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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 31, pp. 1--10.\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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/31\hfil
Integrodifferential equations with nonlocal condition]
{Existence and uniqueness of solutions of nonlinear mixed
integrodifferential equations with nonlocal condition in Banach
Spaces}

\author[M. B. Dhakne, H. L. Tidke  \hfil EJDE-2011/31\hfilneg]
{Machindra B. Dhakne, Haribhau L. Tidke}  % in alphabetical order

\address{Machindra B. Dhakne \newline
Department of Mathematics,
Dr. Babasaheb Ambedkar Marathwada University, \newline
Aurangabad-431 004, India}
\email{mbdhakne@yahoo.com}

\address{Haribhau L. Tidke \newline
Department of Mathematics,
School of Mathematical Sciences,
North Maharashtra University, Jalgaon-425 001, India}
\email{tharibhau@gmail.com}

\thanks{Submitted April 28, 2010. Published February 18, 2011.}
\subjclass[2000]{45N05, 47B38, 47H10}
\keywords{Existence and uniqueness; mild and strong solutions;
\hfill\break\indent  mixed Volterra-Fredholm;
 integrodifferential equation;  Banach fixed point theorem;
 nonlocal condition}

\begin{abstract}
 In this article, we study the existence and uniqueness of mild and
 strong solutions of a nonlinear mixed Volterra-Fredholm
 integrodifferential equation with nonlocal condition in Banach
 spaces. Furthermore, we study continuous dependence of mild
 solutions. Our analysis is based on semigroup theory and Banach
 fixed point theorem.
\end{abstract}

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

\section{Introduction}

Let $X$  be a Banach space with norm $\|\cdot\|$. Let
$B_{r}=\{x\in X: \|x\|\leq r\}\subset X$ be a closed ball in $X$
and $E=C([t_0,t_0+\beta];B_{r})$ denote the complete metric space
with metric
$$
d(x,y)=\|x-y\|_{E}=\sup_{t\in [t_0,t_0+\beta]}\{\|x(t)-y(t)\|:x, y\in E\}.
$$

    Motivated by the work in \cite{b2,b6},  we consider the
nonlinear mixed Volterra-Fredholm integrodifferential equation
\begin{gather}
x'(t)+Ax(t)=f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds,
\int_{t_0}^{t_0+\beta}h(t,s,x(s))ds),\quad
t\in [t_0,t_0+\beta] \label{e1.1}\\
x(t_0)+g(t_1,t_2,\dots ,t_p,x(\cdot)=x_0, \label{e1.2}
\end{gather}
where $0\leq t_0<t_1<t_2<\dots <t_p\leq t_0+\beta$,  $-A$ is the
infinitesimal generator of a $C_0$ semigroup $T(t)$, $t\geq 0$, in
a Banach space $X$ and the nonlinear functions $f:
[t_0,t_0+\beta]\times X\times X\times X\to X$, $g:
[t_0,t_0+\beta]^p\times X\to X$, $k,  h :
[t_0,t_0+\beta]\times[t_0,t_0+\beta]\times X\to X$ and $x_0$ is a
given element of $X$.

    The notion of ``nonlocal condition" has been introduced to
extend the study of the classical initial value problems and it is
more precise for describing nature phenomena than the classical
condition since more information is taken into account, thereby
decreasing the negative effects incurred by a possibly erroneous
single measurement taken at the initial value. The importance of
nonlocal conditions in many applications is discussed in
\cite{a1,b3,b4,b7,b8,d1}. For example, in \cite{d1}, the author
used
\begin{equation}
g(t_1,t_2,\dots ,t_p,x(\cdot))=\sum_{i=1}^p c_ix(t_i), \label{e1.3}
\end{equation}
where $c_i,(i=1,2,\dots ,p)$ are given constants and
$t=0<t_1<\dots<t_p\leq b$ to describe, for instance, the
diffusion phenomenon of a small amount of gas in a transparent
tube can give better result than using the usual local Cauchy
problem with $x(0)=x_0$. In this case, \eqref{e1.3} allows the
additional measurements at $t_{i}$, $i=1,2,\dots p$. Subsequently,
several authors are devoted to studying of nonlocal problems by
using different techniques, see \cite{b1,b5,d2,j1,l1,t1,x1} and
the references given therein.

    The objective of the present paper is to study the existence,
uniqueness and other properties of solutions of the problem
\eqref{e1.1}--\eqref{e1.2}. The main tool employed in our analysis
is based on the Banach fixed point theorem and the theory of
semigroups. Our results extend and improve the correspondence
results in \cite{d3}. We indicate that the method used in this
paper is different from that in \cite{d3}.

     This article is organized as follows. In section 2, we present the
preliminaries and the statement of our main results. Section 3
deals with proof of the theorems. Finally in section 4  we give
example to illustrate the application of our results.


\section{Preliminaries and Main Results}

  Before proceeding to the statement of our main results, we shall
setforth some preliminaries and hypotheses that will be used in
our subsequent discussion.

\begin{definition} \label{def1} \rm
A continuous solution $x$ of the integral equation
\begin{equation}
\begin{aligned}
x(t)&=T(t-t_0)x_0-T(t-t_0)g(t_1,t_2,\dots ,t_p,x(\cdot)) \\
&\quad+\int_{t_0}^t T(t-s)f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta}
h(s,\tau,x(\tau))d\tau)ds,
\end{aligned} \label{e2.1}
\end{equation}
 with $t\in [t_0,t_0+\beta]$,
is said to be a mild solution of
\eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$.
\end{definition}

\begin{definition} \label{def2} \rm
A function $x$ is said to be a strong solution of
\eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$ if $x$ is
differentiable almost everywhere on $[t_0,t_0+\beta]$,
$x'\in L^{1}([t_0,t_0+\beta],X)$ and satisfying
\eqref{e1.1}--\eqref{e1.2} a.e.  on $[t_0,t_0+\beta]$.
\end{definition}

  We list the following hypotheses for our convenience.
\begin{itemize}

 \item[(H1)] There exists a constant $G>0$ such that
\[
\|g(t_1,t_2,\dots ,t_p,x_1(\cdot))-g(t_1,t_2,\dots ,t_p,x_2(\cdot))\|\leq
G \|x_1-x_2\|_{E}
\]
for $x_1, x_2\in E$.

 \item[(H2)] $-A$ is the infinitesimal generator of a $C_0$
 semigroup $T(t)$, $t\geq 0$ in $X$ such that
 \[
 \|T(t)\|\leq M,
 \]
for some $M\geq 1$.

 \item[(H3)] There are constants $L_1, K_1, H_1$ and
 $G_1$ such that
 \begin{gather*}
L_1=\max_{t_0\leq t\leq t_0+\beta}\|f(t,0,0,0)\|,\\
K_1=\max_{t_0\leq s\leq t\leq t_0+\beta}\|k(t,s,0)\|,\\
H_1=\max_{t_0\leq s,  t\leq t_0+\beta}\|h(t,s,0)\|,\\
G_1=\max_{x\in E}\|g(t_1,t_2,\dots ,t_p,x(\cdot))\|.
 \end{gather*}

\item[(H4)] The constants $\|x_0\|, M, G_1,L, K, K_1, H, H_1,
 \beta$ and $r$ satisfy the following two inequalities:
 \begin{gather*}
M[\|x_0\|+G_1+Lr \beta+LKr{\beta}^2+LK_1{\beta}^2+LHr{\beta}^2+LH_1{\beta}^2+L_1\beta]
\leq r,\\
[MG+ML\beta+MLK{\beta}^2+MLH{\beta}^2]<1.
\end{gather*}
 \end{itemize}

    With these preparations we are now in a position to state our
main results to be proved in the present paper.

\begin{theorem}\label{thm1}
Assume that
\begin{itemize}
\item[(i)] hypotheses {\rm (H1)--(H4)} hold,

\item[(ii)] $f:[t_0,t_0+\beta]\times X\times X\times X\to X$ is continuous
in $t$ on $[t_0,t_0+\beta]$ and
there exists a constant $L>0$ such that
\[
\|f(t,x_1,y_1,z_1)-f(t,x_2,y_2,z_2)\|\leq
L(\|x_1-x_2\|+\|y_1-y_2\|+\|z_1-z_2\|),
\]
for $x_i,y_i,z_i\in B_r$, $i=1,2$.

\item[(iii)] $k,  h:[t_0,t_0+\beta]\times [t_0,t_0+\beta]\times X\to X$
are continuous in $s, t$ on $[t_0,t_0+\beta]$ and there
exist positive constants $K,H$ such that
\begin{gather*}
\|k(t,s,x_1)-k(t,s,x_2)\|\leq K(\|x_1-x_2\|), \\
\|h(t,s,x_1)-h(t,s,x_2)\|\leq H(\|x_1-x_2\|),
\end{gather*}
for $x_i,y_i\in B_r$, $i=1,2$.
\end{itemize}
Then problem \eqref{e1.1}--\eqref{e1.2} has a unique mild solution
on $[t_0,t_0+\beta]$.
\end{theorem}


\begin{theorem}\label{thm2}
Assume that
\begin{itemize}
\item[(i)] hypotheses {\rm (H1)--(H4)} hold,

\item[(ii)] $X$ is a reflexive Banach space with norm $\|\cdot\|$ and $x_0\in
 D(A)$,the domain of $A$,

\item[(iii)] $g(t_1,t_2,\dots ,t_p,x(\cdot))\in D(A)$,

\item[(iv)] There exists a constant $L>0$ such that
\begin{align*}
\|f(t_1,x_1,y_1,z_1)-f(t_2,x_2,y_2,z_2)\|&\leq
L(|t_1-t_2|+\|x_1-x_2\|+\|y_1-y_2\|\\
&\quad+\|z_1-z_2\|),
\end{align*}

\item[(v)] There exist constants $K,  H>0$ such that
\begin{gather*}
\|k(t_1,s,x_1)-k(t_2,s,x_2)\|\leq K(|t_1-t_2|+\|x_1-x_2\|), \\
\|h(t_1,s,x_1)-h(t_2,s,x_2)\|\leq H(|t_1-t_2|+\|x_1-x_2\|),
\end{gather*}
\end{itemize}
Then $x$ is a unique strong solution of
\eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$.
\end{theorem}

\begin{theorem}\label{thm3}
Suppose that the functions $f, g, k$ and $h$ satisfy hypotheses
{\rm (H1)-(H4)} and assumptions (ii), (iii) of Theorem \ref{thm1}.
Then, for each pair of elements $x^*_0, x^{**}_0\in X$, and
for the corresponding mild solutions $x_1, x_2$ of  problem
\eqref{e1.1} with
$x_1(t_0)+g(t_1,t_2,\dots ,t_p,x_1(\cdot))=x^*_0$ and
$x_2(t_0)+g(t_1,t_2,\dots ,t_p,x_2(\cdot))=x^{**}_0$, the
inequality
\[
\|x_1-x_2\|_{E}\leq \frac{M}{(1-MG)}\|x^*_0-x^{**}_0\|
\exp{(\frac{ML\beta}{(1-MG)}(1+K\beta+H\beta))}
\]
is true, whenever $G<1/M$.
\end{theorem}

\section{Proofs of theorems}

\begin{proof}[Proof of Theorem \ref{thm1}]
Define an operator $F: E\to E$ by
\begin{align*}
(Fz)(t)&=T(t-t_0)x_0-T(t-t_0)g(t_1,t_2,\dots ,t_p,z(\cdot))\\
&\quad+\int_{t_0}^t T(t-s)f(s,z(s),\int_{t_0}^s
k(s,\tau,z(\tau))d\tau, \int_{t_0}^{t_0+\beta}
h(s,\tau,z(\tau))d\tau)ds,
\end{align*}
for $t\in [t_0,t_0+\beta]$.
Now, we show that $F$ maps $E$ into itself. For
$z\in E, t\in [t_0,t_0+\beta]$ and using hypotheses (H2)-(H4) and
assumptions (ii), (iii), we have
\begin{align*}
&\|(Fz)(t)\|\\
&\leq \|T(t-t_0)x_0\|+\|T(t-t_0)g(t_1,t_2,\dots ,t_p,z(\cdot))\|\\
&\quad+\|\int_{t_0}^t T(t-s)f(s,z(s),\int_{t_0}^s
k(s,\tau,z(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,z(\tau))d\tau)ds\|\\
&\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ \|f(s,z(s),\int_{t_0}^s
k(s,\tau,z(\tau))d\tau,\\
&\quad\int_{t_0}^{t_0+\beta}
h(s,\tau,z(\tau))d\tau)-f(s,0,0,0)\|+\|f(s,0,0,0)\|]ds\\
&\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ L(\|z(s)-0\|+\|\int_{t_0}^s
k(s,\tau,z(\tau))d\tau-0\|\\
&\quad+\|\int_{t_0}^{t_0+\beta}
h(s,\tau,z(\tau))d\tau-0\|)+\|f(s,0,0,0)\|]ds\\
&\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ Lr+L\int_{t_0}^s
\|k(s,\tau,z(\tau))-k(s,\tau,0)+k(s,\tau,0)\|d\tau\\
&\quad+L\int_{t_0}^{t_0+\beta}
\|h(s,\tau,z(\tau))-h(s,\tau,0)+h(s,\tau,0)\|d\tau+L_1]ds\\
&\leq M\|x_0\|+MG_1+M\int_{t_0}^t[ Lr+L\beta
(Kr+K_1)+L\beta(Hr+H_1)+L_1]ds\\
&\leq M[\|x_0\|+G_1+Lr\beta+LKr{\beta}^2+LK_1{\beta}^2
+LHr{\beta}^2+LH_1{\beta}^2+L_1\beta]
\leq r.
\end{align*}
Thus, $F$ maps $E$ into itself.

    Now, for every $z_1, z_2\in E$, $t\in [t_0,t_0+\beta]$ and
using hypotheses (H1), (H2), (H4) and assumptions (ii),
(iii), we obtain
\begin{align*}
&\|(Fz_1)(t)-(Fz_2)(t)\|\\
&\leq \|T(t-t_0)\|\|g(t_1,t_2,\dots ,t_p,z_1(\cdot))-g(t_1,t_2,\dots ,t_p,z_2(\cdot))\|\\
&\quad+\int_{t_0}^t \|T(t-s)\|\|[f(s,z_1(s),\int_{t_0}^s
k(s,\tau,z_1(\tau))d\tau,\int_{t_0}^{t_0+\beta}
h(s,\tau,z_1(\tau))d\tau)\\
&\quad -f(s,z_2(s),\int_{t_0}^s
k(s,\tau,z_2(\tau))d\tau,\int_{t_0}^{t_0+\beta}
h(s,\tau,z_2(\tau))d\tau)]\|ds\\
&\leq MG\|z_1-z_2\|_{E}+\int_{t_0}^t
ML[\|z_1(s)-z_2(s)\|\\
&\quad+\int_{t_0}^s \|k(s,\tau,z_1(\tau))-k(s,\tau,z_2(\tau))\|d\tau\\
&\quad +\int_{t_0}^{t_0+\beta} \|h(s,\tau,z_1(\tau))-h(s,\tau,z_2(\tau))\|d\tau]ds\\
&\leq MG\|z_1-z_2\|_{E}+ML\|z_1-z_2\|_{E} \int_{t_0}^t [1+K\int_{t_0}^s d\tau+H\int_{t_0}^{t_0+\beta} d\tau]ds\\
&\leq MG\|z_1-z_2\|_{E}+ML\|z_1-z_2\|_{E}\beta [1+K\beta+H\beta]\\
&\leq q \|z_1-z_2\|_{E},
\end{align*}
where $q=MG+ML\beta+MLK{\beta}^2+MLH{\beta}^2$ and hence, we obtain
\begin{align*}
\|Fz_1-Fz_2\|_{E}\leq q\|z_1-z_2\|_{E},
\end{align*}
with $0<q<1$. This shows that the the operator $F$ is a
contraction on the complete metric space $E$. By the Banach fixed
point theorem, the function $F$ has a unique fixed point in the
space $E$ and this point is the mild solution of problem
\eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$. This completes
the proof of the Theorem \ref{thm1}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2}]
    All the assumptions of Theorem \ref{thm1}  are being satisfied, then problem
\eqref{e1.1}--\eqref{e1.2} has a unique mild solution belonging to
$E$. Now we will show that $x$ is unique strong
solution of  \eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$.
Take
\begin{gather*}
L_2=\max_{t_0\leq t\leq t_0+\beta}\|f(t,x(t),0,0)\|,\\
K_2=\max_{t_0\leq s\leq t\leq t_0+\beta}\|k(t,s,x(s))\|,\\
H_2=\max_{t_0\leq s,  t\leq t_0+\beta}\|h(t,s,x(s))\|.
\end{gather*}
For $0<\theta<t-t_0$ and $t\in [t_0,t_0+\beta]$, we have
\begin{align*}
&x(t+\theta)-x(t)\\
&=[T(t+\theta-t_0)-T(t-t_0)]x_0\\
&\quad-[T(t+\theta-t_0)-T(t-t_0)]g(t_1,t_2,\dots ,t_p,x(\cdot))\\
&\quad+\int_{t_0}^{t_0+\theta} T(t+\theta-s)f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,x(\tau))d\tau)ds\\
&\quad+\int_{t_0+\theta}^{t+\theta}
T(t+\theta-s)f(s,x(s),\int_{t_0}^s k(s,\tau,x(\tau))d\tau,
\int_{t_0}^{t_0+\beta} h(s,\tau,x(\tau))d\tau)ds\\
&\quad-\int_{t_0}^t T(t-s)f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,x(\tau))d\tau)ds\\
&=T(t-t_0)[T(\theta)-I]x_0-T(t-t_0)[T(\theta)-I]g(t_1,t_2,\dots ,t_p,x(\cdot))\\
&\quad+\int_{t_0}^{t_0+\theta} T(t+\theta-s)[f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta}h(s,\tau,x(\tau))d\tau)\\
&\quad-f(s,x(s),0,0)+f(s,x(s),0,0)]ds\\
&\quad+\int_{t_0}^{t}
T(t-s)[f(s+\theta,x(s+\theta),\int_{t_0}^{s+\theta}
k(s+\theta,\tau,x(\tau))d\tau,\\
&\quad \int_{t_0}^{t_0+\beta} h(s+\theta,\tau,x(\tau))d\tau)
 -f(s,x(s),\\
 &\quad \int_{t_0}^s k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta}
h(s,\tau,x(\tau))d\tau)]ds.
\end{align*}
Using the assumptions and the fact
$\|[T(\theta)-I]x\|=\theta\|Ax\|+o(\theta)$, we obtain
\begin{align*}
&\|x(t+\theta)-x(t)\|\\
&\leq M[\theta \epsilon_1+\theta\|Ax_0\|]+M[\theta \epsilon_{2}+\theta\|Ag(t_1,t_2,\dots ,t_p,x(\cdot))\|]\\
&\quad+\int_{t_0}^{t_0+\theta} M[\|f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\\
&\quad\int_{t_0}^{t_0+\beta} h(s,\tau,x(\tau))d\tau)
-f(s,x(s),0,0)\|+\|f(s,x(s),0,0)\|]ds\\
&\quad+\int_{t_0}^{t}M[\|f(s+\theta,x(s+\theta),\int_{t_0}^{s+\theta}
k(s+\theta,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s+\theta,\tau,x(\tau))d\tau)\\
&\quad-f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta}
h(s,\tau,x(\tau))d\tau)\|]ds\\
&\leq M[\theta \epsilon_1+\theta\|Ax_0\|]+M[\theta
\epsilon_{2}+\theta\|Ag(t_1,t_2,\dots ,t_p,x(\cdot))\|]\\
&\quad+\int_{t_0}^{t_0+\theta}
ML[\int_{t_0}^s K_2d\tau+\int_{t_0}^{t_0+\beta} H_2d\tau]ds\\
&\quad+M\int_{t_0}^{t_0+\theta} L_2 ds+\int_{t_0}^{t}ML[\theta+\|x(s+\theta)-x(s)\|\\
&\quad+\int_{t_0}^{s}K(s+\theta-s|+\|x(\tau)-x(\tau)\|)d\tau\\
&\quad+\int_{s}^{s+\theta} K_2d\tau+\int_{t_0}^{t_0+\beta}
H(|s+\theta-s|+\|x(\tau)-x(\tau)\|)d\tau]ds\\
&\leq M[\theta \epsilon_1+\theta\|Ax_0\|]+M[\theta
\epsilon_{2}+\theta\|Ag(t_1,t_2,\dots ,t_p,x(\cdot))\|]+MLK_2\theta\beta+MLH_2\theta \beta\\
&\quad+ML_2\theta+ML\theta\beta+ML\int_{t_0}^{t}\|x(s+\theta)-x(s)\|ds\\
&\quad+MLK\theta{\beta}^2+MLK_2\theta\beta+MLH\theta{\beta}^2\\
&\leq P\theta+ML\int_{t_0}^{t}\|x(s+\theta)-x(s)\|ds,
\end{align*}
where $\epsilon_1, \epsilon_{2}>0$ and
\begin{align*}
P&=M[\epsilon_1+\|Ax_0\|+\epsilon_{2}+\|Ag(t_1,t_2,\dots ,t_p,x(\cdot))\|+LK_2\beta+LH_2\beta\\
&\quad+L_2+L\beta+LK{\beta}^2+LH{\beta}^2+LK_2\beta,
\end{align*}
which is independent of $\theta$ and $t\in [t_0,t_0+\beta]$.

Thanks to Gronwall's inequality, we obtain
\[
\|x(t+\theta)-x(t)\|\leq P\theta e^{ML\beta}, \quad\text{for }
t\in [t_0,t_0+\beta].
\]
Therefore, $x$ is Lipschitz continuous on $[t_0,t_0+\beta]$. The
Lipschitz continuity of $x$ on $[t_0,t_0+\beta]$ combined with
$(iv)$ and $(v)$ of Theorem \ref{thm2} implies
\begin{align*}
t\to f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds,\int_{t_0}^{t_0+\beta}
h(t,s,x(s))ds)
\end{align*}
is Lipschitz continuous on $[t_0,t_0+\beta]$. By
\cite[Corollary 4.2.11]{p1}, we observe that the equation
\begin{gather*}
y'(t)+Ay(t)=f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds,
\int_{t_0}^{t_0+\beta} h(t,s,x(s))ds),\quad t\in [t_0,t_0+\beta]\\
y(t_0)=x_0-g(t_1,t_2,\dots ,t_p,x(\cdot))
\end{gather*}
has a unique strong solution $y(t)$ on $[t_0,t_0+\beta]$
satisfying the equation
\begin{align*}
y(t)&=T(t-t_0)x_0-T(t-t_0)g(t_1,t_2,\dots ,t_p,x(\cdot))\\
&\quad+\int_{t_0}^t T(t-s)f(s,x(s),\int_{t_0}^s
k(s,\tau,x(\tau))d\tau,\int_{t_0}^{t_0+\beta} h(s,\tau,x(\tau))d\tau)ds\\
&=x(t),\quad t\in [t_0,t_0+\beta].
\end{align*}
Consequently, $x(t)$ is the strong solution of initial value
problem \eqref{e1.1}--\eqref{e1.2} on $[t_0,t_0+\beta]$. This
completes the proof of Theorem \ref{thm2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
    Suppose that  $x_1(t)$ and $x_2(t)$ satisfy
\eqref{e1.1} on $[t_0,t_0+\beta]$ with
$x_1(t_0)+g(t_1,t_2,\dots ,t_p,x_1(\cdot))=x^*_0$ and
$x_2(t_0)+g(t_1,t_2,\dots ,t_p,x_2(\cdot))=x^{**}_0$, respectively
and $x_1, x_2\in E$. Using the equation \eqref{e2.1}, hypotheses
{\rm (H1)--(H4)} and assumptions (ii), (iii),  we obtain
\begin{align*}
&\|x_1(t)-x_2(t)\|\\
&\leq M\|x^*_0-x^{**}_0\|+MG\|x_1-x_2\|_{E}
 +\int_{t_0}^t ML\Big[\|x_1(s)-x_2(s)\|\\
&\quad +\int_{t_0}^s K\|x_1(\tau)-x_2(\tau)\|d\tau
 +\int_{t_0}^{t_0+\beta} H\|x_1(\tau)-x_2(\tau)\|d\tau \Big]ds\\
&\leq M\|x^*_0-x^{**}_0\|+MG\|x_1-x_2\|_{E}\\
&\quad+\int_{t_0}^t ML\Big[\|x_1(s)-x_2(s)\|+\int_{t_0}^s
K\sup_{\tau\in [t_0,s]}\|x_1(\tau)-x_2(\tau)\|d\tau\\
&\quad+\int_{t_0}^{t_0+\beta} H\sup_{\tau\in [t_0,t_0+\beta]}
\|x_1(\tau)-x_2(\tau)\|d\tau \Big]ds\\
&\leq M\|x^*_0-x^{**}_0\|+MG\|x_1-x_2\|_{E}+\int_{t_0}^t
ML\big[1+\beta K+\beta H\big]\|x_1-x_2\|_{E}ds.
\end{align*}
Therefore, we obtain
\[
\|x_1-x_2\|_{E}\leq
\frac{M}{(1-MG)}\|x^*_0-x^{**}_0\|+\int_{t_0}^t
\frac{ML\beta}{(1-MG)}(1+K\beta+H\beta)\|x_1-x_2\|_{E}ds.
\]
Using Gronwall's inequality, we obtain
\[
\|x_1-x_2\|_{E}\leq \frac{M}{(1-MG)}\|x^*_0-x^{**}_0\|
\exp{(\frac{ML\beta}{(1-MG)}(1+K\beta+H\beta))},
\]
provided that $G<\frac{1}{M}$. From this inequality, it follows
that the continuous dependence of solutions depends upon the
initial data. This completes the proof of the Theorem \ref{thm3}.
\end{proof}


\section{Application}

 To illustrate the applications of some of our main
results, we consider the  nonlinear mixed Volterra-
Fredholm partial integrodifferential equation
\begin{equation}
\begin{gathered}
w_{t}(u,t)-w_{uu}(u,t)=P(t,w(u,t),\int_0^t k_1(t,s,w(u,s))ds,
\int_0^{\beta} h_1(t,s,w(u,s))ds),\\
 0< u< 1,  \quad  0\leq t\leq \beta
\end{gathered}\label{e4.1}
\end{equation}
with initial and boundary conditions
\begin{gather}
w(0,t)=w(1,t)=0,\quad 0\leq t \leq \beta,\label{e4.2}\\
w(u,0)+\sum_{i=1}^p w(u,t_i)=w_0(u),\quad
0<t_1<t_2<\dots <t_p\leq\beta. \label{e4.3}
\end{gather}
where $P: [0,\beta] \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$,  $k_1,
 h_1: [0,\beta]\times [0,\beta]\times \mathbb{R}\to \mathbb{R}$ are continuous
functions. We assume that the functions $P, k_1$ and $h_1$ in
\eqref{e4.1}--\eqref{e4.3} satisfy the following conditions:

\begin{enumerate}

 \item  There exists a constant $G^*>0$ such that
\[
|\sum_{i=1}^p w(u,t_i)-\sum_{i=1}^p w(v,t_i)|\leq G^* \sup_{t\in
[0,\beta]}|u(t)-v(t)|
\]
for $u, v\in E_1=C([0,\beta]; B^*_{r^*})$, where
$B^*_{r^*}=\{x\in R:|x|\leq r^*\}$.

\item  There are constants $L^*_1, K^*_1, H^*_1$ and  $G^*_1$ such that
\begin{gather*}
L^*_1=\max_{0\leq t\leq \beta}|P(t,0,0,0)|,\\
K^*_1=\max_{t_0\leq s\leq t\leq t_0+\beta}|k_1(t,s,0)|,\\
H^*_1=\max_{t_0\leq s, t\leq t_0+\beta}|h_1(t,s,0)|,\\
G^*_1=\max_{x\in E_1}|\sum_{i=1}^p w(u,t_i)|,\quad 0<u<1.
\end{gather*}

 \item  $P:[0,\beta]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous in
$t$ on $[0,\beta]$ and
there exists a constant $L^*>0$ such that
\[
|P(t,x_1,y_1,z_1)-P(t,x_2,y_2,z_2)|\leq
L^*(|x_1-x_2|+|y_1-y_2|+|z_1-z_2|),
\]
for $x_i,y_i,z_i\in B^*_{r^*}$, $i=1,2$.

\item $k,  h:[0,\beta]\times [0,\beta]\times \mathbb{R}\to \mathbb{R}$ are
continuous in $s, t$ on $[0,\beta]$ and there exist respectively
constants $K^*,  H^*>0$ such that
\begin{gather*}
|k_1(t,s,x_1)-k_1(t,s,x_2)|\leq K^*(|x_1-x_2|),\\
|h_1(t,s,x_1)-h_1(t,s,x_2)|\leq H^*(|x_1-x_2|),
\end{gather*}
for $x_i,y_i\in B^*_{r^*}$, $i=1,2$.

\item $-A$ is the infinitesimal generator of a $C_0$
 semigroup $T(t)$, $t\geq 0$ in $X$ such that
 \[
 \|T(t)\|\leq M^*,
 \]
for some $M^*\geq 1$.

\item The constants $|w_0(u)|, M^*, G^*_1,L^*, K^*,
K^*_1, H^*, H^*_1,
 \beta$ and $r$ satisfy the following two inequalities:
 \begin{align*}
&M^*[|w_0(u)|+G^*_1+L^*r \beta+L^*K^*r{\beta}^2+L^*K^*_1{\beta}^2\\
&+L^*H^*r{\beta}^2+L^*H^*_1{\beta}^2+L^*_1\beta]\leq r^*,
\end{align*}
and
\[
[M^*G^*+M^*L^*\beta+M^*L^*K^*{\beta}^2+M^*L^*H^*{\beta}^2]<1.
\]
\end{enumerate}

  First, we reduce the equations \eqref{e4.1}--\eqref{e4.3}
into \eqref{e1.1}--\eqref{e1.2} by making suitable choices of
$A, f, g, k$ and $h$. Let $X=L^2[0,1]$. Define the operator
$A: X \to X$ by $Az = -z''$ with domain $D(A)=\{z \in X : z,   z'$ are
absolutely continuous, $ z''\in X$ and $z(0)=z(1)=0\}$. Define
the functions $f:[0,\beta]\times X\times X\times X\to X$,
$k:[0,\beta]\times [0,\beta]\times X\to X$,
$h:[0,\beta]\times [0,\beta]\times X \to X$ and
$g:[0,\beta]^p \times X\to X$ as follows
\begin{gather*}
f(t,x,y,z)(u)=P(t,x(u),y(u),z(u)),\\
k(t,s,x)(u)=k_1(t,s,x(u)),\\
h(t,s,x)(u)=h_1(t,s,x(u)),\\
g(t_1,t_2,\dots ,t_p,x(\cdot)u=\sum_{i=1}^p w(u,t_i)
\end{gather*}
for $t\in [0,\beta], x,y,z\in X $ and $0<u<1$. Then the above
problem \eqref{e4.1}--\eqref{e4.3} can be formulated abstractly as
nonlinear mixed Volterra-Fredholm integrodifferential equation in
Banach space $X$:
\begin{gather}
x'(t)+Ax(t)=f(t,x(t),\int_{t_0}^t k(t,s,x(s))ds,
\int_{t_0}^{t_0+\beta}h(t,s,x(s))ds),\quad
t\in [t_0,t_0+\beta] \label{e4.4}\\
x(t_0)+g(t_1,t_2,\dots ,t_p,x(\cdot)=x_0. \label{e4.5}
\end{gather}
Since all the hypotheses of the Theorem \ref{thm1} are satisfied, the
Theorem \ref{thm1} can be applied to guarantee the mild solution of the
nonlinear mixed Volterra-Fredholm partial integrodifferential
equations \eqref{e4.1}--\eqref{e4.3}.


\subsection*{Acknowledgements}
We are grateful to Professor Julio G. Dix and to the anonymous
referee for their helpful comments that improved this article.


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\end{document}