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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 51, 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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/51\hfil Existence of solutions]
{Existence of solutions for systems of self-referred and
hereditary differential equations}

\author[U. V. Le,   L. T. T.  Nguyen \hfil EJDE-2008/51\hfilneg]
{Ut Van Le, Lan T. T. Nguyen}  % in alphabetical order

\address{Ut Van Le \newline
International Language Academy, Vietnam}
\email{levanut@gmail.com, utlev@yahoo.com}

\address{Lan T. T. Nguyen\newline
 Department of  Graduate Studies, University  CT, Vietnam}
\email{nguyenttlan@gmail.com}

\thanks{Submitted February 5, 2008. Published April 10, 2008.}
\subjclass[2000]{47J35, 45G10}
\keywords{Self-referred and hereditary differential equations; \hfill\break\indent
 linear recursive schemes; existence of solutions}

\begin{abstract}
 In this paper we investigate the existence of solutions of a system
 of self-referred and hereditary differential equations.
 The initial data are assumed to be lower semi-continuous.
 We also formulate some open questions.
\end{abstract}

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

\section{Introduction}

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 $t$. The following system of
differential equations
 \begin{equation}\label{1.1}
 \begin{gathered}
 \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),\\
 u(x,0)=u_0(x),\quad v(x,0)=v_0(x).
 \end{gathered}
 \end{equation}
serves as a mathematical model for the evolution of two
reasonings. It relates self-reference and heredity.

Hereditary phenomena with memories depending on past time
histories have an extensive literature. Differential equations
modelling such phenomena have been considered by several authors
\cite{Fichera3,Fichera1,Fichera2,Fichera4,MMEP2,EP1,PU1,Volterra}.
 The main idea in finding a mathematical model
consists of formalizing mathematically a constitutive law of a
given physical phenomenon. In particular, phenomena whose
evolution depends on their states have been studied in
\cite{SSC,Eder,Pelczar1, Si1,Si2}.

Phenomena depending on their past history and with
unknown constitutive laws have attracted considerable interest
recently.
To study these phenomena,  Miranda and  Pascali \cite{MMEP2}
introduced a new class of functional differential equations. The
mathematical model of these phenomena can be described in the
following way. Let $A:X \to \mathbb{R}$ and $B:X\to \mathbb{R}$ be
two functionals, where $X$ is a space of functions. We consider
the equation
\begin{equation}\label{1.2}
    (Au)(x,t) = u\big(Bu(x,t),t\big),
\end{equation}
where $u(x,t)$ is unknown function satisfying some initial
conditions at $t=0$. If $Bu$ is a ``hereditary'' operator, for
example
$$
Bu(x,t)=\int _0^t u(x,s)ds,
$$
 then (\ref{1.2}) is said
to be of a hereditary and self-referred type. In particular,
Miranda and  Pascali \cite{MMEP1} proved a local existence and
uniqueness for equations of the type
$$
\frac{\partial ^2}{\partial t^2}u(x,t)=k_1 u\Big( \frac{\partial ^2}{\partial
t^2}u(x,t)+k_2 u(x,t),t \Big),
$$
where $k_i$ are given real numbers or real valued functions $k_i=k_i(x,t)$.


The article \cite{MMEP2} also contains some results on the existence
and uniqueness of local solutions for the  integral equations:
\begin{gather}
    u(x,t)=u_0(x)+\int_0^t u\Big( \int _0^\tau u(x,s)ds,\tau
\Big)d\tau,  \label{1.3}\\
 u(x,t)=u_0(x)+\int _0^t
u\Big( \frac{1}{\tau} \int _0^\tau u(x,s)ds,\tau \Big)d\tau,
\label{1.4}\\
 u(x,t)=u_0(x)+\int _0^t u\Big( \int _0^\tau
\frac{1}{2\delta(s)} \int_{x-\delta(s)}^{x+\delta(s)}
u(\varepsilon ,s)d\varepsilon ds,\tau \Big)d\tau, \label{1.5}
\end{gather}
for $t\geq 0,x\in \mathbb{R}$. These equations are equivalent to
the initial value problems:
\begin{gather}
    \begin{gathered}
        \frac{\partial}{\partial t}u(x,t)=u\Big( \int _0^tu(x,s)ds,t\Big),\\
        u(x,0)=u_0(x),
    \end{gathered}\label{1.6}
\\
    \begin{gathered}
        \frac{\partial}{\partial t}u(x,t)= u\Big(\frac{1}{t}\int _0^t u(x,s)ds, t\Big),\\
        u(x,0)=u_0(x),
    \end{gathered}\label{1.7}
\\
    \begin{gathered}
        \frac{\partial}{\partial t}u(x,t)= u\Big( \int_0^t \frac{1}{2\delta(s)} \int_{x-\delta(s)}^{x+\delta(s)}u(\xi,\tau)d\xi
        d\tau,t\Big),\\
        u(x,0)=u_0(x),
    \end{gathered}\label{1.8}
\end{gather}
 for $t\geq 0,x\in \mathbb{R}$, where $u_0$ is a real valued
bounded Lipschitz function and $\delta$ is a given function
 satisfying suitable conditions.

Later,  Pascali and L\^{e} \cite{PU1}, obtained an
existence result for (\ref{1.3}) under less restrictive
assumptions than in \cite{MMEP2}.
Also Pascali  \cite{EP1} obtained the results on the
existence and uniqueness for \eqref{1.1} in the case when
$u_0,v_0$ are bounded and Lipschitz functions. The main goal of
this paper is to obtain the existence of solutions of
\eqref{1.1} under weaker assumptions than in \cite{EP1}. In our
approach we use some ideas from \cite{PU1}.

\section{Existence theorems}

We consider the  system of integral equations
\begin{equation}\label{2.1}
\begin{gathered}
    u(x,t) =u_0(x)+\int_0^t u\Big(v\Big(\int_0^s
    u(x,\tau)d\tau, s\Big), s\Big)ds,
    \\
    v(x,t) =v_0(x)+\int_0^t v\Big(u\Big(\int_0^s
    v(x,\tau)d\tau, s\Big), s\Big)ds.
\end{gathered}
\end{equation}
It is assumed throughout this paper that:
\begin{itemize}
\item[(A1)]  $u_0,v_0$ are non-negative,
\item[(A2)]   $u_0,v_0$ are non-decreasing,
\item[(A3)]   $u_0,v_0$ are bounded,
\item[(A4)]   $u_0,v_0$ are lower semicontinuous
\end{itemize}
 It is worth pointing out that $u_0,v_0$, in this paper, are only
 lower semicontinuous, while Lipschitz continuity was required
in \cite{EP1}. The first part of this section will be devoted to the proof
 of the existence of the solutions for \eqref{2.1}.

\begin{theorem}\label{t1}
    Let {\rm (A1)--(A4)} hold. Then there exist two functions
$u,v:\mathbb{R}\times [0, +\infty) \to \mathbb{R}$ such that
    \begin{itemize}
\item[(B1)] $u$, $v$ are non-negative,
\item[(B2)] $u$, $v$ are non-decreasing with respect to each of
        variables,
\item[(B3)] $u$, $v$ are bounded on the sets $\mathbb{R}\times [0, +\infty)$,
\item[(B4)] $u$, $v$ are lower semicontinuous with respect to $x$,
for all fixed  $t$ in $[0,+\infty)$,
\item[(B5)] $u$, $v$ are Lipschitz with respect to $t\in [0,+\infty)$,
uniformly with respect to $x \in \mathbb{R}$,
    \end{itemize}
    satisfying \eqref{2.1}.
\end{theorem}

\begin{proof}
    The basic idea of the proof is to associate with \eqref{2.1}
    linear recursive schemes as follows
    \begin{equation}\label{2.2}
    \begin{gathered}
        u_1 (x,t) = u_0(x) +\int_0^t u_0\big(v_0\big(u_0(x)s\big)\big)ds,\\
        v_1 (x,t) = v_0(x) +\int_0^t v_0\big(u_0\big(v_0(x)s\big)\big)ds,\\
        u_{n+1}(x,t) = u_0 (x) +\int_0^t u_n\Big(
        v_n\Big(\int_0^su_n(x,\tau)d\tau,s\Big),s\Big)ds,\\
        v_{n+1}(x,t) = v_0 (x) +\int_0^t v_n\Big( u_n
\Big(\int_0^sv_n(x,\tau)d\tau,s\Big),s\Big)ds,
    \end{gathered}
    \end{equation}
for all $x\in \mathbb{R}$, $t\in [0,+\infty)$, $ n\in \mathbb{N}$.
    It follows from (A1) that
    \begin{equation}\label{2.3}
    \begin{gathered}
        u_1(x,t) \geq u_0 (x)\geq 0,\\
        v_1(x,t) \geq v_0 (x)\geq 0,
    \end{gathered}
    \end{equation}
     for all $x \in \mathbb{R}$, $t\in [0,+\infty)$.
According to (A2) and \eqref{2.2}$_{1,2}$, for $t_2>t_1,x_2>x_1$,
for all $t_1,t_2\in [0,+\infty)$, $x_1,x_2\in \mathbb{R}$, we can
     conclude that
    \begin{equation}\label{2.4}
    \begin{gathered}
        u_1 (x,t_2)\geq u_1(x,t_1),\quad \forall x\in \mathbb{R},\\
        u_1(x_2,t)\geq u_1(x_1,t), \quad \forall t\in [0,+\infty),
    \end{gathered}
    \end{equation}
and
    \begin{equation}\label{2.5}
    \begin{gathered}
        v_1 (x,t_2)\geq v_1(x,t_1),\quad \forall x\in \mathbb{R},\\
        v_1(x_2,t)\geq v_1(x_1,t), \quad \forall t\in [0,+\infty).
    \end{gathered}
    \end{equation}
Consequently, (\ref{2.4}) and (\ref{2.5}) imply that $u_1,v_1$ are
    non-decreasing with respect to $x$ and $t$, separately.
    Moreover, \eqref{2.2}$_{1,2}$ and (A3) yield
    \begin{equation}\label{2.6}
    \begin{gathered}
        0\leq u_1 (x,t) \leq (1+t)\|u_0\|_{L^{\infty}},\\
        0\leq v_1 (x,t) \leq (1+t)\|v_0\|_{L^{\infty}},
    \end{gathered}
    \end{equation}
 for all $x\in \mathbb{R}$, $t\in [0,+\infty)$.
Hence, $u_1,v_1$ are uniformly bounded with respect to $x$,
for every $t\in [0,+\infty)$.
    Furthermore, the Lipschitz property of $u_1,v_1$ with respect
    to $t$ (uniformly with respect to $x$) and their lower
semi-continuity with respect to $x$ are
    easily deduced.

    Combining \eqref{2.3}--\eqref{2.6} and with the aid of the induction on
    $n$, we can show that
    \begin{equation}\label{2.7}
    \begin{gathered}
        0\leq u_n(x,t)\leq u_{n+1}(x,t)\leq e^t
        \|u_0\|_{L^{\infty}},\\
        0\leq v_n(x,t)\leq v_{n+1}(x,t)\leq e^t
        \|v_0\|_{L^{\infty}},
    \end{gathered}
    \end{equation}
  for all $x\in\mathbb{R}$, $t\geq 0$, for all $n\in \mathbb{N}$.
    Thus, $u_n,v_n$ are non-negative in $\mathbb{R}\times [0,+\infty)$. Moreover $u_n,v_n$ are non-decreasing with respect to $x$ and
    $t$ separately, Lipschitz on $t \in [0,+\infty)$, uniformly bounded with
    respect to $x$. In addition, it follows easily that they are lower semicontinuous on $x$ for every
    $t\in [0,+\infty)$ as well. Thanks to the above properties of $u_n,v_n$, it is
    a simple matter to deduce the existence of both $u_{\infty}, v_{\infty}$ such that
    \begin{equation}\label{2.8}
    \begin{gathered}
        u_{\infty}(x,t) = \lim_{n\to+\infty} u_n (x,t) \quad
\Big(= \sup_{n\in\mathbb{N}}[u_n(x,t)]\Big),\\
        v_{\infty}(x,t) = \lim_{n\to+\infty} v_n (x,t) \quad
\Big(= \sup_{n\in\mathbb{N}}\left[v_n(x,t)\right]\Big).
    \end{gathered}
    \end{equation}
However, easy computations show that
     \begin{itemize}
\item[(C1)] $u_{\infty},v_{\infty}$ are non-negative on
 $\mathbb{R}\times [0,+\infty)$,
\item[(C2)] $u_{\infty},v_{\infty}$ are non-decreasing with respect
to each of  variables,
\item[(C3)] $u_{\infty},v_{\infty}$ are bounded on the sets
$\mathbb{R}\times [0,  +\infty)$, namely
$0 \leq u_{\infty}(x,t) \leq e^t  \|u_0\|_{L^{\infty}}$,
$0 \leq v_{\infty}(x,t) \leq e^t \|v_0\|_{L^{\infty}}$,
\item[(C4)] $u_{\infty},v_{\infty}$ are lower semicontinuous with
 respect to $x$, for all fixed $t\in [0,+\infty)$,
\item[(C5)] $u_{\infty},v_{\infty}$ are Lipschitz with respect to
$t\in [0,+\infty)$ uniformly with respect to  $x\in \mathbb{R}$.

\end{itemize}
    Next, it remains to prove that $u_{\infty}$ and $v_{\infty}$ satisfy
    \eqref{2.1}. As a matter of fact, for every $x\in \mathbb{R}$
    and every $t\in [0,+\infty)$, it is evident that the following
    integrals exist
    \begin{equation}\label{2.9}
    \begin{gathered}
        \int_0^tu_{\infty}(x,s)ds,\\
        \int_0^tv_{\infty}(x,s)ds,\\
        \int_0^t
        u_{\infty}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
        \int_0^tv_{\infty}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds.
    \end{gathered}
    \end{equation}
    Applying \eqref{2.2}$_{3,4}$ and (\ref{2.8}), we immediately obtain that
    \begin{equation}\label{2.10}
    \begin{gathered}
        u_{n+1}(x,t) - u_0 (x)\leq \int_0^t u_{\infty}\Big(
        v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
        v_{n+1}(x,t) - v_0 (x)\leq \int_0^t v_{\infty}\Big(
        v_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds.
    \end{gathered}
    \end{equation}
    We now observe that ``$\leq$'' in (\ref{2.10}) may be replaced
 by either ``$\geq$'' or ``$=$''.
    Indeed, by using (\ref{2.7}), for $n,p\in \mathbb{N}$, the
    following inequalities obviously hold
    \begin{equation}\label{2.11}
    \begin{gathered}
        u_{n+p}\Big(v_{n+p}\Big(\int_0^tu_{n+p}(x,s)ds,s\Big),t\Big)\geq
            u_{n}\Big(v_{n+p}\Big(\int_0^tu_{n+p}(x,s)ds,s\Big),t\Big),\\
        v_{n+p}\Big(u_{n+p}\Big(\int_0^tv_{n+p}(x,s)ds,s\Big),t\Big)\geq
            v_{n}\Big(u_{n+p}\Big(\int_0^tv_{n+p}(x,s)ds,s\Big),t\Big).
    \end{gathered}
    \end{equation}
    Applying the lower semi-continuity of both
    $u_n,v_n$, we can deduce from (\ref{2.8}) and (\ref{2.11}) that
    \begin{equation}\label{2.12}
    \begin{aligned}
   &\lim_{p\to+\infty}\int_0^tu_{n+p}\Big(v_{n+p}
\Big(\int_0^su_{n+p}(x,\tau)d\tau,s\Big),s\Big)ds\\
 &\geq \int_0^tu_{n}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)
d\tau,s\Big),s\Big)ds,\\
&\lim_{p\to+\infty}\int_0^tv_{n+p}\Big(u_{n+p}
\Big(\int_0^sv_{n+p}(x,\tau)d\tau,s\Big),s\Big)ds\\
&\geq \int_0^tv_{n}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),
s\Big)ds.
\end{aligned}
    \end{equation}
    Clearly,the above inequalities now become
    \begin{equation}\label{2.13}
    \begin{gathered}
        \lim_{p\to+\infty}{\lim}\Big(u_{n+p+1}(x,t)-u_0(x)\Big)\geq
            \int_0^tu_{n}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
        \lim_{p\to+\infty}{\lim}\Big(v_{n+p+1}(x,t)-v_0(x)\Big)\geq
            \int_0^tv_{n}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds.
    \end{gathered}
    \end{equation}
    Combining (\ref{2.8}) with (\ref{2.13}), it is easy to check
    that
     \begin{equation}\label{2.14}
    \begin{gathered}
        u_{\infty}(x,t)-u_0(x)\geq
            \int_0^tu_{n}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
       v_{\infty}(x,t)-v_0(x)\geq
            \int_0^tv_{n}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds.
    \end{gathered}
    \end{equation}
   Letting $n\to+\infty$, (\ref{2.14}) is rewritten as
    \begin{equation}\label{2.15}
    \begin{gathered}
        u_{\infty}(x,t)-u_0(x)\geq
            \int_0^tu_{\infty}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
       v_{\infty}(x,t)-v_0(x)\geq
            \int_0^tv_{\infty}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds.
    \end{gathered}
    \end{equation}
    Consequently, (\ref{2.10}) and (\ref{2.15}) make it obvious that
    \begin{equation}\label{2.16}
    \begin{gathered}
        u_{\infty}(x,t)-u_0(x)=
            \int_0^tu_{\infty}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
       v_{\infty}(x,t)-v_0(x)=
            \int_0^tv_{\infty}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,
    \end{gathered}
    \end{equation}
    or
     \begin{equation}\label{2.17}
    \begin{gathered}
        u_{\infty}(x,t)=u_0(x)+
            \int_0^tu_{\infty}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds,\\
       v_{\infty}(x,t)=v_0(x)+
            \int_0^tv_{\infty}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds.
    \end{gathered}
    \end{equation}
    Finally, it is easily seen from (\ref{2.17}) that
    $u_{\infty},v_{\infty}$ satisfy \eqref{2.1}, and on account of
    (C1)--(C5), the proof of Theorem \ref{t1} is complete.
\end{proof}

Now,  to study the solutions of  \eqref{1.1}, it is
convenient to rewrite (\ref{2.17}) as
 \begin{equation}\label{2.18}
    \begin{gathered}
        u_{\infty}(x,t)-\int_0^tu_{\infty}\Big(v_{\infty}\Big(\int_0^su_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds=u_0(x),\\
       v_{\infty}(x,t)-
            \int_0^tv_{\infty}\Big(u_{\infty}\Big(\int_0^sv_{\infty}(x,\tau)d\tau,s\Big),s\Big)ds=v_0(x),
    \end{gathered}
    \end{equation}
for all $x\in\mathbb{R},t\in [0,+\infty)$. The advantage of
considering the left sides of the equations in (\ref{2.18}) lies
in the fact that these functions are differential with respect to
$t$ for all fixed $x$, and their derivatives are equal to zero.
Moreover, repeated application of (C3) and (C5), for every
fixed $x$, there exist
$\frac{\partial}{\partial t}u_{\infty}(x,t),
\frac{\partial}{\partial t}v_{\infty}(x,t)$ for
a.e. $t$. Differentiating (\ref{2.18}) with respect to $t$ yields
\begin{equation}\label{2.19}
 \begin{gathered}
    \frac{\partial }{\partial t}u_{\infty}(x,t)
    = u_{\infty}\Big(v_{\infty}\Big(\int_0^t u_{\infty}(x,s)ds, t\Big), t\Big), \\
    \frac{\partial }{\partial t}v_{\infty}(x,t)
     = v_{\infty}\Big(u_{\infty}\Big(\int_0^t v_{\infty}(x,s)ds, t\Big), t\Big).
 \end{gathered}
 \end{equation}
 This leads to the following theorem.

 \begin{theorem}\label{t2}
    Under assumptions {\rm (A1)--(A4)}, there exist two functions
$u,v:\mathbb{R}\times [0,+\infty) \to  \mathbb{R}$ which satisfy
(B1)--(B5), such     that
    \begin{equation}\label{2.20}
    \begin{gathered}
        \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),\\
        u(x,0)=u_0(x),\quad v(x,0)=v_0(x),
    \end{gathered}
    \end{equation}
    for $x\in \mathbb{R}$, a.e. $t\in [0,+\infty)$.
\end{theorem}

An interesting point of Theorem \ref{t2} is that the
solutions $(u,v)$  are Lipschitz (with respect to the time variable)
although the initial datum is not.

\section{Some open problems}

In this section, we state some open problems that the readers may find
interesting. Since there have been just a modest number of publications
related to the system  \eqref{1.1}, many open questions have been left.
As a matter of fact, we can study the following:
\begin{itemize}
\item The uniqueness of the solution of \eqref{1.1} under the
    above assumptions; it seems hard.
\item Numerical solutions  of the mentioned systems (see \cite{LM}).
\item Further problems of more general systems; for example,
    \begin{equation}\label{2.21}
    \begin{gathered}
        \frac{\partial }{\partial t}u(x,t)
        = u\Big(f(u,v,x,t), t\Big), \\
        \frac{\partial }{\partial t}v(x,t)
        = v\Big(g(u,v,x,t), t\Big),\\
        u(x,0)=u_0(x),v(x,0)=v_0(x).
    \end{gathered}
    \end{equation}
\end{itemize}

\subsection*{Acknowledgments}
    The authors would like to thank Professor Julio G. Dix, Managing Editor
of the Electron. J. Differential Equations,
and to Professor Jan Chabrowski from the University of Queensland
for their valuable help in the presentation of this paper.
 The authors are also  grateful to Professor Eduardo Pascali
from the University of Salento for the kindness and generosity
he has shown in helping us with our research, in terms of his
unstintingly helpful input both in ideas and in materials that
have benefited us.
This paper is dedicated to L\^{e} V. \'{U}t's mother on  her 75th
 birthday.

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