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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 90, 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-2012/90\hfil Existence of solutions]
{Existence of solutions for a fractional neutral integro-differential
equation with unbounded delay}

\author[B. de Andrade, J. P. C. dos Santos \hfil EJDE-2012/90\hfilneg]
{Bruno de Andrade, Jos\'e Paulo Carvalho dos Santos}  % in alphabetical order

\address{Bruno de Andrade \newline
Departamento de Matem\'atica, ICMC,
Universidade de S\~ao Paulo\\
 S\~ao Carlos-SP, CEP. 13569-970, Brazil}
\email{bruno00luis@gmail.com}

\address{Jos\'e Paulo Carvalho dos Santos\newline
Instituto de Ci\^encias Exatas,  Universidade Federal de Alfenas\\
Alfenas-MG,  CEP. 37130-000, Brazil}
\email{zepaulo@unifal-mg.edu.br}

\thanks{Submitted December 12, 2011. Published June 5, 2012.}
\subjclass[2000]{34K30, 35R10, 47D06}
\keywords{Integro-differential equations; resolvent of operators; unbounded delay}

\begin{abstract}
 In this article, we study  the existence of mild solutions for
 fractional neutral integro-differential equations with infinite delay.
\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{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this article, we  study  the existence of mild solutions for the neutral
fractional integral evolutionary equation
\begin{gather}\label{eqf1}
    D^{\alpha}_t( x(t)+f(t,x_t)) =   Ax(t)+ \int_0^{t} B(t-s)x(s)ds+ g(t,x_t),
 \quad t>0,   \\
\label{eqf2}   x_0=\varphi , \quad x'(0)=x_1,
 \end{gather}
where  $ \alpha \in (1,2)$;  $A, (B(t))_{t\geq0}$   are closed linear
operators defined on a common domain  which is dense in a Banach
space $X$, $D^{\alpha}_t h(t)$ represent the  Caputo derivative of $\alpha>0$
defined by
$$
D^{\alpha}_t h(t)  :=
 \int_0^{t} g_{n-\alpha}(t-s) \frac{d^{n}}{ds^{n}} h(s) ds,
 $$
where $n$ is the smallest integer greater than or equal to $\alpha$  and
 $g_{\beta}(t):= \frac{t^{\beta-1}}{\Gamma(\beta)}, t>0,
 \beta\geq0$. The history $x_{t}: (-\infty,0] \to X$ given by
$x_{t}(\theta)=x(t+\theta)$ belongs to some abstract phase  space
$\mathcal{B}$ defined axiomatically and  $ f,g:I\times \mathcal{B}\to X $ are
appropriate functions.

 The literature related  to ordinary   neutral functional differential
equations is  very extensive and we refer  the reader to
the Hale and Lunel book \cite{HA1} and the references therein.
Partial neutral differential equations  arise, for instance, in
the  transmission line theory, see Wu and Xia \cite{wu4} and the
study of material with fanding memory, see
\cite{Gurtin1,Nunziato1}.  In the paper \cite{HH2}, Hernandez and
Henriquez, study the existence of mild and strong solutions for
the partial neutral system
\begin{gather} \label{e3}
\frac{d}{d t} \big(x(t) + g(t,x_{t})\big)
=   Ax(t)+ f(t,x_t),   \quad t \in I=[0,a],   \\
\label{e4} x_0 = \varphi  ,
\end{gather}
where     $A:D(A)\subset X\to X $  is  a  generator of analytic
semigroup  and   the history $x_{t}: (-\infty,0] \to X$ given by
$x_{t}(\theta)=x(t+\theta)$ belongs to some abstract phase  space
$\mathcal{B}$ defined axiomatically and  $ f,g:I\times \mathcal{B}\to X $ are
appropriate functions. Very recently, Hernandez et al,
\cite{Jose3}, study the existence of mild, strong and classical
solutions for the integro-differential neutral systems
\begin{gather} \label{eq1}
  \frac{d}{dt} (x(t) +
f(t,x_t))=  Ax(t)+ \int_0^{t} B(t-s)x(s) ds + g(t,x_{t}), \quad
  t \in I=[0,b],  \\
  x_0 = \varphi,\quad \varphi \in  \mathcal{B} ,
\end{gather}
where  $ A:D(A)\subset X\to X$ and
 $B(t):D(B(t))\subset X\to X$, $t\geq 0$,  are closed linear
 operators; $(X, \| \cdot\| ) $ is a Banach  space;  the history
$x_{t}: (-\infty,0] \to X$, defined by $x_t(\theta) := x(t
+\theta)$ belongs  to an abstract phase space $\mathcal{B}$ defined
axiomatically and $ f,g : I \times \mathcal{B} \to X$  are
appropriated functions. In the paper \cite{DosSantos},   Dos
Santos et al. study the existence of mild and classical solutions
for the partial neutral systems with unbounded delay
\begin{gather} \label{e1}
  \frac{d}{d t}[x(t) + \int_{-\infty}^{t} N(t-s)x(s)
ds ] =    Ax(t)+ \int_{-\infty}^{t} B(t-s)x(s) d s + f(t,x_t), \;   t \in  [0,a],  \\
\label{e2} x_0 =  \varphi, \quad  \varphi\in   \mathcal{B} ,
\end{gather}
where  $ A, B(t)$ for $t \geq 0$  are closed linear  operators
defined on a common domain  $D(A)$ which is dense in $X$,  $N(t)$
($t \geq 0$)   are bounded linear operators on $X$,  without to
use many of the strong restrictions considered  in the literature.
To the best of our knowledge, the existence of mild solutions for
abstract fractional partial evolutionary integral  equations with
unbounded delay is an untreated topic in the literature and this
fact is the main motivation of the present work.

\section{Preliminaries}\label{preliminaries}

In what follows we recall some definitions, notation and results
that we need in the sequel.  Throughout this paper,
 $(X, \| \cdot\| ) $ is a Banach space and
 $A, B(t)$, $t\geq 0$, are
closed linear operators defined on a common domain
$\mathcal{D}=D(A)$ which is dense in  $X$. The notation
 $[D(A)]$ represents the domain of  $A$  endowed
with the graph norm.  Let $(Z,\| \cdot\| _{Z})$ and   $(W,\|
\cdot \| _{W})$ be Banach spaces. In this paper,  the notation
$\mathcal{L}(Z,W)$ stands for the Banach space of bounded linear
operators  from $Z$ into $W$ endowed with the uniform  operator
topology  and we abbreviate this notation  to $\mathcal{L}(Z)$
when $Z = W$. Furthermore, for appropriate functions
 $K :[0,\infty)\to Z$  the symbol $\widehat{K}$
denotes the Laplace transform of $K$. Thesymbol $B_r(x,Z)$
stands for the closed ball with center at $x$ and radius $r>0$ in
$Z$.  On the other hand, for a bounded function
$\gamma :[0,a]\to Z$ and $ t\in [0,a]$,
 the symbol  $\|\gamma \|_{Z,t}$ is given  by
 \begin{equation} \label{notation1}
\|\gamma\|_{Z,t} =\sup\{\| \gamma(s)\|_{Z} : s \in [0,t]\},
\end{equation}
and we  simplify  this  notation to $\|\gamma\|_{t}$  when no
confusion about the space  $Z$ arises.

To obtain our results,  we  assume that the abstract  fractional
integro-differential  problem
 \begin{gather}
\label{eqa1} D^{\alpha}_t x(t) =   Ax(t)+ \int_0^{t} B(t-s)x(s)ds,   \\
\label{eqa2}  x(0) =  z\in X, \quad x'(0)=0,
\end{gather}
has an associated  $\alpha$-resolvent operator  of bounded linear
operators $(\mathcal{R}_{\alpha}(t))_{t\geq 0}$ on $X$.

\begin{definition}\label{D3}  \rm
 A  one parameter family of bounded linear operators
$(\mathcal{R}_{\alpha}(t))_{t\geq 0} $  on $X$  is called a
$\alpha$-resolvent operator  of  \eqref{eqa1}-\eqref{eqa2} if the
following conditions are satisfied.
\begin{itemize}
\item [(a)] The function $\mathcal{R}_{\alpha}(\cdot): [0, \infty)
\to \mathcal{L}(X)$ is strongly continuous
and $\mathcal{R}_{\alpha}(0)x=x$ for all $x\in X$ and $\alpha \in (1,2)$.

\item [(b)] For  $x \in D(A) $,  $\mathcal{R}_{\alpha}(\cdot)x \in C([0,\infty),
[D(A)]) \bigcap C^{1}((0,\infty),X)$,
 and
\begin{gather}\label{eqrp1}
 D^{\alpha}_t  \mathcal{R}_{\alpha}(t) x
=  A \mathcal{R}_{\alpha}(t)x + \int_0^{t} B(t-s) \mathcal{R}_{\alpha} (s) x d s, \\
\label{eqrp2}  D^{\alpha}_t  \mathcal{R}_{\alpha}(t)x
= \mathcal{R}_{\alpha}(t) A x + \int_0^{t} \mathcal{R}_{\alpha}(t-s)
B(s)x d s,
\end{gather}
for every  $t\geq 0 $.
\end{itemize}
\end{definition}

The existence of a $\alpha$-resolvent operator for problem
\eqref{eqa1}-\eqref{eqa2} was studied  in \cite{ACJ}. In this work
we consider the following conditions.
\begin{itemize}

\item[(P1)]   The operator $A : D(A)\subseteq X \to X $
 is a closed linear operator with $[D(A)]$
dense in $X$. Let $\alpha \in (1,2)$, for some $\phi_0 \in (0,
\frac{\pi}{2}]$  for each $\phi<\phi_0$  there is positive
constants $C_0=C_0(\phi)$ such that  $ \lambda \in \rho(A) $
  for each
 $$
\lambda \in \Sigma_{ 0,\alpha \vartheta } = \{ \lambda \in \mathbb{C}: \lambda\neq
0, \, | \arg(\lambda) | < \alpha \vartheta \},
$$
where $\vartheta=\phi+ \frac{\pi}{2}$ and  $ \| R(\lambda,A) \|
\leq \frac{C_0}{| \lambda |} $ for all  $\lambda \in \Sigma_{0,\alpha \vartheta }$.

\item[(P2)] For all  $t\geq 0$, $B(t):D(B(t)) \subseteq X \to X $
is a closed linear operator, $D(A) \subseteq D(B(t)) $ and
$B(\cdot)x $ is  strongly measurable on $(0,\infty) $ for each $x
\in D(A)$.  There exists  $b(\cdot) \in
L_{\rm loc}^{1}(\mathbb{R}^{+}) $ such that $\widehat{b}(\lambda)$ exists
for $Re(\lambda) > 0$ and  $\| B(t) x \| \leq b(t) \| x \|_1 $
for all  $t>0 $ and  $x \in D(A)$. Moreover, the operator valued
function  $\widehat{B} : \Sigma_{0, \pi/2} \to
\mathcal{L}([D(A)],X)$ has an analytical extension (still denoted
by $\widehat{B}$) to  $\Sigma_{ 0,\vartheta }$ such that $\|
\widehat{B}(\lambda) x \| \leq \|\widehat{B}(\lambda)\| \, \|x\|_1$ for
all $x \in D(A)$, and $\|\widehat{B}(\lambda)\| = O(\frac{  1}{|
\lambda | })$,
as $|\lambda | \to \infty$.

\item[(P3)]  There exists a subspace   $D \subseteq D(A) $   dense in
$[D(A)] $   and   positive constants  $C_{i}$, $i=1,2$, such that
$A(D) \subseteq D(A) $, $\widehat{B}(\lambda)(D) \subseteq D(A) $,
 $\|A \widehat{B}(\lambda) x \| \leq   C_1 \|x\|$ for
every $x\in D$ and  all $ \lambda \in \Sigma_{0, \vartheta}$.
\end{itemize}

In the sequel,  for $r>0$ and
 $ \theta \in (\frac{\pi}{2}, \vartheta )$,
 $$
\Sigma_{r, \theta}= \{ \lambda \in \mathbb{C}: \lambda\neq 0, |
\lambda| >r, \,| arg(\lambda) | < \theta  \} ,
$$
for $\Gamma_{r,\theta },\Gamma^{i}_{r,\theta }$, $i=1,2,3$, are the
paths
\begin{gather*}
\Gamma^{1}_{r,\theta }=\{ t e^{i\theta}: t \geq r \},\\
\Gamma^{2}_{r,\theta }=\{ re^{i\xi}:  -\theta \leq \xi \leq \theta \},\\
\Gamma^{3}_{r,\theta }=\{ t e^{-i\theta}: t \geq r\},
\end{gather*}
 and $\Gamma_{r,\theta }=\bigcup_{i=1}^{3}\Gamma^{i}_{r,\theta }$ oriented
counterclockwise. In addition, $\rho_{\alpha}(G_{\alpha})$ are the sets
\[
\rho_{\alpha}(G_{\alpha})=\{ \lambda \in \mathbb{C}: G_{\alpha}(\lambda):=
\lambda^{\alpha-1}(\lambda^{\alpha} I - A - \widehat{B}(\lambda) )^{-1} \in
\mathcal{L}(X)\}.
\]

We  now define the operator family $(\mathcal{R}_{\alpha}(t))_{t\geq
0} $  by
\begin{equation}\label{defnresolv1}
\mathcal{R}_{\alpha}(t) = \begin{cases}
\frac{1}{2\pi i }
\int_{\Gamma_{r,\theta}} e^{\lambda t}  G_{\alpha}(\lambda)  d\lambda,  & t>0,   \\
I, & t=0.
\end{cases}
\end{equation}

\begin{lemma}[{\cite[Lemma 2.2]{ACJ}}] \label{lemacd}
There  exists   $r_1>0$ such that $\Sigma_{r_1, \vartheta}
\subseteq \rho_{\alpha}(G_\alpha) $ and the function
$G_{\alpha}:\Sigma_{r_1,\vartheta} \to \mathcal{L}(X)$ is analytic. Moreover,
\begin{equation}
G_{\alpha}(\lambda) =  \lambda^{\alpha-1}R(\lambda^{\alpha},A)[ I -
\widehat{B}(\lambda)R(\lambda^{\alpha},A)]^{-1}, \label{eq1b}
\end{equation}
and there exist constants $M_{i}$ for  $i=1,2$ such that
\begin{gather}
\|  G_{\alpha}(\lambda) \| \leq  \frac{M_1}{| \lambda |} , \label{eq2} \\
\|  A G_{\alpha}(\lambda) x \| \leq   \frac{M_2}{| \lambda  |}
\| x \|_1, \,\, x \in D(A), \label{eq3}  \\
\| A G_{\alpha}(\lambda) \| \leq   \frac{M_4}{| \lambda |^{1-\alpha}},
\label{eq4}
\end{gather}
for every  $\lambda \in \Sigma_{r_1 , \vartheta}$.
\end{lemma}

The following result was established in \cite[Theorem 2.1]{ACJ}.

\begin{theorem} \label{teo2}
Assume that conditions {\rm (P1)--(P3)} are fulfilled. Then
there exists a unique $\alpha$-resolvent operator for  problem
\eqref{eqa1}-\eqref{eqa2}.
\end{theorem}


\begin{theorem}[{\cite[Lemma 2.5]{ACJ} }]
 The function $\mathcal{R}_{\alpha}  :[0, \infty) \to
\mathcal{L}(X)$ is strongly continuous and  $\mathcal{R}_{\alpha}  :(0, \infty)
\to \mathcal{L}(X)$ is uniformly  continuous.
\end{theorem}

In what follows,  we assume that the conditions
(P1)--(P3) are satisfied.
We consider now the non-homogeneous problem
\begin{gather} \label{eh1}
    D^{\alpha}_t x(t) =   Ax(t)+ \int_0^{t} B(t-s)x(s)
ds+ f(t), \quad  t \in [0,a],   \\
 \label{eh2} x(0) = x_0,  \quad  x'(0)=0,
 \end{gather}
where $\alpha \in (1,2) $ and $f\in L^{1}([0,a],  X) $.  In the
sequel, $\mathcal{R}_{\alpha}(\cdot)$ is the operator  function
defined by
\eqref{defnresolv1}.
 We begin by  introducing the following concept of classical
solution.

 \begin{definition}\label{D1}\rm
A function $x : [0,a] \to X $, $0<a$,  is called  a classical solution of
  \eqref{eh1}-\eqref{eh2} on $[0,a] $  if
$ x \in  C([0,a],  [D(A)]) \cap C([0,a], X), g_{n-\alpha}\ast x \in
C^1([0,a], X), n=1,2$, the condition \eqref{eh2} holds and the
equations \eqref{eh1} is verified on $[0,a]$.
 \end{definition}

\begin{definition}\label{defsen} \rm
Let $\alpha \in (1,2)$, we define the family $(\mathcal{S}_{\alpha}
(t))_{t\geq 0}$ by $$\mathcal{S}_{\alpha} (t)x := \int_0^t
g_{\alpha-1}(t-s) \mathcal{R}_{\alpha}(s)x ds,$$ for each $t\geq 0$.
\end{definition}

\begin{lemma}[{\cite[Lemma 3.9]{ACJ}}] \label{expests}
 If the function $\mathcal{R}_{\alpha} (\cdot)$ is  exponentially bounded
in  $\mathcal{L}(X)$, then $\mathcal{S}_{\alpha} (\cdot)$ is  exponentially bounded
in  $\mathcal{L}(X)$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 3.10]{ACJ}}] \label{lemaest1}
If the function $\mathcal{R}_{\alpha} (\cdot)$ is  exponentially bounded
in  $\mathcal{L}([D(A)])$, then $\mathcal{S}_{\alpha} (\cdot)$ is
exponentially bounded  in  $\mathcal{L}([D(A)])$.
\end{lemma}

We now establish a  variation of constants formula  for the
solutions of  \eqref{eh1}-\eqref{eh2}.

\begin{theorem}[{\cite[Theorem 3.2]{ACJ} }] \label{formvarconstnr}
 Let $z\in D(A)$. Assume that $f \in C([0,a],X)$  and  $u(\cdot) $ is a
 classical  solution of \eqref{eh1}--\eqref{eh2} on $[0, a]$. Then
\begin{equation} \label{variation}
u(t) = \mathcal{R}_{\alpha}(t) z  +  \int_0^{t}
\mathcal{S}_{\alpha}(t-s) f(s) \, d s,  \quad t \in [0,a].
\end{equation}
\end{theorem}

It is clear from the preceding definition that
$\mathcal{R}_{\alpha}(\cdot)z$ is a solution of problem
\eqref{eqa1}-\eqref{eqa2}  on $(0, \infty)$ for $z \in D(A)$.

\begin{definition}\label{D2}  \rm
Let $f\in L^{1}([0,a],  X) $. A function
$u \in C([0,a],X) $ is called  a mild solution of \eqref{eh1}-\eqref{eh2}
if
\[
u(t) = \mathcal{R}_{\alpha}(t)z + \int_0^{t} \mathcal{S}_{\alpha}(t-s)
f(s)\, d s, \quad t\in[0,a].
\]
\end{definition}

The next results  are proved in \cite{ACJ} and \cite{BCJ}.

\begin{theorem}[{\cite[Theorem 3.3]{ACJ} }] \label{regulenr}
 Let  $z \in D(A) $ and $f \in
C([0,a],X)$. If $f \in L^1([0,a], [D(A)]) $   then the mild
solution of  \eqref{eh1}-\eqref{eh2}
 is a classical  solution.
 \end{theorem}

\begin{theorem}[{\cite[Theorem 3.4]{ACJ}}]  \label{teo2b}
Let  $z \in D(A) $ and $f \in C([0,a],X)$. If $f \in  W^{1,1}([0,a],X)$,
 then the mild  solution of  \eqref{eh1}--\eqref{eh2}
 is a classical  solution.
 \end{theorem}

\begin{lemma}[{\cite[Lemma 2.3]{BCJ}}]   \label{leresol1}
If $R(\lambda_0^{\alpha}, A)$ is compact
for some $\lambda_0^{\alpha} \in \rho(A)$, then $\mathcal{R}_{\alpha}(t) $
and $\mathcal{S}_{\alpha}(t)$ are compact for all $t> 0$.
\end{lemma}

 We will herein define the phase space
$\mathcal{B}$ axiomatically, using ideas and notations developed
in~\cite{HMN}. More precisely, $\mathcal{B}$ will denote the vector
space of functions defined from $(-\infty,0]$ into $X$ endowed
with a seminorm denoted $\|\cdot\|_{\mathcal{B}}$ and such that the
following axioms hold:
\begin{itemize}
\item[(A)] If $x:(-\infty,\sigma+b)\to X$,
$b>0,\sigma\in \mathbb{R}$, is continuous on $[\sigma,\sigma +b)$
and $x_{\sigma}\in \mathcal{B}$, then for every $t\in
[\sigma,\sigma+b)$ the following conditions hold:
\begin{itemize}
\item[(i)] $x_{t}$ is in $\mathcal{B}$.
 \item[(ii)] $\|x(t)\| \leq H \| x_{t}\|_{\mathcal{B}}$.
\item[(iii)] $\| x_{t}\|_{\mathcal{B}} \leq K(t-\sigma)
\sup\{\|  x(s)\|:\sigma\leq s\leq t\}+
 M(t-\sigma)\| x_{\sigma}\|_{\mathcal{B}}$,
\end{itemize}
 where $H>0$ is a constant; $ K,M:[0,\infty) \to
[1,\infty)$, $K(\cdot)$ is continuous, $M(\cdot)$ is locally
bounded and $H,K,M$ are independent of $x(\cdot)$.

 \item[(A1)] For the function $x(\cdot)$ in $(\mathbf{A})$,
the function $t\to x_{t}$ is continuous from  $[\sigma,\sigma+b)$ into $\mathcal{B}$.

\item[(B)] The space $\mathcal{B}$ is complete.
\end{itemize}

\begin{example} \label{example1} \rm
(The phase space $C_r \times L^p(g,X)$)
  Let $r \geq 0$, $1 \leq p < \infty$ and let
$g:(-\infty,-r] \to \mathbb{R}$ be a nonnegative
measurable function which satisfies the conditions (g-5),  (g-6)
in the terminology  of \cite{HMN}.    Briefly, this means that
$g$ is locally integrable and  there exists a non-negative,
 locally bounded function $\gamma$ on $(- \infty, 0]$ such that
$g(\xi+\theta) \leq \gamma(\xi) g(\theta)$,
 for all $ \xi \leq 0$ and $ \theta \in (- \infty , -r)
\setminus N_{\xi }$, where $N_{\xi} \subseteq (- \infty, -r)$ is a
set with Lebesgue measure zero.  The space  $\; C_r \times
L^p(g,X)$ consists of all classes of functions $\; \varphi : (-
\infty , 0] \to X $ such that $ \varphi $ is continuous on
$[- r,0]$, Lebesgue-measurable, and $\; g \| \varphi \| ^p $
is Lebesgue  integrable on $ (- \infty , -r )$.
  The seminorm in $\; C_r\times L^p(g,X)$ is defined by
$$
\|  \varphi \|_{\mathcal{B}} : =
\sup \{ \| \varphi (\theta ) \| : -r\leq \theta \leq 0 \}
 +\Big( \int_{- \infty }^{-r} g(\theta ) \|
 \varphi (\theta ) \|^p
d \theta \Big)^{1/p}.
$$
The space  $\;\mathcal{B} = C_r \times L^p(g;X) $ satisfies axioms
(A), (A-1), (B).   Moreover, when $ r=0$ and $p=2$, we can
take $H = 1$, ${ M(t) = \gamma(-t)^{1/2}}$ and
${K(t) = 1 + \left(\int_{-t}^{0} g(\theta) \,d \theta
\right)^{1/2}}$, for $t \geq 0$. (see \cite[Theorem 1.3.8]{HMN}
for details).
\end{example}

For additional  details concerning  phase space we refer the reader to
 \cite{HMN}.

\section{Neutral problem}

In the next result we denote by $(-A)^{\vartheta}$ the fractional
power of the operator $-A$, (see \cite{PA} for details).

 \begin{lemma}\label{estpr1}
 Suppose that the conditions {\rm (P1)--(P3)}  are satisfied.
 Let $\alpha \in (1,2)$ and $\vartheta \in (0,1)$ such that
$\alpha \vartheta \in (0,1)$, then there exists  positive number
$C$ such that
\begin{gather}\label{10}
\| (-A)^{\vartheta}\mathcal{R}_{\alpha}(t) \|
\leq    Ce^{rt}t^{- \alpha \vartheta},  \\
\label{desps} \| (-A)^{\vartheta}\mathcal{S}_{\alpha}(t) \|
\leq   Ce^{rt} t^{ \alpha( 1-\vartheta)-1},
\end{gather}
for all  $ t>0$.
\end{lemma}


\begin{proof}
Let  $\vartheta \in (0,1)$. From \cite[Theorem 6.10]{PA}, there exist
 $C_{\vartheta}>0$  such that
   $$ \| (-A)^{\vartheta} x
  \| \leq C_{\vartheta} \| Ax \|^{\vartheta} \| x \|^{1-\vartheta},
\quad x \in D(A).
$$
 From $G_{\alpha}(\cdot)$ is valued  $D(A)$,  for all  $ x \in X$
  \begin{equation} \label{despontencia}
\begin{aligned}
 \| (-A)^{\vartheta}G_{\alpha}(\lambda)x \|
 &\leq  C_{\vartheta}\| AG_{\alpha}(\lambda)x \|^{\vartheta} \| G_{\alpha}(\lambda)x \|^{1-\vartheta}
  \\
& \leq  C_{\vartheta}\frac{M_3^{\vartheta}}{| \lambda |^{\vartheta - \alpha \vartheta}}
\| x \|^{\vartheta}
 \frac{M_1^{1-\vartheta}}{ | \lambda |^{1-\vartheta}} \| x \|^{1-\vartheta}
 \\
& \leq  \frac{M_{\vartheta}}{ | \lambda |^{1-\alpha \vartheta}} \| x \|,
\end{aligned}
\end{equation}
where  $M_{\vartheta}$ is independent of   $\lambda$. From
\eqref{despontencia}, we obtain for $t \geq 1$, make the change
$\lambda t = \gamma$. From the
  Cauchy's theorem we obtain that
  \begin{align*}
\|  (-A)^{\vartheta} \mathcal{R}(t) \|
&\leq  \|\frac{1}{2\pi i } \int_{ \Gamma_{r,\theta}}
 e^{ \gamma} (-A)^{\vartheta}G (t^{-1}\gamma ) t^{-1} d\gamma \|   \\
&\leq \frac{M_{\vartheta}}{\pi } \int_r^{\infty} e^{s \cos{\theta}}
  \frac{t^{-1} d s }
 {(t^{-1}s)^{1-\alpha\vartheta}}
+ \frac{M_{\vartheta}}{2\pi } \int_{-\theta}^{\theta} e^{ r \cos{\xi}}
\frac{t^{-1}r d \xi}{(t^{-1} r)^{1-\alpha  \vartheta}}
\\
&\leq  \Big( \frac{M_{\vartheta}}{\pi r^{1-\alpha \vartheta}
|\cos{\theta}|} + \frac{ M_{\vartheta} \theta  r^{\alpha \vartheta} }{\pi}
\Big) \frac{e^{r t}}{t^{\alpha \vartheta}} \\
&\leq  C \frac{e^{r t}}{t^{\alpha \vartheta}}.
\end{align*}
 On the other hand,   using that $G(\cdot)$ is analytic on
$\Sigma_{r, \theta}$,   for $t\in (0,1)$   we obtain
\begin{align*}
 \| (-A)^{\vartheta} \mathcal{R}(t)\|
&=  \|  \frac{1}{2\pi i } \int_{\Gamma_{\frac{r}{t},\theta}}
e^{\lambda t} (-A)^{\vartheta}G (\lambda)  d\lambda \|   \\
&\leq \frac{M_{\vartheta}}{\pi} \int_{ \frac{r}{t} }^{\infty} e^{ts
\cos{\theta}} \frac{d s }{s^{1-\alpha \vartheta}}  +
\frac{M_{\vartheta}}{2\pi} \int_{-\theta}^{\theta}
e^{r \cos{\xi}}  \frac{ r t^{-1} d \xi}{r^{1- \alpha \vartheta}t^{\alpha \vartheta -1}}  \\
&\leq \frac{M_{\vartheta}}{\pi} \int_r^{\infty} e^{u \cos{\theta}}
\frac{ t^{-1}d u }{u^{1- \alpha \vartheta}t^{\alpha \vartheta-1}}  +
\frac{M_{\vartheta}}{2\pi } \int_{-\theta}^{\theta}  e^{r \cos{\xi}}  \frac{ r t^{-1} d \xi}{r^{1-\alpha \vartheta}t^{\alpha \vartheta -1} }\\
&\leq  \Big( \frac{M_{\vartheta}}{\pi r^{1-\alpha \vartheta}
|\cos{\theta}|} + \frac{ M_{\vartheta} \theta  r^{\alpha \vartheta} }{\pi} e^{r} \Big)
\frac{1}{t^{\alpha \vartheta}} \\
&\leq  C \frac{e^{r t}}{t^{\alpha \vartheta}}.
\end{align*}
By the definition of $(\mathcal{S}_{\alpha} (t))_{t\geq 0}$, we obtain
\begin{align*}
 \| (-A)^{\vartheta} \mathcal{S}_{\alpha}(t)\|
&\leq    \int_0^t g_{\alpha-1}(t-s) \| (-A)^{\vartheta}  \mathcal{R}_{\alpha}(s) \| ds  \\
&\leq  \int_0^t g_{\alpha-1}(t-s) C e^{rs}s^{-\alpha \vartheta}  ds  \\
&\leq  e^{rt} \int_0^t \frac{(t-s)^{\alpha -2}}{\Gamma(\alpha-1)} C s^{-\alpha \vartheta} ds  \\
&\leq  \frac{e^{rt}}{\Gamma(\alpha-1)} \int_0^t (t-s)^{\alpha -2} C s^{-\alpha \vartheta} ds \\
&\leq  \frac{e^{rt}}{\Gamma(\alpha-1)} \int_0^t (t-s)^{(\alpha -1)-1} C
s^{(1-\alpha \vartheta)-1}  ds.
\end{align*}
From inequality  \cite[ 6.24]{PA}, we obtain
\[
 \| (-A)^{\vartheta} \mathcal{S}_{\alpha}(t)\|
\leq  \frac{e^{rt} \Gamma(1-\alpha\vartheta)}{\Gamma(\alpha-\alpha \vartheta)}C t^{\alpha(1-
 \vartheta)-1}
 \leq   C e^{rt} t^{\alpha(1- \vartheta)-1}  .
\]
\end{proof}

\begin{remark} \label{rmk1} \rm
If $ \widehat{B}(\lambda)(-A)^{-\vartheta} y = (-A)^{-\vartheta} \widehat{B}(\lambda) y$ for
 $y \in [D(A)]$. We can see that for $\vartheta \in (0,1)$ and
$x \in [D((-A)^{\vartheta})]$
\begin{align*}
(-A)^{\vartheta}G_{\alpha}(\lambda)x
 &= \lambda^{\alpha-1}(-A)^{\vartheta}
R(\lambda^{\alpha},A)[ I - \widehat{B}(\lambda)R(\lambda^{\alpha},A)]^{-1}x \\
&= \lambda^{\alpha-1}(-A)^{\vartheta} R(\lambda^{\alpha},A)[ I -
\widehat{B}(\lambda)R(\lambda^{\alpha},A)]^{-1}(-A)^{-\vartheta}
(-A)^{\vartheta}x.
\end{align*}
Since
$$
\widehat{B}(\lambda)R(\lambda^{\alpha},A)(-A)^{-\vartheta}(-A)^{\vartheta}x
= (-A)^{-\vartheta} \widehat{B}(\lambda)R(\lambda^{\alpha},A)(-A)^{-\vartheta}x,
$$
we obtain
\begin{align*}
(-A)^{\vartheta}G_{\alpha}(\lambda)x
&= \lambda^{\alpha-1}(-A)^{\vartheta}
R(\lambda^{\alpha},A)(-A)^{-\vartheta}[ I -
\widehat{B}(\lambda)R(\lambda^{\alpha},A)]^{-1} (-A)^{\vartheta}x  \\
&= G_{\alpha}(\lambda)(-A)^{\vartheta}x.
\end{align*}
 As consequences of before it is  easy to see that
$$
(-A)^{\vartheta}\mathcal{R}_{\alpha}(t)x=\mathcal{R}_{\alpha}(t)(-A)^{\vartheta}x
\quad\text{and}\quad
(-A)^{\vartheta}\mathcal{S}_{\alpha}(t)x=\mathcal{S}_{\alpha}(t)(-A)^{\vartheta}x,
$$
if  $x \in [D((-A)^{\vartheta})]$.
\end{remark}

If $x \in C(I;X)$ we define  $\overline{x}:(-\infty,b] \to X$ is the extension of
$x$ to $(-\infty,b]$ such that $\overline{x}_0=\varphi$.
In the sequel we introduce the following conditions:
 \begin{itemize}
\item [(H1)] The following conditions are satisfied.
\begin{itemize}
\item[(a)] $ B(\cdot )x\in C(I,X)$ for every $ x \in [D((-A)^{1-\vartheta})]$.
\item[(b)] There is function   $\mu (\cdot) \in L^1(I;\mathbb{R}^{+})$, such that
 \[
  \| B(s)\mathcal{S}_{\alpha}(t)\|_{\mathcal{L}([D((-A)^{\vartheta})],X)}
\leq  M \mu(s) t ^{\alpha \vartheta -1}, \quad   0 \leq s<t \leq b.
\]
\end{itemize}

\item[(H2)]  The function $f(\cdot)$ is $(-A)^{\vartheta}$-valued,
  $f: I \times \mathcal{B} \to
 [D((-A)^{-\vartheta})]$ is continuous  and there  exists $L_f$  such that for all
$ (t_i,\psi_j)\in  I\times  \mathcal{B}$,
\begin{equation} \label{lipcf}
 \| (-A)^{\vartheta}f(t_1,\psi_1) - (-A)^{\vartheta}
f(t_2,\psi_2) \| \leq  L_f (| t_1 - t_2| + \|\psi_1 - \psi_2\|_{\mathcal{B}}).\,
\end{equation}

\item[(H3)]  The function
   $g:I \times  {\mathcal{B}}\to X$ satisfies the following properties.
\begin{itemize}
\item[(a)] The function $g(\cdot,\psi):I\to X$ is strongly measurable
for every  $\psi \in \mathcal{B}$.

\item[(b)] The function  $g(t,\cdot) : \mathcal{B} \to X $  is
continuous for each $t \in I $.

\item[(c)] There exists an integrable
function  $m_g : I \to [0, \infty )$  and a continuous
nondecreasing function $ \Omega_g:[0,\infty)\to (0,\infty)$ such
that
\[
\| g(t,\psi )\| \leq  m_g(t)\Omega_g(\| \psi\|_{\mathcal{B}}
 ), (t,\psi)\in I\times \mathcal{B}.
\]
\end{itemize}
\end{itemize}

\begin{remark}\label{remark2}\rm
In the rest of this section,
$ M_{b}$ and $K_{b} $ are  the constants
$M_{b}= \sup_{s\in [0,b]}  M(s) $ and
$K_{b}=\sup_{s\in [0,b]}K(s)$.
\end{remark}

\begin{definition} \rm
A  function  $u:(-\infty, b] \to X$,
$0<b\leq a$, is  called a mild solution  of \eqref{eq1} on
$[0,b]$, if  $u_0 = \varphi$; $ u|_{[0,b]} \in C([0,b]:X)$; the
function
$  \tau \to A\mathcal{S}_{\alpha}(t-\tau)f(\tau,u_\tau)$ and
 $  \tau \to \int_0^{\tau} B(\tau - \xi )\mathcal{S}_{\alpha}(t-\tau)f(\xi
,u_{\xi}) d \xi$ is integrable on $[0,t)$ for all $t\in (0,b]$ and
for $t \in [0,b]$,
 \begin{equation} \label{solfraca}
\begin{split}
 u(t)&=\mathcal{R}_{\alpha}(t)(\varphi(0)+f(0,\varphi))-f(t,u_{t})
-\int_0^{t}A\mathcal{S}_{\alpha}(t-s)f(s,u_{s})ds   \\
&\quad -  \int_0^{t} \int_0^{s}B(s- \xi)\mathcal{S}_{\alpha}(t-s)
f(\xi, u_{\xi}) d\xi ds+
\int_0^{t}\mathcal{S}_{\alpha}(t-s)g(s,u_{s})ds.
\end{split}
\end{equation}
\end{definition}

\begin{theorem}\label{teoleray}
Let  conditions  {\rm (H1), (H2), (H3)} hold.
If
$$
K_b \Big[ L_f\Big( \| (-A)^{- \vartheta }\| +
\frac{Mb^{\alpha \vartheta}}{\alpha \vartheta} +   \frac{Mb^{\alpha
\vartheta}}{\alpha \vartheta} \int_0^b \mu(\xi)d\xi \Big)
+ M \liminf_{\xi \to \infty} \frac{\Omega_g(\xi)}{\xi} \int_0^b
m_g(s)ds\Big]  <1,
$$
then there exists a mild solution of \eqref{eq1} on $ [0,b]$.
\end{theorem}

\begin{proof}
 Consider the space $S(b)=\{ u \in C(I; X):
u(0)=\varphi(0) \}$ endowed with the uniform convergence topology
and  define the operator $\Gamma: S(b) \to S(b)$ by
\begin{align*}
 \Gamma x(t)
&= \mathcal{R}_{\alpha}(t)(\varphi(0)+f(0,\varphi)) -f(t,\overline{x}_t)
-\int_0^{t}A\mathcal{S}_{\alpha}(t-s)f(s,\overline{x}_s)ds   \\
&\quad - \int_0^{t} \int_0^{s}B(s-
\xi)\mathcal{S}_{\alpha}(t-s)f(\xi, \overline{x}_{\xi}) d\xi ds +
\int_0^{t}\mathcal{S}_{\alpha}(t-s)g(s,\overline{x}_{\rho(t,\overline{x}_s)})ds,
\end{align*}
for $ t \in [0,b]$.
From our assumptions, it is easy to see that $\Gamma S(b) \subset S(b)$.

Let $\bar{\varphi}:(-\infty,b]\to X$ be the extension of $\varphi$
to $(-\infty,b] $ such that $\bar{\varphi}(\theta)=\varphi(0)$ on
$I$. We prove that there exists $r>0$ such that
$\Gamma (B_r(\bar{\varphi}|_{I},S(b)))\subseteq
B_r(\bar{\varphi}|_{I},S(b))$.
If this property is false, then for every $r>0$ there exist
$x^r\in B_r(\bar{\varphi}|_{I},S(b))$ and
$t^r\in I$ such that $r<\|\Gamma x^r(t^r)-\varphi(0)\|$. Then, we
find that
\begin{align*}
&\| \Gamma x^r(t^r)-\varphi(0) \|  \\
& \leq \| \mathcal{R}_{\alpha}(t^r)(\varphi(0)+f(0,\varphi)) \|
 + \| f(t^r,\overline{x^r}_{t^r}) \| \\
&\quad + \int_0^{t^r}\| (-A)^{1- \vartheta}\mathcal{S}_{\alpha}(t^r-s) \|
 \| (-A)^{ \vartheta}f(s,\overline{x^r}_s)\| ds \\
&\quad +  \int_0^{t^r} \int_0^{s} \| B(s-
\xi)\mathcal{S}_{\alpha}(t^r-s)f(\xi, \overline{x^r}_{\xi}) \|  d\xi
ds + \int_0^{t} \| \mathcal{S}_{\alpha}(t-s) \| \|
g(s,\overline{x}_s)\| ds \\
&\leq  \| \mathcal{R}_{\alpha}(t^r)\varphi(0)-\varphi(0) \|
 + \| \mathcal{R}_{\alpha}(t^r)f(0,\varphi) - f(0,\varphi)\|  \\
&\quad + \| (-A)^{- \vartheta }\| \|(-A)^{ \vartheta }  f(t^r, \overline{(x^r)}_{t^r}) - (-A)^{ \vartheta }  f(0,\varphi) \|\\
&\quad + \int_0^{t^r} M(t^r-s)^{\alpha \vartheta-1} \| (-A)^{\vartheta }
f(s^r, \overline{(x^r)}_{s}) - (-A)^{ \vartheta } f(0,\varphi) \| ds \\
&\quad + \int_0^{t^r} M(t^r-s)^{\alpha \vartheta-1} \| (-A)^{
\vartheta }  f(0,\varphi) \| ds \\
&\quad +  \int_0^{t^r} \int_0^s \mu(s-\xi) M(t^r-s)^{\alpha \vartheta-1}
 \| (-A)^{ \vartheta } f({\xi}^r,
\overline{(x^r)}_{{\xi}}) - (-A)^{ \vartheta } f(0,\varphi) \| d\xi ds \\
&\quad + \int_0^{t^r} \int_0^s \mu(s-\xi)M(t^r-s)^{\alpha \vartheta-1} \| (-A)^{ \vartheta } f(0,\varphi) \| d\xi ds \\
&\quad + M \int_0^{t^r} m_g(t^r-s)\Omega_{g}(\|\overline{x^r}_{s}\|_{\mathcal{B}}) ds
 \\
& \leq   (M+1)H \|
\varphi\|_{\mathcal{B}} +  \| \mathcal{R}_{\alpha}(t^r)f(0,\varphi) -
f(0,\varphi)\|    \\
&\quad+ \Big( \frac{Mb^{\alpha \vartheta}}{\alpha
\vartheta}  +   \frac{Mb^{\alpha \vartheta}}{\alpha \vartheta}
\int_0^b \mu(\xi)d\xi  \Big) \|  (-A)^{ \vartheta }   f(0,\varphi)\|
\\
&\quad +  \| (-A)^{ -\vartheta }\|   L_f \left(  t^r  + \|
\overline{(x^r)}_{t^r} - \varphi \|_{\mathcal{B}} \right) \\
&
\quad + \int_0^{t^r}
M(t^r-s)^{\alpha \vartheta-1} L_f\left( s + \| \overline{(x^r)}_{s}
- \varphi \|_{\mathcal{B}} \right) ds \\
&\quad + \int_0^{t^r} \int_0^s
\mu(s-\xi)M(t^r-s)^{\alpha \vartheta-1} L_f\left( \xi + \|
\overline{(x^r)}_{{\xi}^r} - \varphi \|_{\mathcal{B}} \right) d\xi ds \\
&\quad + \Omega_g
 \left(K_br+ (M_b+ HK_b+1)\|\varphi\|_{\mathcal{B}} \right)\int_0^{b}m_g(s)ds \\
&\leq  (M+1)H \|
\varphi\|_{\mathcal{B}} +  \| R(t^r)f(0,\varphi) - f(0,\varphi)\|    \\
&\quad +  \Big(  \frac{Mb^{\alpha \vartheta}}{\alpha \vartheta}  +
\frac{Mb^{\alpha \vartheta}}{\alpha \vartheta}
\int_0^b \mu(\xi)d\xi   \Big) \|  (-A)^{ \vartheta }   f(0,\varphi)\|
\\
  &\quad + \Big(  \| (-A)^{- \vartheta}\| + \frac{Mb^{\alpha
\vartheta}}{\alpha \vartheta} +   \frac{Mb^{\alpha \vartheta}}{\alpha
\vartheta}
\int_0^b \mu(\xi)d\xi \Big) ( L_f (b+ (M_b +HK_b +1) \| \varphi\|_{_{\mathcal{B}}}) )
\\
&\quad +  \Big(  \| (-A)^{- \vartheta}\| + \frac{Mb^{\alpha
\vartheta}}{\alpha \vartheta} +   \frac{Mb^{\alpha \vartheta}}{\alpha
\vartheta}
\int_0^b \mu(\xi)d\xi \Big) L_f K_br  \\
 &\quad + \Omega_g
 \left(K_br+ (M_b+ HK_b+1)\|\varphi\|_{\mathcal{B}}\right)\int_0^{b}m_g(s)ds,
\end{align*}
where $i_c:Y \to X$ represents the continuous inclusion of $Y$ on
$X$. Therefore
\[
1\leq K_b \Big[ L_f\Big( \| (-A)^{- \vartheta }\| +
\frac{Mb^{\alpha \vartheta}}{\alpha \vartheta} +   \frac{Mb^{\alpha
\vartheta}}{\alpha \vartheta} \int_0^b \mu(\xi)d\xi \Big) + M
\liminf_{\xi \to \infty} \frac{\Omega_g(\xi)}{\xi} \int_0^b
m_g(s)ds\Big],
\]
which contradicts our assumption.

Let $r>0$ be  such that
$\Gamma(B_r(\bar{\varphi}|_{I},S(b)))\subseteq
B_r(\bar{\varphi}|_{I},S(b))$.  In the sequel, $r^{*}$ and
$r^{**}$ are the numbers defined by $r^*:=(K_br+ (M_b+
HK_b+1)\|\varphi\|_{\mathcal{B}})$ and
$r^{**}:=\Omega_g(r^*)\int_0^{b}m_g(s)ds$.

  To prove that $ \Gamma$ is  a condensing operator, we introduce
the decomposition   $\Gamma=\Gamma_1+\Gamma_2$,  where, for $ t \in I$,
\begin{gather*}
\begin{aligned}
 \Gamma_1 x(t)
&=\mathcal{R}_{\alpha}(t)(\varphi(0)+f(0,\varphi)) -f(t,\overline{x}_t)
-\int_0^{t}A \mathcal{S}_{\alpha}(t-s)f(s,\overline{x}_s)ds  \\
&\quad  - \int_0^{t} \int_0^{s}B(s-
\xi)\mathcal{S}_{\alpha}(t-s)f(\xi, \overline{x}_{\xi}) d\xi ds,
\end{aligned} \\
 \Gamma_2 x (t) =
\int_0^{t}\mathcal{S}_{\alpha}(t-s)g(s,\overline{x}_{s})ds\,.
\end{gather*}


 On the other hand, for $ u, v
\in B_r(\bar{\varphi}|_{I},S(b))$  and $t\in [0,b]$ we see that
\begin{align*}
& \| \Gamma_1 u(t)-  \Gamma_1 v(t) \|  \\
&\leq  \| (-A)^{- \vartheta }\| \| (-A)^{ \vartheta } f(t,\overline{u}_t)
  - (-A)^{ \vartheta } f(t, \overline{v}_t) \| \\
 &\quad + \int_0^{t} \| (-A)^{1-\vartheta}\mathcal{S}_{\alpha}(t-s) \| \|
(-A)^{ \vartheta }  f(s,\overline{u}_s)  - (-A)^{ \vartheta }
f(s, \overline{v}_s) \|_{Y} ds \\
&\quad + \int_0^{t} \int_0^{s}  \| B(s- \xi)\mathcal{S}_{\alpha}(t-s)
 f(\xi,\overline{u}_\xi)  - f(\xi, \overline{v}_\xi) \| d\xi ds  \\
 & \leq   \| (-A)^{ -\vartheta } \| L_{f}K_b \| u - v  \|_{b}
 + L_{f} K_b \int_0^t M(t-s)^{\alpha \upsilon -1} ds \| u - v  \|_{b}   \\
& \quad + L_{f} K_b \int_0^t \int_0^s \mu(s-\xi) M(t-s)^{\alpha \vartheta -1}
d \xi ds \| u - v
\|_{b}, \\
&\leq    L_{f} K_b  \left( \| (-A)^{ -\vartheta } \| +
\frac{Mb^{\alpha \vartheta}}{\alpha \vartheta } +  \frac{Mb^{\alpha
\vartheta}}{\alpha \vartheta } \int_0^b \mu(\xi)d\xi \right) \| u -
v \|_{b},
\end{align*}
which show  that  $\Gamma_1(\cdot)  $ is a contraction on
$B_r(\bar{\varphi}|_{I},S(b))$.


Next we prove that $\Gamma_2(\cdot)$ is a completely continuous function from
$B_r(\bar{\varphi}|_{I},S(b))$ to
$B_r(\bar{\varphi}|_{I},S(b))$.

\noindent\textbf{Step 1.} The set $\Gamma_2(B_r(\bar{\varphi}|_{I},S(b))(t)$
is relatively compact on $X$ for every $t\in [0,b]$.
The case $t=0$ is trivial.  Let $0< \epsilon < t <b$. From the
assumptions, we can fix numbers $0= t_0 < t_1 < \cdots < t_{n}
= t-\epsilon$
such that $\| \mathcal{S}_{\alpha}(t-s)- \mathcal{S}_{\alpha}(t-s')\| \leq \epsilon   $
if $ s,s'\in  [t_{i},t_{i+1}]$, for some $i=0,1,2,\cdots,n-1$.
let $ x \in  B_r(\bar{\varphi}|_{I},S(b))$.
  Under theses conditions,  from  the mean value theorem for the
Bochner Integral (see \cite[ Lemma 2.1.3]{Ma})  we see that
\begin{align*}
 \Gamma_2 x(t)&= \sum_{i =1}^{n} \int_{t_{i -1}}^{t_{i}}
\mathcal{S}_{\alpha}(t-t_{i}) g(s,\overline{x}_{s}) ds \\
 &\quad +\sum_{i =1}^{n} \int_{t_{i -1}}^{t_{i}}
(\mathcal{S}_{\alpha}(t-s)-\mathcal{S}_{\alpha}(t-t_{i}))
 g(s,\overline{x}_{\rho(t,\overline{x}_s)})ds \\
&\quad +  \int_{t_{n}}^{t}
\mathcal{S}_{\alpha}(t-s)g(s,\overline{x}_{s}) ds \\
 &\in  \sum_{i =1}^{n} (t_{i}-t_{i-1})
 \overline{{\rm co}(\{ \mathcal{S}_{\alpha}(t-t_i)g(s,\psi):
\psi \in B_{r^{*}}(0,\mathcal{B}), s\in [0,b]   \})}\\
&\quad + \epsilon \  r ^{**}+ M \Omega_g(r^{*}) \int_{t-\epsilon}^{t}m_g(s) ds\\
&\in  \sum_{i =1}^{n} (t_{i}-t_{i-1})
 \overline{{\rm co}(\{  W^2_{r
 ^{*}}( t-t_i )  \})}
 + \epsilon B_{r^{**}}(0,X)+ C_{\epsilon},
\end{align*}
where  $\operatorname{diam} (C_{\epsilon})\to 0 $ when  $\epsilon \to 0$. This prove that
 $\Gamma_2( B_{q}(0,S(b)))(t)$ is totally bounded and hence
relatively compact in $X$ for every  $t\in [0,b]$.

\noindent\textbf{Step 2.}  The set  $\Gamma_2(
B_r(\bar{\varphi}|_{I},S(b)))$ is equicontinuous on  $ [0,b]$.
Let $0<\epsilon<t<b$ and  $0<\delta<\epsilon$ such that
$ \| \mathcal{S}_{\alpha}(s) -\mathcal{S}_{\alpha}(s') \| \leq \epsilon $ for every
$s,s' \in [ \epsilon, b]$ with $| s-s'|\leq \delta$.
 Under these conditions, for $ x \in  B_r(\bar{\varphi}|_{I},S(b))$ and
 $0< h \leq \delta $ with $t+h \in [0,b]$, we obtain
 \begin{align*}
& \| \Gamma_2 x(t+h)- \Gamma_2 x(t)\|  \\
&\leq  \int_0^{t-\epsilon} [\mathcal{S}_{\alpha}(t+h-s)-
\mathcal{S}_{\alpha}(t-s)] g(s,\overline{x}_{s}) ds \\
&\quad + \int_{t-\epsilon}^{t}[\mathcal{S}_{\alpha}(t+h-s)
 - \mathcal{S}_{\alpha}(t-s) ]g(s,\overline{x}_{s}) ds
 +  \int_{t}^{t+h}\mathcal{S}_{\alpha}(t+h-s)g(s,\overline{x}_{s}) ds\\
& \leq  \epsilon  r^{**} + 2M \Omega(r^{*}) \int_{t-\epsilon}^{t}
m_g(s) ds +   M \Omega(r^{*}) \int_t^{t+h} m_g(s) ds
\end{align*}
which shows  that the set of functions
$\Gamma_2( B_r(\bar{\varphi}|_{I},S(b)))$ is right equicontinuity at
$t\in (0,b)$. A similar procedure permit to prove  the right
equicontinuity at zero and the left equicontinuity at
$t\in (0,b]$. Thus, $\Gamma_2(  B_r(\bar{\varphi}|_{I},S(b)))$ is
equicontinuous. By using a similar procedure  to proof of
\cite[Theorem 2.3]{Hern3}, we prove that that $\Gamma_2(\cdot)$
is continuous on $ B_r(\bar{\varphi}|_{I},S(b))$, which completes
the proof that  $ \Gamma_2(\cdot)$ is completely continuous.

 To complete the prove that $\Gamma_1(\cdot)$ is continuous,
let  $ (x^n)_{n \in\mathbb{N}}$ be a sequence in $ B_r(\bar{\varphi}|_{I},S(b))$
and  $ x \in  B_r(\bar{\varphi}|_{I},S(b))$  such that  $x^{n}\to x $ in
$ B_r(\bar{\varphi}|_{I},S(b))$.  From the phase space  axioms we
infer that
$\overline{(x^{n})}_{s}\to \overline{x}_{s}$ uniformly for $s\in I$ as
$n\to \infty$.
  Consequently, from \eqref{lipcf},
$\| (-A)^{-\vartheta}f(s, \overline{(x^{n})}_{s})- (-A)^{-\vartheta}
f(s,\overline{x}_{s})\| \to 0$, uniformly on
$[0,b]$ as $n\to\infty$.
Now, a standard application of the Lebesgue dominated convergence
Theorem permits to conclude that $\Gamma_1(\cdot)$ is continuous
on $ B_r(\bar{\varphi}|_{I},S(b))$. The existence of a mild
solution for \eqref{eq1} is now a consequence of
\cite[Theorem 4.3.2]{Ma}. This completes the proof.
\end{proof}

\section{Applications}\label{aplications}

To complete this paper, we  discuss   the existence of solutions for
the partial integro-differential system
\begin{gather}
\begin{aligned}
&\frac{\partial^{\alpha}}{\partial t^{\alpha}} \Big( u(t, \xi) +
\int_{-\infty}^{t}\,\int_0^{\pi} b(t-s,\eta,\xi) u(s,\eta)d \eta
ds \Big)\\
&= \frac{\partial^{2}} {\partial \xi^{2}}  u(t, \xi) +
\int_0^t (t-s)^{\delta} e^{-\gamma(t-s)} \frac{\partial^{2}}
{\partial \xi^{2}} u(s, \xi)
 ds  + \int_{-\infty}^{t} a_0(s-t) u(s, \xi)ds,\\
&\quad  (t,\xi)   \in I\times [0,\pi],
\end{aligned}  \label{eqexemcseg1}
\\
u(t, 0)   =   u(t, \pi)  =  0,\quad  t \in [ 0,b],  \\
u(\theta, \xi)   =  \phi(\theta, \xi), \quad
 \theta \leq 0,\; \xi\in [0,\pi]. \label{eqexemcseg2}
\end{gather}
Where $\frac{\partial^{\alpha}}{\partial t^{\alpha}} =D^{\alpha}_t$,
$ \alpha \in (1,2)$. To treat this system in the abstract form
\eqref{eqf1}-\eqref{eqf2}, we choose the space
$ X = L^{2}([0, \pi])$, $\mathcal{B} = C_0 \times L^p(g,X)$ is the space
 introduced in Example \ref{example1} and $A:D(A) \subseteq X\to X$ is the
operator defined by $ A x= x^{\prime \prime} $, with domain
 $ D(A) = \{ x \in X :x^{\prime \prime} \in X, x(0) = x(\pi) = 0 \}$. It
is well known that $A$ is the infinitesimal generator of an
analytic semigroup $(T(t))_{t\geq 0}$ on $X$. Moreover,  $A$ has a
discrete spectrum with eigenvalues of the form $- n^{2},n \in
\mathbb{N}$, and corresponding normalized eigenfunctions given by
$ z_{n} (\xi) := (\frac{2}{\pi})^{1/2}\sin (n \xi)$ and
the following properties hold
\begin{itemize}
\item[(a)] $\{z_{n} : n \in \mathbb{N} \}$ is an orthonormal basis of $X$.

\item[(b)] For
$x\in  X, {T(t)x = \sum_{n=1}^{\infty} e^{-n^{2}t} \langle x, z_{n}\rangle
z_{n}}$.

\item[(c)] For $\alpha \in (0,1)$, the fractional power
$(-A)^{\alpha}: D((-A)^{\alpha}) \subset X \to
X$ of $A$ is given by $(-A)^{\alpha}x= \sum_{n=1}^{\infty}
n^{2\alpha} \langle x, z_{n} \rangle z_n$, where $D((-A)^{\alpha})
=\{ x \in X: (-A)^{\alpha}x \in X \}$.
\end{itemize}
 Hence, $A$ is sectorial of type and
  the properties (P1)  hold. We also consider  the
operator $B(t): D(A) \subseteq X \to X$, $t\geq 0 $,
 $ B(t)x  = t^{\delta}e^{-\gamma t} A x $ for $x\in D(A)$.
 Moreover, it is easy to see that  conditions   (P2)-(P3)
 in Section \ref{preliminaries} are satisfied with
 $b(t)= t^{\delta}e^{-\gamma t} $ and   $D=C_0^{\infty}([0,\pi])$, where
  $C_0^{\infty}([0,\pi])$
is  the space of infinitely differentiable functions that
vanish at $\xi =0$ and $\xi = \pi$. From the Lemma \ref{estpr1}
it is easy to see that condition
 (H1) is satisfies.

 In the sequel, we  assume that
$\varphi(\theta)(\xi)=\phi(\theta,\xi) $ is a function in $\mathcal{B} $
and that the  following conditions are verified.
\begin{itemize}
\item[(i)]   The functions  $a_0:\mathbb{R}\to
\mathbb{R} $ are continuous and $
   L_g:=   \left(\int_{-\infty}^{0} \frac{(a_0(s))^{2}}{g
(s)}ds \right)^{1/2}<\infty$.

\item[(ii)]   The functions  $\rho_i:[0,\infty) \to
[0,\infty), i=1,2$, are continuous.

 \item[(iii] The functions
   $   b(s, \eta, \xi)$,
   $ {\frac{\partial b(s, \eta, \xi)} {\partial \xi} }$
are  measurable,  $b(s,\eta, \pi) = b(s,\eta,\, 0) = 0$ for all
$(s,\eta)$ and
\[
 L_f:= \max \{(\int_0^{\pi} \int_{-\infty}^{0}
\int_0^{\pi} g^{-1}(\theta)\Big(\frac{\partial^{i}}{\partial\xi
^{i}} b(\theta, \eta, \xi)\Big)^2 d\eta d\theta d\xi
)^{1/2}:i=0,1 \}<\infty.
\]
\end{itemize}



Defining the operators $f, g  :I \times \mathcal{B}\to X$ by
\begin{gather*}
f(\psi)(\xi)  =    \int_{-\infty}^{0}\int_0^{\pi}
b(s,\eta,\xi) \psi (s,\eta)d\eta ds ,  \\
g(\psi)(\xi)  = \int_{-\infty}^{0} a_0(s) \psi(s, \xi)ds.
\end{gather*}
we can transform  \eqref{eqexemcseg1}-\eqref{eqexemcseg2} into the
abstract system  \eqref{eqf1}-\eqref{eqf2}. Moreover, $f, g$ are
bounded linear operators with
$\| f(\cdot)\|_{\mathcal{L}(\mathcal{B},X)} \leq L_f$ and
$\| g(\cdot)\|_{\mathcal{L}(\mathcal{B},X)} \leq L_g$.
 Moreover, a straightforward estimation using (ii) shows that
 $f(I \times \mathcal{B}) \subset D((-A)^{1/2})$  and
$\| (-A)^{1/2}f \|_{\mathcal{L}(\mathcal{B},X)} \leq L_f $. The
following result is a direct consequence of Theorem
\ref{teoleray}.

\begin{proposition}\label{coroexemplo1}
 If
$$
\Big( 1 + \int_{-b}^{0} g(\theta) \,d \theta )  \Big)
\Big( L_f\Big(\| (-A)^{-1/2}\|
+C_{1/2}\sqrt{b}+ C_{1/2}\sqrt{b} \int_0^{b}a(s)ds \Big)+
L_g \Big)<1,
$$
then there exist a mild solutions of
\eqref{eqexemcseg1}--\eqref{eqexemcseg2}.
\end{proposition}

\subsection*{Acknowledgments}
Bruno de Andrade is partially supported by grant  BEX 5549/11-6 
from CAPES - Brazil.
J. dos Santos is partially supported grant CEX-APQ-00476-09 from 
FAPEMIG - Brazil.

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