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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 273, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/273\hfil Fractional integro-differential equations]
{Impulsive neutral fractional integro-differential equations with state
dependent delays and integral condition}

\author[J. Dabas, G. R. Gautam \hfil EJDE-2013/273\hfilneg]
{Jaydev Dabas, Ganga Ram Gautam}  % in alphabetical order

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

\address{Ganga Ram Gautam \newline
Department of Applied Science and Engineering,
IIT Roorkee, Saharanpur Campus \\
Saharanpur-247001, India}
\email{gangaiitr11@gmail.com}

\thanks{Submitted May 3, 2013. Published December 17, 2013.}
\subjclass[2000]{26A33, 34K05, 34A12, 34A37, 26A33}
\keywords{Fractional order differential equation;
nonlocal condition; \hfill\break\indent contraction; impulsive condition}

\begin{abstract}
 In this article, we establish the existence of a solution for an
 impulsive neutral fractional integro-differential state dependent
 delay equation subject to an integral boundary condition.
 The existence results are proved by applying the classical fixed
 point theorems. An example is presented to demonstrate the application
 of the results established.
\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:1}

Let $X$ be a Banach space and $PC_t:=PC([-d,t];X), d>0,0\le t\le T<\infty$, 
be a Banach space of all such functions $\phi:[-d,t]\to X$, which are 
continuous everywhere except for a finite number of
points $t_i$, $i=1,2,\dots,m$, at which $\phi(t_i^+)$ and $\phi(t_i^-)$
exists and $\phi(t_i)=\phi(t_i^-)$, endowed with the norm
$$
\|\phi\|_t=\sup_{-d\leq s\leq t}\|\phi(s)\|_X,\;\phi\in PC_t,
$$
 where $\|\cdot \|_X$ is the norm in $X$. 

In this article we study an impulsive neutral fractional
integro-differential equation of the form
\begin{gather}
\label{1.1}  \begin{aligned}
&D^{\alpha}_t \Big[x(t)+\int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\Big]\\
&=f(t,x_{\rho(t,x_t)}, B(x)(t)), \quad t\in J=[0,T],\;T<\infty,\; t\neq t_k, 
\end{aligned} \\
\label{1.2} \Delta x(t_k)=I_k(x(t_k^-)),\quad 
\Delta x'(t_k)=Q_k(x(t_k^-)),\quad k=1,2,\dots,m,\\
\label{1.4} x(t)=\phi (t),\quad t\in[-d,0],\\
\label{1.5} ax'(0)+bx'(T)=\int_0^Tq(x(s))ds,\quad a+b\neq 0,\; b\neq0,
\end{gather}
where $x'$ denotes the derivative of $x$ with respect to $t$ and 
$D^{\alpha}_t$, $\alpha \in (1,2)$ is Caputo's derivative.
The functions $f:J\times PC_0 \times X\to X$, 
$g:J\times PC_0\to X$, and $q:X\to X$
are given continuous functions where $PC_0=PC([-d,0],X)$ and
for any $x\in PC_T=PC([-d,T],X)$,
$t\in J$, we denote by $x_t$ the element of $PC_0$ defined by
$x_t(\theta)=x(t+\theta),\;\theta\in[-d,0]$.
In the impulsive conditions for $0= t_0<t_1<\dots<t_m<t_{m+1}= T$,
$Q_k,I_k\in C(X,X)$,
$(k=1,2,\dots,m)$, are continuous and bounded functions. We have 
$\Delta x(t_k)=x(t_k^{+})-x(t_k^{-})$ and 
$\Delta x'(t_k)=x'(t_k^{+})-x'(t_k^{-})$. The term $Bx(t)$ is given by
\begin{equation}
Bx(t)=\int_0^tK(t,s)x(s)ds,
\end{equation}
where $K\in C(D,\mathbb{R}^+)$,
is the set of all positive functions which are continuous on $D=\{(t,s)\in
\mathbb{R}^2:0\leq s\leq t<T\}$ and $B^*=\sup_{t\in[0,t]}\int_0^tK(t,s)ds
< \infty$.


The study of fractional differential equations has been gaining importance
in recent years due to the fact that fractional order derivatives  provide
a tool for the description of memory and hereditary properties of various
phenomena. Due to this fact, the fractional order models are capable to
describe more realistic situation than the integer order models.
Fractional differential equations have been  used in many field like
fractals,
chaos,  electrical engineering, medical science, etc. In recent years, we
have seen considerable development on the topics of fractional
differential equations, for instance, we refer to the articles
\cite{8,10,25,26}.

The theory of impulsive differential equations of integer order is well
developed and has applications in mathematical modelling, especially in
dynamics of populations subject to abrupt changes as well as other
phenomena such as harvesting, disease, and so forth.
For general theory and  applications of fractional order differential
equations with impulsive conditions, we refer the reader to the references
\cite{1,7,11,16,20,21,27,28,29}.


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{4,5,6,13,166,16} and the
references therein. On the other hand, we know that the delay arises
naturally in systems due to the transmission of signal or
the mechanical transmission. Moreover, the study of fractional order
problems involving
various types of delay (finite, infinite and state dependant) considered
in Banach spaces has been receiving attention, see
\cite{2,3,8,12,14,15,17,18,19,22,23,24} and references cited in these
articles.

In \cite{16} authors have established the existence and uniqueness of a
solution for the following system
\begin{equation}
\begin{gathered}
D^{\alpha}_t 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}
the results are proved by using the contraction and Krasnoselkii's fixed
point theorems.
This paper is motivated from some recent papers treating the boundary
value problems for impulsive fractional
differential equations \cite{4,5,13,16,29}.

To the best of our knowledge, there is no work available in literature on
impulsive neutral fractional integro-differential equation
 with state dependent delay and with an integral boundary condition.  In
this article,  we first establish a general
framework to find a solution to  system \eqref{1.1}--\eqref{1.5} and then by using
classical fixed point theorems we proved the existence and uniqueness
results.


\section{Preliminaries}\label{pre:2}

In this section, we shall introduce notations, definitions,
preliminary results which are required to establish our main
results. We continue to use the function spaces introduced in the earlier
section. For the following definitions we refer to the reader to see the
monograph of Podlunby \cite{26}.

\begin{definition}\label{def2.1} \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 $0\le\alpha< 1$, then
\begin{equation}
D^{\alpha}_tf(t)={1\over \Gamma(1-\alpha)}\int_0^t(t-s)^{-\alpha}f^{(1)}(s)ds.
\end{equation}
\end{definition}

\begin{definition} \label{def2.2} \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{def2.1e}
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{lemma}[ \cite{29}]\label{fr}
For $\alpha >0$, the general solution of fractional differential equations
$D^{\alpha}_tx(t)=0$
is given by $x(t)=c_0+c_1t+c_2t^2+c_3t^3+\dots+c_{n-1}t^{n-1}$
where
$c_i\in \mathbb{R}$, $i=0,1,\dots,n-1$, $n=[\alpha]+1$ and $[\alpha]$
represent the integral part of the real number $\alpha$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.6]{20}}] \label{fr1}
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 following
fractional integral equation
\begin{equation}
x(t)=\int_0^t{(t-s)^{\alpha-1}\over
\Gamma(\alpha)}h(s)ds-\int_0^w{(w-s)^{\alpha-1}\over
\Gamma(\alpha)}h(s)ds+x_0-c(t-w),
\end{equation}
if and only if $x$ is a solution of the following fractional Cauchy problem
\begin{equation}
D^{\alpha}_tx(t)=h(t),\quad t\in J,\; x(w)=x_0,\;w\ge0.
\end{equation}
\end{lemma}

As a consequence of Lemma \ref{fr} and Lemma \ref{fr1} we have the
following result.

\begin{lemma} 
Let $\alpha\in (1,2)$ and $f:J\times PC_0\times X\to X$ be
continuously differentiable function.
A piecewise continuous differential function $x(t):(-d,T]\to X$ is
a solution of system \eqref{1.1}--\eqref{1.5} if and only if $x$ 
satisfied the integral equation
\begin{equation}  \label{sol}
x(t)=\begin{cases}
\phi(t), \quad t\in[-d,0],\\
\phi(0)-\int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds
+\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^Tq(x(s))ds\\
-\sum_{i=1}^kQ_i(x(t_i^-))
+\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds \\
 -\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds\Big\}
\\
+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds, \quad  t\in[0,t_1],\\
\dots \\
\phi(0)+\sum_{i=1}^kI_i(x(t_i^-))+\sum_{i=1}^k(t-t_i)Q_i(x(t_i^-))\\
-\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds
+\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^Tq(x(s))ds\\
-\sum_{i=1}^kQ_i(x(t_i^-))
+\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds \\
-\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds\Big\}
\\
+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds, \quad t\in(t_k, t_{k+1}].
\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
If $t\in [0,t_1]$, then
\begin{equation} \label{t1}
\begin{gathered}
D^{\alpha}_t [x(t)+\int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds]=
f(t,x_{\rho(t,x_t)}, B(x)(t)),\\
x(t)=\phi (t),\;t\in[-d,0].
\end{gathered}
\end{equation}
Taking the Riemann-Liouville fractional integral of \eqref{t1}
and using the Lemma \ref{fr1}, we have
\begin{equation} \label{1.7}
\begin{gathered}
x(t)+\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
=a_0+b_0t+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds,
\end{gathered}
\end{equation}
using the initial condition, we get $a_0=\phi(0)$, then
\eqref{1.7} becomes
\begin{equation} \label{g1}
\begin{aligned}
&x(t)+\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&=\phi(0)+b_0t+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds.
\end{aligned}
\end{equation}
Similarly, if $t\in (t_1, t_2]$, then
\begin{gather}
\label{1.9} D^{\alpha}_t [x(t)+\int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds]=
f(t,x_{\rho(t,x_t)}, B(x)(t))\\
\label{g8} x(t_1^+)=x(t_1^-)+I_1(x(t_1^-)),\\
\label{g9} x'(t_1^+)=x'(t_1^-)+Q_1(x(t_1^-)).
\end{gather}
Again apply the Riemann-Liouville fractional integral operator on
\eqref{1.9} and using the lemma \ref{fr1}, we obtain
\begin{equation} \label{g10} 
\begin{aligned}
& x(t)+\int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
& = a_1+b_1t+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds,
\end{aligned}
\end{equation}
rewrite \eqref{g10} as
\begin{equation} \label{g11} %\label{1.10a}
\begin{aligned}
&x(t_1^+)+\int_0^{t_1}
\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&= a_1+b_1t_1+\int_0^{t_1}\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}
 f(s,x_{\rho(s,x_s)},B(x)(s))ds,
\end{aligned}
\end{equation}
due to impulsive condition \eqref{g8} and the fact that $x(t_1)=x(t_1^-)$,
we may write \eqref{g11} as
\begin{equation} \label{g12} %\label{1.10a2}
\begin{aligned}
& x(t_1)+I_1(x(t_1^-))+\int_0^{t_1}
\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&= a_1+b_1t_1+\int_0^{t_1}\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}
 f(s,x_{\rho(s,x_s)},B(x)(s))ds.
\end{aligned}
\end{equation}
Now from \eqref{g1}, we have
\begin{equation} \label{g13}
\begin{aligned}
&x(t_1)+\int_0^{t_1} \frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}
g(s,x_{\rho(s,x_s)})ds\\& =\phi(0)+b_0t_1+\int_0^{t_1}
\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},B(x)(s))ds.
\end{aligned}
\end{equation}
From \eqref{g12} and \eqref{g13}, we get
$a_1=\phi(0)+b_0t_1-b_1t_1+I_1(x(t^-_1))$, hence \eqref{g11} can be
written as
\begin{equation}  \label{g14}
\begin{aligned}
& x(t)+\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&=\phi(0)+b_0t_1+b_1(t-t_1)+I_1(x(t^-_1))
+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},B(x)(s))ds.
\end{aligned}
\end{equation}
On differentiating  \eqref{g10} with respect to $t$ at $t=t_1$, and
incorporate second impulsive condition \eqref{g9}, we obtain
\begin{equation}  \label{g15}
\begin{aligned}
&x'(t_1^-)+Q_1(x(t_1^-))+\int_0^{t_1}
\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds\\
&=b_1 +\int_0^{t_1}\frac{(t_1-s)^{\alpha-2}}{\Gamma(\alpha-1)}
f(s,x_{\rho(s,x_s)},B(x)(s))ds,
\end{aligned}
\end{equation}
Now differentiating \eqref{g1}, with respect to $t$ at $t=t_1$,
we get
\begin{equation}  \label{g16}
\begin{aligned}
& x'(t_1)+\int_0^{t_1}
\frac{(t_1-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds\\
&=b_0+\int_0^{t_1}\frac{(t_1-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds.
\end{aligned}
\end{equation}
From  \eqref{g15} and \eqref{g16}, we obtain
$b_1=b_0+Q_1(x(t_1^-))$. Thus, \eqref{g14} become
\begin{equation} \label{1.8}
\begin{aligned}
& x(t)+\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&=\phi(0)+b_0t+I_1(x(t_1^-)) +(t-t_1)Q_1(x(t_1^-))\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds.
\end{aligned}
\end{equation}
Similarly, for $t\in (t_2,t_3]$, we can write the solution of the problem as
\begin{align*} %\label{1.8}
& x(t)+\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&=\phi(0)+b_0t+I_1(x(t_1^-))+I_2(x(t_2^-))
 +(t-t_1)Q_1(x(t_1^-))+(t-t_2)Q_2(x(t_2^-))\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds.
\end{align*}
In general, if $t\in (t_k, t_{k+1}]$, then we have the  result
\begin{equation} \label{1.8b}
\begin{aligned}
&x(t)+\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds\\
&=\phi(0)+b_0t+\sum_{i=1}^kI_i(x(t_i^-))
 +\sum_{i=1}^k(t-t_i)Q_i(x(t_i^-)) \\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds.
\end{aligned}
\end{equation}
Finally, we use the integral boundary condition
$ax'(0)+bx'(T)=\int_0^Tq(x(s))ds$, where $x'(0)$ calculated from
\eqref{g1} and $x'(T)$ from  \eqref{1.8}. On simplifying, we get the
following value of the constant $b_0$,
\begin{align*}
b_0&=\frac{b}{a+b}\Big\{\frac{1}{b}\int_0^Tq(x(s))ds
 -\sum_{i=1}^mQ_i(x(t_i^-))
 +\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds \\
&\quad -\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds\Big\}.
\end{align*}
On summarizing, we obtain the desired integral equation \eqref{sol}.
Conversely, assuming that $x$ satisfies \eqref{sol}, by a direct
computation, it follows that the solution given in \eqref{sol} satisfies
system \eqref{1.1}--\eqref{1.5}. This completes the proof of the lemma.
\end{proof}


\section{Existence result}\label{exir:3}

The function $ \rho :J\times PC_0 \to [-d ,T]$  is continuous and
$\phi (0)\in PC_0$. Let
the function t$\to {\varphi}_t$  be well defined and continuous
from the set
$\Re(\rho^-)=\{\rho(s,\psi ):(s,\psi)\in [0,T]\times PC_0\}$ into $PC_0$.
Further, we introduce the following assumptions to establish our results.
\begin{itemize}
\item[(H1)] There exist positive constants $L_{f1}, L_{f2}, L_q$ and
$L_g$, such that
\begin{gather*}
\|f(t,\psi ,x)-f(t,\chi, y)\|_X \leq  L_{f1}\|\psi
-\chi\|_{PC_0}+L_{f2}\|x-y\|_X,  \\
\|g(t,\psi)-g(t,\chi)\|_X \leq  L_g\|\psi-\chi\|_{PC_0}, t \in J,
\forall\; \psi,\chi \in PC_0,\;\forall\;x, y \in X,\\
\|q(x)-q(y)\|_X \leq L_q\|x-y\|_X,\;\forall x,y\in X.
\end{gather*}
\item[(H2)] There exist positive constants $L_Q, L_I, L_q$,
such that
\[
\|Q_k(x)-Q_k(y)\|_X\leq L_Q\|x-y\|_X,\quad 
\|I_k(x)-I_k(y)\|_X\leq L_I\|x-y\|_X.
\]
\item[(H3)] The  functions $Q_k, I_k, q$ are bounded continuous and there
exist positive constants $C_1, C_2, C_3$,
such that
\[
\|Q_k(x)\|_X\leq C_1, \quad \|I_k(x)\|_X\leq C_2,\quad  
\|q(x)\|_X\leq C_3,\quad \forall x\in X.
\]
\end{itemize}
Our first result is based on the Banach contraction theorem.

\begin{theorem}\label{thm} 
Let the assumptions {\rm (H1)--(H2)} are satisfied with
\begin{align*}
\triangle&= \Big\{m(L_I+TL_Q)+\frac{T^{\alpha}L_g}{\Gamma (\alpha+1)}
 +\frac{bT}{a+b}\Big(\frac{TL_q}{b}+mL_Q\\
&\quad +\frac{T^{\alpha-1}L_g}{\Gamma(\alpha)}
+\frac{T^{\alpha-1}(L_{f1}+L_{f2}B^*)}{\Gamma
(\alpha)}\Big)+\frac{T^{\alpha}(L_{f1}+L_{f2}B^*)}{\Gamma
(\alpha+1)}\Big\}<1.
\end{align*}
Then \eqref{1.1}--\eqref{1.5} has a unique solution.
\end{theorem}

\begin{proof}
 We transform problem \eqref{1.1}--\eqref{1.5} into a fixed point
problem. Consider the operator $P:PC_T\to PC_T$ defined by
\[
Px(t)=\begin{cases}
\phi(t), \quad  t\in [-d,0],\\
\phi(0)-\int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds
+\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^Tq(x(s))ds \\
-\sum_{i=1}^mQ_i(x(t_i^-)) 
+\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds\\
-\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds\Big\}\\
+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds, \quad  t\in[0,t_1]\\
\dots \\
\phi(0)+\sum_{i=1}^kI_i(x(t_i^-))+\sum_{i=1}^k(t-t_i)Q_i(x(t_i^-))\\
-\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds
+\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^Tq(x(s))ds-\sum_{i=1}^mQ_i(x(t_i^-))\\
+\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds
-\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds\Big\}\\
+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds, \quad t\in(t_k, t_{k+1}].
\end{cases}
\]
Let $x, x^* \in PC_T$ and $t\in [0,t_1]$. Then
\begin{align*}
& \|P(x)-P(x^*)\|_X\\
&\leq \int_0^t 
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|g(s,x_{\rho(s,x_s)})
-g(s,x^*_{\rho(s,x^*_s)})\|_Xds\\
&\quad +\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^T\|q(x(s))-q(x^*(s))\|_Xds
+\sum_{i=1}^m\|Q_i(x(t_i^-))-Q_i(x^*(t_i^-))\|_X\\
&\quad +\int_0^T
\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}\|g(s,x_{\rho(s,x_s)})
 -g(s,x^*_{\rho(s,x^*_s)})\|_Xds\\
&\quad +\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}\|f(s,x_{\rho(s,x_s)},
B(x)(s))-f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\Big\}
\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|f(s,x_{\rho(s,x_s)},
B(x)(s))-f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\\
&\leq \Big\{\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_g+\frac{bT}{a+b}\Big(\frac{T}{b}L_q+mL_Q+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}L_g \\
& \quad +\frac{T^{\alpha-1}}{\Gamma
(\alpha)}(L_{f1}+L_{f2}B^*)\Big)+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}(L_{f1}+L_{f2}B^*)\Big\}\|x-x^*\|_{PC_T}.
\end{align*}
In a similar way for $t\in (t_k,t_{k+1}]$, we have
\begin{align*}
& \|P(x)-P(x^*)\|_X\\
& \leq \sum_{i=1}^k\|I_i(x(t_i^-))-I_i(x^*(t_i^-))\|_X
 +\sum_{i=1}^k(t-t_i)\|Q_i(x(t_i^-))-Q_i(x^*(t_i^-))\|_X
\\
&\quad\times  \int_0^t
\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|g(s,x_{\rho(s,x_s)})
 -g(s,x^*_{\rho(s,x^*_s)})\|_Xds\\
& +\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^T\|q(x(s))-q(x^*(s))\|_Xds
 +\sum_{i=1}^m\|Q_i(x(t_i^-))-Q_i(x^*(t_i^-))\|_X \\
&\quad +\int_0^T
\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}\|g(s,x_{\rho(s,x_s)})
 -g(s,x^*_{\rho(s,x^*_s)})\|_Xds\\
& \quad +\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}\|f(s,x_{\rho(s,x_s)},
B(x)(s))-f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\Big\}
\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|f(s,x_{\rho(s,x_s)},
B(x)(s))-f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\\
&\leq \Big\{mL_I+mTL_Q+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_g+\frac{bT}{a+b}
 \Big(\frac{T}{b}L_q+mL_Q+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}L_g \\
& \quad +\frac{T^{\alpha-1}}{\Gamma
(\alpha)}(L_{f1}+L_{f2}B^*)\Big)+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}(L_{f1}+L_{f2}B^*)\Big\}\|x-x^*\|_{PC_T}\\
&\leq \Delta \|x-x^*\|_{PC_T}.
\end{align*}
Since $\Delta<1$, implies that the map $P$ is a contraction map and
therefore has a unique fixed
point $x\in PC_T$, hence  system \eqref{1.1}--\eqref{1.5} has a unique solution on
the interval $[-d,T]$.
This completes the proof of the theorem.
\end{proof}

Our second result is based on 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 operators such that
\begin{itemize}
\item[(i)] $Px+Qy\in B$, whenever $x,y\in B$.
(ii) $P$ is compact and continuous. 
\item[(iii)] $Q$ is a contraction mapping.
Then there exists $z\in B$ such that $z=Pz+Qz$.
\end{itemize} 
\end{theorem}

\begin{theorem}\label{k2}
Let the function $f, g$ be continuous for every $t \in [0, T]$,
and satisfy the assumptions {(H1)--(H3)} with
\begin{align*}
\Delta &= \Big\{\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_g+\frac{bT}{a+b}\Big(\frac{T}{b}L_q+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}L_g+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}(L_{f1}+L_{f2}B^*)\Big)\\
&\quad  +\frac{T^{\alpha}}{\Gamma
(\alpha+1)}(L_{f1}+L_{f2}B^*)\Big\}<1.
\end{align*}
Then  system \eqref{1.1}--\eqref{1.5} has at least one solution on $[-d,T]$.
\end{theorem}

\begin{proof} Choose 
\begin{align*}
r&\geq \Big[\|\phi(0)\|+mL_Ir+mTL_Qr+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_gr+\frac{bT}{a+b}(\frac{T}{b}L_qr+mL_Qr\\
&\quad +\frac{T^{\alpha-1}}{\Gamma (\alpha)}L_gr 
+\frac{T^{\alpha-1}}{\Gamma(\alpha)}
(L_{f1}r+L_{f2}B^*r))+\frac{T^{\alpha}}{\Gamma (\alpha+1)}(L_{f1}r+L_{f2}B^*r)\Big].
\end{align*}
Define $PC_T^r=\{x\in PC_T:\|x\|_{PC_T}\leq r\}$, then $PC_T^r$
is a bounded, closed convex subset in $PC_T$. Consider the operators
$N:PC_T^r\to PC_T^r$ and $P:PC_T^r\to PC_T^r$ for 
$t\in J_k=(t_k, t_{k+1}]$, defined by
\begin{gather}
N(x)= \phi(0)+\sum_{i=1}^kI_i(x(t_i^-))+\sum_{i=1}^k(t-t_i)Q_i(x(t_i^-))
-\frac{bt}{a+b}\sum_{i=1}^mQ_i(x(t_i^-))
\\
\begin{aligned}
P(x)
&= \frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^Tq(x(s))ds+\int_0^T
\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}g(s,x_{\rho(s,x_s)})ds \\
& \quad -\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds\Big\}\\
&\quad -\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}g(s,x_{\rho(s,x_s)})ds
 +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,x_{\rho(s,x_s)},
B(x)(s))ds.
\end{aligned}
\end{gather}
We complete the proof in the following steps:

\noindent\textbf{Step 1.} Let $x, x^* \in PC_T^r$ then,
\begin{align*}
 \|N(x)+P(x^*)\|_X
& \leq \|\phi(0)\|_X+\sum_{i=1}^k\|I_i(x(t_i^-))\|_X
 +\sum_{i=1}^k(t-t_i)\|Q_i(x(t_i^-))\|_X\\
&\quad +\frac{bt}{a+b}\sum_{i=1}^m\|Q_i(x(t_i^-))\|_X
 +\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^T\|q(x^*(s))\|_Xds \\
&\quad +\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|g(s,x^*_{\rho(s,x^*_s)})\|_Xds \\
&\quad  +\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\Big\} \\
&\quad +\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|g(s,x^*_{\rho(s,x^*_s)})\|_Xds
\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|f(s,x^*_{\rho(s,x^*_s)},
B(x^*)(s))\|_Xds
\\
& \leq\Big[\|\phi(0)\|+mC_2+mTC_1+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_gr+\frac{bT}{a+b}(\frac{T}{b}C_3\\
&\quad +mC_1+\frac{T^{\alpha-1}}{\Gamma (\alpha)}L_gr 
+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}(L_{f1}r+L_{f2}B^*r))\\
&\quad +\frac{T^{\alpha}}{\Gamma
(\alpha+1)}(L_{f1}r+L_{f2}B^*r)\Big]\leq r.
\end{align*}
Which shows that $PC_T^r$ is closed with respect to both the maps.

\noindent\textbf{Step 2.} $N$ is continuous. Let $x_n \to x$ be sequence 
in $PC_T^r$, then for each $t\in J_k$
\begin{align*}
& \|N(x_n)-N(x)\|_X\\
& \leq\sum_{i=1}^k\|I_i(x_n(t_i^-))-I_i(x(t_i^-))\|_X
 +\sum_{i=1}^k(t-t_i)\|Q_i(x_n(t_i^-))-Q_i(x(t_i^-))\|_X\\
&\quad +\frac{bt}{a+b}\sum_{i=1}^m\|Q_i(x_n(t_i^-))-Q_i(x(t_i^-))\|_X.
\end{align*}
Since the functions $Q_k$ and $I_k$, $k=1,\dots,m$, are
continuous, hence $\|N(x_n)-N(x)\|\to 0$, as $n\to\infty$. Which implies
that the mapping $N$ is continuous on $PC_T^r$.

\noindent\textbf{Step 3.} 
The fact that the mapping $N$ is uniformly bounded is a
consequence of the following inequality. For each 
$t\in J_k$, $k=0,1,\dots,m$ and for each $x\in PC_T^r$, we have
\begin{align*}
\|N(x)\|_X 
&\leq \|\phi(0)\|_X+\sum_{i=1}^k\|I_i(x(t_i^-))\|_X
+\sum_{i=1}^k(t-t_i)\|Q_i(x(t_i^-))\|_X \\
&\quad +\frac{bt}{a+b}\sum_{i=1}^m\|Q_i(x(t_i^-))\|_X\\
&\leq \|\phi(0)\|+mC_2+mTC_1+\frac{bT}{a+b}mC_1.
\end{align*}


\noindent\textbf{Step 4.} 
Now, to show that $N$ is equi-continuous, let
$l_1,l_2\in J_k$, $t_k\leq l_1<l_2\leq t_{k+1}$,
$k=1,\dots,m$, $x\in PC_T^r$, we have
\[
\|N(x)(l_2)-N(x)(l_1)\|_X
\leq (l_2-l_1)\sum_{i=1}^k\|Q_i(x(t_i^-))\|_X
+\frac{b(l_2-l_1)}{a+b}\sum_{i=1}^m\|Q_i(x(t_i^-))\|_X.
\]
As $l_2\to l_1$, then $\|N(x)(l_2)-N(x)(l_1)\|\to 0$ implies that $N$ is
an equi-continuous map.
Combining the Steps 2 to 4, together with the Arzela Ascoli's theorem, we
conclude that the operator $N$ is compact.

\noindent\textbf{Step 5.} Now, we show that $P$ is a contraction mapping. Let
$x,x^*\in PC_T^r$ and $t\in J_k$, $k=1,\dots,m$, we have
\begin{align*}
& \|P(x)-P(x^*)\|_X\\
&\leq \int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}
 \|g(s,x_{\rho(s,x_s)})-g(s,x^*_{\rho(s,x^*_s)})\|_Xds\\
&\quad +\frac{bt}{a+b}\Big\{\frac{1}{b}\int_0^T\|q(x(s))-q(x^*(s))\|_Xds \\
&\quad +\int_0^T \frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|g(s,x_{\rho(s,x_s)})-g(s,x^*_{\rho(s,x^*_s)})\|_Xds\\
& \quad .+\int_0^T\frac{(T-s)^{\alpha-2}}{\Gamma(\alpha-1)}
 \|f(s,x_{\rho(s,x_s)}, B(x)(s))-f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\Big\}
\\
&\quad +\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\|f(s,x_{\rho(s,x_s)},
B(x)(s))-f(s,x^*_{\rho(s,x^*_s)}, B(x^*)(s))\|_Xds\\
&\leq \Big\{\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_g+\frac{bT}{a+b}\Big(\frac{T}{b}L_q+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}L_g \\
&\quad +\frac{T^{\alpha-1}}{\Gamma
(\alpha)}(L_{f1}+L_{f2}B^*)\Big)+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}(L_{f1}+L_{f2}B^*)\Big\}\|x-x^*\|_{PC_T^r}\\
& \leq \Delta \|x-x^*\|_{PC_T^r}
\end{align*}
As $\Delta<1$, it implies that $P$ is a contraction map. Thus all the
assumptions of the Krasnoselkii's theorem are satisfied. Hence we have
that the set $PC_T^r$  has a fixed point which is the solution of system
\eqref{1.1}--\eqref{1.5} on $(-d,T]$. This completes the proof of the theorem.
\end{proof}

\section{Example}\label{exm:4} 

Consider the following example to demonstrate the application of the
results established.
\begin{gather*}
D^{\alpha}_t[x(t)+\int_0^t\frac{1}{47}x(t-\sigma(x)ds]
=\frac{e^tx(t-\sigma(x(t)))}{25+x^2(t-\sigma(x(t)))}
 +\int_0^t\cos(t-s)\frac{xe^s}{4+x}ds, \\
 t\in [0, T],\; t\neq t_i,
\\
\Delta  x(t_i)=\int_{-d}^{t_i}\frac{\gamma _{i}(t_i-s)x(s)}{25}ds,\quad 
\Delta  x'(t_i)=\int_{-d}^{t_i}\frac{\gamma _{i}(t_i-s)x(s)}{9}ds,
\\
x(t)=\phi(t), t\in (-d, 0],\quad x'(0)+x'(T)=\int_0^T\sin(\frac{1}{4}x(s))ds,
\end{gather*}
where $\gamma _{i}\in C([0,\infty ),X)$, $\sigma \in C(X,[0,\infty ))$, 
$0< t_1< t_2<\dots <t_n <T$. Set $\gamma >0$, and  choose $PC^{\gamma}$ as
$$
PC^{\gamma}=\{\phi \in PC((0,\infty], X): \lim_{t
\to-d}e^{\gamma t }\phi (t)\,\text{exist}\}
$$
with the norm 
$\|\phi\|_{\gamma}=\sup_{t \in (0, \infty]}e^{\gamma t }|\phi (t)|, \, 
\phi \in PC^{\gamma}$.
We set
\begin{gather*}
\rho(t,\varphi )=  t-\sigma (\varphi (0)),\quad
   (t,\varphi )\in J\times PC^{\gamma},\\
f(t,\varphi)= {{e^t(\varphi)}\over{25+(\varphi)^2}}, \quad 
(t,\varphi )\in J\times PC^{\gamma},\\
g(t,\varphi)= {\varphi\over 47}ds,\quad \varphi\in PC^{\gamma},
\\
B(x)(t)=  \int _0^t\cos(t-s){x e^s\over(4+x )}ds,\quad
(t,x )\in I\times PC^{\gamma},
\\
Q_k(x(t_k))= \int_{-d}^{t_i}\frac{\gamma _{i}(t_i-s)x(s)}{25}ds,\\
I_k(x(t_k))= \int_{-d}^{t_i}\frac{\gamma _{i}(t_i-s)x(s)}{9}ds.
\end{gather*}
We can see that all the assumptions of Theorem \ref{thm} are satisfied
with
\begin{gather*}
|f(t,\varphi)-f(t,\chi)| \leq  e^t{\|\varphi -\chi \|\over25} \quad \forall t\in
J,\varphi,\chi \in PC^{\gamma},
\\
|B(x)-B(y)| \leq  e^t {\|x -y \|\over4} \quad \forall t\in
J,\; x,y \in  PC^{\gamma},
\\
|g(t,\varphi)-g(t,\chi)| \leq \frac{1}{47}\|\varphi-\chi\|,\quad \forall t\in
J,\; \varphi,\chi \in  PC^{\gamma},\\
|Q_k(x(t_k))-Q_k(y(t_k))| \leq  {\gamma }^*\frac{1}{25}\|x-y\|,\quad x,y\in X,\\
|I_k(x(t_k))-I_k(y(t_k))| \leq  {\gamma }^*\frac{1}{9}\|x-y\|,\quad x,y\in X,\\
|q(x)-q(y)| \leq  \frac{1}{4}\|x-y\|,\quad x,y\in X.
\end{gather*}
Further, we observe that
\begin{align*}
&\Big\{mL_I+mTL_Q+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}L_g+\frac{bT}{a+b}\Big(\frac{T}{b}L_q+mL_Q+\frac{T^{\alpha-1}}{\Gamma
(\alpha)}L_g \\
&\quad +\frac{T^{\alpha-1}}{\Gamma
(\alpha)}(L_{f1}+L_{f2}B^*)\Big)+\frac{T^{\alpha}}{\Gamma
(\alpha+1)}(L_{f1}+L_{f2}B^*)\Big\}\\
&\approx 0.513 \gamma^*+0.534<1.
\end{align*}
We fix $\gamma^*=\int_{-d}^t\gamma_i(t_i-s)ds<0$, 
$0<t_1<t_2<t_3<1$, $\alpha =3/2$, $T=1$. This implies that there exists 
a unique solution of the considered problem in this section.

\subsection*{Acknowledgements} 
The research work of J. Dabas has been partially
supported by the Sponsored Research \& Industrial Consultancy, Indian
Institute of Technology Roorkee, project No.IITR/SRIC/247/F.I.G(Scheme-A).

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\end{document}