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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 06, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/06\hfil Quasilinear integrodifferential equation]
{Existence of solutions for quasilinear delay integrodifferential
equations with nonlocal conditions}

\author[K. Balachandran, F. P. Samuel \hfil EJDE-2009/06\hfilneg]
{Krishnan Balachandran, Francis Paul Samuel}  % in alphabetical order

\address{Krishnan Balachandran \newline
 Department of Mathematics,
 Bharathiar University,
 Coimbatore 641 046, India}
\email{balachandran\_k@lycos.com}

\address{Francis Paul Samuel \newline
 Department of Mathematics,
 Bharathiar University,
 Coimbatore 641 046, India}
\email{paulsamuel\_f@yahoo.com}


\thanks{Submitted September 26, 2008. Published January 6, 2009.}
\subjclass[2000]{34G20, 47D03, 47H10, 47H20}
\keywords{Semigroup; mild and classical solution; Banach fixed point theorem}

\begin{abstract}
 We prove the existence and uniqueness of mild and classical solution to
 a quasilinear delay integrodifferential equation with nonlocal condition.
 The results are obtained by using $C_0$-semigroup and the Banach
 fixed point theorem.
\end{abstract}

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

\section{Introduction}

The  existence of solution to evolution equations with
nonlocal conditions in Banach space was studied first
by Byszewski \cite{B5}. In that paper, he  established the existence
and uniqueness of mild, strong and classical solutions of the
 nonlocal Cauchy problem
\begin{gather}
u'(t)+Au(t) = f(t,u(t)), t \in (0,a]\label{1e1}\\
u(0)+g(t_1,t_2,\dots ,t_p,u(t_1),u(t_2)\dots ,u(t_p)= u_0,\label{1e2}
\end{gather}
 where $0 < t_1 <  \dots < t_p \leq a$, $-A$ is the infinitesimal
generator of a $C_0$-semigroup in a Banach space $X$, $u_0 \in X$ and
$f: [0,a] \times X \to X$, $g:[0,a]^p \times  X^p \to X$ are given functions.
The symbol $g(t_1,\dots ,t_p, u(\cdot))$\ is used in the sense that in
the place of ``$\cdot$'' we can substitute only elements of the set
$(t_1,\dots ,t_p)$. For example
\[
g(t_1,\dots ,t_p, u(\cdot))= C_1 u(t_1)+\dots +C_p u(t_p),
\]
 where $C_i$ ($i =1,2\dots ,p$) are given constants.
Subsequently many authors extended the work to various kind of nonlinear
evolution equations \cite{B2,B3,B6,B7}.

 Several authors  have studied the existence of solutions
of abstract quasilinear evolution equations in Banach space
\cite{A1,B4,D1,S1}.
 Bahuguna \cite{B1}, Oka \cite{O1} and Oka and Tanaka \cite{O2}
 discussed the existence
of solutions of quasilinear integrodifferential equations in
Banach spaces. Kato \cite{K1} studied the nonhomogeneous evolution
equations and Chandrasekaran \cite{C1} proved the existence of mild
solutions of the nonlocal Cauchy problem for a nonlinear
integrodifferential equation. Dhakne and Pachpatte \cite{D2}
established the existence of a unique strong solution of a
quasilinear abstract functional integrodifferential equation in
Banach spaces. An equation of this type occurs in a nonlinear
conversation law with memory
 \begin{gather}
u(t,x)+\Psi(u(t,x))_x = \int_0^t b(t-s)\Psi(u(t,x))_x\ ds +
f(t,x),\quad  t \in [0,a], \label{1e3}\\
u(0,x)= \phi(x),\quad  x \in \mathbb{R}.\label{1e4}
 \end{gather}
It is clear that if nonlocal condition (\ref{1e2}) is introduced
to (\ref{1e3}), then it will also have better effect than the
classical condition $u(0,x)= \phi(x)$. Therefore, we would like to
extend the results for (\ref{1e1})-(\ref{1e2}) to a class of
integrodifferential equations in Banach spaces.

  The aim of this paper is to prove the existence and uniqueness of mild
and classical solutions of quasilinear delay integrodifferential
equation with nonlocal conditions of the form
 \begin{gather}
u'(t)+A(t,u)u(t)= f(t,u(t),u(\alpha(t)))+\int_0^t
k(t,s,u(s),u(\beta(s)))ds,
 \label{1e5}\\
 u(0)+g(u)= u_0,\label{1e6}
 \end{gather}
where $t \in [0,a]$, $A(t,u)$ is the infinitesimal generator of a $C_0$-semigroup
in a  Banach space $X$,  $u_0 \in X$,
 $f : I \times X \times X \to X$, $k : \Delta \times X \times X \to X$,
$g : C(I : X)\to X$, $\alpha,  \beta:I \to I$
 are given functions.
 Here $I = [0,a]$ and $\Delta = \{(t,s) : 0 \leq s \leq t \leq a\}$.
   The results obtained in this paper are generalizations of the
results given by Pazy \cite{P1}, Kato \cite{K2,K3}
 and Balachandran and Uchiyama \cite{B4}.

\section{Preliminaries}

  Let $X$ and $Y$ be two Banach spaces such that $Y$ is densely and
continuously embedded in $X$. For any Banach spaces $Z$ the norm
of $Z$ is denoted by $\|\cdot\|$ or $\|\cdot \|_{Z}$. The space of
all bounded linear operators from $X$ to $Y$ is denoted by
$B(X,Y)$ and $B(X,X)$ is written as $B(X)$. We recall some
definitions and known facts from Pazy \cite{P1}.

\begin{definition} \label{def2.1} \rm
   Let $S$ be a linear operator in $X$ and let $Y$ be a subspace of $X$.
The operator $\tilde{S}$ defined by
$D(\tilde{S})= \{x\in D(S)\cap Y : Sx \in Y\}$ and $\tilde{S}x = Sx$
for $x \in D(\tilde{S})$ is called the part of $S$ in $Y$.
\end{definition}

\begin{definition} \label{def2.2} \rm
 Let $B$ be a subset of $X$ and for every $0 \leq t \leq a$ and
$b\in B$, let $A(t, b)$ be the infinitesimal generator of a $C_0$
semigroup $S_t,_b(s), s\geq 0$, on $X$. The family of operators
$\{A(t, b)\}, (t, b)\in I \times  B$, is stable if there are constants
$M\geq 1$ and $\omega$ such that
\begin{gather*}
\rho(A(t, b)) \supset (\omega, \infty) \quad\text{for }
 (t, b) \in I \times  B, \\
\|\prod_{j =1}^k R(\lambda : A(t_j, b_j))\|  \leq  M(\lambda - \omega)^{-k}
\end{gather*}
for $\lambda > \omega$ every finite sequences
$0 \leq t_1 \leq t_2 \leq \dots  \leq t_k\leq a$,
$b_j \in B,\  1 \leq j\leq k$.
 The stability of $\{A(t, b)\}, (t, b) \in I \times B$ implies
(see \cite{P1})
 that
 \[
 \|\prod_{j =1}^k S_{t_j,b_j}(s_j)\| \leq M
\exp \big\{\omega\sum_{j=1}^k s_j\big\},\quad  s_j \geq 0
 \]
 and any finite sequences $0 \leq t_1 \leq t_2 \leq \dots  \leq t_k\leq a$,
$b_j \in B$, $1 \leq j\leq k$.  $k = 1,2,\dots $
\end{definition}

\begin{definition} \label{def2.3} \rm
 Let $S_{t,b}(s), s\geq 0$ be the $C_0$-semigroup generatated by
$A(t, b)$, $(t ,b) \in I \times B$. A subspace $Y$ of $X$ is called
$A(t, b)$-admissible if $Y$ is invariant subspace of $S_{t,b}(s)$
and the restriction of $S_{t,b}(s)$ to $Y$ is a $C_0$-semigroup in $Y$.
\end{definition}

     Let $B\subset X$ be a subset of $X$ such that for every
$(t, b) \in I \times  B$, $A(t, b)$ is the infinitesimal generator
 of a $C_0$-semigroup $S_{t,b}(s), s \geq0$ on $X$.
We make the following assumptions:
\begin{itemize}
\item[(E1)] The family $\{A(t, b)\},(t, b) \in I \times B$ is stable.
\item[(E2)] $Y$ is $A(t, b)$-admissible for $(t, b)\in I \times B$
     and the family $\{\tilde{A}(t, b)\}, (t, b) \in I \times  B$
     of parts $\tilde{A}(t, b)$ of $A(t, b)$ in $Y$, is stable in $Y$.
\item[(E3)] For $(t, b)\in I \times  B$, $D(A(t, b)) \supset Y$, $A(t, b)$
     is a bounded linear operator from $Y$ to $X$ and $t \to A(t, b)$
     is continuous in the $B(Y, X)$ norm $\|.\|$ for every $b\in B$.
\item[(E4)] There is a constant $L > 0$ such that
\[
\|A(t, b_1) - A(t, b_2)\|_Y{_\to}_ X\ \leq L \|b_1 - b_2\|_X
\]
holds for every $b_1, b_2 \in B$ and  $0\leq t \leq a$.\\
\end{itemize}
Let $B$ be a subset of $X$ and $\{A(t, b)\}, (t, b)\in I \times B$ be a
family of operators satisfying the conditions (E1)-(E4).
If $u \in C(I : X)$ has values in $B$ then there is a unique evolution
system $U(t,s;u), 0 \leq s  \leq t  \leq  a$, in $X$ satisfying,
(see \cite[Theorem 5.3.1 and Lemma 6.4.2, pp. 135, 201-202]{P1}
\begin{itemize}
\item[(i)]  $\|U(t,s;u)\| \leq M e^{\omega(t-s)}$   for
  $0 \leq s  \leq t  \leq a$.
where $M$ and $\omega$ are stability constants.
\item[(ii)]  $\frac{\partial^+}{\partial t} U(t,s;u)w =A(s, u(s))U(t,s;u)w$
    for $w\in Y$,  for   $0 \leq s  \leq t  \leq a$.
\item[(iii)] $\frac{\partial}{\partial s}  U(t,s;u)w = -U(t,s;u)A(s, u(s))w$
  for $w\in Y$,  for $0 \leq s\leq t \leq a$.
\end{itemize}
Further we assume that
\begin{itemize}
\item[(E5)] For every $u \in C(I : X)$ satisfying $u(t)\in B$ for
  $0  \leq t  \leq a$, we have
\[
U(t,s;u)Y\subset Y,\quad 0 \leq s\leq t \leq a
\]
and $U(t,s;u)$ is strongly continuous in $Y$ for $0 \leq s\leq t \leq a$.
\item[(E6)] $Y$ is reflexive.
\item[(E7)] For every $(t,b_1, b_2) \in I \times B \times B$,
 $f(t, b_1, b_2) \in Y$.
\item[(E8)] $g: C(I : B) \to Y$ is Lipschitz continuous in $X$ and
bounded in $Y$, that is, there exist constants $G  >  0$   and $G_1  >  0$
such that
\begin{gather*}
\|g(u)\|_Y \leq G, \\
\|g(u) - g(v)\|_Y \leq G_1 \max_{t\in I}  \|u(t) - v(t)\|_X.
\end{gather*}
\end{itemize}
For the conditions (E9) and (E10) let $Z$ be taken as both $X$ and $Y$.
\begin{itemize}
\item[(E9)] $k:\Delta \times Z \to Z$ is continuous and there
exist constants $K_1 > 0$ and $K_2 > 0$ such that
\begin{gather*}
\int_0^t\|k(t,s,u_1,v_1)- k(t,s,u_2,v_2)\|_Z ds
\leq K_1 (\|u_1(t) - u_2(t)+v_1(t)-v_(t)\|_Z), \\
 K_2 = \max\{\int_0^t\|k(t,s,0,0)\|_Z \ ds :(t,s)\in \Delta\}.
\end{gather*}
\item[(E10)]     $f : I \times Z \times Z \to Z$ is continuous and
there exist constants $K_3 > 0$ and $K_4 > 0$ such that
\begin{gather*}
      \|f(t,u_1,v_1) - f(t, u_2,v_2)\|_Z
 \leq K_3 ( \|u_1 - u_2\|_Z + \|v_1 - v_2\|_Z) \\
     K_4 = \max_{t\in I} \|f(t,0,0)\|_Z.
\end{gather*}
     Let us take
$M_0 = \max \{\|U(t,s;u)\|_{B(Z)}, 0 \leq s\leq t \leq a, u\in B\}$.

\item[(E11)] $\alpha, \beta : I \to I$ is absolutely continuous and
there exist constants  $b > 0$ and $c > 0$ such that
$\alpha'(t)\geq b$ \ and $\beta'(t) \geq c$ respectively for $t \in I$.

\item[(E12)]
\begin{gather*}
 M_0\Big[\|u_0\|_Y + G+r[K_3a(1+1/b)+K_1a(1+1/c)]+a(K_4+K_2)\Big]
 \leq r
\\
\begin{aligned}
 q &=\Big[Ka \|u_0\|_Y +GKa+M_0G_1+M_0[K_3a(1+1/b)+K_1a(1+1/c)]\\
&\quad +Ka[r(K_3a(1+1/b)+K_1a(1+1/c))]+a(K_4+K_2)\Big]<1.
\end{aligned}
\end{gather*}
\end{itemize}

 Next we prove the existence of local classical solutions of the
quasilinear problem \eqref{1e5}--\eqref{1e6}.

        For a mild solution of \eqref{1e5}--\eqref{1e6} we mean a
function $u\in C(I : X)$ with values in B and $u_0 \in X$ satisfying
the integral equation
\begin{equation}
\begin{aligned}
u(t)&=U(t,0;u)u_0-U(t,0;u)g(u)+\int_0^tU(t,s;u)[f(s,u(s),u(\alpha(s)))\\
&\quad +\int_0^s k(s,\tau,u(\tau),u(\beta(\tau)))d\tau]ds.
\end{aligned} \label{2e1}
\end{equation}
A function $u \in C(I : X)$ such that $u(t)\in D(A(t, u(t)))$ for
$t \in(0, a]$, $u\in C^1((0, a]: X)$ and satisfies \eqref{1e5})--\eqref{1e6}
in $X$ is called a classical solution of (\ref{1e5}) -(\ref{1e6}) on $I$.
Further there exists a constant $K  >  0$ such that for every
$u, v \in C(I : X)$ with  values  in $B$ and every $w \in Y$ we have
\begin{equation}\label{2e2}
 \|U(t,s;u)w - U(t,s;v)w\| \leq K \|w\|_Y \int_s^t \|u(\tau)
 - v(\tau)\|d\tau.
\end{equation}

\section{Existence Result}

\begin{theorem} \label{thm3.1}
        Let $u_0 \in Y$ and let $B = \{u\in X: \|u\|_Y \leq r\}$, $r > 0$.
If the assumptions {\rm (E1)--(E12)} are satisfied, then
\eqref{1e5}--\eqref{1e6} has a unique classical solution
$u \in C([0,a]:Y)\cap C^1((0,a]:X)$
\end{theorem}

 \begin{proof}
Let $S$ be a nonempty closed subset of $C([0,a]:X)$ defined by
$ S = \{u:u\in C([0, a]: X),\|u(t)\|_Y \leq r \ for\  0\leq t\leq a\}$.
 Consider a mapping $P$ on $S$ defined by
\begin{align*}
 (Pu)(t) &= U(t,0;u)u_0 - U(t,0;u)g(u)+\int_0^t U(t,s;u)\Big[f(s,u(s),
  u(\alpha(s)))\\
&\quad +\int_0^sk(s,\tau, u(\tau), u(\beta(\tau)))d\tau\Big]ds.
\end{align*}
         We claim that $P$ maps $S$ into $S$. For $u\in S$, we have
\begin{align*}
 & \|Pu(t)\|_Y \\
&= \|U(t,0;u)u_0 - U(t,0;u)g(u)+ \int_0^t U(t,s;u)\Big[f(s,u(s),u(\alpha(s)))\\
    &\quad + \int_0^s k(s,\tau, u(\tau)u(\beta(\tau)))d\tau\Big]ds\|\\
    &\leq \|U(t,0;u)u_0\|+\|U(t,0;u)g(u)\|\\
    &\quad + \int_0^t\ \|U(t,s;u)\|\Big[\|f(s,u(s),u(\alpha(s)))- f(s,0,0)\|
     + \|f(s,0,0)\|\\
    &\quad + \|\int_0^s [k(s,\tau, u(\tau),u(\beta(\tau)))- k(s,\tau,0,0)]
    d\tau\|+ \|\int_0^s k(s,\tau,0,0)d\tau\|\Big]ds.
\end{align*}
Using  assumptions (E8)-(E11), we get
\begin{align*}
         \|Pu(t)\|_Y
&\leq  M_0 \|u_0\|_Y+M_0G+\int_0^tM_0\Big[ K_3(\|u(s)\|+\|u(\alpha(s))\|)
 +K_4   \\
&\quad +  \int_0^s K_1(\|u(s)\|+u(\beta(\tau))\|)d\tau
       + \int_0^s K_2d\tau\Big]ds  \\
&\leq  M_0 \|u_0\|_Y + M_0G+M_0\Big[ K_3ar+ K_3
  \int_0^t \|u(\alpha(s))\|(\alpha'(s)/b)ds \\
&\quad +  K_4a + K_1ar+K_1\int_0^t(\|u(\beta(s))\|(\beta'(s)/c) ds+K_2a\Big]
  \\
 &\leq  M_0 \|u_0\|_Y+M_0G+M_0\Big[K_3ar+(K_3/b)\int_{\alpha (0)}^{\alpha (t)} \|u(s)\|ds+K_4a \\
&\quad +  K_1ar+(K_1/c)\int_{\beta (0)}^{\beta (t)}(\|u(s)\| ds+K_2a\Big]\\
&\leq  M_0\Big[\|u_0\|_Y + G+r[K_3a(1+1/b)+K_1a(1+1/c)]+a(K_4+K_2)\Big]\\
\end{align*}
 From  assumption (E12), one gets
    $\|Pu(t)\|_Y     \leq r$.
Therefore $P$ maps $S$ into itself. Moreover, if $u,v \in S$, then
\begin{align*}
&\|Pu(t)- Pv(t)\|\\
&\leq \|U(t,0;u)u_0-U(t,0;v)u_0\|+\|U(t,0;u)g(u)-U(t,0;v)g(v)\|\\
&\quad +  \int_0^t\|U(t,s;u)\Big[f(s,u(s),u(\alpha(s)))
+ \int_0^s k(s,\tau,u(\tau),u(\beta(\tau)))d\tau\Big]\\
&\quad - U(t,s;v)\Big[f(s,v(s),v(\alpha(s)))
+ \int_0^s k(s,\tau,v(\tau),v((\beta(\tau)))d\tau\Big]\|ds \\
&\leq  \|U(t,0;u)u_0-U(t,0;v)u_0\|+\|U(t,0;u)g(u)-U(t,0;v)g(u)\|\\
&\quad - \|U(t,0;v)g(u)-U(t,0;v)g(v)\|\\
&\quad +  \int_0^t\Big\{\Big\|U(t,s;u)\Big[f(s,u(s),u(\alpha(s)))
  +\int_0^sk(s,\tau,u(\tau),u(\beta(\tau)))d\tau\Big]\\
&\quad - U(t,s;v)\Big[f(s,u(s),u(\alpha(s)))
  +\int_0^s k(s,\tau,u(\tau),u((\beta(\tau)))d\tau\Big]\|\\
&\quad + \|U(t,s;v)\Big[f(s,u(s),u(\alpha(s)))
  +\int_0^sk(s,\tau,u(\tau),u((\beta(\tau)))d\tau\Big]\\
&\quad - U(t,s;v)\Big[f(s,v(s),v(\alpha(s)))
  +\int_0^sk(s,\tau,v(\tau),v((\beta(\tau)))d\tau\Big]\|\Big\}ds
\end{align*}
Using  assumptions (E8)-(E12), one can get
\begin{align*}
&\|Pu(t)- Pv(t)\|\\
&\leq Ka\|u_0\|_Y\max_{\tau\in I}\|u(\tau)-v(\tau)\|
 +GKa \max_{\tau\in I}\|u(\tau)-v(\tau)\|\\
&\quad + M_0G_1\max_{\tau\in I}\|u(\tau)-v(\tau)\|\\
&\quad + Ka \max_{\tau\in I}\|u(\tau)-v(\tau)\|\Big[K_3\int_0^t\|u(s)\|ds
  +K_3\int_0^t\|u(\alpha(s))\|(\alpha'(s)/b)ds\\
&\quad + K_4a +K_1ar+K_1\int_0^t\|u(\beta(s))\|(\beta'(s)/c) ds+K_2a\Big]\\
&\quad + M_0\Big[K_3\int_0^t\|u(s)-v(s)\|ds+K_3\int_0^t\|u(\alpha(s))
  -v(\alpha(s))\|(\alpha'(s)/b)ds\\
&\quad + K_1a \max_{\tau\in I}\|u(\tau)-v(\tau)\|+K_1\int_0^t\|u(\beta(s))
  -v(\beta(s))\|(\beta'(s)/c) ds \\
&\leq \Big[Ka\|u_0\|_Y+GKa +M_0G_1+M_0[K_3a(1+1/b)+K_1a(1+1/c)]\\
&\quad +  Ka[r(K_3a(1+1/b)+K_1a(1+1/c))]+a(K_4+K_2)\Big]
  \max_{\tau\in I}\|u(\tau)-v(\tau)\|\\
&= q \max_{\tau\in I} \|u(\tau)-v(\tau)\|
\end{align*}
where $0 < q < 1$.  From this inequality it follows that for any $t\in I$,
\[
 \|Pu(t)-Pv(t)\|\leq q \max_{\tau\in I} \|u(\tau)-v(\tau)\|,
\]
 so that $P$ is a contraction on $S$. From the contraction mapping
theorem it follows that $P$ has a unique fixed point $u\in S$ which is
the mild solution of \eqref{1e5})--\eqref{1e6} on $[0,a]$.
Note that $u(t)$ is in $C(I : Y)$ by (E6) see
\cite[pp. 135, 201-202 Lemma 7.4]{P1}. In fact , $u(t)$ is weakly
continuous as a $Y$-valued  function. This implies that $u(t)$ is
separably valued in $Y$, hence
it is strongly measurable. Then $\|u(t)\|_ Y$ is bounded and measurable
function in $t$. Therefore, $u(t)$ is Bochner integrable
(see e.g. \cite[Chap.V]{Y1}). Using relation $u(t)= Pu(t)$,
we conclude that $u(t)$ is in $C(I :Y)$.

 Now consider the evolution equation
 \begin{gather}
  v'(t)+B(t)v(t)= h(t), \quad  t\in [0, a] \label{3e1}\\
  v(0) = u_0-g(u)\label{3e2}
  \end{gather}
where $B(t)= A(t, u(t))$ and
$h(t) = f(t, u(t),u(\alpha(t)))+\int_0^t k(t,s,u(s),u(\beta(s))ds$,
$t \in [0,a]$ and $u$ is the unique fixed point of $P$ in $S$.
We note that $B(t)$ satisfies
(H1)-(H3) in \cite[Sec. 5.5.3]{P1} and $h\in C(I : Y)$.
 Theorem 5.5.2 of \cite{P1} implies that there exists a unique function
$v\in C(I : Y)$ such that $v \in C^1((0,a], X)$ satisfying
\eqref{3e1} and \eqref{3e2} in $X$ and $v$ is given by
\begin{align*}
v(t)&=U(t,0;u)u_0-U(t,0;u)g(u)+\int_0^t U(t,s;u)[f(s,u(s),
 u(\alpha(s)))\\
&\quad + \int_0^s k(s,\tau,u(\tau),u(\beta(\tau)))d\tau]ds,
\end{align*}
 where $U(t,s;u)$ is the evolution system generated by the family
$\{A(t, u(t))\}, t\in I$ of the linear operators in $X$.
The uniqueness of $v$ implies that $v = u$ on $I$ and hence $u$
is a unique classical solution of \eqref{1e5})--\eqref{1e6}
and $u \in C([0,a] : Y)\cap C^1((0,a] :X)$.
\end{proof}


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