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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 58, pp. 1--22.\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-2016/58\hfil Stability of solutions]
{Stability of solutions to impulsive Caputo fractional
 differential equations}

\author[R. Agarwal, S. Hristova, D. O'Regan \hfil EJDE-2016/58\hfilneg]
{Ravi Agarwal, Snezhana Hristova, Donal O'Regan}

\address{Ravi Agarwal \newline
Department of Mathematics,  Texas A\& M University-Kingsville,
 Kingsville,  TX 78363, USA}
\email{agarwal@tamuk.edu}

\address{Snezhana Hristova \newline
Department of Applied Mathematics,
 Plovdiv University,  Plovdiv, Bulgaria}
\email{snehri@gmail.com}

\address{Donal O'Regan \newline
School of Mathematics, Statistics and Applied Mathematics,
National University of Ireland, Galway, Ireland}
\email{donal.oregan@nuigalway.ie}

\thanks{Submitted December 16, 2015. Published February 25, 2016.}
\subjclass[2010]{34A34, 34A08, 34D20}
\keywords{Stability; Caputo derivative;  Lyapunov functions; impulses; 
\hfill\break\indent fractional differential equations}

\begin{abstract}
 Stability of the solutions to a nonlinear impulsive Caputo fractional
 differential equation is studied using Lyapunov like functions.
 The derivative of piecewise continuous Lyapunov functions among the nonlinear
 impulsive Caputo differential equation of fractional order is defined.
 This definition is a natural generalization of the Caputo  fractional
 Dini derivative of a function.  Several sufficient conditions for  stability,
 uniform stability and asymptotic uniform stability of the solution are
 established. Some examples are given to illustrate the results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

The study of stability for fractional order systems is quite recent.
There are several approaches in the literature to study stability, one of
which is the Lyapunov approach. One of the main difficulties on the
application of a Lyapunov function to fractional order differential
equations is the appropriate definition of its derivative among the
fractional differential equations. We give a brief brief overview of
the literature and we use the so called Caputo fractional Dini
derivative.

The presence of  impulses in fractional differential equations lead to
complications with the concept of the solution. Mainly there are two
different approaches: either keeping the lower limit at the initial time
$t_0$ or change the nature of fractional differential equation by moving
the lower limits of the fractional derivative to the points of impulses.
In this paper the second approach is used. The Caputo fractional Dini derivative
is generalized to  piecewise continuous Lyapunov functions among the studied
nonlinear fractional equations with impulses.  Comparison results using this
definition and scalar impulsive fractional differential equations are presented.
Several sufficient conditions for stability, uniform stability and
asymptotic uniform stability are obtained. Some examples illustrate the obtained
results.

\section{Notes on fractional calculus}

Fractional calculus generalizes the derivative and the integral of a
function to a non-integer order \cite{1,  Lakfde,   podlubny,  kilbas} and
there are several definitions of fractional derivatives and
fractional integrals.
In engineering, the fractional order $q$ is often less than 1, so
we restrict our attention to $q\in (0,1)$.
\smallskip

\noindent (1) The Riemann--Liouville (RL)   fractional derivative  
 of order $q\in(0,1)$
of $m(t)$ is given by (see for example \cite[Section 1.4.1.1]{1})
\begin{equation*}
_{t_0}^{RL}D^{q}m(t)=\frac{1}{\Gamma ( 1-q) }\frac{d}{dt}\int_{t_0}^{t}
(t-s) ^{-q}m(s)ds,\quad  t\geq t_0,
\end{equation*}
where $\Gamma (\cdot)$ denotes the usual Gamma function.
\smallskip

\noindent(2)  The Caputo fractional derivative of order $q\in(0,1)$ is defined
by (see for example \cite[Section 1.4.1.3]{1})
\begin{equation} \label{198}
{}_{t_0}^{c}D^{q}m(t)=\frac{1}{\Gamma ( 1-q)}
\int_{t_0}^{t}( t-s) ^{-q}m^{\prime}(s)ds,\quad  t\geq t_0.
\end{equation}
 The properties of the Caputo derivative
are quite similar to those of ordinary derivatives. Also, the
initial conditions of fractional differential equations with the
Caputo derivative has a clear physical meaning and as a result the
Caputo derivative is usually used in real applications.
\smallskip

\noindent(3) The Grunwald-Letnikov fractional derivative is given
by (see for example \cite[Section 1.4.1.2]{1})
\begin{equation*}
{}_{t_0}^{GL}D^{q}m(t)=\lim_{h\to 0}\frac{1}{h^q}
\sum_{r=0}^{[\frac{t-t_0}{h}]} (-1)^r (qCr)m(t-rh), \quad t\geq t_0,
\end{equation*}
and the Grunwald-Letnikov fractional Dini  derivative  by
\begin{equation} \label{765}
{}_{t_0}^{GL}D^{q}_{+}m(t)=\limsup _{h\to 0+}\frac{1}{h^q}
\sum_{r=0}^{[\frac{t-t_0}{h}]} (-1)^r (qCr)m(t-rh), \quad t\geq t_0,
\end{equation}
where $qCr=\frac{q(q-1)(q-1)\dots (q-r+1)}{r!}$  and
$[\frac{t-t_0}{h}]$ denotes the integer part of the fraction
$\frac{t-t_0}{h}$.

\begin{proposition}[{\cite[Theorem 2.25]{2}}] \label{prop1}
 Let $m\in C^1[t_0,b]$. Then
 $$
{}_{t_0}^{GL}D^{q}m(t)={}_{t_0}^{RL}D^{q}m(t) \quad \text{for } t \in (t_0,b].
$$
\end{proposition}

Also, by \cite[Lemma 3.4]{2} we have
${}_{t_0}^{c}D_{t}^{q}m(t)={}_{t_0}^{RL}D_{t}^{q}m(t)-m(t_0)
\frac{(t-t_0)^{-q}}{\Gamma(1-q)}$.

From the relation between the Caputo fractional derivative and
the Grunwald-Letnikov fractional derivative  using \eqref{765} we
define the Caputo fractional Dini derivative  as
\begin{equation} \label{679}
{}_{t_0}^{c}D^{q}_+ m(t)={}_{t_0}^{GL}D^{q}_{+}[m(t)-m(t_0)],
\end{equation}
i.e.
\begin{equation} \label{10}
\begin{aligned}
&{}_{t_0}^{c}D^{q}_+ m(t)\\
&=\limsup _{h\to 0+}\frac{1}{h^q}\Big[m(t)-m(t_0)
-\sum_{r=1}^{[\frac{t-t_0}{h}]} (-1)^{r+1} (qCr)\big( m(t-rh)-m(t_0)\big)\Big].
\end{aligned}
\end{equation}



\begin{definition}[\cite{devi vlm}] \label{def1} \rm
We say $m\in C^q([t_0,T],\mathbb{R}^n)$ if  $m(t)$ is
differentiable (i.e. $m'(t)$ exists), the Caputo derivative ${}_{t_0}^{c}D^{q}m(t)$
exists  and satisfies \eqref{198} for $t\in[t_0,T]$.
\end{definition}

\begin{remark} \label{rmk1}\rm
Definition \ref{def1} could be extended to any interval $I\subset \mathbb{R}_+$.

If  $m\in C^q([t_0,T],\mathbb{R}^n)$ then $_{t_0}^{c}D_+^{q}m(t)=\  _{t_0}^{c}D^{q}m(t)$.
\end{remark}


\section{Impulses in fractional differential equations}

Consider the initial value problem (IVP) for the system of
\emph{fractional differential equations} (FrDE)
with a Caputo derivative for $0<q<1$,
\begin{equation} \label{100}
{}_{\tau_0}^{c}D^{q}x=f(t,x) \quad
\text{for $t\geq \tau_0$  with }  x(\tau_0)=x_0,
\end{equation}
where $x\in \mathbb{R}^n$, $f\in C[\mathbb{R}_{+}\times \mathbb{R}^{n},\mathbb{R}^{n}]$,
and $(\tau_0,x_0)\in \mathbb{R}_+\times\mathbb{R}^n$ is an arbitrary
initial data.

We suppose that the function $f(t,x)$ is smooth enough on $\mathbb{R}_+\times \mathbb{R}^n$,
such that for any initial data $(\tau_0,x_0)\in \mathbb{R}_+\times \mathbb{R}^n$
the IVP for FrDE \eqref{100} has a solution
$x(t)=x(t;\tau_0,x_0)\in C^q([\tau_0,\infty),\mathbb{R}^n)$.
Some sufficient conditions for the existence of global solutions to \eqref{100}
are given in \cite{BM, Lakfde}.

 The IVP for FrDE \eqref{100} is equivalent to the following integral equation
$$
x(t)=x_0+\frac{1}{\Gamma(q)}\int_{\tau_0}^t (t-s)^{q-1}f(s,x(s))ds \quad
  \text{for }  t \geq \tau_0.
$$
In this article we  assume the  points
$t_i$, $i=1,2,\dots $ are fixed such that $t_1<t_2<\dots$ and
$\lim_{k\to \infty}t_k=\infty$.  Let $\tau\in\mathbb{R}_+$ and define the set
$\Omega_{\tau}=\{k:t_k>\tau\}$.

Consider the initial value problem for the system of
\emph{impulsive fractional differential equations} (IFrDE)
with a Caputo derivative for $0<q<1$,
\begin{equation} \label{1}
\begin{gathered}
 {}_{t_0}^{c}D^{q}x=f(t,x) \quad \text{for } t\geq t_0, \;t\neq t_i,\\
 x(t_i+0)=\Phi_i(x(t_i))\quad \text{for }  i\in \Omega_{t_0},\\
 x(t_0)=x_0,
\end{gathered}
\end{equation}
where $x,x_0\in \mathbb{R}^n$, $f:\mathbb{R}_{+}\times \mathbb{R}^{n} \to \mathbb{R}^{n}$,
$t_0\in \mathbb{R}_+$, $\Phi_i: \mathbb{R}^n\to\mathbb{R}^n$,  $i=1,2,3,\dots$. Without loss
of generality we will assume $0\leq t_0<t_1$.

 \begin{remark} \label{rmk3} \rm
In the literature the second equation in \eqref{1}, the so called impulsive
condition is also  given in the equivalent form
$\Delta x(t_i)=I_i(x(t_i))$,   $i\in \Omega_{t_0}$
 where $\Delta x(t_i)=x(t_i+0)-x(t_i-0)$ and the function
$I_i(x)=\Phi_i(x)-x$ gives the  amount of the jump of the solution at the
point $t_i$.
\end{remark}

Let $J\subset \mathbb{R}_+$ be a given interval and $\Delta\subset\mathbb{R}^n$.
Let $J_{\rm imp}=\{t\in J: t\neq t_k,\, k=1,2,\dots\}$ and introduce
the following classes of functions
\begin{gather*}
C^q(J_{\rm imp},\Delta)=\cup_{k=0}^\infty C^q((t_k,t_{k+1}),\Delta),\quad
C(J_{\rm imp},\Delta)=\cup_{k=0}^\infty C((t_k,t_{k+1}),\Delta),
\\
\begin{aligned}
PC^q(J,\Delta) =\Big\{& u\in C^q(J_{\rm imp},\Delta) : u(t_{k})
=\lim_{t\uparrow t_{k}}u(t)<\infty,
u(t_{k}+0)=\lim_{t\downarrow t_{k}}u(t)<\infty  \\
& u'(t_{k})=\lim_{t\uparrow t_{k}}u'(t)<\infty,
 u'(t_{k}+0)=\lim_{t\downarrow t_{k}}u'(t)<\infty \\
&\text{for all }  k: t_k\in J\Big\},
\end{aligned}\\
\begin{aligned}
PC(J,\Delta)=\Big\{&u\in C(J_{\rm imp},\Delta) :
  u(t_{k})=\lim_{t\uparrow t_{k}}u(t)<\infty,
 u(t_{k}+0)=\lim_{t\downarrow t_{k}} u(t)<\infty  \\
&\text{for all }  k: t_k\in J\Big\}.
\end{aligned}
\end{gather*}

Impulsive fractional differential equations is an important area of
study. There are many qualitative results obtained for equations of
type \eqref{1}. We look at the concept of a solutions to fractional
differential equations with impulses. There are mainly two
viewpoints:
\smallskip

\noindent(V1) using the classical Caputo derivative and working in each
subinterval, determined
by the impulses (see for example \cite{AS,AB,BK,BS,B}).
This approach is based on the idea that on each interval between two
consecutive impulses $(t_k,t_{k+1})$  the solution is determined by
the differential equation of fractional order. Since the Caputo fractional
derivative depends significantly on the initial point (which is different for
the ordinary derivative) it leads to a change of the equation on
each interval $(t_k,t_{k+1})$. This approach neglects the lower
limit of the Caputo fractional derivative   at $t_0$ and moves it to
each impulsive time $t_k$.  Then  the IVP for IFrDE \eqref{1}  is equivalent to
the  integral equation
\begin{equation} \label{666}
 x(t)=   \begin{cases}
x_0+\frac{1}{\Gamma(q)}\int_{t_0}^t  (t-s)^{q-1}f(s,x(s))ds
\quad  \text{for } t \in[t_0,t_1]\\[4pt]
x_0+\frac{1}{\Gamma(q)}\sum_{i=1}^k\int_{t_{i-1}}^{t_i} (t_i-s)^{q-1} f(s,x(s))ds  \\
+\frac{1}{\Gamma(q)}\int_{t_k}^t (t-s)^{q-1}f(s,x(s))ds
+\sum_{i=1}^kI_i(x(t_i-0)) \\
\quad \text{for } t \in (t_k,t_{k+1}],\; k=1,2,3,\dots
\end{cases}
\end{equation}
where $I_k(x)=\Phi_k(x)-x$, $k=1,2,\dots$.

Using approach (V1) the solution
$x(t;t_0,x_0)$ of \eqref{1} is
\begin{equation} \label{663}
 x(t;t_0,x_0)=
  \begin{cases}
  X_0(t;t_0,x_0) &  \text{for }  t \in[t_0,t_1]\\
  X_1(t;t_1, \Phi_1(X_0(t_1;t_0,x_0)))  &  \text{for }  t \in(t_1,t_2]\\
  X_2(t;t_2, \Phi_2(X_1(t_2;t_1,\Phi_2(X_0(t_1;t_0,x_0))) &  \text{for }
 t \in(t_2,t_3]\\
                \dots
  \end{cases}
\end{equation}
\begin{itemize}
\item $X_0(t;t_0,x_0)$ is the solution of IVP for FrDE \eqref{100}
 with $\tau_0=t_0$,
\item $ X_1(t;t_1, \Phi_1(X_0(t_1;t_0,x_0)))$ is the solution of IVP for
 FrDE  \eqref{100} with $\tau_0=t_1$, $ x_0=\Phi_1(X_0(t_1;t_0,x_0))$,
\item $X_2(t;t_2, \Phi_1(X_1(t_2; t_1,\Phi_1(X_0(t_1;t_0,x_0)))$ is the
 solution of IVP for the  FrDE  \eqref{100} with $\tau_0=t_2$,
$x_0=\Phi_2(X_1(t_2;t_1,\Phi_1(X_0(t_1;t_0,x_0))$,
\end{itemize}
and so on.

Viewpoint (V1) and the corresponding equivalent integral
equations are based on the presence of  impulses in the
differential equation (see for example book \cite{LB} and the
cited references therein).
\smallskip


\noindent(V2) Keeping  the lower limit $t_0$ of the Caputo derivative for
all $t\geq t_0$ but considering different initial conditions on each interval
$(t_k,t_{k+1})$ (see for example  \cite{F,F4,F2,F3,F1}).
This approach  is based on the fact that the
restriction of the fractional derivative
${}_{t_0}^{c}D^{q}x(t)$ on any interval $(t_k,t_{k+1})$, $k=1,2,\dots$
does not change.  Then the fractional equation is kept on each
interval between two consecutive impulses with only the initial
condition changed. Then the IVP for the IFrDE \eqref{1} is
equivalent to the following integral equation
(see \cite[formula(10)]{F})
\begin{equation} \label{551}
x(t)=   \begin{cases}
     x_0+\frac{1}{\Gamma(q)}\int_{t_0}^t (t-s)^{q-1}f(s,x(s))ds
\quad  \text{for } t \in[{t_0},t_1] \\[4pt]
    x_0+\frac{1}{\Gamma(q)}\int_{t_0}^t (t-s)^{q-1}f(s,x(s))ds
+\sum_{i=1}^kI_i(x(t_i-0)) \\
\quad  \text{for }  t \in (t_k,t_{k+1}],\; k=1,2,3,\dots
  \end{cases}
\end{equation}
As a result  using approach (V2)  the solution
$x(t;t_0,x_0)$ of \eqref{1} is
\begin{equation} \label{554}
x(t;t_0,x_0)=
  \begin{cases}
      X_0(t;t_0,x_0) \quad  \text{for }  t \in[t_0,t_1] \\[4pt]
    X_0(t;t_0,x_0)+\sum_{j=1}^k\Phi_j(x(t_j;t_0,x_0))   \\
        \quad  \text{for } t \in(t_k,t_{k+1}],\; k=1,2,\dots
  \end{cases}
\end{equation}
where $X_0(t;t_0,x_0)$ is the solution of IVP for  FrDE \eqref{100}
with $\tau_0=t_0$.

\begin{remark} \label{rmk4} \rm
From the above  any solution of \eqref{1} is from the class
$PC^q([t_0,b))$, $b\leq\infty$.

 In the case $f(t,x)\equiv 0$ both formulas \eqref{666} and \eqref{551}
coincide and both approaches (V1) and (V2) are equivalent.
\end{remark}


\begin{example} \label{examp1} \rm
Consider  the initial value problem for the scalar IFrDE
with a Caputo derivative for $0<q<1$,
\begin{equation} \label{311}
\begin{gathered}
{}_{t_0}^{c}D^{q}x=Ax, \quad \text{for } t\geq t_0, \; t\neq  t_i,  \\
x(t_i+0)=\Phi_i( x(t_i-0)) \quad \text{for } i=1,2,\dots ,\\
 x(t_0)=x_0,
\end{gathered}
\end{equation}
where $x\in\mathbb{R}$, $A$ is a given real constant.
\smallskip

\noindent\textbf{Case 1.} Let $\Phi_i( x)=a_i + x$ where $a_i\neq  0$,
$i=1,2,\dots$.
Applying (V1) and  \eqref{666}  we obtain the
solution of \eqref{311}, namely
\begin{equation} \label{61}
\begin{split}
x(t;t_0,x_0)
&=\Big(x_0\prod_{i=1}^{k}E_q(A(t_i-t_{i-1})^q)
 +\sum_{i=1}^ka_i\prod_{j=i+1}^k  E_q(A(t_j-t_{j-1})^q) \Big)\\
&\quad\times E_q(A(t-t_k)^q)\quad \text{for }  t \in (t_k,t_{k+1}],\;
k=0, 1,2,3,\dots,
\end{split}
\end{equation}
where the Mittag-Leffler function (with  one parameter) is defined by
$E_{q}(z)=\sum_{k=0}^{\infty }\frac{z^{k}}{\Gamma (qk+1)}$.

Applying  (V2) and \eqref{554}, we obtain  the
solution of \eqref{311}, namely
\begin{equation} \label{62}
x(t;t_0,x_0)=x_0E_q(A (t-t_0)^q)+\sum_{i=1}^k a_k
\end{equation}
for $t \in (t_k,t_{k+1}]$, $k=0, 1,2,3,\dots$.
In this case it looks like \eqref{61} is closer to  the ordinary
case ($q=1$).
\smallskip

\noindent\textbf{Case 2.}
 Let $\Phi_i( x)=a_i x$ where $a_i\neq  1$, $i=1,2,\dots$ are constants.
Applying (V1) and  \eqref{666}  we obtain the
solution of \eqref{311}, namely
\begin{equation} \label{317}
x(t;t_0,x_0)=x_0\Big(\prod_{i=1}^{k}a_iE_q(A(t_i-t_{i-1})^q)\Big)E_q(A(t-t_k)^q)
\end{equation}
for $t \in (t_k,t_{k+1}]$, $k=0, 1,2,\dots$.

Applying  (V2) and \eqref{554}, using
$\frac{\lambda}{\Gamma(q)}\int_0^t\frac{E_q(\lambda s^q)}{(t-s)^q}ds=E_q(\lambda
t^q)-1$  we obtain  the solution of \eqref{311}, namely
\begin{equation} \label{327}
x(t;t_0,x_0)=x_0\Big(E_q(A (t-t_0)^q)+\sum_{i=1}^k E_q(A
(t_i-t_0)^q)(a_i-1)\prod_{j={i+1}}^ka_j\Big),
\end{equation}
for $t \in (t_k,t_{k+1}]$,
$k=0, 1,2,3,\dots$.
In this case it looks like \eqref{317} is closer to the ordinary
case ($q=1$).
\end{example}

The concept of the FrDE with impulses is rather problematic.
 In \cite{F}, the authors  pointed out that the formula, based on (V1)
of solutions for IFrDE in  \cite{AS},  \cite{BK}  is incorrect and
gave a new formula using approach (V2). In  \cite{F3,F1}
the authors established a general framework to find  solutions for
impulsive fractional boundary value problems and obtained some
sufficient conditions for the existence of solutions to impulsive
fractional differential equations based on (V1). In \cite{ZN} the
authors discussed (V1) and
criticized the viewpoint (V2) in \cite{F,F3,F1}.
Next, in \cite{F4}  the authors considered the counterexample in
\cite{F} and provided further explanations about (V2).
In this article we use approach (V1).

Note if for some natural $k$, a component of the function
$\Phi_{k}:\mathbb{R}^n\to\mathbb{R}^n$,
$\Phi_k=(\Phi_{k,1},\Phi_{k,2},\dots,\Phi_{k,n})$ satisfies the
equality $\Phi_{k,j}(x)= x_j$ where $x\in\mathbb{R}^n:x=(x_1,x_2,\dots,x_n)$,
  then there will be no impulse at the point
$t_k$ for the component $x_j(t)$ of the solution of IFrDE \eqref{1}
and \eqref{666} is not correct in this case. To avoid this confusing
situation in the application of approach (V1), mentioned above we
will assume:
\begin{itemize}
\item[(H1)]  If $x\neq 0$ then  $\Phi_{k,j}(x)\neq  x_j$ for  all
$j=1,2\dots,n$ and   $k=1,2,3,\dots$ where $x\in\mathbb{R}^n$,
$x=(x_1,x_2,\dots,x_n)$ and $\Phi_{k}:\mathbb{R}^n\to\mathbb{R}^n$,
$\Phi_k=(\Phi_{k,1},\Phi_{k,2},\dots,\Phi_{k,n})$.
\end{itemize}
Note that (H1) is  equivalent to
$I_{k,j}(x)\neq 0$ if $x\neq 0$  for all $k=1,2,3,\dots$
and $j=1,2\dots,n$ where $I_k=(I_{k,1},I_{k,2},\dots,I_{k,n})$.

\section{Definitions about stability and Lyapunov functions}

The goal of the article is to study the stability of zero solution
of system IFrDEs \eqref{1}.  We will assume the following condition is satisfied
\begin{itemize}
\item[(H2)] $f(t,0)\equiv 0$ for $t\in \mathbb{R}_+$ and $\Phi_i(0)=0$
for $i=1,2,3\dots$.
\end{itemize}
In the definition below we let $x(t;t_{0},x_{0})\in
PC^q([t_0,\infty),\mathbb{R}^n)$ be any solution of \eqref{1}.

\begin{definition} \label{def2} \rm
  The zero solution of \eqref{1} is said to be
\begin{itemize}
 \item  stable  if for every $\epsilon >0$  and $t_0\in\mathbb{R}_+$ there
 exist  $\delta =\delta (\epsilon,t_0)>0$ such that
 for any $x_0\in  \mathbb{R}^n$
   the inequality  $\|x_0\|<\delta$  implies
$\|x(t;t_0,x_0)\|<\epsilon$ for $t\geq t_0$;

 \item uniformly stable  if for every $\epsilon >0$   there
 exist  $\delta =\delta (\epsilon)>0$ such that
 for  $t_0\in\mathbb{R}_+, x_0\in \mathbb{R}^n$ with $\|x_0\|<\delta$
the inequality $\|x(t;t_0,x_0)\|<\epsilon$ holds for $t\geq t_0$;

 \item   uniformly attractive if for $\beta>0$:  for
 every $\epsilon >0$   there exist $T=T(\epsilon)>0$ such that for any
$t_0\in\mathbb{R}_+, x_0\in \mathbb{R}^n$ with
 $\|x_0\|<\beta$ the inequality $\|x(t;t_0,x_0)\|<\epsilon$ holds for $t\geq t_0+T$;

 \item  uniformly asymptotically stable if the zero solution is uniformly
stable  and uniformly attractive.
 \end{itemize}
  \end{definition}

In this article we use the followings two sets:
\begin{gather*}
\mathcal{K} = \{a\in C[\mathbb{R}_{+},\mathbb{R}_{+}]:a\text{ is strictly increasing and }a(0)=0\}, \\
S(A ) =\{x\in \mathbb{R}^n:\|x\|\leq A \}, \;A>0.
\end{gather*}
Furthermore we  consider the initial value problem for a scalar
FrDE
\begin{equation} \label{222}
\begin{gathered}
 {}_{\tau}^{c}D^{q}u=g(t,u) \quad \text{for }  t\geq \tau, \\
 u(\tau)=u_0,
\end{gathered}
\end{equation}
where $u,u_0\in\mathbb{R}$, $\tau\in\mathbb{R}_+$, $g:\mathbb{R}_+\times \mathbb{R}\to\mathbb{R}$.

Consider also the IVP for scalar impulsive fractional differential
equations
\begin{equation} \label{2}
\begin{gathered}
 {}_{t_0}^{c}D^{q}u=g(t,u) \quad \text{for }  t\geq t_0, \; t\neq  t_i,  \\
  u(t_i+0)=\Psi_i( u(t_i-0))\quad \text{for }  i=1,2,\dots ,\\
 u(t_0)=u_0,
\end{gathered}
\end{equation}
where  $u,u_0\in\mathbb{R}$, $g:\mathbb{R}_+\times \mathbb{R}\to \mathbb{R}$,  $\Psi_i: \mathbb{R}\to \mathbb{R}$,  $i=1,2,\dots$.

For the scalar IFrDE \eqref{2} we consider approach (V1) and
similar to condition (H1) we assume the following conditions
\begin{itemize}
\item[(H3)] If $u\neq 0$ then  $\Psi_{k}(u)\neq  u$ for  all  $k=1,2,3,\dots$.

\item[(H4)] $g(t,0)\equiv 0$ for $t\in\mathbb{R}_+$ and $\Psi_i(0)=0$ for $i=1,2,3,
\dots$.
\end{itemize}

Note the stability of the zero solution of the scalar IFrDE
\eqref{2} is defined in a similar manner to that in Definition \ref{def2}.

 \begin{remark} \label{rmk7} \rm
Note in the case $\Psi_i(u)\equiv u$ for $i=1,2,\dots$ the impulsive
fractional equation \eqref{2}  is reduced to the fractional differential
equation \eqref{222}.
\end{remark}

\begin{example} \label{examp2} \rm
Consider the scalar impulsive Caputo fractional differential
equation  \eqref{311}
where $A<0$, $a_i\in[-1,0)\cup(0,1]$, $i=1,2,3,\dots$ are constants.

According to Example \ref{examp1} the IVP for IFrDE \eqref{311}  has a solution
$x(t;t_0,x_0)$ defined by \eqref{317}. Therefore, applying
$0<E_{q}(A(T-\tau)^{q})\leq 1$ for $T\geq \tau$ we obtain
$| x(t;t_0,x_0) | \leq |
x_{0}| $ which  guarantees that the zero
solution  is uniformly stable.
\end{example}


\begin{example} \label{examp3} \rm
Consider the IVP for the scalar impulsive Caputo fractional differential
equation
\begin{equation} \label{29}
\begin{gathered}
 {}_{t_0}^{c}D^{q}u=0, \quad \text{for } t\geq t_0, \; t\neq  t_i,  \\
  u(t_i+0)= a_i\ u(t_i-0)\quad \text{for }  i\in\Omega_{t_0},\\
 u(t_0)=u_0,
\end{gathered}
\end{equation}
where $a_i\neq 0,1$, $i=1,2,3,\dots$ are
constants and there exists a constant $M>0$ with
$\prod_{i=1}^{\infty}|a_i|\leq M$.

The IVP for IFrDE \eqref{29}  has a solution defined by
$u(t;t_0,v_0)=u_0\prod_{i=1}^{k}a_i$ for $t\in (t_k,t_{k+1}]$, $k=0,1,2,\dots$.
Therefore,   we obtain
$ | u(t;t_{0},u_{0}) | \leq |u_{0}|\prod_{i=1}^{k}|a_i|$ for
$t\in (t_k,t_{k+1}]$ which  guarantees that the zero
solution of \eqref{29} is uniformly stable.

Note the existence of a constant $M>0$ with
$\prod_{i=1}^{\infty}|a_i|\leq M$ is guaranteed if
$a_i\in[-1,0)\cup(0,1),\ i=1,2,3,\dots$.
\end{example}

In this article we  study the connection between the
 stability properties  of the solutions of a nonlinear  system IFrDE \eqref{1}
 and the  stability properties of the zero solution of a corresponding
 scalar IFrDE \eqref{2} or corresponding scalar FrDE \eqref{222}.

We now introduce the class $\Lambda $ of piecewise continuous Lyapunov-like
functions which will be used to investigate the stability of the
system IFrDE \eqref{1}.

\begin{definition} \label{def3} \rm
Let $J\in \mathbb{\ R}_+$ be a given interval, and
$\Delta \subset \mathbb{\ R}^n$, $0\in \Delta$ be a given set. We will
say that the function $V(t,x):J\times \Delta\to \mathbb{R}_{+} $,
$V(t,0)\equiv 0$ belongs to the class $\Lambda(J,\Delta) $ if
\begin{itemize}
\item[(1)] The function $V(t,x)$ is continuous on
$J/\{t_k\in J\}\times \Delta$  and it  is  locally Lipschitzian with
respect to its second argument;

\item[(2)] For each $t_k\in J$ and $x\in \Delta$ there exist finite limits
$$
V(t_k-0,x)=\lim_{t\uparrow t_k}V(t,x),\quad
V(t_k+0,x)=\lim_{t\downarrow t_k}V(t,x)
$$
and the equalities
$V(t_k-0,x)=V(t_k,x)$ are valid.
\end{itemize}
\end{definition}

\begin{remark} \label{rmk8} \rm
When the function $V(t,x)\in \Lambda(J,\Delta)$ is additionally continuous
on the whole interval $J$, we will say  $V(t,x)\in \Lambda^C(J,\Delta)$.
\end{remark}

Lyapunov like functions used to discuss stability for differential
equations require an appropriate definition of the derivative of the
Lyapunov function along the studied differential equations. For
nonlinear Caputo fractional differential equations \eqref{1} the
following types of derivatives of Lyapunov functions along the
nonlinear Caputo fractional  differential equations are used:

- \emph{Caputo fractional
derivative of Lyapunov functions} $^c_{\tau_0}D{^q}{_t}V(t,x(t))$, where $x(t)$
is a solution of the studied fractional differential equation \eqref{100}
\cite{Li1,Li3}. This approach requires the
function to be smooth enough (at least continuously differentiable).
It works well for quadratic Lyapunov functions but in the general case
when the Lyapunov function depends on $t$ it can cause some problems
(see Example \ref{examp5}).

- \emph{Dini fractional  derivative of Lyapunov functions}
\cite{Lakfde,LL} given by
\begin{equation} \label{14}
D_+^qV(t,x)=\limsup_{h\to 0+}\frac{1}{h^q}\big
(V(t,x)-V(t-h,x-h^qf(t,x)\big)
\end{equation}
where $0<q<1$. The Dini fractional
derivative seems to be a natural generalization of the ordinary case
($q=1$). This definition requires only continuity of the Lyapunov
function. However it can be quite restrictive (see Example \ref{examp5}) and it
can present some problems (see Example \ref{examp6}).

- \emph{Caputo fractional Dini derivative of Lyapunov functions}
\cite{AHR,AHR1,AHR2}:
\begin{equation} \label{2011}
\begin{split}
&{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x;\tau_0,x_0) \\
&=\limsup_{h\to 0^{+}} {\frac{1}{h^{q}}}\Big\{
V(t,x)-V(\tau_0,x_0)\\
&\quad -\sum_{r=1}^{[\frac{t-\tau_0}{h}]}(-1)^{r+1}qCr
\Big[ V(t-rh,x-h^{q}f(t,x))-V(\tau_0,x_0)\Big]\Big\}
\end{split}
\end{equation}
for  $t \in (\tau_0,T)$,
where $V(t,x)\in \Lambda ^C ([\tau_0,T),\Delta)$,  $x,x_0\in \Delta$,
and there exists $h_1>0$ such that $t-h\in [\tau_0,T)$, $x-h^{q}f(t,x)\in \Delta $
for $ 0<h\leq h_1$.
The above formula is based on the formula \eqref{10} from fractional calculus.
This definition requires only continuity of the Lyapunov function.


Note in \cite{devi vlm} the authors defined a derivative of a
Lyapunov function and called it the Caputo fractional Dini
derivative of $V(t,x)$ (see \cite[Definition 3.2]{devi vlm}):
\begin{equation} \label{6677}
{}^{c}D_{+}^{q}V(t,x)
=\limsup_{h\to 0^{+}}{\frac{1}{h^{q}}}\big[V(t,x)
-\sum_{r=1}^nV(t-rh,x-h^qf(t,x))-V(t_0,x_0)\big],
\end{equation}
(we feel $(-1)^{r+1}qCr$ is missing in the formula).

The  formula \eqref{6677} is quite different than the the Caputo
fractional Dini  derivative of a function \eqref{10}.
Also, in \cite[Definition 5.1]{devi vlm} the authors
define the Caputo fractional Dini derivative of Lyapunov function
$V(s,y(t,s,x))$ by
\begin{equation} \label{6678}
\begin{split}
&{}^{c}D_{+}^{q}V(t,y(t,s,x))\\
&=\limsup_{h\to 0^{+}} {\frac{1}{h^{q}}}\big[V(t,y(t,s,x))\\
&\quad -\sum_{r=1}^n(-1)^{r+1}qCrV(s-rh,y(t,s-rh,x-h^qF(t,x)))\big].
\end{split}
\end{equation}
Formula \eqref{6678} is also quite different than the Caputo
fractional Dini  derivative of a function  \eqref{10}.

We will use definition \eqref{2011} as the definition of Caputo
fractional Dini derivative of a Lyapunov function.

\begin{example} \label{examp4} \rm
Consider the quadratic  Lyapunov function, i.e. $V(t,x)=x^2$ for $x\in \mathbb{R}$.
Recall the scalar ordinary case ($q=1$), i.e. the ordinary
differential equation $x'=f(t,x)$, $x\in\mathbb{R}$,  and the Dini
derivative of the quadratic Lyapunov function applied to it,
\begin{equation} \label{344}
D_{+}V(t,x(t))=2xf(t,x(t)).
\end{equation}
Let $x\in C^q([\tau_0,T],\mathbb{R})$ be a solution of FrDE\eqref{100}.
Then  the Caputo fractional derivative of the quadratic Lyapunov
function $^c_{\tau_0}D{^q}{_t}(x(t))^2$ exists and the equality
\begin{equation}
\label{341}^c_{\tau_0}D{^q}{_t}(x(t))^2=2x(t)f(t,x(t))
\end{equation}
holds (see for example \cite{DM1}).

Apply \eqref{14} to obtain Dini fractional derivative of the quadratic
Lyapunov function, namely
\begin{equation} \label{342}
\begin{split}
&D_+^qV(t,x(t))\\
&=D_+^q(x(t))^2\\
&=\limsup_{h\to 0+}\frac{1}{h^q}\big
((x(t))^2-(x(t-h)-h^qf(t,x(t-h)))^2\big)\\
&= \limsup_{h\to 0+}\frac{1}{h^q}\big(x(t)-x(t-h)+h^qf(t,x(t-h))\big)
\big(x(t)+x(t-h) \\
&\quad -h^qf(t,x(t-h)\big)\\
&=\limsup_{h\to 0+}\big
(\frac{x(t)-x(t-h)}{h}h^{1-q}+f(t,x(t-h))\big)\big(x(t)+x(t-h)\\
&\quad -h^qf(t,x(t-h)\big)\\
&=2x(t)f(t,x(t)).
\end{split}
\end{equation}

Finally, apply \eqref{2011} to obtain the Caputo fractional derivative
of the quadratic Lyapunov function
 \begin{equation} \label{343}
\begin{split}
&{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x(t);\tau_0,x_0)\\
&=\limsup_{h\to 0^{+}} {\frac{1}{h^{q}}}\Big\{(x(t))^2-x_0^2\\
&\quad -\sum_{r=1}^{[\frac{t-\tau_0}{h}]}(-1)^{r+1}qCr
 \Big[ \Big(x(t-rh)-h^{q}f(t,x(t-rh))\Big)^2 -(x_0)^2\Big]\Big\}\\
&=\limsup_{h\to 0^{+}} {\frac{1}{h^{q}}}
\Big\{\sum_{r=0}^{[\frac{t-\tau_0}{h}]}(-1)^{r}qCr \Big[ (x(t-rh))^2-(x_0)^2\Big]\\
&\quad +\sum_{r=1}^{[\frac{t-\tau_0}{h}]}(-1)^{r}qCr
\Big[\Big(x(t-rh)-h^{q}f(t,x(t-rh))\Big)^2-(x(t-rh))^2 \Big]\Big\}.
\end{split}
\end{equation}
Using \eqref{765}, \eqref{679} and $\limsup_{h\to 0^{+}}
\sum_{r=0}^{[\frac{t-\tau_0}{h}]}(-1)^{r}qCr=0$ we obtain
\begin{equation} \label{3437}
\begin{split}
&{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x(t);\tau_0,x_0)\\
&=_{\tau_0}^{GL}D^{q}_{+}\Big[ (x(t))^2-(x_0)^2\Big] \\
& \quad  -\limsup_{h\to 0^{+}}
\sum_{r=0}^{[\frac{t-\tau_0}{h}]}(-1)^{r}qCr f(t,x(t-rh))
 \Big[2x(t-rh)-h^{q}f(t,x(t-rh))\Big]\\
&=_{\tau_0}^{c}D^{q} (x(t))^2-2\limsup_{h\to 0^{+}}
 \sum_{r=0}^{[\frac{t-\tau_0}{h}]}(-1)^{r}qCr f(t,x(t-rh))x(t-rh)\\
&\quad +\limsup_{h\to 0^{+}} h^{q}\sum_{r=0}^{[\frac{t-\tau_0}{h}]}
 (-1)^{r}qCr f(t,x(t-rh))f(t,x(t-rh))\\
&=_{\tau_0}^{c}D^{q} (x(t))^2= 2x(t)f(t,x(t)).
\end{split}
\end{equation}
From \eqref{341},  \eqref{342} and  \eqref{3437} we see that (in the
scalar case) the above derivatives coincide with the ordinary case
\eqref{344}.
\end{example}

\begin{example} \label{examp5} \rm
Let $V:\mathbb{R}_+\times \mathbb{R}\to\mathbb{R}_+$ be given by
$V(t,x)=m^2(t)x^2$ for $x\in \mathbb{R}$ where $m\in C^1(\mathbb{R}_+,\mathbb{R})$.
Recall the Dini derivative of  the Lyapunov function in the ordinary
case ($q=1$) is
\begin{equation} \label{18}
 D_{+}V(t,x)=2x\ m^2(t)f(t,x)+
\frac{d}{dt}\Big[m^2(t)x^{2}\Big].
\end{equation}
If   $x\in C^q([\tau_0,T],\mathbb{R})$ is a solution of FrDE\eqref{100}, then
to obtain the Caputo fractional derivative
$^c_{\tau_0}D{^q}{_t}\Big(m^2(t)(x(t))^2\Big)$ we need a
multiplication rule from fractional calculus, so it could lead to
some difficulties in calculations of the derivative.

Now, let $(t,x)\in\mathbb{R}_+\times\mathbb{R}$ and apply  formula
\eqref{14} to obtain Dini fractional derivative of $V$, namely
\begin{equation} \label{16}\begin{split}
&D_+^qV(t,x) \\
&=\limsup_{h\to 0^{+}} {\frac{1}{h^{q}}}
\Big[m^2(t)x^{2}-m^2(t-h)\Big(x-h^{q}f(t,x)\Big)^2\Big]
 \\
&=\limsup_{h\to 0^{+}} \Big[\frac{m(t)-m(t-h)}{h}x h^{1-q}+m(t-h)f(t,x)\Big) \\
&\quad \times\Big((m(t)+m(t-h))x-m(t-h)h^{q}f(t,x)\Big)\Big]\\
&=2 x\ m^2(t)f(t,x).
\end{split}
\end{equation}

Now we look at \eqref{18} and \eqref{16}.  Both differ significantly.
In the fractional Dini derivative \eqref{16} one term is missing.
Additionally, the Dini fractional derivative  \eqref{16} is
independent of the order of the differential equation  $q$. However
the behavior of solutions of fractional differential equations
depends significantly on the order $q$.


 Let $ t,\tau_0\in\mathbb{R}_+,\ x,x_0\in\mathbb{R}$.  Now use  \eqref{2011}
to obtain the Caputo fractional Dini derivative of $V$, namely
\begin{equation} \label{177}
\begin{split}
&{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x;\tau_0, x_0) \\
&=\lim_{h\to 0^{+}}\sup {\frac{1}{h^{q}}} \Big[ m^2(t)x^{2}
 -  m^2(\tau_0)x_0^2\sum_{r=0}^{[\frac{t-\tau_0}{h}]}( -1) ^{r}qCr  \\
& \quad  -\sum_{r=1}^{[\frac{t-\tau_0}{h}]}( -1) ^{r+1}qCr\ m^2(t-rh)
 \Big( x-h^{q}f(t,x)\Big)^{2}\Big] \\
&=\lim_{h\to 0^{+}}\sup {\frac{1}{h^{q}}} \Big[ m^2(t)h^{q}f(t,x)(2x-h^{q}f(t,x))
 -  m^2(\tau_0)x_0^2\sum_{r=0}^{[\frac{t-\tau_0}{h}]}( -1) ^{r}qCr  \\
&   +\Big( x-h^{q}f(t,x)\Big)^{2}
 \sum_{r=0}^{[\frac{t-\tau_0}{h}]}( -1) ^{r}qCr m^2(t-rh)\Big].
\end{split}
\end{equation}
Now using \eqref{765} from \eqref{177} we obtain
\begin{equation} \label{17}
{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x;\tau_0, x_0)
=2x\ m^2(t)f(t,x) +{}_{\tau_0}^{RL}D^{q}\big(m^2(t)x^2-x_0^2
m^2(\tau_0)\big).
\end{equation}
Note the Caputo fractional Dini derivative depends not only on the
fractional order $q$ but also on the initial data $(\tau_0,x_0)$ of
\eqref{100} which  is similar to the Caputo fractional derivative of
a function.

 Formula \eqref{17} is similar  to
the ordinary case $q=1$ and formula \eqref{18}   consists of two
terms where the ordinary derivative is replaced by the fractional
one.

It seems that formula \eqref{2011} is a natural
generalization of the one for   ordinary  differential equations.
Also, if the function $V(t,x)\equiv c$, $c$ is  a constant, then
for any $ t,\tau_0\in\mathbb{R}_+,\ x,x_0\in\mathbb{R}$ the equality
${}_{\eqref{100}}^{c}D_{+}^{q}V(t,x;\tau_0, x_0)=0$  holds.
\end{example}

In this article we  use piecewise continuous Lyapunov functions from the
class $\Lambda(J,\Delta) $. We define  the derivative of piecewise
continuous Lyapunov functions using the idea of the Caputo fractional Dini
derivative of a function $m(t)$ given by \eqref{10} and based on \eqref{2011}.
 We define the
\emph{generalized Caputo fractional Dini  derivative} of the function
$V(t,x)\in \Lambda ([t_0,T),\Delta)$ along trajectories of solutions
of IVP for the system IFrDE \eqref{1}  as follows:
\begin{equation} \label{200}
\begin{split}
&{}_{\eqref{1}}^{c}\mathcal{D}_{+}^{q}V(t,x;t_0,x_0) \\
&=\limsup_{h\to 0^{+}} {\frac{1}{h^{q}}}\Big\{
V(t,x)-V(t_0,x_0)\\
&\quad -\sum_{r=1}^{[\frac{t-t_0}{h}]}(-1)^{r+1}qCr
 \Big[ V(t-rh,x-h^{q}f(t,x))-V(t_0,x_0)\Big]\Big\}
\end{split}
\end{equation}
for $t \in (t_0,T): t\neq t_k$,
where $x,x_0\in \Delta$, and there exists $h_1>0$ such that
$t-h\in [t_0,T)$, $x-h^{q}f(t,x)\in \Delta $ for $ 0<h\leq h_1$.


\begin{example} \label{examp6} \rm
 Consider the scalar IFrDE \eqref{29} with  $t_0=0$, $t_k=k$,
$a_k= \frac{1}{\sqrt{2}},\ k=1,2,\dots$, and $u_0=2\sqrt{a}$,
$a>0$ is a constant. According to Example \ref{examp3} the solution of  \eqref{29}  is
$x(t;t_0,u_0)=2\sqrt{\frac{a}{2^k}}$ on $(k,k+1]$, $k=0, 1,2,\dots$.

Consider the IFrDE \eqref{29} with $t_0=0$, $t_k=k$,
$a_k= \frac{1}{2}$, $k=1,2,\dots$, and $u_0=a$. Then IFrDE  \eqref{29}
has an unique solution  $u^+(t;t_0,u_0)=\frac{a}{2^k}$ for
$t\in(k,k+1]$, $k=0,1,2,\dots$.

 Let the Lyapunov function $V:\mathbb{R}_+\times \mathbb{R}\to\mathbb{R}_+$ be given by
$V(t,x)= x^2\sin^2t$. It is locally Lipshitz with respect to its
second argument $x$. According to Example \ref{examp5} and formula \eqref{16}
we  obtain the Dini fractional derivative of $V$, namely
${}^{c}D_{+}^{q}V(t,x)=2x\sin^2(t)f(t,x)\equiv 0$.

  All the conditions in  \cite[Theorem 3.1]{St1}  are
satisfied and therefore, the inequality
$ V(t,x(t;t_0,x_0))\leq u^+(t;t_0,u_0))$ has to be hold
for all  $t\geq t_0$. However, the inequality
\[
V(t,2\sqrt{\frac{a}{2^k}}) = 4\frac{a}{2^k}\sin^2t\leq
\frac{a}{2^k},
\]
 i.e. $\sin^2t\leq \frac{1}{4}$  is not satisfied for all $t\geq 0$.
\end{example}

\section{Comparison results for
scalar impulsive Caputo fractional differential equations}

We use the following results for Caputo fractional Dini derivative
of a continuous Lyapunov function.

\begin{lemma}[Comparison result \cite{AHR}]  \label{lem1}
Assume the following conditions are satisfied:
\begin{itemize}
\item[(1)] The function $x^*(t)=x(t;\tau_0,x_0)\in C^q([\tau_0,\tilde{T}],\Delta)$
is a solution   of the FrDE \eqref{100} where $\Delta \subset \mathbb{R}^n,\ 0\in\Delta$,
$\tau_0,\ \tilde{T}\in\mathbb{R}_+,\ \tau_0<\tilde{T}$ are given constants,
 $x_0\in\Delta$.

\item[(2)] The function $ g \in  C([\tau_0,\tilde{T}]\times \mathbb{R},\mathbb{R})$.

\item[(3)]  The function $V\in \Lambda^C([\tau_0,\tilde{T}],\Delta)$ and
$$
{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x;\tau_0,x_0 )\leq g(t,V(t,x))\quad
 \text{for } (t,x)\in [\tau_0,\tilde{T}]\times \Delta\,.
$$

\item[(4)]  The function $u^*(t)=u(t;\tau_0,u_0)$, $u^*\in C^q([\tau_0,\tilde{T}],\mathbb{R})$,
is the maximal solution of the initial value
   problem \eqref{222} with $\tau=\tau_0$.
\end{itemize}
Then the inequality   $V(\tau_0,x_0)\leq u_0$ implies  $V(t,x^*(t))\leq u^*(t)$
for $t\in [\tau_0,\tilde{T}]$.
\end{lemma}

When $g(t,x)\equiv 0$ in Lemma \ref{lem1} we obtain the following result.

\begin{corollary}[\cite{AHR}] \label{coro1}
 Let  (1) in Lemma \ref{lem1} be satisfied and
$V\in \Lambda^C([\tau_0,\tilde{T}],\Delta)$ be such that for any points
$t \in [\tau_0,\tilde{T}]$, $x\in \Delta$  the inequality
${}_{\eqref{100}}^{c}D_{+}^{q}V(t,x;\tau_0,x_0)\leq 0$
   holds.
Then for $t\in [\tau_0,\tilde{T}]$ the inequality
$V(t,x^*(t))\leq V(\tau_0,x_0) $ holds.
\end{corollary}


If the derivative of the Lyapunov function is negative, the
following result is true.

\begin{lemma}[\cite{AHR}] \label{lem2}
  Let Condition (1) of Lemma \ref{lem1} be satisfied and the function
$V\in \Lambda^C([t_0,\tilde{T}],\Delta)$ be such that  for any points
$t \in [\tau_0,\tilde{T}]$, $x\in \Delta$
the
 $$
{}_{\eqref{100}}^{c}D_{+}^{q}V(t,x ;\tau_0,x_0)\leq -c(\|x\|)\,,
$$
where $c\in \mathcal{K}$.
Then for $t\in [\tau_0,\tilde{T}]$,
\begin{equation} \label{456}
V(t,x^*(t))
\leq V(\tau_0,x_0)-\frac{1}{\Gamma(q)} \int_{\tau_0}^t (t-s)^{q-1}c(\|x^*(s)\|)ds \,.
\end{equation}
\end{lemma}

Now we  prove some  comparison results for the system of IFrDE \eqref{1}
and piecewise continuous Lyapunov functions applying the generalized Caputo
fractional Dini derivative \eqref{200}.
Recall $\lim_{k\to \infty}t_k=\infty$. In this section  we assume
without loss of generality that $0\leq t_0<t_1<T$.

As a comparison scalar equation we use the impulsive Caputo fractional
differential equation \eqref{2} or the Caputo fractional differential
equation \eqref{222}.

\begin{lemma}[Comparison result by scalar IFrDE] \label{lem3}
Assume that the following conditions are satisfied:
\begin{itemize}
\item[(1)] Let conditions {\rm (H1)} and {\rm (H3)} be satisfied for all
$k\in \{i: t_i\in(t_0,T)\}$ where $t_0,\ T\in\mathbb{R}_+,\ t_0<T$
are given  constants.

\item[(2)] The function $x^*(t)=x(t;t_0,x_0)\in PC^q([t_0,T],\Delta)$
is a solution of the IFrDE \eqref{1}
where $\Delta \subset \mathbb{R}^n,\ 0\in\Delta$,   $x_0\in\Delta$.

\item[(3)] The function $ g \in  C([t_0,T]\times \mathbb{R},\mathbb{R})$
and the IVP for the IFrDE \eqref{2} has a
unique maximal solution  $u^*(t)=u(t;t_0,u_0)\in PC^q([t_0,T],\mathbb{R})$.

  \item[(4)] The functions $\Psi_k:\ \mathbb{R}\to \mathbb{R}$,   $k\in \{i:\ t_i\in(t_0,T)\}$,
are nondecreasing.

  \item[(5)]  The function $V\in \Lambda([t_0,T],\Delta)$ and
    \begin{itemize}

\item[(i)] for any $\tau_0 \in [t_0,T)$ and $x_0\in \Delta$,
the inequality
${}_{\eqref{1}}^{c}D_{+}^{q}V(t,x;\tau_0,x_0 )\leq 
g(t,V(t,x))$
for $(t,x)\in [\tau_0,T]\times \Delta$, $t\neq t_k$ holds;

\item[(ii)] for any points $t_k\in(t_0,T)$ and  $ x\in \Delta$
we have
$$
V(t_k+0,\Phi_k(x))\leq \Psi_k(V(t_k,x)).
$$
\end{itemize}
\end{itemize}
   Then the inequality   $V(t_0,x_0)\leq u_0$ implies
$V(t,x^*(t))\leq u^*(t)$ for $t\in [t_0,T]$.
\end{lemma}

\begin{proof}
 We use induction.
Let $t\in[t_0,t_1]$. By Lemma \ref{lem1}  the claim in
    Lemma \ref{lem3} holds  on $[t_0,t_1]$.

Let $t\in (t_1,t_2]\cap[t_0,T]$. Then the function
$\overline{u}_1(t)\equiv
u^*(t)$ is the maximal solution of IVP for FrDE \eqref{222} for
$\tau=t_1$ and
$\overline{u}_1(t_1)=
\Psi_1(u^*(t_1-0))(=\Psi_1(u^*(t_1))) =u^*(t_1+0)$ and the function
$\overline{x}_1(t)\equiv x^*(t)$ is a solution of IVP for FrDE
\eqref{100} for $\tau_0=t_1$ and
$x_0= \Phi_1(x^*(t_1-0))=x^*(t_1+0)$.  Using conditions (4), (5)(ii) and the
above proved inequality
$V(t_1,x^*(t_1))=V(t_1,x^*(t_1-0))\leq u^*(t_1-0)$ we obtain
\begin{equation} \label{4569}
\begin{split}
&V(t_1+0,\overline{x}_1(t_1))=V(t_1+0,x^*(t_1+0))\\
&=V(t_1+0,\Phi_1(x^*(t_1-0))) =V(t_1+0,\Phi_1(x^*(t_1)))\\
&\leq \Psi_1(V(t_1,x^*(t_1)))\leq \Psi_1(u^*(t_1-0))\\
&=u^*(t_1+0)=\overline{u}_1(t_1).
\end{split}
\end{equation}
 By Lemma \ref{lem1}
for $\tau_0=t_1$ and $\tilde{T}=\min\{T,t_2\}$ we obtain
$ V(t,\overline{x}_1(t))\leq \overline{u}_1(t)$ for $t\in [t_1,t_2]\cap[t_0,T]$.
 Therefore, $ V(t, x^*(t))\leq u^*(t)$ for $t\in (t_1,t_2]\cap[t_0,T]$,
i.e. the claim of
Lemma \ref{lem3} holds on $[t_0,t_2]\cap[t_0,T]$.

Continuing this process and an induction argument proves that the claim is
true on $[t_0,T]$.
\end{proof}


\begin{example} \label{examp7} \rm
 Consider  the scalar IFrDE \eqref{29} with  $t_0=0$, $t_k=k$,
$a_k= \frac{1}{\sqrt{2}},\ k=1,2,\dots$, and $u_0=2\sqrt{a}$,
$a>0$ is a constant. According to Example \ref{examp3} the solution of  \eqref{29}  is
$x(t;t_0,u_0)=2\sqrt{\frac{a}{2^k}}$ on $(k,k+1]$, $k=0, 1,2,\dots$.

Consider the IFrDE \eqref{29} with $t_0=0$, $t_k=k$,
$a_k= \frac{1}{2}$, $k=1,2,\dots$, and $u_0=a$.
Then IFrDE  \eqref{29} has an unique solution  $u^+(t;t_0,u_0)=\frac{a}{2^k}$ for
$t\in(k,k+1]$, $k=0,1,2,\dots$.


  Let the Lyapunov function $V:\mathbb{R}_+\times \mathbb{R}\to\mathbb{R}_+$ 
be given by $V(t,x)=x^2\sin^2t$.  By Example \ref{examp5} and formula \eqref{17}
we  obtain the Caputo fractional Dini  derivative of $V$, namely
${}_{\eqref{29}}^{c}D_{+}^{q}V(t,x;0, x_0) =x^2{}_{0}^{RL}D^{q}[\sin^2t]$.
Using $\sin^2t-0.5-0.5 \cos (2t)$ and
${}_{0}^{RL}D^{q}\cos(2t)=2^q\cos(2t+\frac{q\pi}{2})$ it follows that
the inequality
${}_{\eqref{29}}^{c}D_{+}^{q}V(t,x;0, x_0)\leq0$ is not satisfied,
i.e. condition (5)(i) of Lemma \ref{lem3} is not satisfied
for $g(t,x)\equiv 0$ so we cannot claim that the inequality
$V(t,x(t;0,x_0))\leq u^+(t;0,u_0))$ has to be hold for all
$t\geq t_0$, i.e. the application of Lemma \ref{lem3} and the Caputo fractional Dini
derivative does not lead to a contradiction as in \cite{St1}
(compare with Example \ref{examp6}).
\end{example}

The result in Lemma \ref{lem3} is also true on the half line
(recall \cite{AHR} that Lemma \ref{lem1} extends to the half line).

\begin{corollary}  \label{coro2}
 Suppose all the conditions of Lemma \ref{lem3} are satisfied
 with  $[t_0,T]$ replaced by $[t_0,\infty)$. Then
   the inequality  $V(t_0,x_0)\leq u_0$ implies
$V(t,x^*(t))\leq u^*(t)$ for $t\geq t_0$.
\end{corollary}

If $\Psi_k(u)\equiv u$ for all $k=1,2,\dots$, we consider the scalar
FrDE \eqref{222} as a comparison equation.

\begin{lemma}[Comparison result by scalar FrDE] \label{lem4}
Assume
\begin{itemize}
\item[(1)] Condition {\rm (H1)} is fulfilled for all
$k\in \{i:\ t_i\in(t_0,T)\}$ where $t_0,\ T\in\mathbb{R}_+,\ t_0<T$
are given  constants.

\item[(2)] Condition (2) of Lemma \ref{lem3} is fulfilled.

\item[(3)] The function $ g \in  C([t_0,T]\times \mathbb{R},\mathbb{R})$
and the IVP for the FrDE \eqref{222} with $\tau=t_0$ has a
unique maximal solution  $u^*(t)=u(t;t_0,u_0)\in C^q([t_0,T],\mathbb{R})$.

  \item[(4)] The function $V\in \Lambda([t_0,T],\Delta)$,
it satisfies the condition (5)(i) of Lemma \ref{lem3} and
    \begin{itemize}
\item[(ii)] for any points $t_k\in(t_0,T)$ and
$ x\in \Delta$  we have
$$
V(t_k+0,\Phi_k(x))\leq V(t_k,x).
$$
\end{itemize}
\end{itemize}
Then the inequality  $V(t_0,x_0)\leq u_0$ implies  $V(t,x^*(t))\leq u^*(t)$
for $t\in [t_0,T]$.
\end{lemma}

\begin{proof} The proof is similar to the one of Lemma \ref{lem3}
 where the inequality \eqref{4569} is replaced by
    \begin{equation} \label{4567}
\begin{split}
&V(t_1+0,\overline{x}_1(t_1))=V(t_1+0,x^*(t_1+0))\\
&=V(t_1+0,\Phi_1(x^*(t_1-0))) =V(t_1+0,\Phi_1(x^*(t_1)))\\
&\leq V(t_1-0,x^*(t_1-0))\leq u^*(t_1-0)\\
&=u^*(t_1+0)=\overline{u}_1(t_1).
\end{split}
\end{equation}
\end{proof}

The result of Lemma \ref{lem4} is also true on the half line.

\begin{corollary} \label{coro3}
Suppose all the conditions of Lemma \ref{lem4} are satisfied
 with  $[t_0,T]$ replaced by $[t_0,\infty)$. Then
   the inequality  $V(t_0,x_0)\leq u_0$ implies
$V(t,x^*(t))\leq u^*(t)$ for $t\geq t_0$.
\end{corollary}

Recall $\lim_{k\to \infty}t_k=\infty$. In our next result
we assume with loss of generality that $t_p<T \leq t_{p+1}$
for some $p\in \{1,2,\dots \}$.
Next we present a comparison result for negative Caputo fractional Dini derivative.

\begin{lemma} \label{lem5}
  Assume the following conditions are satisfied:
\begin{itemize}
\item[(1)]  Conditions (1) and (2) of Lemma \ref{lem4} are  fulfilled.

\item[(2)]  The function $V\in \Lambda([t_0,T],\Delta)$ and
    \begin{itemize}
\item[(i)] for any $\tau_0 \in [t_0,T)$ and $x_0\in \Delta$,
the inequality
${}_{\eqref{1}}^{c}D_{+}^{q}V(t,x;\tau_0,x_0 )\leq -c(\|x\|)$
for $(t,x)\in [\tau_0,T]\times \Delta$, $t\neq t_k$ holds;

\item[(ii)] for any points $t_k\in(t_0,T)$ and  $ x\in \Delta$
the inequalities $V(t_k+0,\Phi_k(x))\leq V(t_k,x)$ hold.
\end{itemize}
\end{itemize}

Then for $t\in [t_0,T]$ the following inequalities hold:
\begin{equation} \label{45611}
V(t,x^*(t))\leq V(t_0,x_0)-\frac{1}{\Gamma(q)}
\int_{t_0}^t (t-s)^{q-1}c(\|x^*(s)\|)ds
\end{equation}
for $t\in [t_0,t_1]$, and
\begin{equation} \label{4561}
\begin {split}
V(t,x^*(t))
&\leq V(t_0,x_0)-\sum_{i=0}^{k-1}\frac{1}{\Gamma(q)}\int_{t_i}^{t_{i+1}}
 (t_{i+1}-s)^{q-1}c(\|x^*(s)\|)ds\\
&\quad -\frac{1}{\Gamma(q)}\int_{t_k}^t (t-s)^{q-1}c(\|x^*(s)\|)ds
\end{split} \end{equation}
for $t\in (t_k,t_{k+1}^{\star}]$, $k=1,2,\dots,p$;
here $t_{k+1}^{\star}= t_{k+1}$ if $k=1,\dots,p-1$ and
$t_{p+1}^{\star}=T$.
\end{lemma}

\begin{proof}
We use induction. Let $t\in[t_0,t_1]$.
    By Lemma \ref{lem2}  with  $\tau_0=t_0$ and $\tilde{T}=t_1$ inequality
\eqref{45611} holds   on $[t_0,t_1]$.

Let $t\in (t_1,t_2]\cap[t_0,T]$. Then the function
 $\overline{x}_1(t)\equiv x^*(t)$ is a solution of IVP for FrDE \eqref{100}
for $\tau_0=t_1$ and $\overline{x}_1(t_1) = \Phi_1(x^*(t_1-0))
  (= \Phi_1(x^*(t_1))) =x^*(t_1+0)$. Using condition  (2)(ii)  we obtain
\begin{equation} \label{667}
\begin{split}
&V(t_1+0,\overline{x}_1(t_1))=V(t_1+0,x^*(t_1+0))\\
&=V(t_1+0,\Phi_1(x^*(t_1-0)))  =V(t_1+0,\Phi_1(x^*(t_1)))\\
&\leq V(t_1,x^*(t_1))=V(t_1,x^*(t_1-0)).
\end{split}
\end{equation}
By Lemma \ref{lem2} with  $\tau_0=t_1$ and $\tilde{T}=\min\{T,t_2\}$,
inequality \eqref{667}  and inequality \eqref{45611} with $t=t_1$
we obtain
\begin{align*}
V(t,\overline{x}_1(t))
&\leq V(t_1+0,\overline{x}_1(t_1))-\frac{1}{\Gamma(q)}
 \int_{t_1}^t (t-s)^{q-1}c(\|x^*(s)\|)ds \\
&\leq V(t_1,x^*(t_1-0))-\frac{1}{\Gamma(q)}
 \int_{t_1}^t (t-s)^{q-1}c(\|x^*(s)\|)ds \\
&\leq V(t_0,x_0)-\frac{1}{\Gamma(q)}\int_{t_0}^{t_1}
(t_1-s)^{q-1}c(\|x^*(s)\|)ds\\
&\quad -\frac{1}{\Gamma(q)}\int_{t_1}^t
(t-s)^{q-1}c(\|x^*(s)\|)ds.
\end{align*}
Therefore, inequality
\eqref{4561} holds on $(t_1,t_2]\cap[t_0,T]$.
Continuing this process and an induction argument proves the claim is
true on $[t_0,T]$.
\end{proof}

The result in Lemma \ref{lem5} is also true on the half line
(recall \cite{AHR} that Lemma \ref{lem2} extends to the half line).

\begin{corollary} \label{coro4}
Suppose all the conditions of Lemma \ref{lem5} are satisfied  with
$[t_0,T]$ replaced by $[t_0,\infty)$.
 Then   for any $t\geq t_0$ the inequalities \eqref{45611},
 \eqref{4561} (where $k=1,2,\dots,p$
 is replaced by $k=1,2,\dots$)  hold.
\end {corollary}

\begin{remark} \label{rmk9} \rm
In this paper we assumed an infinite number of points
$t_i$,  $i=1,2,\dots $ with $t_1<t_2<\dots$ and
$\lim_{k\to \infty}t_k=\infty$. However it is worth noting that the results in
Section 5 (and elsewhere) hold  if we only consider a finite of
points $t_i$, $i=1,2,\dots,p $ for some $p\in \{1,2,\dots\}$  and
$t_1<t_2<\dots<t_p$.
\end{remark}

\section{Main result}

In this section we obtain sufficient conditions for
stability of the zero solution of nonlinear impulsive Caputo fractional
differential equations.

\begin{theorem} \label{thm1}
 Let the following conditions be satisfied:
\begin{itemize}
\item[(1)] Conditions {\rm (H1)--(H4)} are satisfied.

\item[(2)] The functions $f\in PC(\mathbb{R}_+,\mathbb{R}^n)$,  $\Phi_k :\mathbb{R}^n\to\mathbb{R}^n$,
  $k=1,2,\dots$, are  such that   for any $(t_0,x_0)\in\mathbb{R}_+\times\mathbb{R}^n$
 the IVP for the scalar of IFrDE \eqref{1} has a solution
$x(t;t_0,x_0)\in PC^q([t_0,\infty),\mathbb{R}^n)$.

\item[(3)] The functions $ g \in  C(\mathbb{R}_+\times \mathbb{R}, \mathbb{R})$,
$\Psi_k :\mathbb{R}\to\mathbb{R}$, $k=1,2,\dots$, are such that  for any
$(t_0,u_0)\in\mathbb{R}_+\times\mathbb{R}$ the IVP for the scalar IFrDE \eqref{2} has
a solution $u(t;t_0,u_0)\in PC^q([t_0,\infty),\mathbb{R})$ and in the case of
nonuniqueness the IVP has a unique maximal solution.

\item[(4)] The functions $\Psi_i:\ \mathbb{R}\to \mathbb{R}$,  $i=1,2,\dots$,   are nondecreasing.

\item[(5)] There exists a function $V\in \Lambda (\mathbb{R}_+, \mathbb{R}^n) $  such that
\begin{itemize}
\item[(i)] for any points $t_0\in \mathbb{R}_+$ and
$x, x_0\in \mathbb{R}^n$  we have
$$
{}_{\eqref{1}}^{c}D_{+}^{q}V(t,x ;t_0,x_0)\leq g(t,V(t,x))
$$
for $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$;

\item[(ii)] for any points $t_k,\ k=1,2,\dots$ and  $ x\in \mathbb{R}^n$
we have
$$
V(t_k+0,\Phi_k(x))\leq \Psi_k(V(t_k,x));
$$

\item[(iii)]  $b(\|x\|)\leq V(t,x)$ for $ t\in\mathbb{R}_+$, $x\in\mathbb{R}^n$,
  where  $b \in \mathcal{K}$.
\end{itemize}
\item[(6)] The  zero solution of the scalar IFrDE \eqref{2} is   stable.
\end{itemize}
     Then  the zero solution of the system of  IFrDE \eqref{1} is  stable.
     \end{theorem}

\begin{proof}
Let $\epsilon>0$ and $t_0\in \mathbb{R}_+$ be  given. Without loss of
generality we assume $t_0<t_1$. According to condition (6)    there
exists  $\delta_1 =\delta_1(t_0, \epsilon)>0$ such that   the
inequality  $|u_0|<\delta_1$ implies
\begin{equation} \label{201}
|u(t;t_0,u_0)|<b(\epsilon),\quad t\geq t_0,
\end{equation}
where $u(t;t_0,u_0)$ is a solution of the scalar IFrDE  \eqref{2}. Since
$V(t_0,0)= 0$  there exists $\delta_2=\delta_2(t_0,\delta_1)>0$ such
that $V(t_0,x)<\delta_1$ for $\|x\|<\delta_2$. Let  $x_0\in \mathbb{R}^n$
with   $ \|x_0\|<\delta_2$. Then $V(t_0,x_0)<\delta_1$. Consider any
solution $x^*(t)=x(t;t_0,x_0)\in PC^q([t_0,\infty),\mathbb{R}^n)$ of the
IFrDE \eqref{1} which exists according to condition (2). Now let
$u_0^*=V(t_0,x_0)$.  Then $u_0^*<\delta_1$ and inequality
\eqref{201} holds for the unique maximal  solution
$\overline{u}(t;t_0,u_0^*)$ of the scalar IFrDE \eqref{2} (with
$\tau=t_0$ and $u_0=u_0^*$).

According to Corollary \ref{coro2}  the inequality
$V(t,x^*(t))\leq  \overline{u}(t;t_0,u_0^*)$ holds for $t\geq t_0$.
Then for any $t\geq t_0$ from
condition (5)(iii) and inequality \eqref{201}  we obtain
$$
b(\|x^*(t)\|)\leq V(t,x^*(t))\leq  \overline{u}(t;t_0,u_0^*)<b(\epsilon),
$$
so the result follows.
\end{proof}

If we consider the scalar FrDE \eqref{222} as a comparison equation
then the following result holds.

\begin{theorem} \label{thm2}
Let the following conditions be satisfied:
\begin{itemize}
\item[(1)] Conditions {\rm (H1)--(H2)} are satisfied.
\item[(2)] Conditions (2) and (5) of Theorem \ref{thm1} are satisfied where
the condition (5)(ii) is replaced by
\begin{itemize}
\item[(ii)] for any points $t_k$, $k=1,2,\dots$ and
$ x\in \mathbb{R}^n$ we have
$$
V(t_k+0,\Phi_k(x))\leq V(t_k,x).
$$
\end{itemize}

\item[(3)] The function $ g \in  C(\mathbb{R}_+\times \mathbb{R}, \mathbb{R}),\ g(t,0)\equiv 0$
is such that  for any $(t_0,u_0)\in\mathbb{R}_+\times\mathbb{R}$ the IVP for the scalar
FrDE \eqref{222} has a solution $u(t;t_0,u_0)\in C^q([t_0,\infty),\mathbb{R})$
and in the case of nonuniqueness the IVP has
a unique maximal solution.

\item[(4)] The  zero solution of the scalar FrDE \eqref{222} is   stable.
\end{itemize}
     Then  the zero solution of the system of  IFrDE \eqref{1} is  stable.
     \end{theorem}

The proof of above theorem is similar to the one of Theorem \ref{thm1},
applying Corollary \ref{coro3} instead of  Corollary \ref{coro2}.


Now we present some sufficient conditions for stability of the zero
solution of the IFrDE in the case when the condition for the Caputo fractional
Dini derivative of the Lyapunov function is satisfied only on a
ball.

\begin{theorem} \label{thm3}
Let the following conditions be satisfied:
\begin{itemize}
\item[(1)] Conditions (1)--(4) of Theorem \ref{thm1} are fulfilled.

\item[(2)] There exists a function $V\in \Lambda (\mathbb{R}_+,S(\lambda)) $  such that
    \begin{itemize}
\item[(i)] for any points $t_0\in \mathbb{R}_+$ and
$x, x_0\in S(\lambda)$  we have
$$
{}_{\eqref{1}}^{c}D_{+}^{q}V(t,x ;t_0,x_0)\leq g(t,V(t,x))
$$
for $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$;

\item[(ii)] for any points $t_k$, $k=1,2,\dots$ and  $ x\in S(\lambda)$
we have
$$
V(t_k+0,\Phi_k(x))\leq \Psi_k(V(t_k,x));
$$
\item[(iii)]   $b(\|x\|)\leq V(t,x)\leq a(\|x\|)$ for $ t\in\mathbb{R}_+$, $x\in S(\lambda),$
  where  $a,b \in \mathcal{K}$.
\end{itemize}

   \item[(3)] The  zero solution of the scalar IFrDE \eqref{2}
is  uniformly stable.
\end{itemize}
     Then  the  zero solution of the system of IFrDE \eqref{1}  is  uniformly stable.
\end {theorem}

\begin{proof}
Let $\epsilon\in (0,\lambda]$  and $t_0\in \mathbb{R}_+$  be given. From condition
(3)  of Theorem \ref{thm3} there exists   $\delta_1 =\delta_1 (\epsilon)>0$
such that  for any $\tau_0\geq 0$  the inequality
  $|u_0|<\delta_1$ implies
\begin{equation} \label{2000}
|u(t;\tau_0,u_0)|<b(\epsilon),\quad t\geq \tau_0,
\end{equation}
where
$u(t;\tau_0,u_0)$ is a solution of  \eqref{2}.

Let $\delta_1<\min\{ \epsilon, b(\epsilon)\}$.
From $a \in \mathcal{K}$ there
exists  $\delta_2 =\delta_2 (\epsilon)>0 $ so if $s<\delta_2$ then
$a(s)<\delta_1$.  Let $\delta=\min (\epsilon, \delta_2)$. Choose the
initial value  $x_0\in \mathbb{R}^n$ such  that $\|x_0\|<\delta$.
Therefore $x_0\in S(\lambda)$.  Also, let $u_0^*=V(t_0,x_0)$.
From the choice of the point $u_0^*$ and condition (3)(iii) we obtain $u_0^* \leq
a(\|x_0\|)<a(\delta_2)< \delta_1$.   Let
$x^*(t)=x(t;t_0,x_0), \ t\geq t_0$ be a solution of the IVP for IFrDE \eqref{1}
and $u^*(t;t_0,u^*_0)$ be the maximal solution of the IVP for scalar IFrDE ({2}).
Note $u^*(t;t_0,u^*_0)$ satisfies \eqref{2000}. We now prove that
\begin{equation} \label{9011}
\|x^*(t)\|<\epsilon,\quad t\geq t_0.
\end{equation}
Assume inequality \eqref{9011} is not true.
 Denote  $t^*=\inf \{t> t_0:\ \|x^*(t)\geq \varepsilon\}$. Then
\begin{equation} \label{887}
\|x^*(t)\|<\varepsilon\quad\text{for }t\in[t_0,t^*)\quad\text{and}\quad
\|x^*(t^*)\|=\varepsilon.
\end{equation}
If $t^*\neq t_k$, $k\in\mathbb{Z}_+$  or $t^*=t_p$ for some natural number $p$
and $\|x^*(t_p-0)\|=\varepsilon$ then \eqref{9011} is true. If for a natural number
$p$ we have $t^*=t_p$ and $\|x^*(t_p-0)\|<\varepsilon$, then according to Lemma \ref{lem3}
for $T=t^*$ and $\Delta=S(\lambda)$ we get $V(t,x^*(t))\leq u^*(t;t_0,u^*_0)$
on $[t_0,t^*]$. Then applying condition (3)(iii) and inequality \eqref{2000}
we obtain  $b(\varepsilon)=b(\|x^*(t^*)\|)\leq V(t^*,x^*(t^*))\leq u^*(t^*;t_0,u^*_0)$.
Thus  $\|x^*(t^*)| |\leq b^{-1}(u^*(t^*)) <\varepsilon$  and this
contradicts the choice of $t^*$. Therefore,
 \eqref{9011} holds and then the zero
solution of IFrDE \eqref{1} is uniformly stable.
\end{proof}


\begin{corollary} \label{coro5} Suppose
\begin{itemize}
\item[(1)] Conditions {\rm (H1)--(H2)} are satisfied.
\item[(2)] Condition (2) of Theorem \ref{thm1} is satisfied.
\item[(3)] Condition (3) of Theorem \ref{thm3} is satisfied with
$g(t,x)= Au$ and $\Psi_k(u)= a_ku$ for $k=1,2,\dots$ where
$A\leq 0$ and $a_k\in(0,1)$.
\end{itemize}
     Then  the zero solution of the IFrDE \eqref{1}  is uniformly  stable.
\end{corollary}


The above corollary follows from Example \ref{examp2} (if $A<0$) and
 Example \ref{examp3} (if $A=0$)
and Theorem \ref{thm3}.
If we consider the scalar FrDE \eqref{222} as a comparison equation
then the following result for uniform stability is true:

\begin{theorem} \label{thm4}
Let the following conditions be satisfied:
\begin{itemize}
\item[(1)] Conditions (1) and (3) of Theorem \ref{thm2} are fulfilled.
 \item[(2)]  Condition (2) of Theorem \ref{thm1} is fulfilled.
\item[(3)] There exists a function $V\in \Lambda (\mathbb{R}_+,S(\lambda)) $
 satisfying condition (2)(i) and 2(iii) of Theorem \ref{thm3} and
    \begin{itemize}
\item[(ii)] for any points $t_k$, $k=1,2,\dots$ and  $ x\in S(\lambda)$  we have
$$
V(t_k+0,\Phi_k(x))\leq V(t_k,x);
$$
\end{itemize}
   \item[(4)] The  zero solution of the scalar FrDE \eqref{222} is
uniformly stable.
\end{itemize}
     Then  the  zero solution of the system of IFrDE \eqref{1}  is  uniformly stable.
\end {theorem}

Now we present some sufficient conditions for  uniform asymptotic
stability of the zero solution of a system of nonlinear  IFrDE.

\begin{theorem} \label{thm5}
Let the following conditions be satisfied:
\begin{itemize}
\item[(1)] Conditions {\rm (H1)} and {\rm (H2)} are fulfilled.
\item[(2)] Condition (2)  of Theorem \ref{thm1} is fulfilled.
\item[(3)] There exists a function $V\in \Lambda (\mathbb{R}_+,\mathbb{R}^n) $  
such that
    \begin{itemize}
\item[(i)] for any points $t_0\in \mathbb{R}_+$,  and
$x, x_0\in \mathbb{R}^n$  we have
$$
{}_{\eqref{1}}^{c}D_{+}^{q}V(t,x ;t_0,x_0)\leq -c(\|x\|)
$$
for $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$,
where $c\in \mathcal{K}$;


\item[(ii)] for any points $t_k$, $k=1,2,\dots$ and  $ x\in \mathbb{R}^n$ we have
$$
V(t_k+0,\Phi_k(x))\leq V(t_k,x);
$$

\item[(iii)]   $b(\|x\|)\leq V(t,x)\leq a(\|x\|)$ for $ t\in\mathbb{R}_+$, $x\in \mathbb{R}^n$,
  where  $a,b \in \mathcal{K}$.
\end{itemize}
\end{itemize}
Then  the zero solution  of the system of IFrDE \eqref{1} is uniformly
asymptotically stable.
\end{theorem}

\begin{proof}
 From condition (3)(i) we have ${}_{\eqref{1}}^{c}D_{+}^{q}V(t,x ;t_0,x_0)\leq 0$.
Applying Theorem \ref{thm4} with $g(t,u_0)\equiv 0$  we see that the zero solution of
the system of  IFrDE \eqref{1} is uniformly stable. Therefore, for
the number $\lambda$  there exists $\alpha=\alpha(\lambda)\in (0,\lambda)$ such that
for any $\tilde{t}_0\in\mathbb{R}_+$ and $\tilde{x}_0  \in\mathbb{R}^n$   the
inequality $\|\tilde{x}_0\|<\alpha$ implies
\begin{equation} \label{789}
\|x(t;\tilde{t}_0,\tilde{x}_0)\|<\lambda\quad \text{for }
t\geq \tilde{t}_0
\end{equation}
where $x(t;\tilde{t}_0,\tilde{x}_0)$ is any
solution of IFrDE \eqref{1} (with initial data
$(\tilde{t}_0,\tilde{x}_0)$).


Now we prove the zero solution of IFrDE \eqref{1} is uniformly
attractive. Consider the constant $\beta\in(0,\alpha]$  such that
$b^{-1}(a(\beta))< \alpha$. Let  $\epsilon \in(0,\lambda]$ be an arbitrary
number and  $x^*(t)=x(t;t_0,x_0)$ be any solution of \eqref{1} such
that $\|x_0\|<\beta$, $t_0\in\mathbb{R}_+$. Then
$b(\|x_0\|)\leq a(\|x_0\|)<a(\beta)$, i.e.
$\|x_0\|\leq b^{-1}(a(\beta))<\alpha$ and therefore the
inequality
\begin{equation} \label{88}
\|x^*(t)\|<\lambda\quad \text{for } t\geq t_0
\end{equation}
holds. Choose a constant $\gamma=\gamma(\epsilon)\in (0,\epsilon]$ such
that $a(\gamma)<b(\epsilon)$. Let
$T>\big(\frac{q\Gamma(q)a(\alpha)}{c(\gamma)}\big)^{1/q}$,
$T=T(\epsilon)>0$ and $m\in \{1,2,\dots\}$ with $t_m<t_0+T<t_{m+1}$.
We now prove that
 \begin{equation} \label{4450}
\|x^*(t)\|< \epsilon\quad \text{for } t\geq t_0+T.
\end{equation}
Assume
\begin{equation} \label{4490}
\|x^*(t)\|\geq \gamma\quad \text{for every } t\in [t_0,t_0+T].
\end{equation}
Then from Lemma \ref{lem4} (applied to  the interval $[t_m,t_0+T]$ and $\Delta=\mathbb{R}^n$)
and the inequality
$a^q+b^q\geq (a+b)^q$ for $a,b >0$  we obtain
\begin{align*} %\label{4419}
& V(t_0+T,x^*(t_0+T))\\
& \leq V(t_0,x_0)-\sum_{i=0}^{m-1}\frac{1}{\Gamma(q)}\int_{t_i}^{t_{i+1}}
  (t_{i+1}-s)^{q-1}c(\|x^*(s)\|)ds\\
&\quad -\frac{1}{\Gamma(q)}\int_{t_m}^{t_0+T} (t_0+T-s)^{q-1}c(\|x^*(s)\|)ds\\
& \leq a(\|x_0\|)-\sum_{i=0}^{m-1}\frac{c(\gamma)}{\Gamma(q)}
 \int_{t_i}^{t_{i+1}} (t_{i+1}-s)^{q-1}ds
 -\frac{c(\gamma)}{\Gamma(q)}\int_{t_m}^{t_0+T} (t_0+T-s)^{q-1}ds\\
&<a(\alpha)-\frac{c(\gamma)}{q\Gamma(q)}\Big(\sum_{i=0}^{m-1}(t_{i+1}-t_i)^{q}
 +(T+t_0-t_m)^{q} \Big) \\
&\leq a(\alpha)-\frac{c(\gamma)}{q\Gamma(q)}
\Big(\sum_{i=0}^{m-1}(t_{i+1}-t_i) +(T+t_0-t_m) \Big)^q\\
&= a(\alpha)-\frac{c(\gamma)}{q\Gamma(q)}T^q<0.\
\end{align*}
This contradiction proves the existence of $t^*\in [t_0,t_0+T]$ such
that $\|x^*(t^*)\|<\gamma$.
Now there are two cases to be considered, 
namely $t^* \neq t_k$ for
$k=1,2,\dots $ or $t^*=t_n$ for some $n\in \{1,2,\dots\}$.
\smallskip

\noindent\textbf{Case 1.}
 Let $t^* \neq t_k$ for $k=1,2,\dots $. Without loss of
generality assume there exists $j\in \{1,2,\dots\} $ with $t_
j<t^*<t_{j+1}$. From Corollary \ref{coro3} for any $t\geq t^*$ and
$\Delta=\mathbb{R}^n$ we have
\begin{align*}
V(t,x^*(t))
&\leq V(t^*,x^*(t^*))-\frac{1}{\Gamma(q)}\int_{t^*}^{t}
(t-s)^{q-1}c(\|x^*(s)\|)ds \\
&\leq V(t^*,x^*(t^*))\quad \text{for } t\in [t^*,t_{j+1}]
\end{align*}
and
\begin{align*}
V(t,x^*(t))
& \leq V(t^*,x^*(t^*))-\frac{1}{\Gamma(q)}\Big(\int_{t^*}^{t_{j+1}}
(t_{j+1}-s)^{q-1}c(\|x^*(s)\|)ds  \\
&\quad +\sum_{i=j+1}^{l-1}\int_{t_i}^{t_{i+1}}
(t_{i+1}-s)^{q-1}c(\|x^*(s)\|)ds\\
&\quad +\int_{t_l}^t (t-s)^{q-1}c(\|x^*(s)\|)ds \Big)  \\
&\leq V(t^*,x^*(t^*))\quad \text{for } t\in (t_l,t_{l+1}],\;
l=j+1,j+2,\dots\,.
\end{align*}
Then for any $t\geq t^*$ we obtain
\[ % \label{4427}
b(\|x^*(t)\|)\leq V(t, x^*(t))\leq V(t^*, x^*(t^*))
\leq a(\|x^*(t^*)\|)\leq a(\gamma).
\]
Then  $\|x^*(t)\|\leq b^{-1}(a(\gamma))<\varepsilon$  for any $t\geq t^*$.
\smallskip

\noindent\textbf{Case 2.}
Let $t^*=t_n$ for some $n\in \{1,2,\dots\}$. Applying
Corollary \ref{coro3} for any $t> t^*=t_n$, $t\in (t_l,t_{l+1}]$,
$l=n,n+1,\dots$,  and $\Delta=\mathbb{R}^n$ and obtain
\begin{align*}
V(t,x^*(t))
&\leq V(t_n+0,x^*(t_n+0)) 
 -\frac{1}{\Gamma(q)}\Big(\sum_{i=n}^{l-1}\int_{t_i}^{t_{i+1}}
(t_{i+1}-s)^{q-1}c(\|x^*(s)\|)ds\\
&\quad +\int_{t_l}^t (t-s)^{q-1}c(\|x^*(s)\|)ds \Big)  \\
&\leq V(t_n+0,x^*(t_n+0)).
\end{align*}
Then for any $t> t^*=t_n$ from conditions (2)(ii) and (2)(iii)
we get
\begin{align*}  %\label{44271}
b(\|x^*(t)\|)
&\leq V(t, x^*(t)) \leq V(t_n, x^*(t_n+0))\\
&= V(t_n,\Phi_n(x^*(t_n-0)))\leq V(t_n, x^*(t_n-0))\\
& \leq a(\|x^*(t_n-0)\|)\leq a(\gamma).
\end{align*}
Then  $\|x^*(t)\|\leq b^{-1}(a(\gamma))<\varepsilon$ and therefore \eqref{4450}
holds for all $t> t^*$ (hence for $t\geq t_0+T$).
\end{proof}


\begin{remark} \label{rmk10} \rm
The study of stability of a nonzero solution $x^*(t)$ of the IVP for
IFrDE \eqref{1} could be
easily reduced to studing stability of the zero solution of an
appropriately chosen system of IFrDE.
\end{remark}

\section{Applications}


Consider the generalized Caputo population  model.


\begin{example} \label{examp10} \rm
  Let the points $t_k$,  $t_k<t_{k+1}$, $lim_{k\to\infty}t_k=\infty$ be fixed.
Consider the scalar impulsive Caputo
fractional differential equation
\begin{equation} \label{347}
\begin{gathered}
{}_{0}^{c}D^{q}x=-g(t)x(1+x^2)\quad \text{for }  t\geq t_0,\; t\neq t_k, \;
  k=1,2,\dots,\\
x(t_k+0)=\Phi_k(x(t_k-0)),\quad  k=1,2,3,\dots,
\end{gathered}
\end{equation}
 where $x \in \mathbb{R}$, the functions
$g\in C(\mathbb{R}_+,\mathbb{R}_+): g(t)\geq \frac{1}{2t^q\Gamma(1-q)}$,
$\Phi_k\in C(\mathbb{R},\mathbb{R}):  |\Phi_k(x)|\leq c_k |x|$,
$c_k\in (0,1)$, $k=1,2,\dots$, are given constants.

 Consider the function $V(t,x)=x^2$. Then the inequality
$(\Phi_k(x))^2\leq \Psi_k(x^2)$, $k=1,2,\dots $ holds with
$\Psi_k(x)=c_k^2 x$. The Caputo fractional Dini derivative of the quadratic
function  for $t> 0$, $ t\neq t_k$  is
\begin{equation} \label{6655}
\begin{aligned}
{}_{\eqref{347}}^{c}D_{+}^{q}V(t,x;0,x_0)
&=2x \big(-g(t)x(1+x^2)\big)+(x^2-x_0^2)\frac{1}{t^q\Gamma(1-q)}\\
&\leq x^2\Big(-2g(t)(1+x^2)+ \frac{1}{t^q\Gamma(1-q)}\Big) \\
& \leq -2g(t)x^4\leq 0.
\end{aligned}
\end{equation}
Then by Theorem \ref{thm1}, the trivial solution of IFrDE \eqref{347} is   stable.
\end{example}

\subsection*{Acknowledgments}
This research was partially supported by  the Fund NPD, Plovdiv University,
 No. MU15-FMIIT-008.


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\end{document}