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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 55, 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}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/55\hfil Nonlinear mixed Volterra-Fredholm equations]
{Existence of global solutions to nonlinear mixed
Volterra-Fredholm integrodifferential equations with nonlocal
conditions}

\author[H. L. Tidke\hfil EJDE-2009/55\hfilneg]
{Haribhau L. Tidke}

\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 11, 2008. Published April 21, 2009.}
\subjclass[2000]{45N05, 47G20, 34K30, 47D09}
\keywords{Global solution; Volterra-Fredholm integrodifferential equation;
\hfill\break\indent
 Leray-Schauder alternative; priori bounds; nonlocal condition}

\begin{abstract}
 In this paper, we investigate the  existence of global
 solutions to first-order initial-value problems, with nonlocal
 condition for nonlinear mixed Volterra-Fredholm
 integrodifferential equations in Banach spaces. The technique used
 in our analysis is based on an application of the topological
 transversality theorem known as Leray-Schauder alternative and
 rely on a priori bounds of solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

   Let $\mathbb{R}^{n}$  be the Euclidean $n$-space with norm $\|\cdot\|$.
Let $B=C([0,b],\mathbb{R}^{n})$ be the Banach space of all continuous
functions from $[0,b]$ into $\mathbb{R}^{n}$ endowed with supremum norm
\[
\|x\|_B=\sup\{\|x(t)\|:t\in [0,b]\}.
\]
 Now we study the mixed Volterra-Fredholm integrodifferential equations of the form
 \begin{gather}
x'(t)=f(t,x(t),\int_0^t k(t,s,x(s))ds,\int_0^b h(t,s,x(s))ds
),\quad t\in[0,b],\label{e1.1}\\
  x(0)+g(x)=x_0,\label{e1.2}
\end{gather}
where $f: [0,b]\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times
\mathbb{R}^{n}\to \mathbb{R}^{n}$, is a function, $k,  h :
[0,b]\times [0,b]\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ are
continuous functions and $g: B\to \mathbb{R}^{n}$ is given
function, and $x_0 $ is a given element of $\mathbb{R}^{n}$.

  Several authors have investigated the problems of existence,
uniqueness and other properties of solutions of the special forms
of \eqref{e1.1}-\eqref{e1.2}, see \cite{b1,c1,d1,d2,l1,p1,p2,p3}
and some of the references given therein. The equations of the
form \eqref{e1.1}-\eqref{e1.2} play an important role for abstract
formulation of many initial, boundary value problems of perturbed
differential equations, partial differential equations and partial
integrodifferential equations which arise in various applications
like chemical reaction kinetics. population dynamics, heat-flow in
material with memory, viscoelastic and reaction diffusion
problems.

    The work in nonlocal initial value problem( IVP for short) was
initiated by Byszewski. In \cite{b2} Byszewski using the method of
semigroups and the Banach fixed point theorem proved the existence
and uniqueness of mild, strong and classical solution of first
order IVP. For the importance of nonlocal conditions in different
fields, the interesting reader is referred to \cite{b2} and the
references cited therein. Recently, in an interesting paper
 Dhakne and Kendre \cite{d2} studied the
existence of global solutions to \eqref{e1.1} when $g=0, x(0)=0$.

    The aim of the this paper is to study the existence
of  global solutions to  \eqref{e1.1}-\eqref{e1.2}. The main
tool used in our analysis is based on an application of the
topological transversality theorem known as Leray-Schauder
alternative, rely on a priori bounds of solutions. The interesting
and useful aspect of the method employed here is that it yields
simultaneously the global existence of solutions and the maximal
interval of existence. We are motivated by the work of
Dhakne and  Kendre \cite{d2} and influenced by the work of
Byszewski \cite{b2}.

  The paper is organized as follows: In Section 2, we present
the preliminaries and hypotheses. Section 3 deals with the main
result. Finally, in Section 4, we give an example to illustrate
the application of our theorem.

\section{Preliminaries and Hypotheses}

   Before proceeding to the main result, we shall set forth some
preliminaries and hypotheses that will be used in our subsequent
discussion.

\begin{definition} \label{def1} \rm
Let  $ f \in L^{1}(0,b;\mathbb{R}^{n})$. The function $ x\in B$
given by
\begin{equation}
x(t)=x_0-g(x)+ \int_0^t f(s, x(s),
\int_0^s k(s, \tau, x(\tau))d\tau,
 \int_0^b h(s, \tau,x(\tau))d\tau)ds,
\label{e2.1}
\end{equation}
for $t\in[0,b]$ is called the solution of the initial value problem
\eqref{e1.1}-\eqref{e1.2}.
\end{definition}

    Our results are based  on the following lemma, which is a version
of the topological transversality theorem given by Granas \cite[p. 61]{d3}.

\begin{lemma}[Leray-Schauder Alternative] \label{lem1}
 Let $S$ be a convex subset of a normed linear space $E$ and assume
$0 \in S$. Let $F:S\to S$ be a completely continuous operator, and let
\[
\varepsilon(F)=\{x \in S : x=\lambda Fx\text{ for some }0<\lambda <1\}.
\]
 Then either $\varepsilon(F)$ is unbounded or $F$
has a fixed point.
\end{lemma}

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

\item[(H1)] There exists a constant $G$ such that
\begin{equation*}
\|g(x)\|\leq G,\quad\text{for }x \in \mathbb{R}^{n}.
\end{equation*}

\item[(H2)] There exists a continuous function $p: [0,b] \to
\mathbb{R}_+=[0,\infty)$ such that
\begin{equation*}
\|\int_0^t k(t,s,x(s))ds\|\leq p(t)\|x\|,
\end{equation*}
for every $t,s \in [0,b]$ and $x \in \mathbb{R}^{n}$.

\item[(H3)] There exists a continuous function $q: [0,b] \to
\mathbb{R}_+$ such that
\begin{equation*}
\|\int_0^b h(t,s,x(s))ds\|\leq q(t)\|x\|,
\end{equation*}
for every $t,s \in [0,b]$ and $x \in \mathbb{R}^{n}$.

 \item[(H4)]
There exists a continuous function $l: [0,b] \to \mathbb{R}_+$
such that
\begin{equation*}
\|f(t,x,y,z)\|\leq l(t)K(\|x\|+\|y\|+\|z\|),
\end{equation*}
for every $t\in [0,b]$ and $ x, y, z \in \mathbb{R}^{n}$, where
$K: \mathbb{R}_{+}\to (0,\infty)$ is continuous nondecreasing
function satisfying
\begin{equation*}
K(\alpha(t)x)\leq \alpha(t)K(x),
\end{equation*}
and $\alpha(t)$ is defined as the function $p$.
\end{itemize}

\section{Existence of a Solution}

\begin{theorem} \label{thm1}
Suppose that  {\rm (H1)-(H4)} hold. Then the
initial-value problem \eqref{e1.1}-\eqref{e1.2} has a solution $x$ on
$[0,b]$ provided $b$ satisfies
\begin{equation}
\int_0^b M(s)ds < \int_c^{\infty} \frac{ds}{K(s)},\label{e3.1}
\end{equation}
where $c=\|x_0\|+G$ and $M(t)=l(t)[1+p(t)+q(t)]$ for $t\in [0,b]$.
\end{theorem}

\begin{proof}
To prove the existence of a solution of nonlinear mixed
Volterra-Fredholm integrodifferential equations
\eqref{e1.1}-\eqref{e1.2}, we apply topological transversality
theorem. First we establish the priori bounds on the solutions of
the initial value problem
\begin{equation}
x'(t)=\lambda f(t,x(t),\int_0^t k(t,s,x(s))ds,\int_0^b
h(t,s,x(s))ds),\quad t\in [0,b], \label{e3.2}
\end{equation}
under the initial condition \eqref{e1.2} for $\lambda\in (0,1)$.
Let $x(t)$ be a solution of the problem \eqref{e3.2}-\eqref{e1.2},
then it satisfies the equivalent integral equation
\begin{equation}
x(t)=x_0-g(x)+ \lambda \int_0^t f(s,x(s),\int_0^s
k(s,\tau,x(\tau))d\tau,
 \int_0^b h(s,\tau,x(\tau))d\tau)ds,
\label{e3.3}
\end{equation}
for $t\in [0,b]$. Using \eqref{e3.3}, hypotheses (H1)-(H4) and the
fact that $\lambda \in (0,1)$, we have
\begin{equation} \label{e3.4}
\begin{aligned}
\|x(t)\|&\leq \|x_0-g(x)\|+\|\int_0^t f(s,x(s),\int_0^s
k(s,\tau,x(\tau))d\tau,\int_0^b h(s,\tau,x(\tau))d\tau)ds\| \\
&\leq[\|x_0\|+G]+\int_0^t \|f(s,x(s),\int_0^s
k(s,\tau,x(\tau))d\tau,\int_0^b h(s,\tau,x(\tau))d\tau)\|ds \\
&\leq[\|x_0\|+G]+\int_0^t l(s)K(\|x(s)\|+p(s)\|x(s)\|+q(s)\|x(s)\|)ds \\
&\leq[\|x_0\|+G]+\int_0^t
l(s)(1+p(s)+q(s))K(\|x(s)\|)ds.
\end{aligned}
\end{equation}
Denoting by $u(t)$ the right-hand side of the above inequality,
 we have
\begin{equation*}
u(t)=[\|x_0\|+G]+\int_0^t l(s)(1+p(s)+q(s))K(\|x(s)\|)ds.
\end{equation*}
Then $\|x(t)\|\leq u(t)$ and $u(0)=[\|x_0\|+G]=c$. Therefore,
\begin{gather*}
u(t)=c+\int_0^t l(s)(1+p(s)+q(s))K(\|x(s)\|)ds\\
u(t)\leq c+\int_0^b l(s)(1+p(s)+q(s))K(u(s))ds.
\end{gather*}
Differentiating $u(t)$ and using the fact that K is increasing
continuous,  we get
\begin{equation}
\begin{gathered}
u'(t)\leq l(t)(1+p(t)+q(t))K(u(t))
\leq l(t)(1+p(t)+q(t))K(u(t)), \\
\frac{u'(t)}{K(u(t))}\leq M(t).
\end{gathered}\label{e3.5}
\end{equation}
Integrating  from $0$ to $t$ and using change of variables
$t\rightarrow s=u(t)$ and the condition \eqref{e3.1}, we obtain
\begin{equation}
\int_c^{u(t)}\frac{ds}{K(s)}
\leq \int_0^t M(s)ds
\leq \int_0^b M(s)ds < \int_c^\infty \frac{ds}{K(s)}.
\label{e3.6}
\end{equation}
 From this inequality  and the mean value theorem we observe that
there exists a constant $\gamma$, independent of $\lambda\in(0,1)$
such that $u(t)\leq \gamma$ for $t\in[0,b]$ and hence
$\|x(t)\|\leq \gamma$ for $t\in [0,b]$ and consequently, we have
\begin{equation*}
\|x\|_B =\sup \{\|x(t)\|: t\in[0,b]\}\leq \gamma.
\end{equation*}

  Now, we rewrite \eqref{e1.1}-\eqref{e1.2} as follows:
If $ y\in B$ and
$x(t)=x_0-g(x)+y(t)$, $t\in [0,b]$, where $y(t)$ satisfies
\begin{align*}
y(t)&=\int_0^t f(s,y(s)+x_0-g(y),\int_0^s
k(s,\tau,y(\tau)+x_0-g(y))d\tau,\\
&\quad \int_0^b h(s,\tau,y(\tau)+x_0-g(y))d\tau)ds,\quad t\in [0,b]
\end{align*}
if and only if $x(t)$ satisfies
\begin{equation*}
x(t)=x_0-g(x)+\int_0^t f(s, x(s), \int_0^s k(s, \tau,
x(\tau))d\tau,\int_0^b h(s, \tau, x(\tau))d\tau)ds,\quad
t\in[0,b].
\end{equation*}
We define the operator $F: B_0\to B_0$, $B_0=\{ y\in B:y(0)=0\}$
by
\begin{equation} \label{e3.7}
\begin{aligned}
(Fy)(t)&=\int_0^t f(s,y(s)+x_0-g(y),\int_0^s
k(s,\tau,y(\tau)+x_0-g(y))d\tau,\\
&\quad \int_0^b h(s,\tau,y(\tau)+x_0-g(y))d\tau)ds,\quad t\in [0,b].
\end{aligned}
\end{equation}
Then $F$ is clearly continuous.

   Next, we prove that $F$ is completely continuous. Let $\{w_m\}$ be a
bounded sequence in  $B_0$, i.e. $\|w_m\|_{B}\leq d$ for all $m$,
where $d$ is a positive constant. From the definition of operator
$F$ and using the hypotheses $(H_1)-(H_4)$ and the fact that
$\|w_m\|_B\leq d$, we obtain
\begin{align*}
\|(Fw_m)(t)\|
&\leq \int_0^t \|f(s,w_m(s)+x_0-g(w_m),\int_0^s
k(s,\tau,w_m(\tau)+x_0-g(w_m))d\tau,\\
&\quad \int_0^b h(s,\tau,w_m(\tau)+x_0-g(w_m))d\tau)\|ds\\
&\leq \int_0^t l(s)[1+p(s)+q(s)]K(\|w_m(s)+x_0-g(w_m)\|)ds\\
&\leq \int_0^t l(s)[1+p(s)+q(s)]K(d+c)ds\\
&\leq M^{*}K(d+c)b,
\end{align*}
where $M^{*}=\sup\{M(t):t\in [0,b]\}$. This implies that the set
$\{(Fw_m)(t): \|w_m\|_B\leq d,\quad 0\leq t\leq b\}$ is uniformly
bounded in $\mathbb{R}^{n}$.

    Now we shall show that the sequence $Fw_{m}$ is equicontinuous.
Let $t_1, t_2\in [0,b]$, Then from the definition of operator $F$
and using the hypotheses $(H_1)-(H_4)$ and the fact that
$\|w_m\|_B\leq d$, we have
\begin{equation} \label{e3.8}
\begin{aligned}
&\|(Fw_m)(t_1)-(Fw_m)(t_2)\|  \\
&\leq \int_{t_1}^{t_2}\|f(s,w_m(s)+x_0-g(w_m),\int_0^s
k(s,\tau,w_m(\tau)+x_0-g(w_m))d\tau, \\
&\quad
\int_0^b h(s,\tau,w_m(\tau)+x_0-g(w_m))d\tau)\|ds \\
&\leq \int_{t_1}^{t_2}l(s)[1+p(s)+q(s)]K(\|w_m(s)+x_0-g(w_m)\|)ds \\
&\leq \int_{t_1}^{t_2}l(s)[1+p(s)+q(s)]K(d+c)ds \\
&\leq M^{*}K(d+c)(t_2-t_1),
\end{aligned}
\end{equation}
where $M^{*}=\sup\{M(t):t\in [0,b]\}$. From \eqref{e3.8} we
conclude that $\{Fw_m\}$ is equicontinuous and hence by
Arzela-Ascoli theorem the operator $F$ is completely continuous.

Finally, the set
\begin{equation*}
\varepsilon(F)=\{y \in B_0 : y=\lambda Fy,\; \lambda \in
(0,1)\}
\end{equation*}
is bounded in $B$, since for every $ y\in \varepsilon(F)$, the
function  $x(t)=x_0-g(x)+y(t)$ is a solution of
\eqref{e3.2}-\eqref{e1.2} for which we have proved that
$\|x\|_B\leq \gamma$ and hence $\|y\|_B\leq \gamma+c$.
Consequently, by Lemma \ref{lem1}, the operator $F$ has a fixed
point in $B_0$. This means that the initial value problem
\eqref{e1.1}-\eqref{e1.2} has a solution. This completes the proof
of the theorem.
\end{proof}

\begin{remark} \label{rmk1} \rm
We note that in the special case, if we take $(i)$ $M(t)=1$ in
condition \eqref{e3.1} and the integral on the right side in
\eqref{e3.1} is assumed to diverge, then the solutions of
equations \eqref{e1.1}-\eqref{e1.2} exist for every $b<\infty$.
\end{remark}


\section{Application}

In this section we apply some of the results established in this
paper. First, we consider the partial firs-order differential
equation with nonlocal condition
\begin{gather}
w_t(u,t)=P(t,w(u,t),\int_0^t k_1(t,s,w(u,s))ds,\int_0^b
h_1(t,s,w(u,s))ds),\label{e4.1}\\
w(0,t)=w(\pi,t)=0,\quad 0\leq t \leq b,\label{e4.2}\\
w(u,0)+g(w(u,t))=w_0(u),\quad 0\leq u\leq \pi,\label{e4.3}
\end{gather}
where $P: [0,b] \times \mathbb{R} \times \mathbb{R}\times
\mathbb{R}\to \mathbb{R}$, is a function and
$k_1, h_1: [0,b]\times [0,b] \times \mathbb{R}\to \mathbb{R}$ are continuous.
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 $G$ such that
$|g(x)|\leq G$,  for  $x \in \mathbb{R}$.

\item There exists a nonnegative function $p_1$ defined on
$[0,b]$ such that
\begin{equation*}
\big|\int_0^t k_1(t,s,x)ds\big| \leq p_1(t)|x|
\end{equation*}
for $ t ,s \in [0,b]$ and $ x\in \mathbb{R}$.

\item There exists a nonnegative function $q_1$ defined on $[0,b]$
such that
\begin{equation*}
|\int_0^b h_1(t,s,x)ds| \leq q_1(t)|x|
\end{equation*}
for $ t ,s\in [0,b]$ and $ x\in \mathbb{R}.$

\item There exists
nonnegative real valued continuous function $l_1$ defined on
$[0,b]$ and a positive continuous increasing function $K_1 $
defined on $\mathbb{R}_+$ such that
\begin{equation*}
|P(t,x,y,z)| \leq l_1(t)K_1(|x|+|y|+|z|)
\end{equation*}
for $ t\in [0,b]$ and $ x,y,z\in \mathbb{R}$.

\end{enumerate}
Let us take $X=L^{2}[0,\pi]$. Suppose that
\begin{equation*}
\int_0^b l_1(s)(1+p_1(s)+q_1(s))ds < \int_c^\infty
\frac{ds}{K_1(s)}
\end{equation*}
is satisfied, where $c=\|w_0\|+G$. Define the functions
$f:[0,b]\times X\times X\times X\to X$, $ k, h: [0,b] \times
[0,b]\times X\to X$ as follows
\begin{gather*}
f(t,x,y,z)(u)=P(t,x(u,t),y(u,t),z(u,t)),\\
k(t,s,x)(u)=k_1(t,s,x(u,t))\quad\text{and}\\
h(t,s,x)(u)=h_1(t,s,x(u,t))
\end{gather*}
for $ t\in[0,b], x,y,z\in X $ and $ 0\leq u\leq \pi$. With these
choices of the functions, the equations \eqref{e4.1}-\eqref{e4.3}
can be modelled abstractly as nonlinear mixed Volterra-Fredholm
integrodifferential equation with nonlocal condition in Banach
space $X$:
\begin{gather}
x'(t)=f(t,x(t),\int_0^t k(t,s,x(s))ds,\int_0^b h(t,s,x(s))ds),\quad
t\in [0,b],\label{e4.4}\\
x(0)+g(x)=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 solution of
the nonlinear mixed Volterra-Fredholm partial integrodifferential
equation \eqref{e4.1}-\eqref{e4.3} with nonlocal condition.

\subsection*{Acknowledgements}
The author wishes to express his sincere thanks to the anonymous
referee and Professor J. G. Dix for their helpful comments and
suggestions.


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\end{document}