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

\AtBeginDocument{{\noindent\small 
\emph{Electronic Journal of Differential Equations}, 
Vol. 2009(2009), No. 164, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/164\hfil Existence of solutions]
{Existence of solutions to quasilinear functional differential
equations}

\author[S. Abbas,  D. Bahuguna\hfil EJDE-2009/164\hfilneg]
{Syed Abbas, Dhirendra Bahuguna}  % in alphabetical order

\address{Syed Abbas \newline
Department  of Mathematics and  Statistics, Indian  Institute of
Technology Kanpur,  Kanpur - 208016, India}
\email{sabbas.iitk@gmail.com}

\address{Dhirendra Bahuguna \newline
Department  of Mathematics and Statistics, Indian  Institute of
Technology Kanpur, Kanpur - 208016, India} \email{dhiren@iitk.ac.in}

\thanks{Submitted May 23, 2009 Published December 21, 2009.}
\thanks{Supported by grant SR/S4/MS:581/09 from DST, India}
\subjclass[2000]{47D60, 34A12, 35K90} 
\keywords{$C_0$-semigroup; quasilinear differential equation; mild solution}

\begin{abstract}
 In this article we use the theory of $C_0$-semigroup of
 bounded linear operators to establish the existence and uniqueness
 of a classical solution to a quasilinear functional differential
 equation considered in a Banach space.
\end{abstract}

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


\section{Introduction}

In this article we study the the existence and uniqueness of a
classical solution to the following quasilinear functional
differential equation, considered in a Banach space $X$,
\begin{equation}  \label{eq1}
\begin{gathered}
\frac{du(t)}{dt}+A(t,u(t))u(t) = F(t,u_{t}), \quad t\in [0,T], \\
u_0 =\phi  \quad \text{on} \quad [-\tau,0],
\end{gathered}
\end{equation}
where $u_t(\theta)=u(t+\theta)$, $\theta \in [-\tau,0]$. For $t \in
[0,T]$, we denote by $\mathcal{C}_t$ the Banach space of all
continuous functions from $[-\tau,t]$ to $X$ endowed with the
supremum norm
$$
\|\chi\|_{\mathcal{C}_t}=\sup_{-\tau \le \theta \le
t}\|\chi(\theta)\|_X, \quad \chi \in \mathcal{C}_t.
$$
The function $F(t, \psi)$ is defined on $[0,T] \times \mathcal{C}_0$
to $X.$ Here we see that $u_t \in \mathcal{C}_0$. We assume that for
$u \in C_T,$ $F(\cdot,u_{(\cdot)}):[0,T] \to X$ is a bounded $L^1$
function. Further we assume that there is a subset $B$ of $X$ such
that for $(t,u) \in [0,T] \times \mathcal{C}_T$ with $u(t) \in B$
for $t \in [0,T]$, $A(t,u(t))$ is a linear operator in $X$. Also
$\phi \in \mathcal{C}_0$ is Lipschitz continuous with Lipschitz
constant $L_{\phi}$.

Quasilinear evolution equations forms a very important class of
evolution equations as many time dependent phenomena in physics,
chemistry and biology can be represented by such evolution
equations. For more details on the theory and applications of
quasilinear evolution equations we refer to \cite{caps, ladas,zeid}.

Kato \cite{katos}  considered the  quasilinear evolution equation
\begin{equation} \label{eq01}
\begin{gathered}
\frac{du(t)}{dt}+A(t,u(t))u(t) = G(t,u(t)), \quad t\in (0,T], \\
u(0) =u_0,
\end{gathered}
\end{equation}
in a Banach space and shown the existence of a strong solution under
suitable assumptions on $A$ and $G$. The various cases of equation
\eqref{eq01} have been treated by Amann \cite{amman} in the
interpolation spaces using the theory of analytic semigroups.
Bahuguna \cite{bd} has shown the existence of a classical solution
of the following integrodifferential equation considered in a Banach
space,
\begin{equation} \label{eq11}
\begin{gathered}
\frac{du(t)}{dt}+A(t,u(t))u(t) = K(u)(t)+f(t), \quad t\in [0,T], \\
u(0) =x,
\end{gathered}
\end{equation}
where
$$
K(u)(t)=\int_{0}^{t}a(t-s)k(s,u(s))ds,
$$
and $A(t,w)$ is a linear operator in $X$ for each $(t,w) \in [0,T]
\times W$, $W$ being an open subset of $X$. In this paper we
strengthen the result of \cite{bd} for a functional differential
equation. We show the existence and uniqueness of a classical
solution of \eqref{eq1}.

  \section{Preliminaries}

Let $B(X,Y)$ be the set of all bounded linear operators from $X$ to
$Y$. $B(X,Y)$ is a Banach space with the norm
$$
\|A\|_{B(X,Y)}=\sup_{x\in X,x\neq0} \frac{\|Ax\|_Y}{\|x\|_X}.
$$
We denote $B(X,X)$ by $B(X)$. Let $B$ be a subset of $Y$, where $Y$
is densely and continuously embedded in $X$. Since $Y$ is
continuously embedded in $X$ so it is a subset of $X$ too. A family
$\{A(t,w), (t,w) \in [0,T]\times B\}$ of infinitesimal generators of
a $C_0$-semigroup $S_{t,w}(s)$, $s \ge 0$ on $X$ is called stable if
there exist constants $M \ge 1$ and $w$, known as stability
constants, such that
$$
\rho(A(t,w)) \supset (w,\infty) \quad (t,w) \in [0,T]\times B,
$$
where $\rho(A(t,w))$ is the resolvent set of $A(t,w)$ and
$$
\big\|\prod_{j=1}^{k}R(\lambda:A(t_j,w_j))\big\|_{B(X)} \le
\frac{M}{(\lambda -w)^k} \quad for \ \lambda >w
$$
and every finite sequence
$$
0 \le t_1 \le t_2 \cdots \le t_k \le T, \quad w_j \in B.
$$
 Let $S_{t,w}(s)$, $s \ge 0$ be the $C_0$-semigroup generated by
$A(t,w)$. A subspace $Y$ of $X$ is called $A(t,w)$-admissible if $Y$
is an invariant subspace of $S_{t,w}(s), s \ge 0$, and the
restriction of $S_{t,w}(s)$ to $Y$ is a $C_0$-semigroup in $Y$. We
will use the following hypothesis on $A(t,w)$:
\begin{itemize}
\item[(H1)] There is a subset $B$ in $X$ such that the family
$\{A(t,w),(t,w) \in [0,T] \times B\}$ is stable.

\item[(H2)] $Y$ is $A(t,w)$-admissible for all
 $(t,w)$ in $[0,T]\times B$ and the family\\
 $\{\tilde{A}(t,w),(t,w) \in [0,T] \times
 B\}$ of parts of $A(t,w)$ in $Y$ is stable  in $Y$.

\item[(H3)] For $(t,w) \in [0,T] \times B$, $A(t,w)$ is a
bounded linear operator from $Y$ to $X$ and $A(\cdot,w)$ is
continuous in $B(Y,X)$ i.e. $A(\cdot,w) \in C([0,T],B(Y,X))$ also
$D(A(t,w))\supset Y$.

\item[(H4)] There exists a positive constant $L_A$ such that
$$
\|A(t,w_1)-A(t,w_2)\|_{B(Y,X)} \le L_A \|w_1-w_2\|_Y
$$
for all $(t,w_1),(t,w_2) \in [0,T]\times B$.
\end{itemize}
Next we define an evolution family as follows.

\begin{definition} \label{def2.1} \rm
A two parameter family of bounded linear operators $U(t,s), t\ge s
\ge 0$, on $X$ is called an evolution system if
\begin{itemize}
\item[(i)] $U(s,s)=I$ and $U(t,r)U(r,s)=U(t,s)$, $t\ge r\ge s\ge 0$;
\item[(ii)] $(t,s) \to U(t,s)$ is strongly continuous for
$t\ge s\ge 0$.
\end{itemize}
\end{definition}

If $u \in C([0,T],X)$ and the family $\{A(t,w),(t,w) \in [0,T]\times
X\}$ of operators satisfies (H1)--(H4) then there exists an
evolution system $U_u(t,s)$ (\cite[Theorem 4.6]{pa}) in $X$
satisfying:
\begin{itemize}
\item[(i)]  $\|U_u(t,s)\|_{B(X)} \le M e^{\delta(t-s)}$ for
$t\ge s \ge0$, where $M$ and $\delta$ are the stability constants;

\item[(ii)]  $\frac{\partial ^+}{\partial t}U_u(t,s)w|_{t=s}=A(s,u(s))w$
for $w\in Y$;

\item[(iii)] $\frac{\partial ^+}{\partial
s}U_u(t,s)w|_{t=s}=-U_u(t,s)A(s,u(s))w$ for $w\in Y$.

\end{itemize}
Moreover there exists a constant $C_0 >0$ such that for every $u,v
\in C([0,T],X)$ with values in $B$ and every $y \in Y$ we have
$$
\|U_u(t,s)y-U_v(t,s)y\|_X \le C_0\|y\|_Y
\int_{s}^{t}\|u(\xi)-v(\xi)\|_X\,d\xi.
$$
Now we mention some additional hypotheses.
\begin{itemize}

\item[(H5)] For each $u\in C(\mathbb{R},X)$, we have
$$
U_u(t,s)Y \subset Y, \quad s,t \in \mathbb{R}, s\le t,
$$
 and $U_u(t,s)$ is strongly continuous in $Y$.

\item[(H6)]  Every closed convex and bounded subset of $Y$
is also closed in $X$.

\item[(H7)] There exists a constant $L_F>0$ such that
$$
\|F(t, \phi_1)-F(s, \phi_2)\|_X \le
L_F(|t-s|+\|\phi_1-\phi_2\|_{\mathcal{C}_0})
$$
for all $(t,\phi_1), (s, \phi_2) \in [0,T]\times {\mathcal{C}_0}$.
\end{itemize}

We note that the condition (H6)) is always satisfied if $X$ and $Y$
are reflexive Banach spaces.

\begin{definition} \label{def2.2} \rm
A function $u\in \mathcal{C}_T$ with values in $B$ satisfying
\begin{gather*}
u(t)=U_u(t,0)\phi(0)+\int_{0}^{t}U_u(t,s)F(s,u_s)ds, \quad t \in [0,T]\\
u_0=\phi \quad\text{on } [-\tau,0],
\end{gather*}
is called a mild solution to \eqref{eq1} on $[0,T]$.
\end{definition}

\begin{definition} \label{def2.3} \rm
A function $u\in \mathcal{C}_T$ such that $u(t)\in Y \cap B$ for
$t\in (0,T]$ and $u \in \mathcal{C}^1((0,T],X)$ satisfying the
equation \eqref{eq1} in $X$ is called a classical solution to
\eqref{eq1} on $[0,T]$. Where $\mathcal{C}^1([0,T],X)$, space of all
continuously differentiable functions from $[0,T]$ to $X$ and $Y$ is
a $A(t,w)$-admissible subspace of $X$.
\end{definition}

  \section{Main result}

 In this section we prove the existence and
uniqueness result for a classical solution to \eqref{eq1}. Let
$\tilde{\phi} \in \mathcal{C}_T$ be given by
$\tilde{\phi}(t)=\phi(t)$ for $t \in [-\tau,0]$ and
$\tilde{\phi}(t)=\phi(0)$ for $t \in [0,T]$. Denote
\begin{gather*}
B_r(\phi(0))=\{x \in X : \|x-\phi(0)\|_X \le r\},\\
B_{2r}(\tilde{\phi_0})=\{\chi \in \mathcal{C}_0 :
\|\chi-\tilde{\phi_0}\|_{\mathcal{C}_0} \le 2r\}.
\end{gather*}

\begin{theorem}  \label{thm11}
Let $B$ and $V$ be  open subsets of $X$ and $\mathcal{C}_0$,
respectively, and the family $\{A(t,w)\}$ of linear operators for
$t\in [0,T]$ and $w\in B_r(\phi(0))$ satisfy  assumptions 
{\rm (H1)-(H6)} and $A(t,w)\phi(0) \in Y$ with
$$
\|A(t,w)\phi(0)\|_Y \le C
$$
for all $(t,w) \in [0,T] \times B$. Suppose $F(t,u_t)$ satisfies
$\rm{(H7)}$. Then there exists a unique local classical solution of
\eqref{eq1}.
\end{theorem}

\begin{proof}
 From assumption (H5) for $t \ge s$,
$t,s \in [0,T]$ and $u \in \mathcal{C}([0,T];X)$ with values in $B$,
we have
$$
\|U_u(t,s)\|_{B(Y)} \le C_1.
$$
Take $r >0$ such that $B_r(\phi(0)) \subset B$ and
$B_{2r}(\tilde{\phi_0}) \subset V$. Choose
$$
T_0=\min\Big\{T, \frac{r}{2C_1 C\|\phi(0)\|_X},
\frac{r}{L_F},\frac{r}{2C_1(2L_Fr+N)},
\frac{1}{n\Lambda},\frac{r}{L_{\phi}}\Big \}
$$
where $\Lambda= C_0 \|\phi(0)\|_X +C_1
L_F+C_0(2L_Fr+N)\frac{T_0}{2}$, $n>1$ is any natural number and
$\|F(s,u_0)\|_X \le N$, where $N$ is a positive constant.

Define the set
$$
S=\{\psi \in\mathcal{C}_{T_0}:\psi_0=\phi, \mbox{ for }  t\in
[-\tau,0], \psi(t)\in B_r(\phi(0)), \ t\in [0,T_0]\}.
$$
We easily deduce that $S$ is a closed, convex and bounded subset of
$\mathcal{C}_{T_0}$. Take $\psi \in S$. Now for $\theta \in
[-\tau,0]$ we have the following two cases.

\textbf{Case 1:} If $t+\theta \le 0$ we have
\begin{align*}
\|\psi_t(\theta)-\tilde{\phi_0}(\theta)\|_X
&=\|\psi(t+\theta)-\tilde{\phi}(\theta)\|_X \\
&= \|\phi(t+\theta)-\phi(\theta)\|_X \quad
\text{(by  the definition  of $S$)} \\
&\le L_\phi T_0 \le r.
\end{align*}

\textbf{Case 2:} If $t+\theta \ge 0$ we have
\begin{align*}
\|\psi_t(\theta)-\tilde{\phi_0}(\theta)\|_X
&=\|\psi(t+\theta)-\tilde{\phi}(\theta)\|_X \\
&\le \|\psi(t+\theta)-\phi(0)\|_{X}+||\phi(0)-\phi(\theta)\|_X \\
&\le r+L_\phi (-\theta) \quad
\text{(since $\psi(t+\theta) \in B_r(\phi(0))$)} \\
&\le r+L_\phi t  \\
&\le  r+L_\phi T_0 \le 2r \quad \text{(since $-\theta \le t \le
T_0$).}
\end{align*}
Thus, for $\psi \in S$, $\psi_t \in B_{2r}(\phi)$. Define $G:S \to
S$ by
\[
Gu(t)=
\begin{cases}
U_u(t,0)\phi(0)+\int_{0}^{t}U_u(t,s)F(s,u_s)ds, & t \in
[0,T_0],\\
\phi(t), & t \in [-\tau,0].
\end{cases}
\]
First we show that $G$ is well defined and $Gu(0)=\phi(0)$. For $t
\ge 0$, we have
$$
Gu(t)-\phi(0)=U_u(t,0)\phi(0)-\phi(0)+\int_{0}^{t}U_u(t,s)F(s,u_s)ds.
$$
Taking the norm, we get
\[
\|Gu(t)-\phi(0)\|_X \le \|U_u(t,0)\phi(0)-\phi(0)\|_X
+\int_{0}^{t}\|U_u(t,s)F(s,u_s)\|_X ds.
\]
Integrating (iii), we obtain
$$
U_u(t,0)\phi(0)-\phi(0)=\int_{0}^{t}U_u(t,s)A(s,u(s))\phi(0)ds.
$$
Thus we have
\begin{equation} \label{eq3.1}
\begin{aligned}
\|U_u(t,0)\phi(0)-\phi(0)\|_X
& \le \int_{0}^{t}\|U_u(t,s)A(s,u(s))\|_X\|\phi(0)\|_X ds \\
&\le  C_1CT_0\|\phi(0)\|_X \le \frac{r}{2}.
\end{aligned}
\end{equation}
Also, we have
 %\label{eq3.2}
\begin{align*}
\int_{0}^{t}\|U_u(t,s)F(s,u_s)\|_X ds & \le  C_1
\int_{0}^{t}(\|F(s,u_s)-F(s,u_0)\|_X+\|F(s,u_0)\|_X) ds \\
& \le  C_1 \int_{0}^{t}(\|F(s,u_s)-F(s,\phi)\|_X+\|F(s,\phi)\|_X)
ds \\
& \le  C_1 \int_{0}^{t}(L_F\|u_s-\phi\|_X+N) ds \\
& \le  C_1 (2L_Fr+N)T_0 \le \frac{r}{2},
\end{align*}
using the result that for $u \in S$, $u_s \in B_{2r}(\phi)$. Thus,
for $u \in S$ and $t \ge 0$, we get
$$
\|Gu(t)-\phi(0)\|_X \le r.
$$
So $G$ is well defined. For $u, v \in S$, we consider
\begin{align*}
Gu(t)-Gv(t)
&= U_u(t,0)\phi(0)-U_v(t,0)\phi(0) \\
&\quad + \int_{0}^{t}(U_u(t,s)F(s,u_s)-U_v(t,s)F(s,v_s))ds.
\end{align*}
Let
\begin{equation}
\begin{aligned}
I_1 &=  \|U_u(t,0)\phi(0)-U_v(t,0)\phi(0)\|_X \\
&\leq C_0\|\phi(0)\| \int_{0}^{t}\|u(s)-v(s)\|_X ds \\
&\leq  C_0\|\phi(0)\|_X \|u-v\|_{\mathcal{C}_{T_0}}T_0.
\end{aligned}\label{ceq1}
\end{equation}
Also let
\begin{equation}
\begin{aligned}
I_2 &= \big\|\int_{0}^{t}(U_u(t,s)F(s,u_s)-U_v(t,s)F(s,v_s))ds\big\|_X\\
&\leq \int_{0}^{t}\Big(\|(U_u(t,s)F(s,u_s)-U_u(t,s)F(s,v_s)\|_X\\
&\quad + \|U_u(t,s)F(s,v_s)-U_v(t,s)F(s,v_s))\|_X \Big)ds \\
&\leq  C_1 L_F \int_{0}^{t}\|u_s-v_s\|_{\mathcal{C}_0}ds + C_0
\int_{0}^{t} \|F(s,v_s)\|_X \int_{s}^{t}\|u(\xi)-v(\xi)\|_X d\xi
ds \\
&\leq  C_1 L_F\int_{0}^{t}\sup_{\theta}\|u(s+\theta)-v(s+\theta)\|_X
ds
\\
&\quad + C_0 (2L_F r+N)\int_{0}^{t}\int_{s}^{t}\|u(\xi)-v(\xi)\|_X
d\xi ds
\\
&\leq C_1 L_F T_0\|u-v\|_{\mathcal{C}_{T_0}}+ C_0
(2L_Fr+N)\int_{0}^{t}\int_{0}^{s}\|u(\xi)-v(\xi)\|_X d\xi ds
\\
&\leq  C_1 L_F T_0\|u-v\|_{\mathcal{C}_{T_0}}
+C_0(2L_Fr+N)\|u-v\|_{\mathcal{C}_{T_0}}\frac{T_0^2}{2} \\
&\leq  (C_1 L_F+C_0(2L_Fr+N))\frac{T_0^2}{2}
\|u-v\|_{\mathcal{C}_{T_0}}.
\end{aligned}\label{ceq2}
\end{equation}
Hence from (\ref{ceq1}) and (\ref{ceq2}) we get
\begin{equation}
\begin{aligned}
I_1+I_2
& =  \|Gu(t)-Gv(t)\|_X \\
&\leq \Big(C_0\|\phi(0)\|_X T_0+(C_1
L_F+C_0(2L_Fr+N))\frac{T_0^2}{2}\Big) \|u-v\|_{\mathcal{C}_{T_0}}\\
&\leq  \Lambda T_0 \|u-v\|_{\mathcal{C}_{T_0}} \\
&\le \frac{1}{n}\|u-v\|_{\mathcal{C}_{T_0}}.
\end{aligned} \label{ceq}
\end{equation}
Thus $G$ is a contraction from $S$ to $S$. So, by the Banach
contraction mapping theorem, $G$ has a unique fixed point $u \in S$
which satisfies the integral equation. Hence it is a mild solution
of \eqref{eq1}.  Now, we consider the following evolution equation
\begin{equation}
\begin{gathered}
\frac{dv(t)}{dt}+A(t,u(t))v(t) = F(t,u_t), \quad t \in [0,T_0],\\
u(0) = \phi(0).
\end{gathered} \label{eq2}
\end{equation}
Denote $\tilde{A}(t)=A(t,u(t))$ and $\tilde{F}(t)=F(t,u_t)$, then
equation (\ref{eq2}) can be written as
\begin{equation}
\begin{gathered}
\frac{dv(t)}{dt}+\tilde{A}(t)v(t) = \tilde{F}(t), \quad t \in
[0,T_0], \\
u(0) = \phi(0),
\end{gathered} \label{eqpazy}
\end{equation}
where $u$ is the unique fixed point of $G$ in $S$.

Now we show that $F(\cdot,\chi) \in \mathcal{C}_{T_0}$ for $t,s \in
[0,T_0]$. By assumption (H7) we have
$$
\|F(t,\chi)-F(s,\chi)\|_X \leq L_F|t-s|.
$$
Hence for each $\epsilon>0$ there exists a $\delta>0$ such that if
$|t-s| \le \delta$, implies $\|F(t,\chi)-F(s,\chi)\|_X \le
\epsilon$.

Thus, $F(t,\chi) \in \mathcal{C}_{T_0}$ for a fixed $\chi$. Hence
from Pazy \cite[Theorem 5.5.2]{pa}, we get a unique function $v \in
\mathcal{C}^1((0,T_0],X)$ satisfying (\ref{eqpazy}) in $X$ and $v$
given by
$$
v(t)= U_u(t,0)\phi(0)+\int_{0}^{t}U_u(t,s)F(s,u_s)ds, \quad t \in
[0,T_0].
$$
Where $U_u(t,s), 0 \leq s \leq t \leq T_0$ is the evolution system
generated by the family $\{A(t,u(t))\}$, $t \in [0,T_0]$. The
uniqueness of $v$ implies that $v \equiv u$ on $[0,T_0]$. Thus $u$
is a unique local classical solution of \eqref{eq1}.
\end{proof}

\section{Example}

Let us consider the equation
\begin{equation}
 \frac{du(t)}{dt}+A(t,u(t))u(t)=K(u)(t), \quad t \in [0,T],
\label{exp}
\end{equation}
where
$$
K(u)(t)=\int_{0}^{t}k(t-s)f(s,u(s))ds
$$
and $A(t, u(t))$ satisfies all the required conditions of Theorem
\ref{thm11}. Further let $k: [0,T] \to \mathbb{R}$ and
$f:[0,T]\times B \to X$ be continuous functions, where $B$ is a
subset of $X$. We also assume that $f(\cdot, u(\cdot)):[0,T] \to  X$
is a bounded function and there exists a constant $L_f\ge 0$ such
that
$$
\|f(t,u(s))-f(s,v(s))\|_X \le L_f (|t-s|+\|u(s)-v(s)\|_X).
$$
If we put $t-s=-\eta$ in the second term on the right hand side of
(\ref{exp}) to obtain
\begin{align*}
\int^{t}_{0}k(t-s)f(s,u(s))ds
&= \int^{0}_{-t}k(-\eta)f(t+\eta,u(t+\eta))d\eta \\
&=\int^{0}_{-t}k(-\eta)f(t+\eta,u_t(\eta))d\eta,
\end{align*}
then (\ref{exp}) can be rewritten as
\begin{equation}
\frac{du}{dt}+A(t,u(t))u(t)=F(t,u_t), \label{feq}
\end{equation}
where $F:[0,T]\times \mathcal{C}_0  \to X$ given by
$$
F(t,\phi)=\int_{-t}^0 k(-\eta)f(t+\eta,\phi(\eta))d\eta.
$$
here $k$ is bounded on $[0,T]$; i.e., $\sup_{t\in [0,T]}|k(t)| \le
M_2 <\infty$, for some positive constant $M_2$. For
$(t,\phi),\;(s,\psi)\in [0,T] \times \mathcal{C}_0$, we have
\begin{align*}
&\|F(t,\phi)-F(s,\psi)\|_X\\
& \leq  \big\|\int^{0}_{-t}k(-\eta)f(t+\eta,\phi(\eta))d\eta
 -\int^{0}_{-s}k(-\eta)f(s+\eta,\psi(\eta))d\eta\big\|_X\\
& \leq  \int^{-s}_{-t}|k(-\eta)|\|f(t+\eta,\phi(\eta))\|_Xd\eta \\
&\quad + \int^{0}_{-s}|k(-\eta)|\|f(t+\eta,\phi(\eta))
 -f(s+\eta,\psi(\eta))\|_Xd\eta \\
&\leq M_2M_1|t-s|+M_2TL_f(|t-s|+\|\phi(\eta)-\psi(\eta)\|_X)
\\
&\leq M_2M_1|t-s|+M_2TL_f(|t-s|+\|\phi-\psi\|_{\mathcal{C}_0})
\\
& \leq  L_F(|t-s|+\|\phi-\psi\|_{\mathcal{C}_0}),
\end{align*}
where $L_F=M_2M_1+TM_2L_f$ and $\|f(t,\phi)\|_X \le M_1$ for some
positive constant $M_1$.

Thus all the conditions of theorem \ref{thm11} are satisfied, so we
may apply the results established in the earlier sections to ensure
the existence and uniqueness  of the solution.

\subsection*{Acknowledgments}
 The authors would like to thanks the anonymous
referees for their constructive comments and suggestions which
helped us to improve the original manuscript considerably.


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