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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 31, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/31\hfil Stability for a non-local problem]
{Stability for a non-local non-autonomous
system of fractional order differential equations \\ with delays}

\author[A. M. A. El-Sayed, F. M. Gaafar,  E. M. A. Hamadalla\hfil 
EJDE-2010/31\hfilneg]
{Ahmed M. A. El-Sayed, Fatma  M. Gaafar,  Eman M. A. Hamadalla}
 % in alphabetical order

\address{Ahmed M. A. El-Sayed \newline
Faculty of Science,  Alexandria University,  Alexandria,  Egypt}
\email{amasayed@hotmail.com}

\address{Fatma  M. Gaafar  \newline
Faculty of Science, Damanhour, Alexandria University, Alexandria,
Egypt}
\email{gaafarfatma@yahoo.com}

\address{Eman M. A. Hamadalla \newline
Faculty of Science,  Alexandria University,  Alexandria,  Egypt}
\email{emanhamdalla@hotmail.com}


\thanks{Submitted October 12, 2009. Published February 26, 2010.}
\subjclass[2000]{34A12, 34A30, 34D20}
\keywords{Riemann-Liouvile derivatives; nonlocal
non-autonomous system; \hfill\break\indent
time-delay system; stability analysis}

\begin{abstract}
 In this article, we establish sufficient conditions for the
 existence, uniqueness and uniformly stability of solutions
 for a class of nonlocal non-autonomous system of fractional-order
 delay differential equations with several delays.
\end{abstract}

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

\section{Introduction}

Let $ x(t)=(x_1(t),x_2(t),\dots ,x_n(t))'$, where $'$
denoted the transpose of the matrix. Let $ \alpha \in(0,1]  $ and
$ i=1,2,\dots ,n$.  Consider the nonlocal problem
\begin{gather}
D^{\alpha}x_i(t) = \sum_{j=1}^n a_{ij}(t)
 x_j(t) + \sum_{j=1}^n b_{ij}(t)
 x_j(t-r_j) + h_i(t) , \quad   t> 0 \label{1}\\
x(t)=\Phi(t) \quad  \text{for }    t<0 ,\quad\text{and}\quad
         \lim_{t\to 0^-}
 \Phi(t) = O \label{2}\\
I^{\beta} x(t)|_{t=0}=O ,  \quad   \beta \in  (0,1] \label{3}
\end{gather}
where $D^{\alpha}$ denoted the Riemann-Liouville derivative of
order $\alpha$;
$A(t) = (a_{ij}(t))_{n \times n}$,
$B(t) = (b_{ij}(t))_{n \times n}$,
$H(t) = (h_i(t))_{n \times 1} $,
$ \Phi(t) = (\phi_i(t))_{n \times 1}$ are given
matrices; $O$ is the zero matrix;  $r_j\geq 0 $ are constants.

Fractional differential equations has been studied by various
researchers because they appear in various fields:
physics, mechanics, engineering,
electrochemistry, economics;
see for example  \cite{71}-\cite{7},
 \cite{54}-\cite{75} and references therein.

In this work, we discuss the existence, uniqueness and stability of
solution of the non-autonomous time-varying delay
 system \eqref{1}-\eqref{3}.
Abd El-Salam and El-Sayed \cite{1} proved the existence of a unique
uniformly stable solution for the non-autonomous system
\[
^cD_a^\alpha x(t) = A(t) x(t) + f(t)
\quad         x(0) = x^0 ,  \quad t>0
\]
where $^cD_a^\alpha$ is the Caputo fractional derivatives (see
\cite{5}-\cite{6}), $A(t)$ and $f(t)$ are continuous matrices.
El-Sayed \cite{11} proved the existence and uniqueness of the solution
$u(t)$ of the problem
\begin{gather*}
^cD^{\alpha}_a u(t) + CD_a^{\beta}u(t-r)
= Au(t) + Bu(t-r), \quad 0\leq\beta\leq\alpha\leq1\\
u(t) = g(t),\quad t\in[a-r,a], \; r>0
\end{gather*}
by the method of steps, where $A,B,C$ are bounded operators defined
on a Banach space $X$. Zhang  \cite{2} established the existence
of a  unique solution for the delay fractional differential equation
\[
D^{\alpha} x(t)=A_0x(t) + A_1x(t-r) + f(t), \quad
  t>0,  \quad    x(t)=\phi(t), \quad  t\in [-r,0]
\]
by the method of steps, where $A_0, A_1$ are constant matrices.
a study of finite time stability was shown there.

Here we prove the existence of a unique solution
for \eqref{1}-\eqref{3}, of the  form
\[
x_i(t) = \begin{cases}
\phi_i(t)  ,   &t < 0\\
0  ,  & t = 0\\
I^\alpha \{\sum_{j=1}^n a_{ij}(t)  x_j(t) + \sum_{j=1}^n b_{ij}(t)
 x_j(t-r_j) + h_i(t)\}    , &  t > 0.
\end{cases}
\]
This solution is in $C((-\infty,T])$, $T<\infty$, and is
uniformly stable.


\section{Preliminaries}

In this section, we introduce notation, definitions, and preliminary
facts which are used thought this paper.

\begin{definition} \label{def1} \rm
 The fractional (arbitrary) order
integral of a function $f \in L_1[a,b] $ of order $ \alpha \in
R^+ $ is defined by
\[
I^{\alpha}_a
 f(t) = \int_a^t \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)} f(s)\, ds.
\]
where $\Gamma$ is the gamma function; see \cite{4,5,54,6}.
\end{definition}

\begin{definition} \label{def2}\rm
 The Riemann-liouville fractional
(arbitrary) order derivatives of order $\alpha \in (n-1,n) $ of
the function $f$ is defined by
\[
D^{\alpha}_a f(t) = \frac{d^n}{dt^n} I^{n-\alpha}_a f(t) =
\frac{1}{\Gamma(n-\alpha)}  (\frac{d}{dt})^n \int_a^t (t -
s)^{n-\alpha-1} f(s) ds    ,  \quad   t \in [a,b];
\]
see \cite{4,5,54,6}.
\end{definition}

The concept of stability can be related to that of continuous
dependence of solution on their initial value.
Consider the non-autonomous linear system
\begin{equation}\label{5}
x'(t) = A(t) x(t)
\end{equation}
with the initial condition
$x(t_0) = x^0$.

\begin{definition} \label{def3}\rm
  The solution $x=0$ of \eqref{5} is
called stable  if for any $\epsilon>0,t_0\geq
0$, there exist $ \delta(\epsilon,t_0)>0 $ such that
$\|x(t,t_0,x^0)\|<\epsilon$ for $t\geq t_0$ as soon as
$\|x^0\|<\delta$. And the solution $x=0$ of \eqref{5} will be
called uniformly stable if $ \delta(\epsilon,t_0)$ can be chosen
independent of $t_0: \delta(\epsilon,t_0)\equiv \delta(\epsilon)$;
see \cite{3}.
\end{definition}


\section{Existence and Uniqueness}

Let $ X = ( C_{n}(I) , \| \cdot \|_{1} ) $,  where $C_n (I)$ be the
class of  continuous column $n$-vectors functions.
For $ x \in C_n[0,T] $, define the norm
$ \| x \| = \sum_{i=1}^n \sup_{t\in [0,T]} \{ e^{-Nt} |x_i(t)| \}$.
For a matrix $B$ define the norm
$\| B \| = \sum_{i=1}^n |b_i| = \sum_{i=1}^n\sup_{t,j}
| b_{ij} |$.

\begin{theorem} \label{thm1}
 Let $ a_{ij}, b_{ij}, h_i, \phi_i$ be in $C(I) $.
Then there exist a unique solution $ x \in X $ of
\eqref{1}-\eqref{3}
\end{theorem}

\begin{proof}
 For $t>0$, equation \eqref{1} can be written as
\[
\frac{d}{dt} I^{1-\alpha} x_i(t) =\sum_{j=1}^n a_{ij}(t)
 x_j(t) + \sum_{j=1}^n b_{ij}(t)  x_j(t-r_j) + h_i(t)
\]
integrating both sides of the above equation, we obtain
\[
I^{1-\alpha} x_i(t) - I^{1-\alpha} x_i(t)|_{t=0} = \int_0^t
 \{\sum_{j=1}^n a_{ij}(t)  x_j(t)+\sum_{j=1}^n b_{ij}(s)
 x_j(s-r_j) + h_i(s)\} \,ds
\]
then
\[
 I^{1-\alpha} x_i(t) = \int_0^t
 \{\sum_{j=1}^n a_{ij}(t)  x_j(t)+\sum_{j=1}^n b_{ij}(s)
 x_j(s-r_j) + h_i(s)\}\, ds\,.
\]
Applying the operator by $I^\alpha$, on both sides,
\[
I x_i(t) = I^{\alpha+1} \big\{\sum_{j=1}^n a_{ij}(t)
 x_j(t) + \sum_{j=1}^n b_{ij}(t)  x_j(t-r_j) + h_i(t)\big\}
\]
differentiating both side, we obtain
\begin{equation}\label{4}
x_i(t) = I^\alpha \{\sum_{j=1}^n a_{ij}(t)
 x_j(t) + \sum_{j=1}^n b_{ij}(t)
 x_j(t-r_j) + h_i(t)\},  \quad   i=1,2,\dots ,n
\end{equation}
Now let $F : X  \to  X $, defined by
\[
Fx_i = I^{\alpha} \big\{ \sum_{j=1}^n a_{ij}(t)  x_j(t) + \sum_{j=1}^n
b_{ij}(t)  x_j(t-r_j) + h_i(t)\big\}
\]
then
\begin{align*}
|Fx_i-Fy_i|
&= |I^{\alpha} \{ \sum_{j=1}^n a_{ij}(t)
 \{x_j(t)-y_j(t)\} +    \sum_{j=1}^n b_{ij}(t)
 \{x_j(t-r_j)-y_j(t-r_j)\}\}|\\
&= \big| \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\{
\sum_{j=1}^na_{ij}(s) \{x_j(s)-y_j(s)\}\\
&\quad + \sum_{j=1}^n b_{ij}(s)
   \{x_j(s-r_j)-y_j(s-r_j)\}\}d s\big| \\
&\leq \int_0^t \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
\sum_{j=1}^n |a_{ij}(s)|
    | x_j(s) - y_j(s) | d s \\
   &\quad + \int_0^t \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
    \sum_{j=1}^n |b_{ij}(s)|
    | x_j(s-r_j) - y_j(s-r_j) | d s \\
&\leq \sum_{j=1}^n \sup_{t, \forall j} |a_{ij}(t)| \int_0^t
\frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
    | x_j(s) - y_j(s) | d s \\
  &\quad  + \sum_{j=1}^n \sup_{t, \forall
j} |b_{ij}(t)| \int_0^t \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
    | x_j(s-r_j) - y_j(s-r_j) | d s\\
&\leq \sum_{j=1}^n a_i \int_0^{t} \frac{(t -
s)^{\alpha-1}}{\Gamma(\alpha)}
| x_j(s) - y_j(s) | d s \\
&\quad  +\sum_{j=1}^n b_i \int_0^{r_j} \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
| x_j(s-r_j) - y_j(s-r_j) | d s\\
&\quad + \sum_{j=1}^n b_i \int_{r_j}^t \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
    | x_j(s-r_j) - y_j(s-r_j) | d s\\
  &\leq a_i \sum_{j=1}^n \int_0^{t} \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
| x_j(s) - y_j(s) | d
s\\
&\quad  + b_i \sum_{j=1}^n \int_{r_j}^t \frac{(t - s)^{\alpha-1}}{\Gamma(\alpha)}
    | x_j(s-r_j) - y_j(s-r_j) | d s
\end{align*}
and
\begin{align*}
&e^{-Nt}|Fx_i-Fy_i|\\
&\leq  a_i \sum_{j=1}^n\int_{0}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s)} e^{-Ns}|x_j(s)-y_j(s)|d s
\\
&\quad + b_i \sum_{j=1}^n\int_{r_j}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s+r_j)} e^{-N(s-r_j)}|x_j(s-r_j)-y_j(s-r_j)|d s \\
&\leq a_i \sum_{j=1}^n \sup_t\{e^{-Nt}|x_j(t)-y_j(t)|\}
\int_{0}^{t}  \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s)} d s\\
&\quad + b_i \sum_{j=1}^n\int_{0}^{t-r_j} 
 \frac{(t-\theta-r_j)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-\theta)} e^{-N\theta}|x_j(\theta)-y_j(\theta)|d \theta \\
&\leq a_i \sum_{j=1}^n
 \sup_t\{e^{-Nt}|x_j(t)-y_j(t)|\} \frac{1}{N^\alpha} \int_0^{Nt}
 \frac{u^{\alpha-1} e^{-u}}{\Gamma(\alpha)}  du\\
&\quad +b_i \sum_{j=1}^n
 \sup_t\{e^{-Nt}|x_j(t)-y_j(t)|\} \int_{0}^{t-r_j}
\frac{(t-\theta-r_j)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-\theta)} d \theta \\
&\leq \frac{a_i}{N^\alpha}\|x-y\|+b_i\sum_{j=1}^n
\sup_t\{e^{-Nt}|x_j(t)-y_j(t)|\}\int_0^{t-r_j}\frac{u^{\alpha-1}}{\Gamma(\alpha)}
 e^{-Nu}e^{-Nr_j} du\\
&\leq \frac{a_i}{N^\alpha}\|x-y\|+b_i\sum_{j=1}^n
\sup_t\{e^{-Nt}|x_j(t)-y_j(t)|\}
\frac{e^{-Nr_j}}{N^\alpha}\int_0^{N(t-r_j)}
\frac{u^{\alpha-1}e^{-u}}{\Gamma(\alpha)}  du\\
&\leq \frac{a_i}{N^\alpha} \|x - y\| + b_i \sum_{j=1}^n
 \sup_t\{e^{-Nt}|x_j(t)-y_j(t)|\} \frac{e^{-Nr_j}}{N^\alpha}\\
&\leq \frac{a_i}{N^\alpha} \|x - y\| + \frac{b_i}{N^\alpha}
\sum_{j=1}^n
 \sup_te^{-Nt}|x_j(t)-y_j(t)|\\
&\leq \frac{a_i+b_i}{N^\alpha} \|x - y\|\,.
\end{align*}
Then
\begin{align*}
\| F x - F y \|
&=  \sum_{i=1}^n \sup_t  e^{-Nt} |Fx_i-Fy_i|\\
&\leq \sum_{i=1}^n \frac{a_i+b_i}{N^\alpha} \| x - y \| \\
&\leq  \frac{\|A\| + \|B\|}{N^\alpha} \| x - y \|.
\end{align*}
Now choose $N$ large enough such that $\frac{\|A\| +
\|B\|}{N^\alpha} < 1$, so the map $ F : X \to X $ is a
contraction and it has a fixed point $ x=F x $ and hence, there
exist a unique column vector
$x \in X$ which is the solution of the integral equation \eqref{4}.


We now prove the equivalence between the integral equation \eqref{4}
and the nonlocal problem \eqref{1}-\eqref{3}.
Indeed, since $ x  \in C_n(I) $ and
$ I^{1-\alpha}x(t) \in C_n(I)$ applying the operator
 $ I^{1-\alpha} $ on both sides of \eqref{4}, we obtain
\begin{align*}
I^{1-\alpha} x_i(t)&= I^{1-\alpha} I^\alpha \{\sum_{j=1}^n
a_{ij}(t) x_j(t)+\sum_{j=1}^n b_{ij}(t)
x_j(t-r_j)+h_i(t)\},  \quad   i=1,2,\dots ,n      \\
&= I\{\sum_{j=1}^n a_{ij}(t) x_j(t) + \sum_{j=1}^n b_{ij}(t)
 x_j(t-r_j)+h_i(t)\}\,.
\end{align*}
Differentiating both sides,
\[
D I^{1-\alpha} x_i (t) = D I \{\sum_{j=1}^n a_{ij}(t) x_j(t) +
\sum_{j=1}^n b_{ij}(t)  x_j(t-r_j) + h_i(t)\},.
\]
Then
\[
D^{\alpha} x_i(t) = \sum_{j=1}^n a_{ij}(t) x_j(t)+
\sum_{j=1}^n b_{ij}(t)  x_j(t-r_j) + h_i(t) ,  \quad t> 0
\]
which proves the equivalence of \eqref{4} and \eqref{1}.

We want to prove that  $ \lim_{t\to 0^+} x_i = 0$.
Since $  x_j(s), a_{ij}(s), h_i(s)  $ are continuous on
$[0,T]$,  there exist constants $l_j, L_j, m_i, M_i$ such
that $l_j\leq a_{ij}(s)x_j(s)\leq L_j$ and
$ m_i \leq  h_i(s) \leq M_i$.
We have
\[
I^\alpha \{\sum_{j=1}^n a_{ij}(t)
 x_j(t) + h_i(t)\}=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\big\{\sum_{j=1}^n a_{ij}(s)
 x_j(s) + h_i(s)\big\} d s
\]
which implies
\begin{align*}
\big\{\sum_{j=1}^nl_j+m_i\big\}
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\,d s
&\leq I^\alpha\{\sum_{j=1}^n a_{ij}(t) x_j(t)+h_i(t)\}\\
& \leq \{\sum_{j=1}^nL_j+M_i\}\int_0^{t}
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} \,ds
\end{align*}
which in turn implies
\[
\{\sum_{j=1}^nl_j+m_i\} \frac{t^{\alpha}}{\Gamma(\alpha+1)} \leq
I^\alpha\{\sum_{j=1}^n a_{ij}(t) x_j(t)+h_i(t)\}  \leq
\{\sum_{j=1}^n L_j+M_i\} \frac{t^{\alpha}}{\Gamma(\alpha+1)}
\]
and
\[
\lim_{t\to 0^+} I^\alpha\{\sum_{j=1}^n a_{ij}(t)
x_j(t)+h_i(t)\} = 0.
\]

Since $ b_{ij}(s),  \phi_j(s-r_j) $ are continuous on $[0,r_j]$,
 there exist constants $ k_j, K_j $ such that
$ k_j \leq  b_{ij}(s) \phi_j(s-r_j) \leq K_j $.
Also $ b_{ij}(s),  x_j(s-r_j) $  are continuous on $[r_j,T]$,
then there exist a
constants $ k^*_j, K^*_J $ such that $ k^*_j \leq
 b_{ij}(s) x_j(s-r_j)\leq K^*_j $.
Let $ k = \min_{\forall j} \{k_j,  k^*_j\} $ and
$ K = \max_{\forall j} \{K_j,K^*_j\}, $ we have
\begin{align*}
&I^\alpha\sum_{j=1}^n b_{ij}(t) x_j(t-r_j)\\
&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\sum_{j=1}^n b_{ij}(s)    x_j(s-r_j)d s\\
&= \int_0^{r_j}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\sum_{j=1}^n
b_{ij}(s)
\phi_j(s-r_j)d s+\int_{r_j}^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\sum_{j=1}^n b_{ij}(s)  x_j(s-r_j)d s
\end{align*}
which implies
\begin{align*}
&\sum_{j=1}^nk_j\int_0^{r_j}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
d s+\sum_{j=1}^n
k^*_j\int_{r_j}^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} \,d s\\
&\leq I^\alpha\sum_{j=1}^n b_{ij}(t)x_j(t-r_j)  \\
&\leq \sum_{j=1}^n K_j \int_0^{r_j}
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} d s+\sum_{j=1}^n K^*_j
\int_{r_j}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\,ds
\end{align*}
which implies
\begin{align*}
&\sum_{j=1}^n k_j
\Big(\frac{t^{\alpha}}{\Gamma(\alpha+1)}
-\frac{(t-r_j)^{\alpha}}{\Gamma(\alpha+1)}\Big)
+\sum_{j=1}^n k^*_j\frac{(t-r_j)^{\alpha}}{\Gamma(\alpha+1)} \leq
I^\alpha\sum_{j=1}^n b_{ij}(t) x_j(t-r_j) \\
& \leq \sum_{j=1}^n K_j
\Big(\frac{t^{\alpha}}{\Gamma(\alpha+1)}
-\frac{(t-r_j)^{\alpha}}{\Gamma(\alpha+1)}\Big)
+\sum_{j=1}^n K^*_j \frac{(t-r_j)^{\alpha}}{\Gamma(\alpha+1)}\,.
\end{align*}
Then
\[
k \frac{t^{\alpha}}{\Gamma(\alpha+1)} \leq I^\alpha\sum_{j=1}^n
b_{ij}(t) x_j(t-r_j) \leq K \frac{t^{\alpha}}{\Gamma(\alpha+1)}
\]
and
\[
\lim_{t\to 0^+} I^\alpha\sum_{j=1}^n b_{ij}(t)
x_j(t-r_j) = 0.
\]
Then from \eqref{4} $ \lim_{t\to 0^+} x_i = 0$.
\end{proof}

Now for $t \in (-\infty , T], T<\infty$, the solution of
\eqref{1}-\eqref{3} takes the form
\[
x_i(t) = \begin{cases}
\phi_i(t)  ,  & t < 0\\
0   , &  t = 0\\
I^\alpha \{\sum_{j=1}^n a_{ij}(t)  x_j(t) + \sum_{j=1}^n b_{ij}(t)
 x_j(t-r_j) + b_i(t)\}   ,  & t > 0
\end{cases}
\]

\section{Stability}

In this section we study the stability of the solution of the
nonlocal problem \eqref{1}-\eqref{3}.

\begin{definition} \label{def5} \rm
The solution of the non-autonomous linear
system \eqref{1} is stable if for any $ \epsilon>0$, there exist
$\delta>0 $ such that for any two solutions
$ x(t) = (x_1(t),x_2(t),\dots ,x_n(t))' $ and
$ \widetilde{x}(t) = (\widetilde{x}_1(t),
\widetilde{x}_2(t),\dots ,\widetilde{x}_n(t))' $
with the initial conditions \eqref{2}-\eqref{3}
and  $\{ I^\beta {\widetilde{x}}(t)|_{t=0} = 0$,
$ \beta \in (0,1] , \widetilde{x}(t) = \widetilde{\Phi}
(t) $ for $t<0$   and $\lim_{t\to 0} \widetilde{\Phi}(t) = O\}$,
respectively, one has
             $\|\Phi(t) - \widetilde{\Phi}(t)\| \leq \delta$,
then
$\|x(t) - \widetilde{x}(t)\| < \epsilon $ for all $ t\geq0$.
\end{definition}

\begin{theorem} \label{thm2}
 The solution of the nonlocal delay system \eqref{1}-\eqref{3}
is uniformly stable.
\end{theorem}

\begin{proof}
 Let $ x(t) $ and $ \widetilde{x}(t) $ be two
solutions of the system \eqref{1} under the conditions
\eqref{2}-\eqref{3} and
$I^\beta \widetilde{x}(t)|_{t=0} = 0$,
$\widetilde{x}(t) = \widetilde{\Phi}(t)$,  $t<0$ and
$\lim_{t\to 0}  \widetilde{\Phi}(t) = O$,
respectively.
Then for $t>0$, from \eqref{4}, we have
\begin{align*}
|x_i-\widetilde{x}_i|
&= | I^{\alpha} \{ \sum_{j=1}^n a_{ij}(t)
 ( x_j(t) - \widetilde{x}_j(t))   +  \sum_{j=1}^n b_{ij}(t)
 ( x_j(t-r_j) - \widetilde{x}_j(t-r_j)\}|\\
&\leq \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\{\sum_{j=1}^n|a_{ij}(s)|
   |x_j(s)-\widetilde{x}_j(s)|\\
&\quad +\sum_{j=1}^n|b_{ij}(s)|
   |x_j(s-r_j)-\widetilde{x}_j(s-r_j)|\}d s \\
   &\leq \sum_{j=1}^n \sup_{t, \forall
j} |a_{ij}(t)| \int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
    |x_j(s)-\widetilde{x}_j(s)| d s \\
&\quad + \sum_{j=1}^n \sup_{t, \forall j} |b_{ij}(t)| \int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
    |x_j(s-r_j)-\widetilde{x}_j(s-r_j)| d s \\
&\leq \sum_{j=1}^n a_i \int_0^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
|\phi_j(s)-\widetilde{\phi}_j(s)|\,ds \\
&\quad +\sum_{j=1}^n b_i
\int_0^{r_j}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
|\phi_j(s-r_j)-\widetilde{\phi}_j(s-r_j)|d s \\
&\quad + \sum_{j=1}^n b_i \int_{r_j}^t
 \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
    |x_j(s-r_j) - \widetilde{x}_j(s-r_j)| d s
  \end{align*}
and
\begin{align*}
&e^{-Nt} |x_i-\widetilde{x}_i|\\
&\leq a_i\sum_{j=1}^n\int_{0}^{t}
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s)} e^{-Ns}|x_j(s)-\widetilde{x}_j(s)|d
s\\
&\quad  +
b_i\sum_{j=1}^n\int_{0}^{r_j}
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s+r_j)} e^{-N(s-r_j)}|\phi_j(s-r_j)-\widetilde{\phi}_j(s-r_j)|d
s\\
&\quad  + b_i\sum_{j=1}^n\int_{r_j}^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s+r_j)} e^{-N(s-r_j)}|x_j(s-r_j)-\widetilde{x}_j(s-r_j)|d s \\
&\leq a_i \sum_{j=1}^n \sup_t\{e^{-Nt}|x_j(t)-\widetilde
x_j(t)|\} \int^{t}_0  \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-s)} d s \\
&\quad  + b_i \sum_{j=1}^n\int^{0}_{-r_j} 
 \frac{(t-\theta-r_j)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-\theta)} e^{-N\theta}|\phi_j(\theta)-\widetilde{\phi}_j(\theta)|d \theta \\
&\quad  + b_i \sum_{j=1}^n\int_{0}^{t-r_j} 
 \frac{(t-\theta-r_j)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-\theta)} e^{-N\theta}|x_j(\theta)-\widetilde{x}_j(\theta)|d \theta \\
&\leq \frac{a_i}{N^\alpha} \|x_j(t)-\widetilde x_j(t)\|
\int_0^{Nt}
 \frac{u^{\alpha-1} e^{-u}}{\Gamma(\alpha)}  du\\
&\quad + b_i \sum_{j=1}^n
 \sup_t\{e^{-Nt}|\phi_j(t)-\widetilde{\phi}_j(t)|\} \int^{0}_{-r_j}
\frac{(t-\theta-r_j)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-\theta)} d \theta \\
&\quad  + b_i \sum_{j=1}^n
 \sup_t\{e^{-Nt}|x_j(t)-\widetilde{x}_j(t)|\} \int_{0}^{t-r_j}
\frac{(t-\theta-r_j)^{\alpha-1}}{\Gamma(\alpha)}
 e^{-N(t-\theta)} d \theta \\
& \leq \frac{a_i}{N^\alpha} \|x_j(t)-\widetilde x_j(t)\|\\
& \quad + b_i\sum_{j=1}^n
\sup_t\{e^{-Nt}|\phi_j(t)-\widetilde{\phi}_j(t)|\}
\frac{e^{-Nr_j}}{N^\alpha}\int^{Nt}_{N(t-r_j)} \frac{u^{\alpha-1}
e^{-Nu}}{\Gamma(\alpha)}
 du\\
&\quad + b_i \sum_{j=1}^n
 \sup_t\{e^{-Nt}|x_j(t)-\widetilde{x}_j(t)|\} \frac{e^{-Nr_j}}{N^\alpha}
\int_0^{N(t-r_j)}
 \frac{u^{\alpha-1} e^{-u}}{\Gamma(\alpha)}  du\\
&\leq \frac{a_i}{N^\alpha}\|x_j(t)-\widetilde
x_j(t)\|+\frac{b_i}{N^\alpha}\sum_{j=1}^n
 e^{-Nr_j} \sup_t\{e^{-Nt}|x_j(t)-\widetilde{x}_j(t)|\}\\
&\quad +\frac{b_i}{N^\alpha}\sum_{j=1}^n
 e^{-Nr_j} \sup_t\{e^{-Nt}|\phi_j(t)-\widetilde{\phi}_j(t)| \}
\\
&\leq \frac{a_i}{N^\alpha}\|x_j(t)-\widetilde
x_j(t)\|+\frac{b_i}{N^\alpha}\sum_{j=1}^n
\sup_t\{e^{-Nt}|x_j(t)-\widetilde{x}_j(t)|\}\\
&\quad  + \frac{b_i}{N^\alpha}\sum_{j=1}^n
\sup_t\{e^{-Nt}|\phi_j(t)-\widetilde{\phi}_j(t)|\}
\\
&\leq  \frac{a_i + b_i}{N^\alpha} \| x-\widetilde{x} \| +
 \frac{b_i}{N^\alpha}\| \Phi-\widetilde{\Phi} \|\,.
\end{align*}
Then
\begin{align*}
\|x-\widetilde{x}\|
&= \sum_{i=1}^n \sup_t e^{-Nt} |x_i-\widetilde{x}_i\\
& \leq \sum_{i=1}^n \frac{a_i+b_i}{N^\alpha}
\|x-\widetilde{x}\|
+\sum_{i=1}^n \frac{b_i}{N^\alpha} \|\Phi-\widetilde{\Phi}\|\\
&\leq  \frac{\| A \|+\|B\|}{N^\alpha} \| x-\widetilde{x} \| +
\frac{\|B\|}{N^\alpha} \|\Phi-\widetilde{\Phi}\|;
\end{align*}
i.e.,
\[
     \Big(1 - \frac{\| A \|+\| B \|}{N^\alpha}\Big)
 \| x-\widetilde{x} \|
 \leq  \frac{\| A \|}{N^\alpha} \| \Phi-\widetilde{\Phi} \|
\]
and
\[
\| x-\widetilde{x} \| \leq  \Big(1 -
\frac{\|A\|+\|B\|}{N^\alpha} \Big)^{-1}
\|\Phi-\widetilde{\Phi}\|;
\]
therefore, for $\delta>0$ such that
$\|\Phi-\widetilde{\Phi}\|<\delta$, we can find
$ \epsilon=\big(1-\frac{\|A\|+\|B\|}{N^\alpha} \big)^{-1}\delta  $
such that $  \| x-\widetilde{x} \| \leq \epsilon $ which proves that
the solution $ x(t) $ is uniformly stable.
\end{proof}

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\end{document}