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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 229, 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}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/229\hfil Impulsive fractional differential equations]
{Integral boundary-value problem for impulsive fractional functional
differential equations with infinite delay}

\author[A. Chauhan, J. Dabas, M. Kumar \hfil EJDE-2012/229\hfilneg]
{Archana Chauhan, Jaydev Dabas, Mukesh Kumar}  % in alphabetical order

\address{Archana Chauhan \newline
Department of Mathematics, 
Motilal Nehru National Institute of Technology,
Allahabad - 211 004, India}
\email{archanasingh.chauhan@gmail.com}

\address{Jaydev Dabas \newline
Department of Applied Science and Engineering, 
IIT Roorkee,
Saharanpur Campus, Saharanpur-247001, India}
\email{jay.dabas@gmail.com}

\address{Mukesh Kumar \newline
Department of Mathematics, 
Motilal Nehru National Institute of technology,
Allahabad - 211 004, India}
\email{mukesh@mnnit.ac.in}

\thanks{Submitted May 29, 2012. Published December 18, 2012.}
\subjclass[2000]{26A33, 34K05, 34A12, 34A37}
\keywords{Fractional differential equation;
integral boundary condition; \hfill\break\indent
impulsive conditions; infinite delay}

\begin{abstract}
 In this article, we establish a general framework for finding
 solutions for impulsive fractional integral boundary-value problems.
 Then, we prove the existence and uniqueness of solutions by applying
 well known fixed point theorems. The obtained results are illustrated
 with an example for their feasibility.
\end{abstract}

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



\section{Introduction}\label{intro}
 The purpose of this article is to establish the
existence and uniqueness of solution to an integral boundary-value
problem for impulsive fractional functional integro-differential
equation with infinite delay of the form:
\begin{equation} \label{ME}
\begin{gathered}
^cD_t^{\alpha}x(t)=f(t,x_t,Bx(t)),\quad t\in J=[0,T],\; t\neq t_k,\\
\Delta x(t_k)=Q_k(x(t_k^{-})),\quad k=1,2,\dots,m,\\
\Delta x'(t_k)=I_k(x(t_k^{-})),\quad k=1,2,\dots,m,\\
x(t)=\phi(t),\quad t\in(-\infty,0], \\
ax'(0)+bx'(T)=\int_0^Tq(x(s))ds,
\end{gathered}
\end{equation}
where $T>0$, $\alpha\in(1,2)$, $a,b\in \mathbb{R}$ such that $a+b\neq
0$. $^cD_t^\alpha$ is the Caputo fractional derivative. The
functions $f:J\times \mathfrak{B}_h\times X\to X$ and
$q:X\to X$ are given functions that satisfy certain assumptions, where
$\mathfrak{B}_h$ is a phase space defined in details in Section $2$.
Here $0= t_0<t_1<\dots<t_m<t_{m+1}= T$,
$Q_k,I_k\in C(X,X)$,
$(k=1,2,\dots,m)$, are bounded functions,
$\Delta x(t_k)=x(t_k^{+})-x(t_k^{-})$ and
$\Delta x'(t_k)=x'(t_k^{+})-x'(t_k^{-})$. We assume that
$x_t:(-\infty,0]\to X$, $x_t(s)=x(t+s)$, $s\leq 0$, belong to
an abstract phase space $\mathfrak{B}_h$. The term $Bx(t)$ is given
by
\[
Bx(t)=\int_{0}^{t}K(t,s)x(s)ds,
\]
where $K\in C(D,\mathbb{R}^{+})$, the set of all positive functions which are
continuous on $D=\{(t,s)\in \mathbb{R}^2:0\leq s\leq t<T\}$ and
$B^{*}=\sup_{t\in[0,t]}\int_{0}^{t}K(t,s)ds<\infty$.

Fractional differential equations have attracted considerable
interest because of their ability to model complex phenomena. Due to
the extensive applications of fractional differential equations in
engineering and science, research in this area has grown
significantly all around the world. For more details about
fractional calculus and fractional differential equations we refer
the interested readers to the books by Podlubny \cite{ip},  Hilfer
\cite{HH} and the papers \cite{rpam,BA1,BA3,DL6,DL4,DL3,ZT,gw,xz}
and references there in.

The impulsive differential equations arising from the real world
problems to describe the dynamics of processes in which sudden,
discontinuous jumps occurs. Such processes are naturally seen in
biology, physics, engineering, etc. Due to their significance, many
authors have been established the solvability of impulsive
differential equations. For the general theory and applications of
such equations we refer the interested reader to see the papers
\cite{BA3,ZT,xz} and references therein.

Integral boundary conditions have various applications in applied
fields such as blood flow problems, chemical engineering,
thermoelasticity, underground water flow, population dynamics etc.
For a detailed description of the integral boundary conditions, we
refer the reader to some recent papers \cite{BA2,BA1,BA3,AB} and the
references therein. On the other hand, we know that delay arises
naturally in practical systems due to the transmission of signal or
the mechanical transmission. Moreover, the Cauchy problem for
various delay equations in Banach spaces has been receiving more and
more attention during the past decades, see
\cite{DL6,DL2,DL4,DL3,DL1} and for boundary value problem with
infinite delay one can see these papers\cite{BD1,BD2} and references
therein.

Recently, Michal Fec˘kan et al  \cite{MYJ} gave a counter example
to show that the formula of solutions for impulsive fractional
differential equations used in previous papers are incorrect. Since
the authors used that the Caputo derivative ${}^cD_t^\alpha$
restricted on $(a,b]$, $0<a<b$, is ${}^cD_{a,t}^\alpha$, but
unfortunately it does not hold. In \cite{MYJ}, the authors
introduced a correct formula of solutions for a impulsive Cauchy
problem with Caputo fractional derivative. In \cite{JYM}, the author
discussed some existence results for boundary value problems for
impulsive fractional differential equations. However, the theory of
boundary value problem for impulsive fractional differential
equations is still in the initial stages. Our work is motivated by
these papers \cite{MYJ,TW,JYM}.


To the best of our knowledge, this is the first paper dealing with
integral boundary value problem involving impulsive nonlinear
integro-differential equations of fractional order $\alpha\in(1,2)$
with infinite delay. We organize the rest of this paper as follows:
in Section \ref{sec:2}, we present some necessary definitions and preliminary
results that will be used to prove our main results. The proofs of
our main results are given in Section \ref{sec:3}.
Section \ref{sec:4} contains an
illustrative example.


\section{Preliminaries and assumptions} \label{sec:2}

In this section, we shall introduce some basic
definitions, properties and lemmas which are required for
establishing our results. Let $(X,\|\cdot\|_X)$ be a real Banach space.

To describe fractional order functional differential equations with
infinite delay, we need to discuss the abstract phase space
$\mathfrak{B}_h$ in a convenient way (see for instance in
\cite{xz}). Assume that $h:(-\infty,0]\to(0,\infty)$ is a
continuous functions with $l=\int_{-\infty}^0 h(t)dt<\infty$. For
any $a>0$, we define $\mathfrak{B}=\{\psi:[-a,0]\to X$ such
that $\psi(t)$ is bounded and measurable$\}$ and equip the space
$\mathfrak{B}$ with the norm
$$
\|\psi\|_{[-a,0]}=\sup_{s\in[-a,0]}\|\psi(s)\|_X,\quad \forall\; \psi \in
\mathfrak{B}.
$$

Let us define $\mathfrak{B}_h=\{\psi:(-\infty,0]\to X$,
 such that for any
$c>0$, $\psi |_{[-c,0]}\in \mathfrak{B}$ and
$\int_{-\infty}^0 h(s)\|\psi\|_{[s,0]}ds<\infty\}$.
If $\mathfrak{B}_h$ is endowed
with the norm
$$
\|\psi\|_{\mathfrak{B}_h}=\int_{-\infty}^0 h(s)\|\psi\|_{[s,0]}ds,\quad
 \forall\;\psi \in \mathfrak{B}_h,
$$
then it is clear that $(\mathfrak{B}_h,\|\cdot\|_{\mathfrak{B}_h})$ is a
Banach space. Now we consider the space
\begin{align*}
\mathfrak{B}_h'
&= \{x:(-\infty,T]\to X\text{ such that } x|_{J_k}\in C(J_k,X)
\text{  and there exist}\\
&x(t_k^{+})\ \mbox{ and } x(t_k^{-})\text{ with }
x(t_k)=x(t_k^{-}),\; x_0=\phi \in \mathfrak{B}_h,k=1,\dots,m\},
\end{align*}
where $x|_{J_k}$ is the restriction of $x$ to
$J_k=(t_k,t_{k+1}],k=0,1,2,\dots,m$. Set $\|\cdot\|_{B_h'}$ to be a
seminorm in $\mathfrak{B}_h'$ defined by
$$
\|x\|_{\mathfrak{B}_h'}=\sup\{\|x(s)\|_X:s \in[0,T]\}
+\|\phi\|_{\mathfrak{B}_h}, x\in \mathfrak{B}_h'.
$$
Assume that $x\in \mathfrak{B}_h'$, then for
 $t\in J,\;x_t\in \mathfrak{B}_h$. Moreover,
$$
l\|x(t)\|_X\leq\|x_t\|_{\mathfrak{B}_h}\leq
l\sup_{0<s<t}\|x(s)\|_X+\|x_0\|_{\mathfrak{B}_h},
$$
where $l=\int_{-\infty}^0h(t)dt$.

\begin{definition} \label{def2.1} \rm
The Riemann-Liouville fractional integral operator for
order $\alpha > 0$, of a function $f:\mathbb{R}_+\to \mathbb{R}$
and $f\in L^1(\mathbb{R}_+,X)$
is defined by
\begin{equation} \label{e2.1}
J_t^0f(t)=f(t),\;J_t^{\alpha}f(t)={1\over
\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}f(s)ds,\quad \alpha>0,\;t>0,
\end{equation}
where $\Gamma(\cdot)$ is the Euler gamma function.
\end{definition}

\begin{definition} \label{def2.2} \rm
 Caputo's derivative of order $\alpha$ for a function $f:
[0,\infty)\to \mathbb{R}$ is defined as
\begin{equation}
D^\alpha_tf(t)={1\over \Gamma(n-\alpha)}\int_0^t(t-s)^{n-\alpha-1}f^{(n)}(s)ds
=J^{n-\alpha}f^{(n)}(t),
\end{equation}
for $n-1\leq\alpha<n,\;n\in N$. If $0<\alpha\le 1$, then
\begin{equation}
D^{\alpha}_tf(t)={1\over
\Gamma(1-\alpha)}\int_0^t(t-s)^{-\alpha}f^{(1)}(s)ds.
\end{equation}
Obviously, Caputo's derivative of a constant is equal to zero.
\end{definition}

\begin{definition} \label{def2.3} \rm
A function $x\in \mathfrak{B}_h'$ is said to be a solution of the
problem-\eqref{ME} if $x$ satisfies the differential equation
${}^cD_t^\alpha x(t)=f(t,x_t,Bx(t))$ a.e. on
$J\setminus \{t_1,\dots,t_m\}$ and the
following conditions:
\begin{equation}
\begin{gathered}
\Delta x(t_k)=Q_k(x(t_k^{-})),\quad k=1,2,\dots,m,\\
\Delta x'(t_k)=I_k(x(t_k^{-})),\quad  k=1,2,\dots,m,\\
x(t)=\phi(t),\quad t\in(-\infty,0], \\
ax'(0)+bx'(T)=\int_0^Tq(x(s))ds.
\end{gathered}
\end{equation}
\end{definition}

\begin{lemma}[{\cite[lemma 2.5]{JYM}}] \label{lem2.4}
For $\alpha>0$, the general solution of fractional differential
equation ${}^cD_t^\alpha x(t)=0$ is given by
$x(t)=c_0+c_1t+c_2t^2+\dots+c_{n-1}t^{n-1}$, where $c_i\in
\mathbb{R}$, $ i=0,1,2,\dots,n-1$ $(n=[\alpha]+1)$ and $[\alpha]$
denotes the integer part of the real number $\alpha$.
\end{lemma}
Note that
\begin{equation}
x(t)=x_0-ct-\int_0^\omega\frac{(\omega-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)ds
+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)ds
\end{equation}
is the solution of the Cauchy problem
\begin{equation}
\begin{gathered}
{}^cD_t^\alpha x(t)= h(t),\quad t\in J,\; \alpha\in(1,2),\\
x(0)= x_0-\int_0^\omega\frac{(\omega-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)ds.
\end{gathered}
\end{equation}
Now, we can obtain the following result.

\begin{lemma}[{\cite[lemma 2.6]{MYJ}}] \label{lem2.5}
Let $\alpha\in(1,2)$, $c\in \mathbb{R}$ and $h:J\to \mathbb{R}$ be
continuous function. A function $x\in C(J,\mathbb{R})$ is a solution
of the  fractional integral equation \eqref{IE}
\begin{equation}\label{IE}
x(t)=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)ds
-\int_0^\omega\frac{(\omega-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)ds
+x_0-c(t-\omega)
\end{equation}
 if and only if $x$ is a solution of the  fractional Cauchy problem
 \begin{equation}
\begin{gathered}
{}^cD_t^\alpha x(t)= h(t),\quad t\in J,\\
x(\omega)= x_0,\quad \omega\geq 0.
\end{gathered}
\end{equation}
\end{lemma}

\begin{lemma}  \label{lem2.6}
Let $\alpha\in(1,2)$ and $f:J\times \mathfrak{B}_h\times X\to \mathbb{R}$
 be continuously differentiable function.
A piecewise continuously differentiable function $x\in \mathfrak{B}_h'$
is a solution of system \eqref{ME} if and only
if $x\in \mathfrak{B}_h'$ is a solution of the fractional integral equation
\begin{equation}\label{int-sol}
x(t)\\
=\begin{cases}
\phi(t),
\quad\text{if } t\in(-\infty,0],
\\[3pt]
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)
-\frac{bt}{a+b}\sum_{i=1}^mI_i(x(t_i^{-}))\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_s,Bx(s))ds
+\frac{t}{a+b}\int_0^Tq(x(s))ds, \\
\quad \text{if }t\in[0,t_1],\\
\dots, \\
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)
+\sum_{i=1}^k(t-t_i)I_i(x(t_i^{-}))\\
+\sum_{i=1}^kQ_i(x(t_i^{-}))-\frac{bt}{a+b}\sum_{i=1}^mI_i(x(t_i^{-}))\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_s,Bx(s))ds
+\frac{t}{a+b}\int_0^Tq(x(s))ds, \\
\quad \text{if }t\in (t_k,t_{k+1}],
\end{cases}
\end{equation}
where $k=1,\dots,m$.
\end{lemma}

\begin{proof}
 Assume $x$ satisfies  \eqref{ME}. If $t\in[0,t_1]$, then
\begin{equation}\label{L1}
{}^cD_t^\alpha x(t)=f(t,x_t,Bx(t)),\;t\in(0,t_1],\; x(0)=\phi(0).
\end{equation}
By using Lemma \ref{lem2.5}, we can write the solution of \eqref{L1} as
\begin{equation}
x(t)=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)-ct.
\end{equation}
If $t\in(t_1,t_2]$, then
\begin{equation}
\begin{gathered}
{}^cD_t^\alpha x(t)=f(t,x_t,Bx(t)),\;t\in(t_1,t_2],\\
x(t_1^+)=x(t_1^-)+Q_1(x(t_1^-)),\;x'(t_1^+)=x'(t_1^-)+I_1(x(t_1^-)).
\end{gathered}
\end{equation}
Again by lemma \ref{lem2.5}, we have the following form of the solution
\begin{align*}
x(t)
&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds
-\int_0^{t_1}\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds\\
&\quad +x(t_1^-)+Q_1(x(t_1^-))-d(t-t_1)\\
&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds
+\phi(0)-ct_1+Q_1(x(t_1^-))-d(t-t_1).
\end{align*}
Since $x'(t_1^+)=x'(t_1^-)+I_1(x(t_1^-))$, we obtain
$d=c-I_1(x(t_1^-))$. Thus
\begin{align*}
x(t)=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds
+\phi(0)+(t-t_1)I_1(x(t_1^-))+Q_1(x(t_1^-))-ct.
\end{align*}
If $t\in(t_2,t_3]$, then by similar way using the lemma \ref{lem2.5}, we have
\begin{align*}
x(t)
&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds-\int_0^{t_2}\frac{(t_2-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds\\
&\quad +x(t_2^-)+Q_2(x(t_2^-))-e(t-t_2).\\
&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)-ct_2+(t_2-t_1)I_1(x(t_1^-))\\
&\quad +Q_1(x(t_1^-))+Q_2(x(t_2^-))-e(t-t_2).
\end{align*}
Since $x'(t_2^+)=x'(t_2^-)+I_2(x(t_2^-))$, we obtain
$e=c-I_1(x(t_1^-))-I_2(x(t_2^-))$. Thus
\begin{align*}
x(t)&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds
 +\phi(0)+(t-t_1)I_1(x(t_1^-))\\
&\quad +(t-t_2)I_2(x(t_2^-))+Q_1(x(t_1^-))+Q_2(x(t_2^-))-ct.
\end{align*}
Similarly, if $t\in(t_k,t_{k+1}]$, then again from lemma \ref{lem2.5}, we
have
\begin{align*}
x(t)&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)+\sum_{i=1}^k(t-t_i)I_i(x(t_i^-))\\
&\quad+\sum_{i=1}^kQ_i(x(t_i^-))-ct.
\end{align*}
By using the integral boundary condition
$ax'(0)+bx'(T)=\int_0^Tq(x(s))ds$, we obtain
\begin{align*}
c&= {b\over a+b}\int_0^T{(T-s)^{\alpha-1}\over\Gamma(\alpha-1)}f(s,x_s,Bx(s))ds\\
&\quad +{b\over a+b}\sum_{i=1}^mI_i(x(t_i^-))-{1\over a+b}\int_0^Tq(x(s))ds.
\end{align*}
Thus for $t\in[0,t_1]$,
\begin{align*}
x(t)&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds
 +\phi(0)-\frac{bt}{a+b}\sum_{i=1}^mI_i(x(t_i^{-}))\\
&\quad-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
f(s,x_s,Bx(s))ds+\frac{t}{a+b}\int_0^Tq(x(s))ds,
\end{align*}
and for $t\in(t_k,t_{k+1}]$, $k=1,2,\dots,m$, we have
\begin{align*}
x(t)
&=\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)
+\sum_{i=1}^k(t-t_i)I_i(x(t_i^{-}))\\
&\quad+\sum_{i=1}^kQ_i(x(t_i^{-}))-\frac{bt}{a+b}\sum_{i=1}^mI_i(x(t_i^{-}))\\
&\quad -\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 f(s,x_s,Bx(s))ds
+\frac{t}{a+b}\int_0^Tq(x(s))ds.
\end{align*}
Conversely, assume that $x$ satisfies \eqref{int-sol}. By a direct
computation, it follows that the solution given in \eqref{int-sol}
satisfies the system \eqref{ME}. This completes the proof of the
lemma.
\end{proof}

Further we introduce the following assumptions to establish our
results.
\begin{itemize}
\item[(H1)] There exists constants $\mu_1,\mu_2>0$, such that
\[
\|f(t,\varphi,x)-f(t,\psi,y)\|_X
\leq\mu_1\|\varphi-\psi\|_{\mathfrak{B}_h}
+\mu_2\|x-y\|_X, \]
$t\in J$, $\varphi,\psi\in \mathfrak{B}_h$, $x,y\in X$.

\item[(H2)] The function $q:X\to X$ is
continuous and there exists constant $L_q>0$, such that
\[
\|q(x(s))-q(y(s))\|_X\leq L_q\|x-y\|_X.
\]

\item [(H3)] For each $k=1,\dots,m$, there exists $\overline{L},L>0$, such
that
\begin{gather*}
\|Q_k(x)-Q_k(y)\|_X\leq\overline{L}\|x-y\|_X,\quad \forall x,y\in X.\\
\|I_k(x)-I_k(y)\|_X\leq L\|x-y\|_X,\quad \forall x,y\in X.
\end{gather*}

\item[(H4)] The function $f:J\times \mathfrak{B}_h\times X\to X$ is
continuous and there exist two continuous functions
$\mu_1,\mu_2:J\to(0,\infty)$ such that
$\|f(t,\psi,x)\|_X\leq\mu_1(t)\|\psi\|_{\mathfrak{B}_h}+\mu_2(t)\|x\|_X$
and
$\mu_1^*=\sup_{t\in[0,T]}\mu_1(t),\mu_2^*=\sup_{t\in[0,T]}\mu_2(t)$.

\item[(H5)] The functions $q:X\to X$, $I_k:X\to X$ and
$Q_k:X\to X,\;k=1,\dots,m$
are continuous and  there exist constants $C,\rho,\Omega$ such that
$\|q(x)\|_X\leq C,\;x\in X$,
$\rho=\max_{1\leq k\leq m,x\in B_r}\{\|I_k(x)\|_X\}$ and
$\Omega=\max_{1\leq k\leq m,x\in B_r}\{\|Q_k(x)\|_X\}$.
\end{itemize}


\section{Existence and uniqueness results}\label{sec:3}

\begin{theorem}\label{thm3.1}
Suppose that the assumptions {\rm (H1)--(H3)} hold and
\[
\Lambda=\Big[\frac{(a+(1+\alpha)b)(\mu_1l+\mu_2B^*)T^\alpha}{(a+b)
\Gamma(\alpha+1)}+\frac{(a+2b)LTm+L_qT^2}{a+b}
+\overline{L}m\Big]<1.
\]
Then  \eqref{ME} has an unique solution.
\end{theorem}

\begin{proof} Consider the operator $N: B_h'\to B_h'$
defined by
\[
Nx(t)
=\begin{cases}
\phi(t), \quad \text{if } t\in(-\infty,0],
\\
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)
 -\frac{bt}{a+b}\sum_{i=1}^mI_i(x(t_i^{-}))
\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_s,Bx(s))ds
 +\frac{t}{a+b}\int_0^Tq(x(s))ds,\\
\quad \text{if } t\in[0,t_1],\\
\dots, \\
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_s,Bx(s))ds+\phi(0)
 +\sum_{i=1}^k(t-t_i)I_i(x(t_i^{-}))\\
 +\sum_{i=1}^kQ_i(x(t_i^{-}))-\frac{bt}{a+b}\sum_{i=1}^mI_i(x(t_i^{-}))
\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_s,Bx(s))ds
+\frac{t}{a+b}\int_0^Tq(x(s))ds,\\
\quad \text{if } t\in (t_k,t_{k+1}],
\end{cases}
\]
where $k=1,2,\dots,m$. Let $y(.):(-\infty,T]\to X$ be the
function defined by
\[
y(t)=\begin{cases}\phi(t),&  t\in (-\infty,0];\\
0,&  t\in J,\end{cases}
\]
then $y_0=\phi$.
For each $z\in C([0,T],\mathbb{R})$ with $z(0)=0$, we denote
\[
\overline{z}(t)=\begin{cases}
0,&  t\in (-\infty,0];\\
z(t),&  t\in J.
\end{cases}
\]
If $x(.)$ satisfies  \eqref{int-sol} then we can
decompose $x(\cdot)$ as $x(t)=y(t)+\overline{z}(t)$, which implies
$x_t=y_t+\overline{z}_t$ for $t\in J$ and the function $z(\cdot)$
satisfies
\[
z(t)=\begin{cases}
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,y_s+\overline{z}_s,
 B(y(s)+\overline{z}(s)))ds+\phi(0)\\
-\frac{bt}{a+b}\sum_{i=1}^mI_i(z(t_i^{-}))+\frac{t}{a+b}\int_0^T
 q(y(s)+\overline{z}(s))ds\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds,\;
& t\in[0,t_1],
\\
\dots, \\
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,y_s+\overline{z}_s,
 B(y(s)+\overline{z}(s)))ds+\phi(0)\\
+\sum_{i=1}^k(t-t_i)I_i(z(t_i^{-}))+\sum_{i=1}^kQ_i(z(t_i^{-}))\\
-\frac{bt}{a+b}\sum_{i=1}^mI_i(z(t_i^{-}))
 +\frac{t}{a+b}\int_0^Tq(y(s)+\overline{z}(s))ds\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds,\;
& t\in (t_k,t_{k+1}],
\end{cases}
\]
where $k=1,2,\dots,m$. Set $\mathfrak{B}_h''=\{z\in
\mathfrak{B}_h'$ such that $z_0=0\}$ and let
$\|\cdot\|_{\mathfrak{B}_h''}$ be the seminorm in
$\mathfrak{B}_h''$ defined by
\[
\|z\|_{\mathfrak{B}_h''}=\sup_{t\in
J}\|z(t)\|_X+\|z_0\|_{\mathfrak{B}_h}= \sup_{t\in J}\|z(t)\|_X, \;
z\in \mathfrak{B}_h''.
\]
Thus $({\mathfrak{B}_h''},\|\cdot\|_{{\mathfrak{B}_h''}})$ is a
Banach space. We define the operator
$P:{\mathfrak{B}_h''}\to {\mathfrak{B}_h''}$ by
\[
Pz(t)=\begin{cases}
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds+\phi(0)\\
-\frac{bt}{a+b}\sum_{i=1}^mI_i(z(t_i^{-}))+\frac{t}{a+b}
 \int_0^Tq(y(s)+\overline{z}(s))ds\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds,\;
& t\in[0,t_1],
\\
\dots, \\
\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds+\phi(0)\\
+\sum_{i=1}^k(t-t_i)I_i(z(t_i^{-}))+\sum_{i=1}^kQ_i(z(t_i^{-}))\\
-\frac{bt}{a+b}\sum_{i=1}^mI_i(z(t_i^{-}))+\frac{t}{a+b}
 \int_0^Tq(y(s)+\overline{z}(s))ds\\
-\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds,\;
& t\in (t_k,t_{k+1}],
\end{cases}
\]
where $k=1,2,\dots,m$. It is clear that the operator $N$ has a
unique fixed point if and only if $P$ has a unique fixed point. So
let us prove that $P$ has a unique fixed point.  Let $z,z^{*}\in
\mathfrak{B}_h''$ and $t\in [0,t_1]$ we have
\begin{align*}
&\|(Pz)(t)-(Pz^{*})(t)\|_X\\
&\leq\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 \|f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))
-f(s,y_s+\overline{z}_s^*,B(y(s)\\
&\quad +\overline{z}^*(s)))\|_X\,ds
 +\frac{bt}{a+b}\sum_{i=1}^m\|I_i(z(t_i^{-}))-I_i(z^*(t_i^{-}))\|_X\\
&\quad +\frac{t}{a+b}\int_0^T\|q(y(s)+\overline{z}(s))-q(y(s)+\overline{z}^*(s))\|_X\,ds\\
&\quad +\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))
 -f(s,y_s+\overline{z}_s^*,B(y(s)\\
&\quad +\overline{z}^*(s))\|_X\,ds\\
&\leq\Big[\frac{(\mu_1l+\mu_B^*)(a+(1+\alpha)b)T^\alpha}{(a+b)
 \Gamma(\alpha+1)}+\frac{T(bLm+L_qT)}{a+b}\Big]\|z-z^*\|_{\mathfrak{B}_h''}.
\end{align*}
If $t\in(t_k,t_{k+1}]$, $k=1,2,\dots,m$, then
\begin{align*}
&\|(Pz)(t)-(Pz^*)(t)\|_X\\
&\leq \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 \|f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))
 -f(s,y_s+\overline{z}_s^*,B(y(s)\\
&\quad +\overline{z}^*(s)))\|_X\,ds
 +\sum_{i=1}^k(t-t_i)\|I_i(z(t_i^{-}))-I_i(z^*(t_i^{-}))\|_X
 \\
&\quad +\sum_{i=1}^k\|Q_i(z(t_i^{-}))-Q_i(z^*(t_i^{-}))\|_X\\
&\quad +\frac{bt}{a+b}\sum_{i=1}^m\|I_i(z(t_i^{-}))-I_i(z^*(t_i^{-}))\|_X
\\
&\quad +\frac{t}{a+b}\int_0^T\|q(y(s)+\overline{z}(s))-q(y(s)
 +\overline{z}^*(s))\|_X\,ds\\
&\quad +\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))
-f(s,y_s+\overline{z}_s^*,B(y(s)\\
&\quad +\overline{z}^*(s)))\|_X\,ds\\
&\leq\Big[\frac{(a+(1+\alpha)b)(\mu_1l+\mu_2B^*)T^\alpha}{(a+b)
 \Gamma(\alpha+1)}+\frac{(a+2b)LTm+L_qT^2}{a+b}
+\overline{L}m\Big]\|z-z^*\|_{\mathfrak{B}_h''}.
\end{align*}
Thus for all $t\in[0,T]$, we have the estimate
\begin{align*}
&\|P(z)-P(z^*)\|_{B_h''}\\
&\leq \Big[\frac{(a+(1+\alpha)b)(\mu_1l+\mu_2B^*)T^\alpha}{(a+b)
 \Gamma(\alpha+1)}+\frac{(a+2b)LTm+L_qT^2}{a+b}
+\overline{L}m\Big]\|z-z^*\|_{B_h''}\\
&\leq\Lambda\|z-z^*\|_{\mathfrak{B}_h''}.
\end{align*}
Since $\Lambda<1$,  the map $P$ is a contraction map and
has a unique fixed point
 $z\in \mathfrak{B}_h''$, which is obviously a solution of the system
 \eqref{ME} on $(-\infty,T]$. This completes the proof of the theorem.
\end{proof}

Our second existence result is based on the following Krasnoselkii's
fixed point theorem.

\begin{theorem} \label{kraski}
Let $B$ be a closed convex and nonempty subset of a Banach space
$X$. Let $P$ and $Q$ be two operator such that
(i) $Px+Qy\in B$, whenever $x,y\in B$.
(ii) $P$ is compact and continuous. (iii) $Q$ is a contraction mapping.
Then there exists $z\in B$ such that $z=Pz+Qz$.
\end{theorem}

\begin{theorem}
Suppose that  assumptions {\rm  (H1), (H4), (H5)} are satisfied with
$$
\Delta=\frac{(\mu_1l+\mu_2B^*)(a+(1+\alpha)b)T^\alpha}{(a+b)
 \Gamma(\alpha+1)}<1.
$$
Then  \eqref{ME} has at least one solution on $(-\infty,T]$.
\end{theorem}

\begin{proof} Choose
\begin{align*}
r&\geq \Big[\|\phi(0)\|+(\rho T+\Omega)m+\frac{(b\rho m+TC)T}{a+b}\\
&\quad +\frac{(\mu_1^*(\|\phi\|+lr)+\mu_2^*B^*r)(a+(1+\alpha)b)T^\alpha}{(a+b)
 \Gamma(\alpha+1)}\Big].
\end{align*}
Define $B_r=\{z\in \mathfrak{B}_h'':\|z\|_{\mathfrak{B}_h''}\leq r\}$,
 then $B_r$ is a bounded, closed convex subset in $\mathfrak{B}_h''$. Let
$P_1:B_r\to B_r$ and $P_2:B_r\to B_r$ be defined as
\begin{align*}
(P_1z)(t)
&= \phi(0)+\frac{t}{a+b}\int_0^Tq(y(s)+\overline{z}(s))ds
 +\sum_{i=1}^kQ_i(z(t_i^-))\\
&\quad -\frac{bt}{a+b}\sum_{i=1}^mI_i(z(t_i^-))
 +\sum_{i=1}^k(t-t_i)I_i(z(t_i^-)),\quad t\in J_k.\\
(P_2z)(t)
 &= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,y_s+\overline{z}_s,B(y(s)
 +\overline{z}(s)))ds\\
&\quad -\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))ds,\quad t\in J_k,
\end{align*}
where $J_0=[0,t_1]$ and $J_k=(t_k,t_{k+1}]$, $k=1,\dots,m$.
Now, we proceed the proof in following steps:

\textbf{Step 1.} Let $z,z^*\in B_r$, then we show that $P_1z+P_2z^*\in
B_r$ for $t\in J_k,k=0,1,\dots,m$. We have
\begin{equation}\label{L2l}
\begin{aligned}
&\|(P_1z)(t)+(P_2z^*)(t)\|_X\\
&\leq \|\phi(0)\|_X+\frac{t}{a+b}\int_0^T\|q(y(s)+\overline{z}(s))\|_X\,ds\\
&\quad +\sum_{i=1}^k\|Q_i(z(t_i^-))\|_X
 +\frac{bt}{a+b}\sum_{i=1}^m\|I_i(z(t_i^-))\|_X+\sum_{i=1}^k(t-t_i)\|I_i(z(t_i^-))
 \|_X\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 \|f(s,y_s+\overline{z}^*_s,B(y(s)+\overline{z}^*(s)))\|_X\,ds\\
&\quad+\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|f(s,y_s+\overline{z}^*_s,B(y(s)+\overline{z}^*(s)))\|_X\,ds,
\end{aligned}
\end{equation}
we estimate the inequality \eqref{L2l}, by using (H4) and (H5), as
\begin{align*}
\|(P_1z)(t)+(P_2z^*)(t)\|_X
&\leq \Big[\|\phi(0)\|+(\rho T+\Omega)m+\frac{(b\rho m+TC)T}{a+b}\\
&\quad  +\frac{(\mu_1^*(\|\phi\|+lr)+\mu_2^*B^*r)(a+(1+\alpha)b)T^\alpha}
 {(a+b)\Gamma(\alpha+1)}\Big],
\end{align*}
which implies that $\|P_1z+P_2z^*\|_{B_h''}\leq r$.

\textbf{Step 2.} Now, we shall show that the mapping $(P_1z)(t)$ is
continuous on $B_r$. For this purpose, let $\{z^n\}_{n=1}^\infty$ be
a sequence in $B_r$ with $\lim z^n\to z$ in $B_r$. Then for
$t\in J_k,k=0,1,\dots,m$, we have
\begin{align*}
&\|(P_1z^n)(t)-(P_1z)(t)\|_X\\
&\leq\frac{t}{a+b}\int_0^T\|q(y(s)+\overline{z}^n(s))-q(y(s)
 +\overline{z}(s))\|_X\,ds\\
&+\sum_{i=1}^k\|Q_i(z^n(t_i^-))-Q_i(z(t_i^-))\|_X
 +\frac{bt}{a+b}\sum_{i=1}^m\|I_i(z^n(t_i^-))-I_i(z(t_i^-))\|_X\\
&+\sum_{i=1}^k(t-t_i)\|I_i(z^n(t_i^-))-I_i(z(t_i^-))\|_X.
\end{align*}
Since the functions $q,Q_k, I_k$, $k=0,1,\dots,m$, are
continuous, hence $\lim_{n\to\infty}P_1z^n=P_1z$ in $B_r$. Which
implies that the mapping $P_1$ is continuous on $B_r$.

\textbf{Step 3} $(P_1z)(t)$ is uniformly bounded follows by the
following inequality. For $t\in J_k,k=0,1,\dots,m$, we have
\begin{align*}
\|(P_1z)(t)\|_X
&\leq \|\phi(0)\|_X+\frac{t}{a+b}\int_0^T\|q(y(s)+\overline{z}(s))\|_X\,ds
+\sum_{i=1}^k\|Q_i(z(t_i^-))\|_X\\
&\quad +\frac{bt}{a+b}\sum_{i=1}^m\|I_i(z(t_i^-))\|_X
+\sum_{i=1}^k(t-t_i)\|I_i(z(t_i^-))\|_X\\
&\leq \|\phi(0)\|_X+\frac{T(TC+b\rho m)}{a+b}+\Omega m+\rho mT.
\end{align*}

\textbf{Step 4} To show that $P_1(B_r)$ is equicontinuous. Let
$\tau_1,\tau_2\in J_k$, $t_k\leq \tau_1<\tau_2\leq
t_{k+1}$, $k=0,1,\dots,m$, $z\in B_r$, we have
\begin{align*}
\|(P_1z)(\tau_2)-(P_2z)(\tau_1)\|_X
&\leq (\tau_2-\tau_1)[\frac{1}{a+b}\int_0^T\|q(y(s)+\overline{z}(s))\|_X\,ds\\
&\quad +\sum_{i=1}^k\|I_i(z(t_i^-))\|_X
 +\frac{b}{a+b}\sum_{i=1}^m\|I_i(z(t_i^-))\|_X],
\end{align*}
which implies that $P_1(B_r)$ is equicontinuous. Finally, combing
Step 2 to Step 4 together with the Ascol's theorem, we conclude that
the operator $P_1$ is a compact.

\textbf{Step 5.} Now, we show that $P_2$ is a contraction mapping. Let
$z,z^*\in B_r$ and $t\in J_k,\;k=0,1,\dots,m$, we have
\begin{align*}
&\|(P_2z)(t)-(P_2z^*)(t)\|_X\\
&\leq\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
\|f(s,y_s+\overline{z}_s,B(y(s)+\overline{z}(s)))
-f(s,y_s+\overline{z}^*_s,B(y(s)\\
&\quad +\overline{z}^*(s)))\|_X\,ds
 +\frac{bt}{a+b}\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|f(s,y_s+\overline{z}_s,B(y(s)\\
&\quad +\overline{z}(s)))
-f(s,y_s+\overline{z}^*_s,B(y(s)+\overline{z}^*(s)))\|_X\,ds\\
&\leq \frac{(\mu_1l+\mu_2B^*)(a+(1+\alpha)b)T^\alpha}{(a+b)\Gamma(\alpha+1)}
\|z-z^*\|_{B_h''}\\
&\leq\Delta\|z-z^*\|_{\mathfrak{B}_h''},
\end{align*}
where
$$
\Delta=\frac{(\mu_1l+\mu_2B^*)(a+(1+\alpha)b)T^\alpha}{(a+b)\Gamma(\alpha+1)}.
$$
As $\Delta<1$, then $P_2$ is a contraction map. Thus all the
assumptions of the Theorem \ref{kraski} are satisfied and the
conclusion of the Theorem \ref{kraski} implies that the system
\eqref{ME} has at least one solution on $(-\infty,0]$. This
completes the proof of the theorem.
\end{proof}

\section{Application} \label{sec:4}
We consider the  model
\begin{equation}\label{ex}
\begin{gathered}
\begin{aligned}
{}^CD_t^{3/2}u(t)
&=\frac{1}{(t+9)^2}\int_{-\infty}^0e^{2\theta}\sin(\|u(t+\theta)\|_X)d\theta\\
&\quad +\frac{1}{(t+7)^2}\sin\big(\|\int_0^t(t-s)u(s)ds\|_X\big),
\quad t\in[0,1],\; t\neq t_i, i=1,2,3,
\end{aligned} \\
u(t)=\phi(t),\quad t\in(-\infty,0],
\\
\Delta u(t_i)=\int_{-\infty}^0e^{2\theta}\frac{\|u(t_i+\theta)\|_X}
 {25+\|u(t_i+\theta)\|_X}d\theta,
\\
\Delta u'(t_i)=\int_{-\infty}^0e^{2\theta}\frac{\|u(t_i+\theta)\|_X}
 {27+\|u(t_i+\theta)\|_X}d\theta,\\
u'(0)+u'(1)=\int_0^1\sin({1\over2}\|u(s)\|_X)ds,
\end{gathered}
\end{equation}
where $X$ is a real Banach space, $0<t_1<t_2<t_3<1$ are prefixed
numbers and $\phi\in \mathfrak{B}_h$. Let $h(s)=e^{2s},s<0$ then
$l=\int_{-\infty}^0h(s)ds=1/2$ and define
$\|\phi\|_{\mathfrak{B}_h}=\int_{-\infty}^0h(s)
\sup_{s<\theta<0}\|\phi(\theta)\|_X\,ds$.
Hence for $t\in[0,1]$ and $\phi\in \mathfrak{B}_h$, we set
\begin{gather*}
f(t,\phi,Bu(t))
=\frac{1}{(t+9)^2}\int_{-\infty}^0h(\theta)\sin(\|\phi(\theta)\|_X)d\theta
+\frac{1}{(t+7)^2}\sin(\|Bu(t)\|_X),\\
Q_i(\phi)= \int_{-\infty}^0h(\theta)\frac{\|\phi(\theta)\|_X}{25+\|\phi(\theta)\|_X}d\theta,\\
I_i(\phi)= \int_{-\infty}^0h(\theta)\frac{\|\phi(\theta)\|_X}{27+\|\phi(\theta)\|_X}d\theta,
\end{gather*}
where $Bu(t)=\int_0^t(t-s)u(s)ds$, now
$B^*=\sup_{t\in[0,1]}\int_0^t(t-s)ds={1\over2}<\infty$. Then the
above equations \eqref{ex} can be written in the abstract form as
\eqref{ME}. Moreover,
\begin{gather*}
\|f(t,\phi,Bu(t))-f(t,\psi,Bv(t))\|_X
\leq {1\over81}\|\phi-\psi\|_{\mathfrak{B}_h}+{1\over49}\|Bu(t)-Bv(t)\|_X,\\
\|Q_i(\phi)-Q_i(\psi)\|_X \leq {1\over25}\|\phi-\psi\|_{\mathfrak{B}_h},\\
\|I_i(\phi)-I_i(\psi)\|_X \leq {1\over27}\|\phi-\psi\|_{\mathfrak{B}_h},\\
\|q(u)-q(v)\|_X \leq {1\over2}\|u-v\|_X,
\end{gather*}
therefore, (H1),(H2) and (H3) are satisfied with
$\mu_1=1/81$, $\mu_2=1/49$, $L_q=1/2$,
$L=1/27$, $\overline{L}=1/25$.
Further,
\[
\frac{(a+(1+\alpha)b)(\mu_1l+\mu_2B^*)T^\alpha}{(a+b)\Gamma(\alpha+1)}
+\frac{(a+2b)LTm+L_qT^2}{a+b}
+\overline{L}m \approx 0.558<1.
\]
Thus, all the assumptions of
Theorem \ref{thm3.1} are satisfied. Hence, the impulsive fractional
 boundary-value problem \eqref{ex} has a unique solution.

\subsection*{Acknowledgements}
The authors are grateful to the anonymous referees for their valuable
suggestions that led to the improvement of the original manuscript.
J. Dabas has been partially supported by Sponsored
 Research \& Industrial Consultancy, Indian Institute of Technology Roorkee,
 project IITR/SRIC/247/F.I.G (Scheme-A).

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\end{document}