and defined for all $t \geq  0$. Then it is also a solution of the problem
\[
x' = A(t)x + B(t)x, x(t_0) = x_0.
\]
Therefore, by the variation of constants formula,
\[
{x(t) = Y(t)Y}^{-1}(t_0{)x}_0+\int_{t_0}^t
{Y(t)Y}^{-1}{(s)B(s)x(s)\,ds}
\]
or, for $t, t_0\geq  0$,
\begin{align*}
\Psi (t)x(t) &= \Psi (t)Y(t)Y^{-1}(t_0)\Psi^{-1}(t_0)\Psi (t_0)x_0 \\
&\quad+ \int_{t_0}^t\Psi (t)Y(t)Y^{-1}(s)\Psi ^{-1}
(s)\Psi (s)B(s)\Psi ^{-1}(s)\Psi (s)x(s)\,ds\,.
\end{align*}
 From the above conditions, it results that
\[
\|\Psi (t)x(t)\|
\leq  K\|\Psi (t_0)x(t_0)\|  +
K \int_{t_0}^t | \Psi (s)B(s)\Psi ^{-1}{(s)}| \| \Psi (s)x(s)\| \,ds,
\]
for $t \geq  t_0 \geq  0$.
Therefore, by Gronwall's inequality,
\[
\| \Psi {(t)x(t)}\| \leq  K\| \Psi
(t_0{)x(t}_0)\| {e}^{K
\int_{t_0}^t| \Psi {(s)B(s)}\Psi ^{-1}{(s)}| \,ds},
\quad\text{for } t \geq  t_0\,.
\]
Thus, putting $L = \int_0^{\infty }| \Psi (t)B(t)\Psi ^{-1}(t)| \,dt$,
we have
\[
\| \Psi {(t)x(t)}\| \leq  K \| \Psi
(t_0{)x(t}_0)\| {e}^{{KL}},\quad\text{for all } t \geq  t_0\geq  0.
\]
This inequality shows that the system \eqref{e3} is $\Psi$-uniformly stable on
$\mathbb{R}_{+}$.
The proof is complete.
\end{proof}

\subsection*{Remark}

The above theorem generalizes a results of Caligo \cite{c1}, Conti \cite{c4}
in connection with uniform stability. \smallskip

If the linear system \eqref{e2} is only $\Psi$-stable, then the linear
system \eqref{e3} can not be $\Psi$-stable. This is shown by the next
example transformed after an example due to Perron \cite{p1}.

\begin{example} \label{ex2} \rm
Let $a \in\mathbb{R}$ be such that $1 \leq  2a < 1 + e^{-\pi }$
and let
\[
{A(t)}=\begin{pmatrix}
-{ a} & 0 \\
0 & \sin \ln ({t + 1})+\cos \ln ({t + 1})-2{a}
\end{pmatrix}
\]
Then
\[
{Y(t)}=\begin{pmatrix}
e^{-{a}({t+1})} & 0 \\
0 & e^{({t+1})[ \sin \ln ({t+1})-2{a}] }
\end{pmatrix}.
\]
is a fundamental matrix for the homogeneous system \eqref{e2}.

Let
$\Psi (t)= \begin{pmatrix}
e^{{a}({t+1})} & 0 \\
0 & 1
\end{pmatrix}$.
We have
\[
\Psi {(t)Y(t)}=\begin{pmatrix}
1 & 0 \\
0 & e^{{(t+1})[ \sin \ln ({t+1})-2{a}] }
\end{pmatrix}.
\]
Because $|\Psi (t)Y(t) |$ is bounded on $\mathbb{R}_{+}$, it
follows that the system \eqref{e2} is $\Psi$-stable on $\mathbb{R}_{+}$.
For 0 $\leq $ s $\leq $ t < $\infty $, we have
\[
| \Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{(s)}|
=\begin{pmatrix}
1 & 0 \\
0 & e^{{f(t) - f(s)}}
\end{pmatrix},
\]
where $f(t) = (t+1)\sin\ln(t+1) - 2at$.

It is easy to see that $\lim_{n\to \infty }[f(t_{n}e^{\alpha }-1)
- f(t_{n}-1)] = \infty $, where $t_{n}=e^{(8n+1)\frac{\pi }{4}}$ and
$\alpha =\arccos \frac{1+e^{-\pi }}{\sqrt{2}}$.
Thus, $| \Psi (t)Y(t)Y^{-1}(s)\Psi ^{-1}(s)| $ is not
bounded for $0 \leq  s \leq  t < \infty $.
 From Theorem 1, it follows that the system \eqref{e2} is not $\Psi$-uniformly
stable on $\mathbb{R}_{+}$.
\end{example}

If we take
\begin{equation*}
{B(t)}=
\begin{pmatrix}
0 & 0 \\
e^{-a({t+1})} & 0
\end{pmatrix}
,
\end{equation*}
then
\begin{equation*}
{Y}_{1}{(t)}=
\begin{pmatrix}
e^{-a({t+1})} & 0 \\
e^{({t+1})[ \sin \ln ({t+1})-2{a}] }\int_{1}^{{t+1}}e^{-s\sin \ln s}{\,ds} &
e^{({t+1})[ \sin \ln ({\ t+1})-2{a}] }
\end{pmatrix}
\end{equation*}
is a fundamental matrix for the perturbed system \eqref{e3}. We have
\begin{equation*}
\Psi {(t)Y}_{1}{(t)}=
\begin{pmatrix}
1 & 0 \\
e^{({t+1})[ \sin \ln ({t+1})-2{a}] }\int_{1}^{{t+1}}e^{-s\sin \ln s}{\,ds} &
e^{({t+1})[ \sin \ln ({\ t+1})-2{a}] }
\end{pmatrix}
.
\end{equation*}
Let $\alpha \in (0,\pi /2)$ be such that $\cos \alpha > (2a-1)e^{\pi }$. Let
$t_{n}=e^{(2n-\frac{1}{2})\pi }$ for $n = 1,2\dots $. For $t_{n}\leq s \leq
t_{n}e^{\alpha }$ we have $s \cos \alpha \leq -s \sin \ln s \leq s$ and
hence
\begin{align*}
&e^{t_{n}e^{\pi }(\sin \ln t_{n}e^{\pi }-2a)}\int_{1}^{t_{n}e^{\pi}}
e^{-s\sin \ln s}\,ds \\
&> e^{t_{n}e^{\pi }(\sin \ln t_{n}e^{\pi }-2a)} \int_{t_{n}}^{t_{n}e^{\alpha
}}e^{-s\sin \ln s}\,ds \\
&> e^{t_{n}e^{\pi }(1-2a)}\int_{t_{n}}^{t_{n}e^{\alpha }} e^{s\cos \alpha
}\,ds \\
&= e^{t_{n}[ (1-2a)e^{\pi }+\cos \alpha] }\big( {e} ^{t_{n}(e^{\alpha
}-1)\cos \alpha }{\ - 1}\big) \cos ^{-1}\alpha \to \infty
\end{align*}
This shows that $| \Psi (t)Y_{1}(t) |$ is unbounded on $\mathbb{R}_{+}$. It
follows that the equation \eqref{e3} is not $\Psi$-stable on 
$\mathbb{R}_{+}$. Finally, we have 
$\int_0^{\infty }| \Psi (s)B(s)\Psi ^{-1}(s) | \,ds <+\infty $.

Also, the Theorem 3 is no longer true if we require that $\Psi (t)B(t)\Psi
^{-1}(t) \to 0$ as $t \to \infty$, instead of the condition
\begin{equation*}
\int_0^{\infty }| \Psi {(s)B(s)}\Psi ^{-1}{(s)}| \,ds < +\infty .
\end{equation*}
This is shown by the next example, adapted from an example in Cesari \cite
{c2}.

\begin{example} \label{ex3}\rm
Consider the system \eqref{e2} with
\[
A(t) = \begin{pmatrix}
0 & 1 \\
-1 & -\frac{2}{{t+1}}
\end{pmatrix}.
\]
Then
\[
Y(t) = \begin{pmatrix}
\frac{{\sin(t+1)}}{{t+1}} & \frac{{\cos(t+1)}}{{t+1}} \\
\frac{{(t+1)\cos(t+1) - \sin(t+1)}}{{(t+1)}^{2}} &
-\frac{{(t+1)\sin(t+1) + \cos(t+1)}}{{(t+1)}^{2}}
\end{pmatrix}.
\]
is a fundamental matrix for the homogeneous system \eqref{e2}.

Let
$\Psi {(t) = }\begin{pmatrix}
{t+1} & 0 \\
0 & {t+1}
\end{pmatrix}$.
We have
\begin{align*}
&\Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}(s)\\
& =\begin{pmatrix}
\frac{{(s+1)\cos(t-s) + \sin(t-s)}}{{s+1}} & \sin {(t-s)} \\
\frac{{(t-s)\cos(t-s) - (ts+t+s+2)\sin(t-s)}}{{(t+1)(s+1)}}
& \frac{{(t+1)\cos(t-s) - \sin(t-s)}}{{t+1}}
\end{pmatrix},
\end{align*}
for $0 \leq  s \leq  t < \infty $.
It is easy to see that the system \eqref{e2} is $\Psi$-uniformly stable on
$\mathbb{R}_{+}$.
\end{example}

Now, we consider the system \eqref{e3} with
\begin{equation*}
B(t) =
\begin{pmatrix}
0 & 0 \\
0 & \frac{2}{{t+1}}
\end{pmatrix}
.
\end{equation*}
Then
\begin{equation*}
\widetilde{{Y}}(t) =
\begin{pmatrix}
\sin {t} & \cos {t} \\
\cos {t} & -\sin {t}
\end{pmatrix}
.
\end{equation*}
is a fundamental matrix for the perturbed system \eqref{e3}. We have
\begin{equation*}
\Psi (t)\widetilde{{Y}}(t) = (t+1)
\begin{pmatrix}
\sin {t} & \cos {t} \\
\cos {t} & -\sin {t}
\end{pmatrix}
.
\end{equation*}
It follows that the system \eqref{e3} is not $\Psi$-(uniformly) stable on 
$\mathbb{R}_{+}$. Finally, we have
\begin{equation*}
\int_0^{\infty }| \Psi (s)B(s)\Psi ^{-1}(s)| \,ds = +\infty \quad \text{and}%
\quad \lim_{t\to \infty }| \Psi (t)B(t)\Psi ^{-1}(t)| = 0.
\end{equation*}

\begin{theorem} \label{thm4}
Suppose that:
\begin{enumerate}
\item There exist a continuous function
$\varphi  : \mathbb{R}_{+}\to  (0,\infty )$ and a positive constant
$M$ such that the fundamental matrix $Y(t)$ of the system \eqref{e2} satisfies
the condition
\[
\int_0^t\varphi {(s)}| \Psi {(t)Y(t)Y}^{-1}{(s)}
\Psi ^{-1}{(s)}|  \,ds \leq  M, \quad \forall t \geq  0
\]

\item $B(t)$ is a continuous $n \times  n$ matrix function on
$\mathbb{R}_{+}$ such that
\[
\sup_{{t}\geq 0} \varphi ^{-1}{(t)}| \Psi{(t)B(t)}\Psi ^{-1}{(t)}|
\]
is a sufficiently small number.
\end{enumerate}
Then the linear system \eqref{e3} is $\Psi$-stable on $\mathbb{R}_{+}$.
\end{theorem}

\begin{proof}
 From the first assumption of theorem it follows that there exists a positive
constant $N$ such that
\[
| \Psi {(t)Y(t)}|  \leq  N, \quad \forall t \geq  0.
\]
The solution of \eqref{e3} with initial condition $x(t_0) = x_0$ is unique
and defined for all $t \geq  0$. Then it is also a solution of the problem
\[
x' = A(t)x + B(t)x, \quad   x(t_0) = x_0.
\]
Therefore, by the variation of constants formula,
\[
{x(t) = Y(t)Y}^{-1}(t_0{)x}_0+\int_{{t}
_0}^t{Y(t)Y}^{-1}(s)B(s)x(s)\,ds, \quad t \geq  0\,.
\]
Hence,
\begin{align*}
\| \Psi {(t)x(t)}\|
&\leq \| \Psi {(t)Y(t)Y}^{-1}(t_0)\Psi ^{-1}(t_0)\Psi
(t_0{)x}_0\| \\
&\quad + \int_{{t}_0}^t\| \Psi {(t)Y(t)Y}^{-1}
{(s)}\Psi ^{-1}{(s)}\Psi {(s)B(s)}\Psi ^{-1}{(s)}\Psi
{(s)x(s)}\| \,ds,
\end{align*}
 for all $t \geq  t_0$.
If we put
\[
b = \sup_{t\geq 0}\varphi ^{-1}(t)| \Psi (t)B(t)\Psi ^{-1}(t)|  < M^{-1},
\]
then, for $T > t_0$ and $t \in  [t_0, T]$, we have
\[
\| \Psi {(t)x(t)}\|
\leq | \Psi {(t)Y(t)}| | {Y}^{-1}(t_0)\Psi ^{-1}(t_0
)| \| \Psi (t_0{)x}_0\|
+ Mb\sup_{{t}_0\leq {t}\leq {T}}\| \Psi {(t)x(t)}\| .
\]
Therefore,
\[
\sup_{{t}_0\leq {t}\leq {T}}\| \Psi {(t)x(t)}\|
\leq { ( 1 - Mb )}^{-1}{N}| {Y}
^{-1}(t_0)\Psi ^{-1}(t_0)| \| \Psi
(t_0{)x}_0\| .
\]
It follows that the system \eqref{e3} is $\Psi$-stable on $\mathbb{R}_{+}$.
The proof is complete.
\end{proof}

\subsection*{Remark}

We can show that the conclusion of Theorem 4 is valid if the condition
\begin{equation*}
\underset{{t}\geq 0}{\sup } \varphi ^{-1}{(t)}| \Psi {(t)B(t)}\Psi ^{-1}{(t)}%
| {\ < M}^{-1}
\end{equation*}
is replaced with the condition
\begin{equation*}
\lim_{{t}\to \infty } \varphi ^{-1}{(t)}| \Psi {(t)B(t)}\Psi ^{-1}{(t)}| =
0\,.
\end{equation*}

Theorem \ref{thm4} is no longer true if we require that the system \eqref{e2}
be $\Psi$-(uniformly) stable on $\mathbb{R}_{+}$ instead of the condition
\begin{equation*}
\int_0^t\varphi {(s)}| \Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{(s)}| \,ds \leq
M,\quad \forall t \geq 0\,.
\end{equation*}
This is shown by the next example.

\begin{example} \label{ex4} \rm
Consider the system \eqref{e2} with $A(t) = O_{2}$. Then, a fundamental
matrix for the system \eqref{e2} is $Y(t) = I_{2}$.
Consider
\[
\Psi (t) = \begin{pmatrix}
1 & 0 \\
0 & \frac{1}{{t+1}}
\end{pmatrix}.
\]
Because
\[
\Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{(s)}=\begin{pmatrix}
1 & 0 \\
0 & \frac{{s + 1}}{{t + 1}}
\end{pmatrix}
\]
is bounded for 0 $\leq  s \leq t < +\infty$, it follows that
the system \eqref{e2} is $\Psi$-uniformly stable on $\mathbb{R}_{+}$.
If we take
\[
B(t) = \begin{pmatrix}
0 & 0 \\
0 & \frac{{a}}{\sqrt{{t+1}}}
\end{pmatrix},
\]
where $a > 0$, then
\[
\widetilde{{Y}}(t) =  \begin{pmatrix}
1 & 0 \\
0 & {e}^{{2a}\sqrt{{t+1}}}
\end{pmatrix} .
\]
is a fundamental matrix for the perturbed system \eqref{e3}.
Because
\[
\Psi (t)\widetilde{{Y}} (t) = \begin{pmatrix}
{1} & {0} \\
{0} & \frac{{e}^{{2a}\sqrt{{t+1}}}}{{t+1}}
\end{pmatrix}
\]
is unbounded on $\mathbb{R}_{+}$, it follows that the perturbed system
\eqref{e3} is not $\Psi$-stable on $\mathbb{R}_{+}$.

Finally, we have $\sup_{{t}\geq 0}| \Psi (t)B(t)\Psi
^{-1}(t)| = a$ and $\lim_{{t}\to \infty }|\Psi (t)B(t)\Psi ^{-1}(t)|= 0$.
\end{example}

\section{$\Psi$-stability of the nonlinear system \eqref{e1}}

The purpose of this section is to study the $\Psi$-(uniform) stability of
trivial solution of \eqref{e1}. Now, we state a hypothesis which we shall
use in various places.

\begin{itemize}
\item[(H0)]  For all $t_0\geq 0$, $x_0\in \mathbb{R}^n$ and $\rho > 0$, if $%
\| \Psi (t_0)x_0\| < \rho $, then there exists a unique solution $x(t)$ on $%
\mathbb{R}_{+}$ of \eqref{e1} such that $x(t_0)=x_0$ and $\| \Psi (t)x(t)
\|\leq \rho $ for all $t $ in $[0,t_0]$.
\end{itemize}

This is a natural hypothesis in studying $\Psi$-stability of system %
\eqref{e1}. In \cite{h1}, this hypothesis is tacitly used in particular case
$\Psi = I_{n}$.

\begin{theorem} \label{thm5}
 Assume that Hypothesis (H0) is satisfied. Assume that
there exist a continuous function
$\varphi :\mathbb{R}_{+}\to (0,\infty)$ and a positive constant
$M$ such that the fundamental matrix $Y(t)$ of the system \eqref{e2}
satisfies the condition
\[
\int_0^t\varphi {(s)}| \Psi {(t)Y(t)Y}^{-1}{(s)}
\Psi ^{-1}{(s)}| { \,ds }\leq  M,\quad \forall t \geq  0.
\]
Also assume that function $F$ satisfies the condition
\[
\| \Psi {(t)F(t,s,x)}\| \leq  f(t,s)\| \Psi {(s)x}\| ,
\]
for 0 $\leq  s \leq  t < \infty $  and for all $x$ in
$\mathbb{R}^n$, where $f$ is a continuous nonnegative function on $D$
such that
\[
\sup_{t\geq 0} \int_0^t\frac{{f(t,s)}}{\varphi {(t)}}\,ds < \frac{1}{M}.
\]
Then, the trivial solution of the system \eqref{e1} is $\Psi$-stable on
$\mathbb{R}_{+}$.
\end{theorem}

\begin{proof} From the second assumption of the theorem, it follows that
there exists a positive constant $N$ such that
\[
| \Psi {(t)Y(t)}| \leq  N, \quad\mbox{for all }t \geq 0\,.
\]
 From the third assumption of the theorem, there exists $q$ such that
\[
\int_0^t\frac{{f(t,s)}}{\varphi {(t)}}\,ds \leq
q < \frac{{1}}{{M}},\quad \mbox{for all }t \geq  0\,.
\]
For a given $\varepsilon > 0$ and $t_0\geq 0$, we
choose
\[
\delta  = \min\{\frac{\varepsilon }{{2}},
\frac{{(1 -qM)}\varepsilon }{{2N}| {Y}^{-1}(t_0)\Psi ^{-1}
(t_0)| }\}.
\]
Let $x_0\in \mathbb{R}^n$ be such that $\|\Psi (t_0)x_0\|  < \delta $.

 From the first assumption of the theorem , there exists a unique solution
$x(t)$ on $\mathbb{R}_{+}$ of the system \eqref{e1} such that $x(t_0) = x_0$
and $\| \Psi (t)x(t) \| \leq \delta $ for all $t \in [0, t_0$].
Suppose that there exists $\tau  > t_0$ such that
\[
\| \Psi (\tau )x(\tau )\|  = \varepsilon \quad\mbox{and}\quad \|
\Psi (t)x(t)\| < \varepsilon \quad\mbox{for }t \in  [t_0,\tau ).
\]
By the classical formula of variation of constants, we have
\begin{align*}
\| \Psi (\tau)x(\tau )\|
&\leq \| \Psi (\tau )Y(\tau )Y^{-1}(t_0)\Psi ^{-1}(t_0)\Psi (t_0{)x}
_0\| \\
&\quad +\int_{{t}_0}^{\tau }| \Psi (\tau {)Y(}\tau
{)Y}^{-1}{(s)}\Psi ^{-1}{(s)}| \int_0^{{s}
}\| \Psi {(s)F(s,u,x(u))}\| {\,du\,ds }\\
&\leq  N| {Y}^{-1}(t_0)\Psi ^{-1}(t_0)| \delta \\
&\quad + \int_{{t}_0}^{\tau }\varphi {(s)}| \Psi (\tau
{)Y(}\tau {)Y}^{-1}{(s)}\Psi ^{-1}{(s)}| \int_0^{
{s}}\frac{{f(s,u)}}{\varphi {(s)}}\| \Psi {(u)x(u)
}\| {\,du\,ds }\\
&\leq {N}| {Y}^{-1}(t_0)\Psi ^{-1}(t_0)| \delta  \\
&\quad + \varepsilon \int_{{t}_0}^{\tau }\varphi {(s)}| \Psi
(\tau {)Y(}\tau {)Y}^{-1}{(s)}\Psi ^{-1}{(s)}|
\int_0^{{s}}\frac{{f(s,u)}}{\varphi {(s)}}{\,du\,ds}\\
&\leq { N}| {Y}^{-1}(t_0)\Psi ^{-1}{
(t}_0)| \delta +\varepsilon {q}\int_{{t}
_0}^{\tau }\varphi {(s)}| \Psi (\tau {)Y(}\tau {)Y
}^{-1}{(s)}\Psi ^{-1}{(s)}| {\,ds }\\
&\leq { N}| {Y}^{-1}(t_0)\Psi ^{-1}{
(t}_0)| \delta +\varepsilon qM \\
&< \varepsilon {(1 - qM) + }\varepsilon qM = \varepsilon ,
\end{align*}
which is a contradiction.
Therefore, the trivial solution of system \eqref{e1} is $\Psi$-stable
on $\mathbb{R}_{+}$.
The proof is complete.
\end{proof}

\begin{corollary}  \label{coro1}
Suppose that $g$ and $h$ are continuous nonnegative functions on
$\mathbb{R}_{+}$ such that
\[
\sup_{{t}\geq {0}} \frac{{g(t)}}{\varphi {(t)}}\int_0^t h(s)\,ds < \frac{{1}}{{M}}.
\]
Then in Theorem \ref{thm5} we can consider $f(t,s) = g(t)h(s)$.
\end{corollary}

\begin{corollary} \label{coro2}
Suppose that $k$ is a continuous nonnegative function on
$\mathbb{R}_{+}$ such that
\[
\sup_{{t}\geq {0}} \frac{{1}}{\varphi {(t)}}\int_0^t k(u)\,du < \frac{{1}}{{M}}.
\]
Then in Theorem \ref{thm5} we can consider $f(t,s) = k(t - s)$.
\end{corollary}

\begin{corollary} \label{coro3}
If in Theorem \ref{thm5}, the third condition is replaced by the
condition:

The function $F$ satisfies: For all $\varepsilon >0$
there exists $\delta(\varepsilon )>0$ such that for all
$x$in
\[
B_{\delta (\varepsilon )}
= \{x \in C_{c}: \sup_{t\geq 0}\| \Psi (t)x(t)\|
\leq \delta (\varepsilon )\}
\]
we have
\[
\| \Psi {(t)F(t,s,x(s))}\| {
}\leq  \varepsilon {f(t,s)}\| \Psi {(s)x(s)}
\| \quad {for } 0 \leq  s \leq  t < +\infty,
\]
where $f$ is a continuous nonnegative function on $D$ such that
\[
\sup_{{t}\geq {0}} \int_0^t\frac{{f(t,s)}}{\varphi {(t)}}{\,ds < +}\infty ,
\]
then the trivial solution of system \eqref{e1} is $\Psi$-stable on
$\mathbb{R}_{+}$.
\end{corollary}

The proof of the above corollary is similar to that of Theorem \ref{thm5}.

\begin{theorem} \label{thm6}
Assume hypothesis (H0) is satisfied.
Assume the function $F$ satisfies
\[
\| \Psi {(t)F(t,s,x)}\|  \leq { f(t,s)}
\| \Psi {(s)x}\| , \quad\mbox{for } 0\leq {s}\leq t< \infty
\]
and for every $x \in \mathbb{R}^n$, where $f$ is
a continuous nonnegative function on $D$ such that
\[
M= \int_0^{\infty}\int_0^t{f(t,s)\,ds\,dt < }\infty .
\]
Also assume the fundamental matrix $Y(t)$ of the system \eqref{e2}
is such that
\[
| \Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{(s)}|  \leq  K
\]
for all $0\leq  s \leq  t < +\infty $, where $K$ is a positive constant.
Then, the trivial solution of \eqref{e1} is $\Psi$-uniformly stable
on $\mathbb{R}_{+}$.
\end{theorem}

\begin{proof}
Let $\varepsilon > 0$  and
$\delta ( \varepsilon )=0.5\varepsilon K^{-1}(1+M)^{-1}e^{-KM}$.
Let $t_0\geq  0$ and $x_0\in \mathbb{R}^n$ be such that
$\| \Psi (t_0)x_0\|  < \delta ( \varepsilon )$. There
exists a unique solution $x(t)$ on $\mathbb{R}_{+}$ of \eqref{e1} such that
$x(t_0)=x_0$ and $\|\Psi (t)x(t)\| \leq \delta( \varepsilon ) $
for all $t \in  [0,t_0]$.
For $t \geq  t_0$, we have
\begin{align*}
&\| \Psi {(t)x(t)}\| \\
&=\| \Psi {(t)Y(t)Y}^{-1}(t_0)\Psi ^{-1}(t_0)\Psi (t_0{)x}_0 \\
&\quad + \int_{t_0}^t\Psi {(t)Y(t)Y}^{-1}{(s)}\Psi ^{-1}{
(s)}\int_0^{s}\Psi {(s)F(s,u,x(u))\,du\,ds}\|  \\
&\leq K \| \Psi (t_0{)x}_0\|  + K
\int_{t_0}^t\int_0^{s}{f(s,u)}\| \Psi {(u)x(u)}
\| {\,du\,ds = K}\| \Psi (t_0{)x}_0\|\\
&\quad + K\int_{t_0}^t\int_0^{t_0}{f(s,u)}\| \Psi {
(u)x(u)}\| {\,du\,ds + K}\int_{t_0}^t\int_{t_0}^{s}{f(s,u)
}\| \Psi {(u)x(u)}\| {\,du\,ds }\\
&\leq  K\delta ( \varepsilon ) (1 + M) + K
\int_{t_0}^t\int_{t_0}^{s}{f(s,u)}\| \Psi {(u)x(u)}
\| \,du\,ds.
\end{align*}
It is easy to see that the function
$Q(t) = \int_{t_0}^t\int_{t_0}^{s}f(s,u)\| \Psi (u)x(u) \| \,du\,ds$
 is continuously differentiable and increasing on $[t_0, \infty )$.
For $t \geq  t_0$, we have
\begin{align*}
Q'(t) &=\int_{{t}_0}^t{f(t,u)}\| \Psi {(u)x(u)}\| \,du\\
&\leq \int_{{t}_0}^{{t}}{f(t,u)[K}\delta (\varepsilon )(1+M) + KQ(u)]\,du\\
&= K\delta (\varepsilon ){(1 + M)}\int_{{t}_0}^t
{f(t,u)\,du + K}\int_{{t}_0}^t{f(t,u)Q(u)\,du.}
\end{align*}
Then
\begin{align*}
&\big[ Q(t)\exp\big(-K\int_{t_0}^t \int_{{t}_0}^{{s}} f(s,u)\,du\,ds\big)
\big] '\\
&= \exp\big(- K\int_{{t}_0}^t\int_{{t}_0}^{{s}} f(s,u)\,du\,ds\big)
\big[ Q'{(t) - KQ(t)}\int_{{t}_0}^t{f(t,u)\,du}\big]  \\
&\leq \exp\big(- K\int_{{t}_0}^t\int_{{t}_0}^{{s}}f(s,u)\,du\,ds\big)\\
&\quad\times \big[ {K}\delta (\varepsilon ){(1+M)}
\int_{{t}_0}^t{f(t,u)\,du+K}\int_{{t}_0}^t
{f(t,u)(Q(u)-Q(t))\,du}\big] \\
&\leq \exp\big(- K\int_{{t}_0}^t\int_{{t}_0}^{{s}}f(s,u)\,du\,ds\big)
\big[K\delta (\varepsilon ){(1 + M)}\int_{{t}_0}^t{f(t,u)\,du}\big] \\
&=\big[ - \delta (\varepsilon ){(1 + M)e}^{-{ K}
\int_{{t}_0}^t\int_{{t}_0}^{{s}}{f(s,u)\,du\,ds}
}\big] '.
\end{align*}
Integrating from $t_0$ to $t$ ($t \geq  t_0$), we have
\[
{Q(t)e}^{-{K}\int_{{t}_0}^t\int_{{t}_0}^{
{s}}{f(s,u)\,du\,ds}} \leq  \delta (\varepsilon ){
(1 + M)}\left[ { 1 }-{ e}^{-{ K}\int_{{t}_0}^{{t}
}\int_{{t}_0}^{{s}}{f(s,u)\,du\,ds}}\right] .
\]
We deduce that
\[
\| \Psi {(t)x(t)}\|  \leq  \delta (
\varepsilon ) {K(1 + M)e}^{{KM}} <
\varepsilon ,\quad \mbox{for all } t \geq  t_0.
\]
This proves that the trivial solution of \eqref{e1} is $\Psi$-uniformly
stable on $R_{+}$.
The proof is complete.
\end{proof}

\begin{corollary} \label{coro4}
Suppose that $g$ and $h$ are continuous nonnegative functions on $R_{+}$
such that
\[
\int_0^{\infty }{g(t)}\int_0^t h(s)\,ds\,dt < +\infty .
\]
Then in Theorem \ref{thm6} we can consider $f(t,s) = g(t)h(s)$.
\end{corollary}

\subsection*{Remark}

Theorem \ref{thm6} generalizes a result of Hara, Yoneyama and Itoh \cite{h1}.

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\end{document}