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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 07, 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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/07\hfil Self-referred and hereditary system]
{Existence of solutions to a self-referred and hereditary
system of differential equations}
\author[E. Pascali\hfil EJDE-2006/07\hfilneg]
{Eduardo Pascali}

\address{Eduardo Pascali \hfill\break
Department of Mathematics ``Ennio De Giorgi'',
University of Lecce, C. P. 193, 73100, Lecce, Italy}
\email{eduardo.pascali@unile.it}

\date{}
\thanks{Submitted June 13, 2005. Published January 16, 2006.}
\subjclass[2000]{47J35, 45G10}
\keywords{Non-linear evolution systems; Hereditary systems}

\begin{abstract}
 We establish the  existence and uniqueness of a local solution
 for the system of differential equations
 \begin{gather*}
 \frac{\partial }{\partial t}u(x,t)
 = u\Big(v\Big(\int_0^t u(x,s)ds, t\Big), t\Big) \\
 \frac{\partial }{\partial t}v(x,t)
 = v\Big(u\Big(\int_0^t v(x,s)ds, t\Big), t\Big).
 \end{gather*}
 with given initial conditions $u(x,0)=u_0(x)$ and $v(x,0)=v_0(x)$.
\end{abstract}

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


\section{Introduction}

Equations representing self-reference phenomena have been
written of the form
\begin{equation}\label{geneq}
Au(x,t)=u(Bu(x,t),t),
\end{equation}
where $A$, $B$ are functionals on a real function space.
The existence and uniqueness of solutions to this equation
have been studied by several authors. The particular case
when the variable $x$ does not appear explicitly was studied
in \cite{EE,JG,JG1}.
More general cases have been studien in \cite{MP1,MP2,MP3}.
In  \cite{MP1},
$$
Au(x,t)=\frac{\partial}{\partial t}u(x,t)\quad\mbox{and}\quad
\quad Bu(x,t)=\int _0^t u(x,s)ds,
$$
where $B$ can be interpreted as a ``memory'' functional.
In \cite{MP3}, we have considered the equation
$$
\frac{\partial ^2}{\partial t^2}u(x,t)=k_1 u\Big(
\frac{\partial ^2}{\partial t^2}u(x,t)+k_2u(x,t),t\Big)
$$
where $k_i$ are nonnegative real numbers, or bounded regular real functions.
In this paper we establish the existence and uniqueness of local solutions
for the system of functional differential equations
\begin{gather*}
\frac{\partial }{\partial t}u(x,t) =
u\Bigl(v\Big(\int_0^t u(x,s)ds, t\Big), t\Bigr)
\\
\frac{\partial }{\partial t}v(x,t) =
v\Bigl(u\Big(\int_0^t v(x,s)ds, t\Big), t\Bigr).
\end{gather*}
This system can be considered a model for the evolution of 
two reasonings, as follows:
If  $x$ is an event, $t$ is the time, and $u(x,t), v(x,t)$
are two reasonings about $x$ at time $t$, then 
the term $v(\int_0^t u(x,s)ds, t)$ can be considered as a
 ``criticism'' of $v$ over all previous reasonings
of $u$ on $x$, up to time $t$. 


\section{The main result}
In this section we prove the following theorem.

\begin{theorem} \label{thm2.1}
Let $u_0,v_0:\mathbb{R}\to \mathbb{R}$ be bounded and Lipschitz continuous.
Then, there exist $T_0 > 0$ and two real bounded and
Lipschitz continuous functions $u_\infty ,v_\infty:\mathbb{R} \times [0,T_0]\to
\mathbb{R}$ such that
\begin{gather*}
\frac{\partial}{\partial t}u_\infty(x,t)
=u_\infty \Big( v_\infty\Big(\int _0^t u_\infty (x,\tau)d\tau,t\Big),t
\Big) \\
\frac{\partial}{\partial t}v_\infty(x,t)
=v_\infty \Big(u_\infty\Big(\int _0^t v_\infty (x,\tau)d\tau,t\Big),t
\Big) \\
u_\infty(x,0)=u_0(x), \quad v_\infty(x,0)=v_0(x)
\end{gather*}
for all $x\in \mathbb{R}$ and all $t\in [0,T_0]$. Moreover the functions
$u_\infty ,v_\infty$ are unique.
\end{theorem}

\begin{proof}
Let $u_0,v_0$ be given, and let $L_0, M_0>0$ be such that
$$
|u_0(x)-u_0(y)|\leq L_0|x-y|,\quad |v_0(x)-v_0(y)|\leq M_0|x-y|.
$$
for all $x,y\in \mathbb{R}$.
Define the sequences of functions $(u_n)_n,(v_n)_n$, for all $x\in R$
and $t>0,$ as follows:
\begin{gather*}
u_1(x,t)=u_0(x)+\int_0^t u_0\big(v_0(u_0(x)\tau)\big)d\tau,
\\
v_1(x,t)=v_0(x)+\int_0^t v_0\big(u_0(v_0(x)\tau)\big)d\tau,
\\
u_{n+1}(x,t)=
u_0(x)+\int_0^t u_n\Big(v_n\Big(\int_0^\tau u_n(x,s)ds,\tau\Big),\tau\Big)d\tau,
\\
v_{n+1}(x,t)=
v_0(x)+\int_0^t v_n\Big(u_n\Big(\int_0^\tau v_n(x,s)ds,\tau\Big),\tau\Big)d\tau.
\end{gather*}
Notice that
\begin{gather} \label{eqa1}
|u_1(x,t)-u_0(x)|\leq \Vert u_0\Vert_{\infty} t \equiv A_1(t)\\
\label{eqa2}
|v_1(x,t)-v_0(x)|\leq \Vert v_0\Vert_{\infty} t \equiv B_1(t),
\end{gather}
for all $x\in \mathbb{R},\ t>0$.
Moreover, using (\ref{eqa1}), (\ref{eqa2}) we have
\begin{align*}
&|u_2(x,t)-u_1(x,t)| \\
&\leq \Big|\int _0^t u_1 (v_1 (\int_0^{\tau} u_1(x,s)ds,\tau)\tau)d\tau
-\int_0^t u_0 (v_0 (u_0(x)\tau))d\tau\Big|
\\
&\leq \Big|\int_0^t u_1(v_1(\int_0^{\tau} u_1(x,s)ds,\tau)\tau)d\tau
-\int_0^t u_0(v_1(\int_0^{\tau} u_1(x,s)ds,\tau))d\tau\Big|
\\
&\quad + \Big| \int_0^t u_0(v_1(\int_0^{\tau} u_1(x,s)ds,\tau))d\tau
-\int_0^t u_0(v_0(u_0(x)\tau))d\tau\Big|
\\
&\leq \int_0^t \Vert u_0\Vert_{\infty}\tau d\tau
+\int_0^t L_0 \Big|v_1(\int_0^{\tau} u_1(x,s)ds,\tau)-v_0(u_0(x)\tau)
\Big|d\tau
\\
&\leq \int_0^t \Vert u_0 \Vert_{\infty}\tau d\tau
 +\int_0^t L_0 \Big[ \Big|v_1 (\int_0^{\tau} u_1(x,s)ds,\tau)
-v_0(\int_0^{\tau} u_1(x,s)ds)\Big|
\\
&\quad + \Big|v_0(\int_0^{\tau} u_1(x,s)ds)-v_0(u_0(x)\tau)\Big|\Big] d\tau
\\
&\leq \int_0^t \Bigl( \Vert u_0\Vert_{\infty}\tau +L_0\Big[
\Vert v_0\Vert_{\infty}\tau + M_0
\int_0^{\tau} \Vert u_0\Vert_{\infty}sds\Big]\Bigr)d\tau
\\
&= \int_0^t \Bigl(A_1 (\tau) + L_0 \Big[B_1(\tau)+ M_0 \int_0^{\tau}
A_1(s)ds\Big]\Bigr)d\tau
\end{align*}
for all $x\in\mathbb{R}$,  and all $t>0$.
In a similar way we prove
$$
|v_2(x,t)-v_1(x,t)|\leq \int_0^t \Bigl(B_1(\tau) + M_0 [A_1(\tau)+
L_0 \int_0^{\tau} B_1(s)ds]\Bigr)d\tau
$$
for all $x\in \mathbb{R}$,  and all $t>0$.
We have also
\begin{align*}
|u_1(x,t)-u_1(y,t)|&\leq L_0|x-y|+ \int_0^t L_0^2M_0|x-y|\tau d\tau
\\
&=\Big(L_0 +\int_0^t {L_0^2}M_0\tau d\tau \Big)|x-y|\equiv L_1(t)|x-y|;
\\
|v_1(x,t)-v_1(y,t)|&\leq M_0|x-y|+ \int_0^t L_0 M_0^2|x-y|\tau d\tau
\\
&=\Bigl(M_0 +\int_0^t L_0 M_0^2 \tau d\tau\Bigr)|x-y|\equiv M_1(t)|x-y|.
\end{align*}
It is easy to prove the  inequality
$$
|u_2(x,t)-u_2(y,t)|\leq \Bigl[L_0 +\int_0^t M_1 (\tau) \frac{1}{2}
\frac{d}{d\tau}(\int_0^{\tau}L_1(s)ds)^2 d\tau\Bigr]|x-y|\,.
$$
Set now
$$
L_2 (t)\equiv L_0 +\int_0^t M_1 (\tau) \frac{1}{2}
\frac{d}{d\tau}\Bigl(\int_0^{\tau} L_1(s)ds\Bigr)^2 d\tau.
$$
Moreover, we remark that
$$
|v_2(x,t)-v_2(y,t)|\leq \Bigl[M_0 +\int_0^t L_1 (\tau) \frac{1}{2}
\frac{d}{d\tau}\Bigl(\int_0^{\tau} M_1(s)ds\Bigr)^2 d\tau\Bigr]|x-y|,
$$
and set
$$
M_2 (t)\equiv M_0 +\int_0^t L_1 (\tau) \frac{1}{2}
\frac{d}{d\tau}\Bigl(\int_0^{\tau} M_1(s)ds\Bigr)^2 d\tau.
$$
We define for all $n$ and $t>0$:
\begin{align*}
A_{n+1} (t) &=\int _0^t \Bigl( A_n (\tau) + L_{n-1}(\tau) [B_n(\tau)+
M_{n-1}(\tau) \int_0^{\tau} A_n(s)ds]\Bigr)d\tau ;
\\
B_{n+1} (t) &=\int _0^t \Bigl(B_n (\tau) + M_{n-1}(\tau) [A_n(\tau)+
L_{n-1}(\tau) \int_0^{\tau} B_n(s)ds]\Bigr)d\tau ;
\\
L_{n+1}(t)&= L_0 +\int_0^t M_n (\tau) \frac{1}{2} \frac{d}{d\tau}
(\int_0^{\tau} L_n(s)ds)^2 d\tau;
\\
M_{n+1}(t)&= M_0 +\int_0^t L_n (\tau) \frac{1}{2} \frac{d}{d\tau}
(\int_0^{\tau} M_n(s)ds)^2 d\tau.
\end{align*}
By induction, it is easily to prove that for all $x\in\mathbb{R},\; t>0$,
\begin{gather} \label{eqa3}
|u_{n+1}(x,t)-u_{n}(x,t)| \leq A_{n+1}(t) \\
\label{eqa4} 
|v_{n+1}(x,t)-v_{n}(x,t)| \leq B_{n+1}(t)
\end{gather}
and, for all $x,y\in\mathbb{R},\; t>0$,
\begin{gather}\label{eqa5}
|u_{n+1}(x,t)-u_{n+1}(y,t)| \leq L_{n+1}(t) |x-y|\\
\label{eqa6}
|v_{n+1}(x,t)-v_{n+1}(y,t)| \leq M_{n+1}(t) |x-y|
\end{gather}
In a very simple way we can prove also that  for all
$x\in\mathbb{R},\; t>0$,
\begin{gather} \label{eqa7}
|u_{n+1}(x,t)|\leq e^t \|u_0\|_{\infty}\\
\label{eqa8}
|v_{n+1}(x,t)|\leq e^t \|v_0\|_{\infty}
\end{gather}
Since
\begin{gather*}
0\leq L_1 (t) =L_0 + M_0 L_0^2 t^2/ 2 \\
0\leq M_1 (t) =M_0 + L_0 M_0^2 t^2/ 2,
\end{gather*}
we can choose
$T_0 >0$ and $h>0$ such that $2h<1$ and for all $t\in [0,T_0]$:
\begin{gather*}
L_0^2 \frac{t^2}{2} \leq 1 ,\\
M_0^2 \frac{t^2}{2} \leq 1,\\
(M_0+L_0)^3 \frac{t^2}{2}\leq M_0\wedge L_0,\\
0\leq (M_0+L_0+1)t +(M_0+L_0)^2 \frac{t^2}{2}\leq h .
\end{gather*}
Then $0\leq L_1(t)$, $M_1(t) \leq M_0 +L_0 \equiv K_0$ for all
$t\in [0, T_0]$.

 From the previous definitions we deduce:
\begin{gather*}
0\leq L_2(t) \leq L_0 +\int_0^t M_1 (\tau) \frac{1}{2}
\frac{d}{d\tau}\Bigl(\int_0^{\tau} L_1(s)ds\Bigr)^2 d\tau \leq L_0 +
K_0^3 \frac{t^2}{2}, \\
0\leq M_2(t) \leq M_0 +\int_0^t L_1 (\tau) \frac{1}{2}
\frac{d}{d\tau}(\int_0^{\tau} M_1(s)ds)^2 d\tau \leq M_0 +
K_0^3 \frac{t^2}{2}.
\end{gather*}
Then we have
$$
0\leq L_2(t) ,\quad  M_2 (t) \leq K_0 \quad \forall t\in [0, T_0],
$$
and hence, by induction,
\begin{gather} \label{eqa9}
0\leq L_n(t) ,\quad M_n (t) \leq M_0 + L_0 \equiv K_0\quad \forall
t\in [0, T_0].
\end{gather}
 From the definitions of $A_n$ e $B_n$, we deduce
\begin{gather*}
0\leq A_{n+1} (t) \leq \int _0^t \Bigl(A_n (\tau) + K_0 B_n(\tau)+
K_0^2 \int_0^{\tau} A_n(s)ds\Bigr)d\tau;
\\
0\leq B_{n+1} (t) \leq \int _0^t \Bigl(A_n (\tau) + K_0 A_n(\tau)+
K_0^2 \int_0^{\tau} B_n(s)ds\Bigr)d\tau.
\end{gather*}
For the continuity of $A_n$ and  $B_n$ in $[0, T_0],$ we deduce:
\begin{gather*}
0\leq A_{n+1} (t) \leq \|A_n\|_{\infty} \Bigl(t+ K_0^2 \frac{t^2}{2}\Bigr)
+ K_0 t \|B_n\|_{\infty};
\\
0\leq B_{n+1} (t) \leq \|B_n\|_{\infty} \Bigl(t+ K_0^2 \frac{t^2}{2}\Bigr)
+ K_0 t \|A_n\|_{\infty}.
\end{gather*}
Now, for all $t\in [0, T_0]$,
$$
0\leq A_{n+1} (t);\quad  B_{n+1} (t) \leq h ( \|A_n\|_{\infty}+
\|B_n\|_{\infty})
$$
Hence, taking the supremum over $t$ and adding the inequalities, we deduce
that the series
$$
\sum (\|A_n\|_{\infty}+\|B_n\|_{\infty})
$$
is convergent; then the same holds for both the series $\sum
\|A_n\|_{\infty}$ and $\sum \|B_n\|_{\infty}$.
\\
 %From well known results, there exist two real functions $u^* $ and $v^*$,
%defined in $\mathbb{R}\times[0, T_0]$, Lipschitz continuous in all the variables,
%such that:  $(u_n)_n$ is uniformly convergent to $u^*$ and $(v_n)_n$ is
%uniformly convergent to $v^*$ in $\mathbb{R}\times[0,T_0]$.
%AUTHOR: PLEASE NAME THE WELL KNOWN RESULTS AND GIVE A REFERENCE.

We remember that $L^{\infty}(\mathbb{R}\times[0, T_0];\mathbb{R})$ is a complete metric space with respect to lagrangian metric; then from the inequalities (\ref{eqa3}), (\ref{eqa4}), applying the Banach-Caccioppoli theorem, we have that
$(u_n)_n$ and $(v_n)_n$ are Cauchy sequences. Hence there exist two real functions $u^* $ and $v^*$, defined in $\mathbb{R}\times[0, T_0]$ such that:  $(u_n)_n$ is uniformly convergent to $u^*$ and $(v_n)_n$ is uniformly convergent to $v^*$ in $\mathbb{R}\times[0,T_0];$ moreover, from (\ref{eqa7}),(\ref{eqa8}) and (\ref{eqa9}), $u^* $ and $v^*$ are Lipschitz continuous in all the variables.

We remark that, for all $n\in N$, $x \in \mathbb{R}$, $t \in [0, T_0]$:
\begin{align*}
&\Big|u_n \Big(v_n \Big(\int_0^t u_n (x, \tau)d\tau, t\Big), t\Big)
 - u^* \Big(v^* \Big(\int_0^t u^* (x, \tau)d\tau, t\Big), t\Big)\Big|\\
&\leq \|u_n -u^*\|_{\infty} + K_0 \|v_n -v^*\|_{\infty} +
K_0^2 t\|u_n -u^*\|_{\infty}.
\end{align*}
Then $u^*$ and $v^*$ verify that for all $x\in \mathbb{R}$ and $t\in [0, T_0]$:
\begin{gather*}
u^* (x,t) = u_0 (x) + \int_0^t u^*\Bigl(v^*(\int_0^{\tau} u^*(x,s)ds,\tau),
\tau\Bigr)d\tau,
\\
v^* (x,t) = v_0 (x) + \int_0^t v^*\Bigl(u^*(\int_0^{\tau} v^* (x,s)ds,\tau),
\tau\Bigr)d\tau,
\end{gather*}
respectively.
Let us now prove the uniqueness. Let $(u_*, v_*)$ be another pair of
solutions and remark that:
\begin{align*}
&\Big|u^*\Bigl(v^*(\int_0^{\tau} u^* (x,s)ds,\tau), \tau\Bigr)
-u_*\Bigl(v_*(\int_0^{\tau}u_* (x,s)ds,\tau), \tau\Bigr)\Big|
\\
&\leq K_0 \Big|v^*(\int_0^{\tau} u^* (x,s)ds,\tau)-
v_*(\int_0^{\tau} u_*(x,s)ds,\tau)\Big|
\\
&\quad+\Big|u^*\Bigl(v_*(\int_0^{\tau} u_* (x,s)ds,\tau), \tau\Bigr)-
u_*\Bigl(v_*(\int_0^{\tau}u_* (x,s)ds,\tau), \tau\Bigr)\Big|
\\
&\leq K_0 \Bigl(K_0 \Big|\int_0^{\tau} u^*(x,s)ds -\int_0^{\tau} u_*(x,s)ds\Big|
\\
&\quad+\Big|v^*(\int_0^{\tau} u^*(x,s)ds,\tau)-v_*(\int_0^{\tau} u^* (x,s)ds,\tau))\Big|
+\|u^* -u_* \|_{\infty}
\\
&\leq (1+K_0^2 t)\|u^* -u_* \|_{\infty} + K_0
\|v^* -v_* \|_{\infty}.
\end{align*}
Therefore,
$$
|u^*(x,\tau)-u_*(x,\tau)|\leq (t+K_0^2 {\frac{t^2}{2}})\|u^* -u_*
\|_{\infty} + K_0 t\|v^* -v_* \|_{\infty}.
$$
In a similar way we can prove the  estimates:
\begin{gather*}
|u^*(x,\tau)-u_*(x,\tau)|
\leq (t+K_0^2 {\frac{t^2}{2}})\|u^* -u_*\|_{\infty} +
K_0 t\|v^* -v_* \|_{\infty},
\\
|v^*(x,\tau)-v_*(x,\tau)|\leq (t+K_0^2 {\frac{t^2}{2}})
\|v^* -v_*\|_{\infty} + K_0 t\|u^* -u_* \|_{\infty}.
\end{gather*}
Then
$$
|u^*(x,\tau)-u_*(x,\tau)|\leq (t+K_0^2 {\frac{t^2}{2}} +K_0 t)
\max (\|u^* -u_* \|_{\infty};\|v^* -v_* \|_{\infty}),
$$
and
$$
|v^*(x,\tau)-v_*(x,\tau)|\leq (t+K_0^2 {\frac{t^2}{2}} +K_0 t)
\max (\|v^* -v_* \|_{\infty};\|u^* -u_* \|_{\infty}).
$$
 From $(t+K_0^2 {\frac{t^2}{2}} +K_0 t)\leq h <1,$ we have
$$
\max(\|u^* -u_* \|_{\infty};\|v^* -v_* \|_{\infty}) < h
\max(\|u^* -u_* \|_{\infty};\|v^* -v_* \|_{\infty}).
$$
Then the uniqueness follows and the proof is complete.
\end{proof}

\section{Some Open Problems}

The previous results and the proposed type of systems can be
investigated and generalized in many different directions. In what
follows, we give some of the problems whose investigation seems
to be interesting.
\begin{itemize}
\item[(A)] The first problem is to investigate the existence
of global solutions, also for Lipschitzian and bounded initial data.

\item[(B)] It could be more difficult to establish existence and
uniqueness for data $u_0, v_0$ bounded and uniformly continuous (or
simply continuous). Moreover, when the global existence is guaranteed,
an interesting problem can be to give particular condition on data
$u_0, v_0$ such that there exists $T^* >0$ for which  $u(x,t)=v(x,t)$
for all $x\in \mathbb{R}$ and $t \geq T^*$.
\end{itemize}


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\bibitem{JG} Jan-Guo Si and Sui Sun Cheng:
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\bibitem{JG1} Jan-Guo Si, Xin-Ping Wang and Sui Sun Cheng:
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\bibitem{MP1} M. Miranda (Jr), E. Pascali:
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\bibitem{MP2} M. Miranda (Jr), E. Pascali:
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\bibitem{MP3} M. Miranda (Jr), E. Pascali:
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\bibitem{VV} V. Volterra:
\emph{Opere Matematiche: Memorie e note}, Vol. V, 1926-1940,
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\end{thebibliography}

\end{document}