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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 29, pp. 1--13.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}


\begin{document}
\title[\hfilneg EJDE-2007/29\hfil $\Psi$-conditional asymptotic stability]
{On the $\Psi$-conditional asymptotic stability of the solutions of a
nonlinear Volterra integro-differential system}

\author[A. Diamandescu\hfil EJDE-2007/29\hfilneg]{Aurel Diamandescu}

\address{Aurel Diamandescu \newline
Department of Applied Mathematics, 
University of Craiova, 13, ``Al. I.
Cuza'' st., 200585, Craiova, Romania}
\email{adiamandescu@central.ucv.ro}

\thanks{Submitted October 4, 2006. Published February 14, 2007.}
\subjclass[2000]{45M10, 45J05}
\keywords{$\Psi$-stability; $\Psi$-conditional asymptotic stability}

\begin{abstract}
We provide sufficient conditions for $\Psi$-conditional asymptotic stability
of the solutions of a nonlinear Volterra integro-differential system.
\end{abstract}

\maketitle

\numberwithin{equation}{section} 
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{definition}[theorem]{Definition} 
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark} 
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

The purpose of this paper is to provide sufficient conditions for 
$\Psi $-conditional asymptotic stability of the solutions of the 
nonlinear Volterra integro-differential system 
\begin{equation}  \label{e1}
x'=A(t)x+\int_0^tF(t,s,x(s))ds
\end{equation}
and for the linear system 
\begin{equation}
x'=[ A(t)+B(t)] x  \label{e2}
\end{equation}
as a perturbed systems of 
\begin{equation}
y'=A(t)y.  \label{e3}
\end{equation}

We investigate conditions on a fundamental matrix $Y(t)$ of the linear
equation \eqref{e3} and on the functions $B(t)$ and $F(t,s,x)$ under which
the solutions of \eqref{e1}, \eqref{e2} or \eqref{e3} are 
$\Psi $-conditionally asymptotically stable on $\mathbb{R}_{+}$. 
Here, $\Psi $ is a
continuous matrix function. The introduction of the matrix function $\Psi $
permits to obtain a mixed asymptotic behavior of the solutions.

The problem of $\Psi $- stability for systems of ordinary differential
equations has been studied by many authors, as e.g. Akinyele \cite{a1,a2},
Constantin \cite{c1,c2}, Hallam \cite{h1}, Kuben \cite{k1}, Morchalo 
\cite{m2}. In these papers, the function $\Psi $ is a scalar continuous function
(and monotone in \cite{a2}, nondecreasing in \cite{c1}).

In our papers \cite{d1,d2,d3}, we have proved sufficient conditions for
various types of $\Psi $-stability of the trivial solution of the equations 
\eqref{e1}, \eqref{e2} and \eqref{e3}. In these papers, the function $\Psi $
is a continuous matrix function.

Recent works for stability of solutions of \eqref{e1} have been by Avramescu 
\cite{a3}, by Hara, Yoneyama and Itoh \cite{h2}, by Lakshmikantham and Rama
Mohana Rao \cite{l1}, by Mahfoud \cite{m1} and others. Coppel's paper 
\cite[Chapter III, Theorem 12]{c3}, \cite{c4} deal with the instability and
conditional asymptotic stability of the solutions of a systems of
differential equations. Sp\"{a}th's paper \cite{s2} and Weyl's paper 
\cite{w1} deal with the conditional stability of solutions of systems of
differential equations. In our papers \cite{d4,d5}, we have proved a
necessary and sufficient conditions for $\Psi $-instability and 
$\Psi $-conditional stability of the equation \eqref{e3} and sufficient 
conditions
for $\Psi $-instability and $\Psi $-conditional stability of trivial
solution of the equations \eqref{e1} and \eqref{e2}.

\section{Definitions, notation and hypotheses}

Let $\mathbb{R}^d$ denote the Euclidean $d$-space. For $x = ( x_1, x_2,\dots
x_d)^T \in R^d$, let $\| x\| = \max \{ | x_1|, |x_2| ,\dots | x_d| \}$ be
the norm of $x$. For a $d\times d$ matrix $A = (a_{ij})$, we define the norm 
$A$ by $| A| = \sup_{\| x\| \leq 1}\|Ax\| $; it is well-known that $| A| =
\max_{1\leq i\leq d } \sum_{j=1}^d| a_{ij}| $.

In the equations \eqref{e1}--\eqref{e3} we assume that $A(t)$ is a
continuous $d\times d$ matrix on $\mathbb{R}_{+} = [0, \infty )$ and 
$F : D \times \mathbb{R}^{d}\to \mathbb{R}^d$, 
$D = \{ (t,s) \in \mathbb{R}^2: 0\leq s \leq t <\infty \}$, is a 
continuous $d$-vector with respect to all variables.

Let $\Psi _i : \mathbb{R}_{+}\to (0,\infty )$, $i = 1, 2, \dots d$, be a
continuous functions and 
\begin{equation*}
\Psi = \mathop{\rm diag }[\Psi _1, \Psi _2,\dots \Psi_d].
\end{equation*}
A matrix $P$ is said to be a projection matrix if $P^{2} = P$. If $P$ is a
projection, then so is $I - P$. Two such projections, whose sum is $I$ and
whose product is $0$, are said to be supplementary.

\begin{definition} \label{def2.1} \rm
 The solution $x(t)$ of \eqref{e1} is said to be $\Psi $-stable on
$\mathbb{R}_{+}$, if for every $\varepsilon  > 0$ and any
$t_0\geq $ 0, there exists a
$\delta =\delta (\varepsilon ,t_0) > 0$ such that any solution
$\widetilde{x}$(t) of \eqref{e1} which
satisfies the inequality
$\| \Psi (t_0)(\widetilde{x}(t_0)-x(t_0))\|
< \delta (\varepsilon,t_0)$
exists and satisfies the inequality $\| \Psi (t)(\widetilde{x}
(t)-x(t))\|  < \varepsilon $ for all $t \geq  t_0$.

Otherwise, is said that the solution x(t)  is  $\Psi$-unstable
on $\mathbb{R}_{+}$.
\end{definition}

\begin{definition} \label{def2.2} \rm
A function $\varphi  : \mathbb{R}_{+}\to \mathbb{R}^d$ is
said to be $\Psi $-bounded on $\mathbb{R}_{+}$ if
$\Psi (t)\varphi (t)$ is bounded on $\mathbb{R}_{+}$.
\end{definition}

\begin{remark} \rm
For $\Psi _{i} = 1$, $i = 1, 2,\dots d$, we obtain the notion
of classical stability, instability and boundedness, respectively.
\end{remark}

\begin{definition} \label{def2.3}\rm
The solution $x(t)$ of \eqref{e1} is said to be $\Psi $-conditionally
stable on $\mathbb{R}_{+}$ if it is not $\Psi $-stable on $\mathbb{R}_{+}$ but
there exists a sequence ($x_{n}(t)$) of solutions of \eqref{e1}
 defined for all $t\geq $ 0 such that
\begin{equation*}
\lim_{n\to \infty }\Psi (t)x_n(t) = \Psi(t)x(t),\quad
\text{uniformly on }R_{+}.
\end{equation*}
If the sequence $x_n(t)$ can be chosen so that
\begin{equation*}
\lim_{t\to \infty } \Psi (t)(x_{n}(t) -x(t)) = 0,\quad\text{for }
n = 1, 2, \dots
\end{equation*}
then $x(t)$ is said to be $\Psi $-conditionally asymptotically stable
on $R_{+}$.
\end{definition}

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

(2) In the same manner as in classical conditional asymptotic stability, we
can speak about $\Psi $-conditional asymptotic stability of a linear
equation.
Indeed, let $x(t)$, $y(t)$ be two solutions of the linear equation \eqref{e3}.
We suppose that $x(t)$ is $\Psi $-conditionally asymptotically stable on
$\mathbb{R}_{+}$.
Then $y(t)$ is $\Psi$-unstable on $\mathbb{R}_{+}$
(see \cite[Theorem 1]{d4}) and
\begin{gather*}
\lim_{n\to \infty }\Psi (t)y_n(t) = \Psi
(t)y(t),\quad \text{uniformly on }\mathbb{R}_{+},
\\
\lim_{t\to \infty }\Psi (t)( y_{n}(t) -y(t) ) = 0,\quad
\text{for }n = 1, 2, \dots
\end{gather*}
where y$_{n}(t) = x_{n}(t) - x(t) + y(t)$, $n \in  N$ are solutions of the
linear equation \eqref{e3}. Thus, all solutions of \eqref{e3} are
$\Psi $-conditionally asymptotically stable on $\mathbb{R}_{+}$.
\end{remark}

\section{$\Psi $-conditional asymptotic stability of linear equations}

In this section we give necessary and sufficient conditions for the 
$\Psi $-conditional asymptotic stability of the linear equation \eqref{e3} and
sufficient conditions for the $\Psi $-conditional asymptotic stability of
the linear equations \eqref{e3} and \eqref{e2}.

\begin{theorem} \label{thm3.1}
The linear equation \eqref{e3} is $\Psi $-conditionally asymptotically
stable on $\mathbb{R}_{+}$ if and only if it
has a $\Psi $-unbounded solution on $\mathbb{R}_{+}$  and a
non-trivial solution $y_0(t)$ such that
$\lim_{t\to \infty }\Psi (t)y_0(t) = 0$.
\end{theorem}

\begin{proof}
Let $Y(t)$ be a fundamental matrix for \eqref{e3}. Suppose that the linear
equation \eqref{e3} is $\Psi $-conditionally asymptotically stable on 
$\mathbb{R}_{+}$. From Definition \ref{def2.3} and \cite[Theorem 3.1]{d1}, it
follows that $|\Psi $(t)Y(t)$|$ is unbounded on $\mathbb{R}_{+}$. Thus, the
linear equation \eqref{e3} has at least one $\Psi $-unbounded solution on 
$\mathbb{R}_{+}$. In addition, there exists a sequence $(y_{n}(t))$ of
non-trivial solutions of \eqref{e3} such that $\lim_{n\rightarrow \infty
}\Psi (t)y_{n}(t)=0$, uniformly on $\mathbb{R}_{+}$ and $\lim_{t\rightarrow
\infty }\Psi (t)y_{n}(t)=0$ for $n=1,2,\dots $. The proof of the
``only if'' part is complete.

Suppose, conversely, that \eqref{e3} has at least one $\Psi $-unbounded
solution on $\mathbb{R}_{+}$ and at least one non-trivial solution $y_0(t)$
such that $\lim_{t\to \infty }\Psi (t)y_0(t) =0$. It follows that the matrix 
$\Psi (t)Y(t)$ is unbounded on $\mathbb{R}_{+}$. Consequently, the linear
equation \eqref{e3} is $\Psi $-unstable on $\mathbb{R}_{+}$ 
(See \cite[Theorem 1]{d4}). On the other hand, $( \frac{1}{n}y_0(t)) $ 
is a sequence of
solutions of \eqref{e3} such that $\lim_{n\to \infty }\frac{1}{n}\Psi
(t)y_0(t) = 0$, uniformly on $\mathbb{R}_{+}$ and 
$\lim_{t\to \infty }\frac{1}{n}\Psi (t)y_0(t) = 0$ for $n \in \mathbb{N}$.
 Thus, the linear equation \eqref{e3} is $\Psi $-conditionally 
 asymptotically stable on $\mathbb{R}_{+}$. The proof is complete.
\end{proof}

We remark that Theorem \ref{thm3.1} generalizes a similar result in
connection with the classical conditional asymptotic stability in \cite{c3}.

The conditions for $\Psi $-conditional asymptotic stability of the linear
equation \eqref{e3} can be expressed in terms of a fundamental matrix for 
\eqref{e3}.

\begin{theorem} \label{thm3.2}
Let $Y(t)$ be a fundamental matrix for \eqref{e3}.
Then, the linear equation \eqref{e3} is $\Psi $-conditionally
asymptotically stable on $\mathbb{R}_{+}$  if and only if there are
satisfied two following conditions:
\begin{itemize}
\item[(a)]  There exists a projection $P_1$ such that
$\Psi (t)Y(t)P_1$ is unbounded on $\mathbb{R}_{+}$;

\item[(b)] there exists a projection $P_2 \neq 0$
such that $\lim_{t\to \infty }\Psi (t)Y(t)P_2 = 0$.
\end{itemize}
\end{theorem}

\begin{proof}
First, we shall prove the sufficiency. From the hypoyhesis (a) and 
\cite[Theorem 1]{d4}, it follows that the linear equation \eqref{e3} is 
$\Psi$-unstable on $\mathbb{R}_{+}$.

Let $y(t)$ be a non-trivial solution on $\mathbb{R}_{+}$ of the linear
equation \eqref{e3}. Let $(\lambda _{n})$ be such that $\lambda _{n}\in 
\mathbb{R}\setminus \{ 1\}$, $\lim_{n\to \infty }\lambda _{n} = 1$ and let 
$(y_{n} )$ be defined by 
\begin{equation*}
y_{n}(t) = Y(t)P_{2} Y^{-1}(0)(\lambda _{n} y(0)) + Y(t)(I
-P_{2})Y^{-1}(0)y(0), t \geq 0.
\end{equation*}
It is easy to see that $y_n(t)$, $n \in N$, are solutions of the linear
equation \eqref{e3}.

For $n \in N$ and $t \geq 0$, we have 
\begin{align*}
\| \Psi (t)y_{n}(t) -\Psi (t)y(t)\| &= \| \Psi (t)Y(t)P_{2}
Y^{-1}(0)((\lambda_{n}- 1)y(0))\| \\
& \leq | \lambda _{n}-1| | \Psi (t)Y(t)P _{2}| \| Y^{-1}(0)y(0)\|
\end{align*}
Thus, 
\begin{gather*}
\lim_{n\to \infty } \Psi (t)y_n(t) = \Psi (t)y(t), \quad\text{uniformly on }
\mathbb{R}_{+}, \\
\lim_{t\to \infty }\Psi (t)( y_{n}(t) - y(t) ) = 0, \quad\text{for }n = 1,
2, \dots .
\end{gather*}
It follows that the linear equation \eqref{e3} is $\Psi $-conditionally
asymptotically stable on $\mathbb{R}_{+}$.

Now, we shall prove the necessity. From $\Psi $-conditional asymptotic
stability on $\mathbb{R}_{+}$ of \eqref{e3}, it follows that $\Psi (t)Y(t)$
is unbounded on $\mathbb{R}_{+}$ (see \cite[Theorem 1]{d4}.

In addition, there exists a non-trivial solution $y_0(t)$ on $\mathbb{R}_{+}$
of \eqref{e3} such that $\lim_{t\to \infty }\Psi (t)y_0(t) = 0$. Thus, there
exists a constant vector $c \neq 0$ such that $\Psi (t)Y(t)c$ is such that 
$\lim_{t\to \infty }\Psi (t)Y(t)c = 0$. Let $c_s = \| c\| $. Let $P_2$ be the
null matrix in which the $s$-th column is replaced with $\| c\| ^{-1}c$.
Thus, $P_2$ is a projection and $\lim_{t\to \infty }\Psi (t)Y(t)P_2 =0$.

The proof is now complete.
\end{proof}

A sufficient condition for $\Psi $-conditional asymptotic stability is given
by the following theorem.

\begin{theorem} \label{thm3.3}
If there exist two supplementary projections
$P_{1}$, $P_{2}$, $P_{i} \neq 0$, and
a positive constant $K$ such that the fundamental matrix
$Y(t)$ of the equation \eqref{e3} satisfies the condition
\begin{equation*}
\int_0^{{t}}| \Psi (t)Y(t)P_1 Y^{-1}(s)
\Psi ^{-1}(s)| ds+\int_{t}^\infty
|\Psi (t)Y(t)P_2 Y^{-1}(s)\Psi ^{-1}(s)| {ds}\leq { K}
\end{equation*}
 for all $t \geq 0$, then, the linear equation \eqref{e3} is
$\Psi $-conditionally asymptotically stable on $\mathbb{R}_{+}$.
\end{theorem}

The proof of the above theorem follows from 
\cite[Theorem 2 and Lemmas 1, 2]{d4}.

\begin{theorem} \label{thm3.4}
Suppose that:
\begin{enumerate}
\item There exist supplementary projections $P_1$, $P_2$,
 $P_i \neq  0$, and a constant $K>0$
 such that the fundamental matrix $Y(t)$ of \eqref{e3} satisfies
 the conditions
\begin{gather*}
| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)|
 \leq  K, \quad\text{for }0 \leq  s \leq t, \\
| \Psi (t)Y(t)P_2 Y^{-1}(s)\Psi ^{-1}(s)|
 \leq  K, \quad\text{for }0 \leq  t \leq s.
\end{gather*}

\item $\lim_{t\to \infty }\Psi (t)Y(t)P_1=0$.

\item $B(t)$ is a $d\times d$ continuous matrix function on
$\mathbb{R}_{+}$  such that
\begin{equation*}
\int_0^\infty | \Psi {(t)B(t)}\Psi^{-1}{(t)}| dt \quad\text{is convergent.}
\end{equation*}

\item The linear equations \eqref{e2} and \eqref{e3} are $\Psi$-unstable
on $\mathbb{R}_{+}$.
\end{enumerate}
Then  \eqref{e2} is $\Psi$-conditionally
asymptotically stable on $\mathbb{R}_{+}$.
\end{theorem}

\begin{proof}
We choose t$_0\geq 0$ sufficiently large so that 
\begin{equation*}
q = K\int_{t_0}^\infty | \Psi {(t)B(t)}\Psi ^{-1} (t)| dt < 1.
\end{equation*}
We put 
\begin{equation*}
S = \{ x : t_0,\infty )\to \mathbb{R}^d:x \text{ is continuous and 
$\Psi$-bounded on }[t_0,\infty ) \}.
\end{equation*}
Define on the set $S$ a norm by 
\begin{equation*}
||| x||| = \sup_{t\geq t_0} \| \Psi (t)x(t)\| .
\end{equation*}
It is well known that $( S, ||| \cdot |||) $ is a Banach real space.

For $x \in S$, we define 
\begin{equation*}
( {Tx}) (t) = \int_{t_0}^t{Y(t)P}_1 Y^{-1}(s)B(s)x(s)ds - \int_t^\infty
Y(t)P_2 Y^{-1} (s)B(s)x(s)ds, \quad t \geq t_0.
\end{equation*}
It is easy to see that $(Tx)(t)$ exists and is continuous for $t \geq t_0$
(see the Proof of \cite[Theorem 3]{d5}). We have 
\begin{align*}
\| \Psi (t)( Tx) (t)\| &\leq {\ K}\int_{t_0}^\infty | \Psi (s)B(s)\Psi
^{-1}(s)| \| \Psi (s)x(s)\| ds \\
&\leq {\ q }\sup_{t\geq t_0}\| \Psi (t)x(t)\| = q||| x||| ,\quad\text{for }t
\geq t_0.
\end{align*}
This shows that $TS \subseteq S$.

On the other hand, $T$ is linear and 
\begin{equation*}
||| Tx_1-Tx_2||| = ||| T(x_1-x_2)||| \leq {\ q}||| x_1-x_2||| .
\end{equation*}

Thus, $T$ is a contraction on the Banach space $( S, ||| \cdot ||| )$.

Now, for every fixed $\Psi$- bounded solution $y$ of \eqref{e3} we define an
operator $S_{ y} : S \to S$, by the relation 
\begin{equation}
{S}_{ y}x(t) = y(t) + Tx(t), \quad t \in [t_0,\infty ).  \label{e4}
\end{equation}
It follows by the Banach contraction principle that $S_y$ has a unique fixed
point in $S$. An easy computation shows that the fixed point $x(t)= S_yx(t)$, 
$t \in [t_0,\infty)$, is a $\Psi$-bounded solution of \eqref{e2}.

Let $S_{2}$, $S_{3}$ be the spaces of $\Psi$-bounded solutions of equations 
\eqref{e2} and \eqref{e3} respectively. We define the mapping $C : S_{3}\to
S_{2}$ in the following way: For every $y \in S_{3}$, $Cy$ will be the fixed
point of the contraction $S_y$.

Now, from $x=Cy$ and $x_{0}=Cy_{0}$, we have that $x=y+Tx$,
 $x_{0}=y_{0}+Tx_{0}$ respectively. We obtain 
\begin{align*}
|||{\ x-x}_{0}|||& \leq |||{\ y-y}_{0}|||+|||{\ Tx-Tx}_{0}||| \\
& \leq |||{\ y-y}_{0}|||+q|||{\ x-x}_{0}|||.
\end{align*}
Thus 
\begin{equation}
|||{\ x-x}_{0}|||\leq {(1-q)}^{-1}|||{\ y-y}_{0}|||.  \label{e5}
\end{equation}
On the other hand, 
\begin{align*}
|||{\ y-y}_{0}|||& =|||{\ x-Tx-x}_{0}{\ +}\text{ }{Tx}_{0}||| \\
& \leq |||{\ x-x}_{0}\text{ }|||\text{ }+\text{ }|||{\ Tx-Tx}_{0}\text{ }|||
\\
& \leq {\ (1+q)}|||{\ x-x}_{0}\text{ }|||.
\end{align*}
Thus, $C$ is homeomorfism.

Now, we prove that if $x,$ $y\in S$ are $\Psi $-bounded solutions of 
\eqref{e2} and \eqref{e3} respectively such that $x=Cy$, then 
\begin{equation*}
\lim_{t\rightarrow \infty }\Vert \Psi {(t)(x(t)-y(t))}\Vert =0.
\end{equation*}
Indeed, for a given $\varepsilon >0$, we choose $t_{1}\geq t_{0}$ so that

\begin{equation*}
{K}\sup_{t\geq t_{0}}\Vert \Psi {(t)x(t)}\Vert \int_{t_{1}}^{\infty }|\Psi {%
(s)B(s)}\Psi ^{-1}(s)|ds<\frac{\varepsilon }{3}.
\end{equation*}
Thus, for $t\geq t_{1}$, we have 
\begin{align*}
& \Vert \Psi {(t)(x(t)-y(t))}\Vert \\
& =\Vert \Psi {(t)(Tx)(t)}\Vert \\
& \leq \int_{t_{0}}^{t}\Vert \Psi (t)Y(t)P_{1}Y^{-1}{\ (s)B(s)x(s)}\Vert ds
\\
& \quad +\int_{t}^{\infty }\Vert \Psi (t)Y(t)P_{2}Y^{-1}(s)\Psi ^{-1}(s)\Psi 
{(s)B(s)}\Psi ^{-1}(s)\Psi {(s)x(s)}\Vert {ds} \\
& \leq |\Psi (t)Y(t)P_{1}|\int_{t_{0}}^{t_{1}}\Vert {\ Y}^{-1}{(s)B(s)x(s)}%
\Vert ds \\
& \quad +K\sup_{t\geq t_{0}}\Vert \Psi {(t)x(t)}\Vert \int_{t_{1}}^{\infty
}|\Psi {(s)B(s)}\Psi ^{-1}(s)|ds \\
& \quad +K\sup_{t\geq t_{0}}\Vert \Psi {(t)x(t)}\Vert \int_{t}^{\infty
}|\Psi {(s)B(s)}\Psi ^{-1}(s)|ds \\
& <|\Psi (t)Y(t)P_{1}|\int_{t_{0}}^{t_{1}}\Vert Y^{-1}{(s)B(s)x(s)}\Vert 
{ds+2}\frac{\varepsilon }{3}.
\end{align*}
Thus and assumption 3, 
\begin{equation}
\lim_{t\rightarrow \infty }\Vert \Psi {(t)(x(t)-y(t))}\Vert =0.  \label{e6}
\end{equation}
>From the hypotheses, \cite[Theorem1 and 2]{d4} it follows that the linear
equation \eqref{e3} is $\Psi $-conditionally asymptotically stable on $%
\mathbb{R}_{+}$.

Let $x(t)$ be a $\Psi $-bounded solution on $\mathbb{R}_{+}$ of \eqref{e2}.
>From the assumption 4, this solution is $\Psi $-unstable on 
$\mathbb{R}_{+}$. Let $y=C^{-1}x$. From Definition \ref{def2.3}, it follows that there
exists a sequence $(y_{n})$ of solutions of \eqref{e3} defined on 
$\mathbb{R}_{+}$ such that 
\begin{gather*}
\lim_{n\rightarrow \infty }\Psi (t)y_{n}(t)=\Psi (t)y(t),\quad \text{%
uniformly on }\mathbb{R}_{+}, \\
\lim_{t\rightarrow \infty }\Psi (t)(y_{n}(t)-y(t))=0,\quad \text{for }%
n=1,2,\dots .
\end{gather*}
Let $x_{n}=Cy_{n}$. From \eqref{e5} it follows that the sequence $(x_{n})$
of solutions of \eqref{e2} defined on $[t_{0},\infty )$ (in fact, defined on 
$\mathbb{R}_{+}$) satisfies the condition 
\begin{equation*}
\lim_{n\rightarrow \infty }\Psi {(t)x}_{n}{(t)=}\Psi (t)x(t),\quad \text{%
uniformly on }[{t}_{0},\infty ).
\end{equation*}
Clearly, 
\begin{equation*}
\lim_{n\rightarrow \infty }{\ x}_{n}{(t}_{0})=x(t_{0}).
\end{equation*}
By the Dependence on initial conditions Theorem (see \cite[Chapter I,
Theorem 3]{c3}), it follows that 
\begin{equation*}
\lim_{n\rightarrow \infty }x_{n}(t)=x(t),\quad \text{uniformly on }[0,t_{0}].
\end{equation*}
Hence, 
\begin{equation*}
\lim_{n\rightarrow \infty }\Psi (t)x_{n}(t)=\Psi (t)x(t),\quad 
\text{uniformly on }[0,t_{0}].
\end{equation*}
Thus, 
\begin{equation*}
\lim_{n\rightarrow \infty }\Psi {(t)x}_{n}{(t)=}\Psi (t)x(t),\quad 
\text{uniformly on }\mathbb{R}_{+}.
\end{equation*}
This shows that the linear equation \eqref{e2} is $\Psi $-conditionally
stable on $\mathbb{R}_{+}$. From \eqref{e6} and 
\begin{equation*}
\Psi (t)(x_{n}(t)-x(t))=\Psi (t)(x_{n}(t)-y_{n}(t))+\Psi
(t)(y_{n}(t)-y(t))+\Psi (t)(y(t)-x(t)),
\end{equation*}
it follows that 
\begin{equation*}
\lim_{t\rightarrow \infty }\Psi (t)(x_{n}(t)-x(t))=0,\quad \text{for }%
n=1,2,\dots .
\end{equation*}
This shows that the linear equation \eqref{e2} is $\Psi $-conditionally
asymptotically stable on $\mathbb{R}_{+}$. The proof is complete.
\end{proof}

\begin{theorem} \label{thm3.5}
Suppose that:
\begin{enumerate}
\item There exist two supplementary projections $P_1$, $P_2$,
 $P_i\neq  0$, and a positive constant $K$ such that the fundamental
matrix $Y(t)$ of the equation \eqref{e3} satisfies the
condition
\begin{equation*}
\int_0^{{t}}| \Psi (t)Y(t)P_{1} Y^{-1}{
(s)}\Psi ^{-1}(s)| ds  + \int_{t}^{\infty
}| \Psi (t)Y(t)P_{2} Y^{-1}(s)\Psi
^{-1}(s)| {ds}\leq { K}
\end{equation*}
for all t $\geq 0$.

\item $B(t)$ is a $d\times d$ continuous matrix function on
$\mathbb{R}_{+}$ such that
\begin{equation*}
\lim_{t\to \infty }| \Psi {(t)B(t)}\Psi^{-1}{(t)}|  = 0.
\end{equation*}
\end{enumerate}
Then, the linear equation \eqref{e2} is $\Psi$-conditionally
asymptotically stable on $\mathbb{R}_{+}$.
\end{theorem}

The proof of the above theorem is similar to the proof of Theorem 
\ref{thm3.4}.

\begin{remark} \rm
The first condition of the above Theorems can certainly be
satisfied if A(t) = A is a d$\times $d real constant matrix which has
characteristic roots with different real parts. In this case, e.g., there
exists an interval $(\alpha , \beta )\subset \mathbb{R}$ such that for
$\lambda \in  (\alpha, \beta)$,
 $\Psi (t) = e^{-\lambda {t}}I_{d}$ and $Y(t)$ can satisfy the first
hypotheses of Theorems.

We have a similar situation if $A(t)$ is a $d\times d$ real continuous
periodic matrix (See \cite[Examples 1, 2]{d5}).

Thus, the above results can be considered as a generalization of a
well-known result in conection with the classical conditional asymptotic
stability.
\end{remark}

\begin{remark} \rm
If in the above Theorems, the linear equation \eqref{e3} is only
$\Psi $-conditionally asymptotically stable on $\mathbb{R}_{+}$,
then the perturbed equation \eqref{e2} can not be $\Psi $-conditionally
asymptotically stable on $\mathbb{R}_{+}$.

This is shown by the next example transformed after an equation due
to Perron \cite{p1}.
\end{remark}

\begin{example} \label{exa1} \rm
Let $a, b \in \mathbb{R}$  such that $0 < 4a< 1$, $b \neq  0$ and
\begin{equation*}
A(t)=\begin{pmatrix}
\sin \ln (t+1)+\cos \ln (t+1)-4a & 0 \\
0 & -2a
\end{pmatrix}.
\end{equation*}
Then, a fundamental matrix for the homogeneous equation \eqref{e3} is
\begin{equation*}
Y(t) = \begin{pmatrix}
e^{(t+1)[\sin \ln (t+1)-4a]} & 0 \\
0 & e^{-2a(t+1)}
\end{pmatrix}.
\end{equation*}
Let
\begin{equation*}
\Psi (t)=\begin{pmatrix}
1 & 0 \\
0 & e^{a(t+1)}
\end{pmatrix}.
\end{equation*}
We have
\begin{equation*}
\Psi (t)Y(t)=\begin{pmatrix}
e^{(t+1)[\sin \ln (t+1)-4a]} & 0 \\
0 & e^{-a(t+1)}
\end{pmatrix}.
\end{equation*}
Let $t_{n}' = e^{(2n+\frac{1}{2})\pi } - 1$
 for $n = 1, 2\dots $.
Since $\lim_{n\to \infty }| \Psi (t_{n}')Y(t_{n}')|  = \infty $,
 it follows that the linear
equation \eqref{e3} is $\Psi$-unstable on $\mathbb{R}_{+}$
(see \cite[Theorem 1]{d4})

 From Theorem \ref{thm3.1} it follows that the linear equation \eqref{e3} is
$\Psi $-conditionally asymptotically stable on $\mathbb{R}_{+}$.
If we take
\begin{equation*}
{B(t) = }\begin{pmatrix}
0 & be^{-2a(t+1)} \\
0 & 0
\end{pmatrix},
\end{equation*}
then, a fundamental matrix for the perturbed equation \eqref{e2} is
\begin{equation*}
\widetilde{Y}(t)=\begin{pmatrix}
be^{(t+1)[\sin \ln (t+1)-4a]}\int_1^{t+1}e^{-s \sin \ln s}
{ds} & e^{(t+1)[\sin \ln (t+1)-4a]} \\
e^{-2a(t+1)} & 0
\end{pmatrix}.
\end{equation*}
We have
\begin{equation*}
\Psi (t)\widetilde{Y}(t)=
\begin{pmatrix}
be^{(t+1)[\sin \ln (t+1)-4a]}\int_1^{t+1}e^{-s \sin \ln s}
{ds} & e^{(t+1)[\sin \ln (t+1)-4a]} \\
e^{-a(t+1)} & 0
\end{pmatrix}.
\end{equation*}
Since $\lim_{n\to \infty }| \Psi (t_n')
\widetilde{Y}(t_n')|  = \infty $, it follows that the
perturbed equation \eqref{e2} is $\Psi$-unstable on $\mathbb{R}_{+}$
(see \cite[Theorem 1]{d4}).

Let $\alpha \in (0,\frac \pi 2)$. Let t$_n = e^{(2n-\frac 12)\pi }$
for $n = 1, 2,\dots$. For $t_n \leq  s \leq  t_ne^\alpha $
 we have $s\cos \alpha \leq  -s \sin \ln s \leq  s$ and hence
\begin{align*}
e^{t_ne^\pi (\sin \ln t_ne^\pi -4a)}\int_1^{t_ne^\pi }e^{-s
\sin \ln s}ds
&> e^{t_ne^\pi (\sin \ln t_ne^\pi
-4a)}\int_{t_n}^{t_ne^\alpha }e^{-s \sin \ln s}{ds }\\
&\geq  e^{t_{n}e^{\pi}(1-4a)}\int_{t_{n}}^{t_{n}e^{\alpha
}}e^{s \cos \alpha }ds \\
&= e^{t_{n}[(1-4a)e^{\pi }+\cos \alpha ]}
\frac{e^{t_{n}(e^{\alpha }-1)\cos \alpha }-1}{\cos \alpha }\to
\infty .
\end{align*}
Thus, the columns of $\Psi (t)\widetilde{Y}(t)$ are unbounded on
$\mathbb{R}_{+}$.
It follows that the perturbed equation \eqref{e2} is not $\Psi $-conditionally
asymptotically stable on $\mathbb{R}_{+}$ (see Theorem \ref{thm3.1}).

Finally, we have $| \Psi (t)B(t)\Psi ^{-1}(t) = be^{-3a(t+1)}$.
Thus, $B(t)$ satisfies the conditions:
\begin{equation*}
\lim_{t\to \infty }| \Psi {(t)B(t)}\Psi ^{-1}
{(t)}|  =0;
\end{equation*}
and
$\int_0^\infty | \Psi {(t)B(t)}\Psi ^{-1}{(t)}| dt$ can be a
sufficiently small number.
\end{example}

\section{$\Psi$-conditional asymptotic stability of the nonlinear equation 
\eqref{e1}}

In this section we give sufficient conditions for the $\Psi $-conditional
asymptotic stability of $\Psi$-bounded solutions of the nonlinear Volterra
integro-differential system \eqref{e1}.

\begin{theorem} \label{thm4.1}
Suppose that:
\begin{enumerate}
\item There exist supplementary projections $P_1$, $P_2$,
$P_i \neq 0$ and a constant $K> 0$ such that the fundamental
 matrix $Y(t)$ of \eqref{e3} satisfies the condition
\begin{equation*}
\int_0^{{t}}| \Psi (t)Y(t)P_1 Y^{-1}(s)
\Psi ^{-1}(s)| ds + \int_{{t}}^\infty|
\Psi (t)Y(t)P_2 Y^{-1}(s)\Psi ^{-1}(s)| {ds}\leq { K}
\end{equation*}
for all $t \geq 0$.

\item The function $F(t,s,x)$ satisfies the inequality
\begin{equation*}
\| \Psi {(t)}\left( {F(t,s,x(s)) }-{ F(t,s,y(s))}
\right) \|  \leq { f(t,s)}\| \Psi (s)\left(
{x(s) }-{ y(s)}\right) \| ,
\end{equation*}
for $0 \leq  s \leq  t < \infty $ and for all continuous and
$\Psi$-bounded functions $x, y : \mathbb{R}_{+}\to \mathbb{R}^d$,
where $f(t,s)$ is a continuous nonnegative function on $D$ such that
\begin{equation*}
F(t,s,0) = 0, \quad
\lim_{t\to \infty }\int_0^t f(t,s)ds = 0,\quad
\sup_{t\geq 0}\int_0^t{f(t,s)ds < K}^{-1}.
\end{equation*}
\end{enumerate}
Then, all $\Psi$-bounded solutions of \eqref{e1} are
$\Psi$-conditionally asymptotically stable on $\mathbb{R}_{+}$.
\end{theorem}

\begin{proof}
Let 
\begin{equation*}
q = K \sup_{t\geq 0} \int_0^t f(t,s)ds < 1.
\end{equation*}
We put 
\begin{equation*}
S = \{ x : \mathbb{R}_{+}\to {R}^d : x\text{ is continuous and } 
\Psi\text{-bounded on }\mathbb{R}_{+}\}.
\end{equation*}
Define on the set $S$ a norm by 
\begin{equation*}
||| {x}||| = \sup_{t\geq 0}\| \Psi {(t)x(t)}\| .
\end{equation*}
It is well-known that ($S$, $||| \cdot ||| $) is a Banach space. For $x \in
S $, we define 
\begin{align*}
\left( {Tx}\right) (t) &= \int_0^t{Y(t)P}_1 Y^{-1}(s)\int_o^s
F(s,u,x(u))\,du\,ds \\
&\quad - \int_t^\infty {Y(t)P}_2 Y^{-1}(s)\int_o^s{\ F(s,u,x(u))\,du\,ds, t }
\geq 0.
\end{align*}
For $0 \leq t \leq v$, we have 
\begin{align*}
&\| \Psi {(t)}\int_{t}^{{v}}{Y(t)P}_{2} Y^{-1} (s)\int_{o}^{s}{
F(s,u,x(u))\,du\,ds}\| \\
&= \| \int_{t}^{{v}}\Psi (t)Y(t)P_{2} Y^{-1} (s)\Psi
^{-1}(s)\int_{o}^{s}\Psi (s)F(s,u,x(u))du\,ds\| \\
&\leq \int_{t}^{{v}}| \Psi (t)Y(t)P_{2} Y^{-1} (s)\Psi ^{-1}(s)|
\int_{o}^{s}\| \Psi {(s)F(s,u,x(u))}\| \,du\,ds \\
&\leq \int_{t}^{{v}}| \Psi (t)Y(t)P_{2} Y^{-1}(s) \Psi ^{-1}(s)| \int_0^{s}{
f(s,u)}\| \Psi {\ (u)x(u)}\| \,du\,ds \\
&\leq \sup_{u\geq 0} \| \Psi {(u)x(u)}\| \int_{t}^{{v}}| \Psi (t)Y(t)P_{2}
Y^{-1}(s)\Psi ^{-1}(s)| \int_0^{s} f(s,u)\,du\,ds \\
&\leq {\ qK}^{-1}\sup_{u\geq 0} \| \Psi {(u)x(u)} \| \int_{t}^{{v}}| \Psi
(t)Y(t)P_{2} Y^{-1}{\ (s)}\Psi ^{-1}(s)| {ds.}
\end{align*}
>From assumption 1, it follows that the integral 
\begin{equation*}
\int_t^\infty {Y(t)P}_2 Y^{-1}(s)\int_o^s{F(s,u,x(u))\,du\,ds}
\end{equation*}
is convergent. Thus, $(Tx)(t)$ exists and is continuous for $t \geq 0$. For 
$x \in S$ and $t \geq 0$, we have 
\begin{align*}
\| \Psi {(t)(Tx)(t)}\| &= \|\int_0^t\Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)
\int_o^s\Psi {(s)F(s,u,x(u))\,du\,ds } \\
&\quad - \int_t^\infty \Psi (t)Y(t)P_2 Y^{-1}(s)\Psi ^{-1}(s)\int_o^s\Psi {
(s)F(s,u,x(u))\,du\,ds}\| \\
& \leq \int_0^t| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)| \int_o^s\| \Psi {
(s)F(s,u,x(u))}\| \,du\,ds \\
&\quad + \int_t^\infty | \Psi (t)Y(t)P_2 Y^{-1}(s) \Psi ^{-1}(s)| \int_o^s\|
\Psi {(s)F(s,u,x(u))} \| \,du\,ds \\
&\leq \int_0^t| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)| \int_0^s{f(s,u)}\|
\Psi {(u)x(u)} \| \,du\,ds \\
&\quad + \int_t^\infty | \Psi (t)Y(t)P_2 Y^{-1}(s) \Psi ^{-1}(s)| \int_o^s{
f(s,u)}\| \Psi {(u)x(u)} \| \,du\,ds \\
&\leq q \sup_{u\geq 0} \| \Psi {(u)x(u)} \| .
\end{align*}
This shows that $TS \subseteq S$. On the other hand, for $x, y \in S$ and 
$t \geq 0$, we have 
\begin{align*}
&\| \Psi {(t)}\left( (Tx)(t) - (Ty)(t) \right) \| \\
&=\| \int_0^{t}\Psi (t)Y(t)P_{1} Y^{-1}(s) \Psi ^{-1}(s)\int_{o}^{s}\Psi
(s)\left( F(s,u,x(u)) - F(s,u,y(u))\right) \,du\,ds \\
&\quad -\int_{t}^{\infty }\Psi (t)Y(t)P_{2} Y^{-1}(s)\Psi ^{-1}
(s)\int_{o}^{s}\Psi (s)\left( F(s,u,x(u)) - F(s,u,y(u)) \right) \,du\,ds\| \\
&\leq \int_0^{t}| \Psi (t)Y(t)P_{1} Y^{-1}(s)\Psi ^{-1}(s)| \int_{o}^{s}\|
\Psi (s)\left( F(s,u,x(u))-F(s,u,y(u))\right) \| \,du\,ds \\
&\quad +\int_{t}^{\infty }\!| \Psi (t)Y(t)P_{2} Y^{-1}(s) \Psi ^{-1}(s)|
\int_{o}^{s}\!\| \Psi (s)\left( F(s,u,x(u)) - F(s,u,y(u))\right) \| \,du\,ds
\\
&\leq \int_0^t| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1} (s)| \int_0^s{f(s,u)}\|
\Psi (u)(x(u) - y(u)) \| \,du\,ds \\
&\quad + \int_t^\infty | \Psi (t)Y(t)P_2 Y^{-1}(s)\Psi ^{-1}(s)| \int_0^s{
f(s,u)}\| \Psi {(u)(x(u) - y(u))}\| \,du\,ds \\
&\leq q \sup_{u\geq 0} \| \Psi {(u)(x(u) - y(u))}\| .
\end{align*}
It follows that 
\begin{equation*}
\sup_{t\geq 0} \| \Psi {(t)}\left( {(Tx)(t) - (Ty)(t)}\right) \| \leq q
\sup_{t\geq 0} \| \Psi {(t)(x(t) - y(t))}\| .
\end{equation*}
Thus, we have 
\begin{equation*}
||| Tx - Ty||| \leq q ||| x - y||| .
\end{equation*}
This shows that $T$ is a contraction of the Banach space 
$( S,||| \cdot |||) $.

As in the Proof of Theorem \ref{thm3.4}, it follows by the Banach
contraction principle that for any function $y \in S$, the integral equation 
\begin{equation}  \label{e7}
x = y + Tx
\end{equation}
has a unique solution $x \in S$. Furthermore, by the definition of $T$, 
$x(t)- y(t)$ is differentiable and 
\begin{equation*}
\left( {x(t) - y(t)}\right) '{\ = A(t)}\left( {x(t) - y(t)}\right)
+ \int_0^{t}F(t,s,x(s))ds, t \geq 0.
\end{equation*}
Hence, if y(t) is a $\Psi$-bounded solution of \eqref{e3}, $x(t)$ is a 
$\Psi$-bounded solution of \eqref{e1}. Conversely, if $x(t)$ is a $\Psi$-bounded
solution of \eqref{e1}, the function $y(t)$ defined by \eqref{e7} is a 
$\Psi$-bounded solution of \eqref{e3}.

Thus, \eqref{e7} establishes a one-to-one correspondence $C$ between the 
$\Psi$-bounded solutions of \eqref{e1} and \eqref{e3}: $x = Cy$.

Now, we consider the analogous equation 
\begin{equation*}
x_0 = y_0 + Tx_0.
\end{equation*}
We get 
\begin{equation}
{(1 - q)}||| {\ x - x}_0 ||| \leq ||| {\ y - y}_0 ||| .  \label{e8}
\end{equation}
Now, we prove that if $x, y \in S$ are $\Psi$-bounded solutions of \eqref{e1}
and \eqref{e3} respectively such that $x = Cy$, then 
\begin{equation}
\lim_{t\to \infty }\| \Psi {(t)( x(t) - y(t) )}\| = 0.  \label{e9}
\end{equation}
For a given $\varepsilon >0$, we can choose $t_1 \geq 0$ such that 
\begin{equation*}
{K}||| x||| \int_0^{t }f(t,s)ds < \frac \varepsilon 2,
\end{equation*}
for $t \geq t_1$. Moreover, since $\lim_{t\to \infty }| \Psi (t)Y(t)P_1| = 0$
(see \cite[Lemma 1]{d4}), there exists a number $t_2\geq t_1$ such that 
\begin{equation*}
{qK}^{-1}| \Psi (t)Y(t)P_1||| | x||| \int_0^{{t}_1}| {P}_1 Y^{-1}(s)\Psi
^{-1} (s)| ds < \frac \varepsilon 2
\end{equation*}
for $t \geq t_2$. We have, for $t \geq t_2$, 
\begin{align*}
&\| \Psi {(t)( x(t) - y(t) )}\| \\
& \leq \int_0^t| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)| \int_o^s\| \Psi {%
(s)F(s,u,x(u))}\| {\,du\,ds +} \\
&\quad + \int_t^\infty | \Psi (t)Y(t)P_2 Y^{-1}(s) \Psi ^{-1}(s)| \int_o^s\|
\Psi {(s)F(s,u,x(u))} \| \,du\,ds \\
&\leq \int_0^t| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)| \int_0^s{f(s,u)}\|
\Psi {(u)x(u)} \| \,du\,ds \\
&\quad + \int_t^\infty | \Psi (t)Y(t)P_2 Y^{-1}(s) \Psi ^{-1}(s)|
 \int_o^s{f(s,u)}\| \Psi {(u)x(u)} \| \,du\,ds \\
&\leq {qK}^{-1}| \Psi (t)Y(t)P_1|\,|||x||| \int_0^{{t}_1}| {P}_1
Y^{-1}(s)\Psi ^{-1}(s)| ds \\
&\quad + ||| x||| \int_{t_1}^{{t}}| \Psi (t)Y(t)P_1 Y^{-1}(s)\Psi ^{-1}(s)| 
\Big( \int_0^s{f(s,u)du}\Big) ds \\
&\quad + ||| x||| \int_{{t}}^\infty | \Psi (t)Y(t)P_2 Y^{-1}(s)\Psi
^{-1}(s)| \Big( \int_0^s{f(s,u)du}\Big) ds < \varepsilon .
\end{align*}
Now, let $x(t)$ be a $\Psi$-bounded solution of \eqref{e1}. This solution is 
$\Psi$-unstable on $\mathbb{R}_{+}$.

Indeed, if not, for every $\varepsilon $ > 0 and any $t_0\geq 0$, there
exists a $\delta =\delta (\varepsilon ,t_0)> 0$ such that any solution 
$\widetilde{{x}}(t)$ of \eqref{e1} which satisfies the inequality $\| \Psi
(t_0) ( \widetilde{{x}}(t_0) - x(t_0) ) \| < \delta (\varepsilon ,t_0)$
exists and satisfies the inequality $\| \Psi (t)(\widetilde{{x}}(t) -
x(t))\| < \varepsilon $ for all $t \geq t_0$.

Let $z_0\in R^{d}$ be such that $P_{1}z_0 = 0$ and $0 < \| \Psi (0)z_0\| <
\delta (\varepsilon ,0)$ and let $\widetilde{x}(t)$ the solution of 
\eqref{e1} with the initial condition $\widetilde{x}(0) = x(0) + z_0$. Then
 $\| \Psi (t)z(t)\| < \varepsilon $ for all $t \geq 0$, where $z(t) = 
\widetilde{x}(t) - x(t)$.

Now we consider the function $y(t) = z(t) - (Tz)(t)$, $t \geq 0$.

Clearly, $y(t)$ is a $\Psi$-bounded solution on $\mathbb{R}_{+}$ of 
\eqref{e3}. Without loss of generality, we can suppose that $Y(0) = I$. It
is easy to see that $P_1y(0) = 0$. If $P_2y(0) \neq 0$, from 
\cite[Lemma 2]{d4}, it follows that 
$\limsup_{t\to \infty }\| \Psi (t)y(t)\| = \infty $,
which is contradictory. Thus, $P_2y(0) = 0$ and then $y(t) = 0$ for 
$t \geq 0 $.

It follows that $z = Tz$ and then $z = 0$, which is a contradiction. This
shows that the solution $x(t)$ is $\Psi$-unstable on $\mathbb{R}_{+}$.

Let $y = x - Tx$. From Theorem \ref{thm3.3}, it follows that there exists a
sequence $(y_{n})$ of solutions of \eqref{e3} defined on $\mathbb{R}_{+}$
such that 
\begin{gather*}
\lim_{n\to \infty }\Psi (t)y_n (t) = \Psi (t)y(t),\quad\text{uniformly on }
\mathbb{R}_{+}, \\
\lim_{t\to \infty }\Psi (t)( y_{n}(t) - y(t) )=0, \quad n = 1,2,\dots .
\end{gather*}
Let $x_{n} = Cy_{n}$. From \eqref{e8} it follows that the sequence $(x_{n})$
of solutions of \eqref{e1} defined on $\mathbb{R}_{+}$ is such that 
\begin{equation*}
\lim_{n\to \infty }\Psi (t)x_n(t) = \Psi (t)x(t), \quad\text{uniformly on }%
\mathbb{R}_{+}.
\end{equation*}
This shows that the solution $x(t)$ is $\Psi $-conditionally stable on 
$\mathbb{R}_{+}$. From \eqref{e9} and 
\begin{equation*}
\Psi (t)( x_{n}(t) - x(t) ) = \Psi (t)( x_{n}(t)- y_{n}(t) ) + \Psi (t)(
y_{n}(t) - y(t) ) + \Psi (t)( y(t) - x(t) ),
\end{equation*}
it follows that 
\begin{equation*}
\lim_{t\to \infty }\Psi (t)( x_{n}(t) - x(t) ) = 0, \quad\text{for } n = 1,
2, \dots .
\end{equation*}
This shows that the solution $x(t)$ is $\Psi $-conditionally asymptotically
stable on $\mathbb{R}_{+}$. The proof is now complete.
\end{proof}

\begin{corollary} \label{coro4.1}
If in Theorem \ref{thm4.1} we assume that $f(t.s) =g(t)h(s)$, where
$g$ and $h$ are nonnegative continuous functions on $\mathbb{R}_{+}$
 such that
\begin{gather*}
\sup_{t\geq 0} {g(t)}\int_0^t h(s)ds < K^{-1} , \\
\lim_{t\to \infty }{g(t)}\int_0^t h(s)ds = 0,
\end{gather*}
then the conclusion of the Theorem remains valid.
\end{corollary}

\begin{corollary} \label{coro4.2}
If in Theorem \ref{thm4.1} we assume that $f(t.s) =g(t)h(s)$, where 
$g$ and $h$ are
nonnegative continuous functions on $\mathbb{R}_{+}$
 such that
\begin{gather*}
I = \int_0^\infty h(s)\,ds \quad\text{is convergent},
\\
\lim_{t\to \infty }g(t) = 0, \quad \sup_{t\geq 0}g(t) < \frac 1{KI},
\end{gather*}
then the conclusion of the Theorem remains valid.
\end{corollary}

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