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\markboth{\hfil Cauchy problem for derivors \hfil EJDE--2001/32}
{EJDE--2001/32\hfil J-F Couchouron, C. Dellacherie, \& M. Grandcolas \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 32, pp. 1--19. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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Cauchy problem for derivors in finite dimension
%
\thanks{ {\em Mathematics Subject Classifications:} 34A12, 34A40, 34A45, 34D05.
\hfil\break\indent
{\em Key words:} derivor, quasimonotone operator, accretive operator,
Cauchy problem, \hfil\break\indent
uniqueness condition.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted December 4, 2000. Published May 8, 2001.} }
\date{}
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\author{
Jean-Fran\c{c}ois Couchouron, Claude Dellacherie, \\
\& Michel Grandcolas}
\maketitle
\begin{abstract}
In this paper we study the uniqueness of solutions to
ordinary differential equations which fail to satisfy
both accretivity condition and the uniqueness condition of
Nagumo, Osgood and Kamke.
The evolution systems considered here are governed by a
continuous operators $A$ defined on $\mathbb{R}^N$ such that
$A$ is a derivor; i.e., $-A$ is quasi-monotone with respect
to $(\mathbb{R}^{+})^N$.
\end{abstract}
\section{Introduction}
For $T>0$, we study the Cauchy Problem (CP)
$$ \begin{gathered}
\dot u(t) +Au(t)=f(t),\quad t\in [0,T]\cr
u(0)=u_{0}, \end{gathered} \eqno(1.1)
$$
where $A$ is a continuous operator
on $ \mathbb{R}^N$ and $f$ belongs to
$L^{1}([0,T]:\mathbb{R}^N)$.
We require in addition that $A$ be a derivor on $ \mathbb{R}^N$
(or equivalently that $-A$ be quasi-monotone
with respect to the cone $ (\mathbb{R^{+}})^N$)
and has an additional order property (see Assumption \textrm{H1T}
in Section 2).
The existence of local solutions of (1.1)
is proved by standard arguments (see \cite{w1} and Lemma 4.2).
For instance, in the continuous case, this local existence comes from the
Peano's Theorem.
So the problem is essentially to prove
the uniqueness of a local solution and the existence of global solutions.
An important remark is that the identity operator minus the limit of
infinitesimal generators of increasing
semigroups is a derivor on the domains of the operators (see remark 2.1.d).
The aim of this paper consists of giving a special converse of this previous
property. General studies of evolution problems
governed by derivors can be found in \cite{b2,d1,d2,w1} (for
existence of extremal solutions of differential inclusions in $ \mathbb{R}^N$)
and in \cite{h2} for the behavior of the flow (stability, etc.) in the
regular case: $A$ is $C^{1}$.
This work establishes uniqueness for the Cauchy Problem and complements
previous studies.
Let us point out that derivors often occur in the theory of production
processes in Economics (for cooperative systems, see \cite{d3,h3}), in
Chemistry \cite{h1}, and in Biology \cite{h1}.
Our uniqueness result given in the sequel applies to these situations.
Notice also that the additional order property; namely,
existence of uniform ascents, (see Definition 2.2)
has obvious interpretations in applications and may be considered
as a special extension of the submarkovian property (see remark 2.2.3)
and \cite{b1}).
Nevertheless the notion of uniform ascents
is a new concept built from the concept of progressions in \cite{d3}.
This ascent notion which extends the usual submarkovian property
seems to lead naturally to the maximum
principle worked in \cite{d3}. Finally, we emphasize that
the ascent notion is the key to obtain a suitable
increasing resolvent (see Proposition 2.4 and Theorem 3.2).
In this paper, the operator $A$ does not satisfy either uniqueness
conditions such as those given by Nagumo, Osgood and Kamke \cite{c2,k1,k2}
nor accretivity conditions, even in a generalized sense as in
\cite{c2,c4,k2}.
We will exhibit in Section 5 a simple example of operator on
$\mathbb{R}^{2}$ which satisfies all our conditions and none of the
uniqueness conditions quoted above. Consequently our framework is
not included in the submarkovian case, since a continuous
submarkovian derivor is accretive in
($\mathbb{R}^N,\|\cdot \|_{\infty}$). Moreover based in our analysis, it
appear that a simple natural-order property can
replace a classical Lipschitz condition about uniqueness in the Cauchy
Problem.
Uniqueness and order-preserving dependence
with respect to the initial value $u_{0}$ are stated in Theorem 3.1.
In the case $f=0$, Theorem 3.2 guarantees the existence of a global solution
and a special form of the Crandall-Ligget exponential formula
\cite[p. 319]{c5} involving suitable selections of the multi-valued operators
$(I+\lambda A)^{-1}$ (while in \cite{d1}
$(I+\lambda A)^{-1}$ is single valued and Lipschitz).
This paper is organized as follows.
Section 2 is devoted to general definitions and preliminaries.
The main results are stated in Section 3, while the proofs
are given in the next section.
Section 5 gives an example in $ \mathbb{R}^{2}$
which demonstrates the need for Theorems 3.1 and 3.2.
Some remarks about the asymptotic behavior follow in Section 6.
\section{Generalities}
We supply $\mathbb{R}^N$ with the usual partial order relation
$u\leq v$ if $u^i\leq v^i$ for all $i=1,\dots ,N$,
where $u^i$ is the $i$-th component. The vector in $\mathbb{R}^N$ whose
components are $C,\dots,C$ is denoted by $C$.
The symbol $\|\cdot \|$ stands for any norm in $\mathbb{R}^N$.
The symbol $\mathbb{N}^{*}$ denotes the set of integers greater than zero.
\paragraph {Definition 2.1}
We say that the map $A$ is a {\bf derivor} on $\mathbb{R}^N$
if it satisfies the condition
\begin{enumerate}
\item[(i)] For each $(u,v)\in (\mathbb{R}^N)^{2}$
and each $i\in \{1,\dots ,N\}$
$$(u\leq v \hbox{ and } u^i=v^i) \quad \hbox{implies that}\quad
A^iu\geq A^iv\eqno (2.1)$$
\end{enumerate}
We say that the map $A$ is a {\bf moderate derivor}
(resp. a {\bf strong derivor})
if, in addition to (i), it satisfies
\begin{enumerate}
\item[(ii)] For each $u \in \mathbb{R}^N$, there exist
$u_{1},u_{2} \in \mathbb{R}^N$
(resp. two sequences $(u_{k})_{k}\to +\infty$, $(v_{k})_{k}\to -\infty$)
such that $u_{1}\leq u \leq u_{2}$ and $Au_{1}\leq 0 \leq Au_{2}$
(see \cite{c2}).
(resp. $ \lim_{k \to +\infty} Au_{k}=+\infty$
and $ \lim_{k \to +\infty} Av_{k}=-\infty$).
\end{enumerate}
The previous notation $ \lim_{k \to +\infty} w_{k}=+\infty$ in $\mathbb{R}^N$
may be interpreted to mean that $ \lim_{k \to +\infty} w_{k}^{j}=+\infty$ in
$\mathbb{R}^N$ for each $j \in \{1,\dots ,N\}$.
The derivor notion coincides with the notion
of quasimonotone operator on $\mathbb{R}^N$, except the sign
(see \cite{b2}, \cite[p. 91]{d1}, \cite{k1}).
In these references, $-A$ is
quasimonotone with respect to $(\mathbb{R^{+}})^N$ if
$$(u\leq v \hbox{ and } x^{*}(u)=x^{*}(v))
\quad\hbox{implies that}\quad x^{*}(-Au)\leq x^{*}(-Av)
\eqno (2.2)$$
for any linear positive form $x^{*}$ on $\mathbb{R}^N$.
Hence, $A$ is a derivor, because
if $x^{*}$ is a linear positive form on $ \mathbb{R}^N$, $x^{*}$
is a linear combination with positive coefficients
of coordinate forms on $\mathbb{R}^N$.
\paragraph{Remark 2.1}
a) Condition (i) in the definition of derivor is automatically fulfilled
for any operator $A$ from $\mathbb{R}$ to $\mathbb{R}$, but it is not in
the case of Condition (ii).
A special case where (ii) holds for
an operator $A$ from
$\mathbb{R}$ to $\mathbb{R}$ is the case where
$A$ is a non-decreasing operator
such that there is $v\in \mathbb{R}$ satisfying $Av=0$.
\\
b) When $A$ is a linear derivor, the reader
can check that Condition (ii) is equivalent to:
there is $u\geq 1$ satisfying $Au\geq 0$.
\\
c) An equivalent form of definition 2.1.(i) is:
$ A^i $ is decreasing with respect to $ x^{j}$ for each
$i\not= j$ with $i,j \in \{1,\dots ,N\}$ (see \cite{h3}).
\\
d) If $P$ is an increasing operator
on $\mathbb{R}^N$, then $A=I-P$ is a derivor.
Therefore, when $(P_{t})$ is an increasing semi-group on
$\mathbb{R}^N$, then $A_{t}=\frac{I-P_{t}}{t}$
with $t>0$ is a derivor and so is $A_{0}$,
defined by $A_{0}u=\lim_{t\downarrow 0}A_{t}u$
(on the domain where this limit exists).
\subsection*{Ascents}
We denote by $V_{K}(u_{0})$ the set of compact neighborhoods of $u_{0}$.
\paragraph{Definition 2.2} We say that a derivor $A$
has a (strict) {\bf uniform ascent}
at $u_{0}$ if there are $V \in V_{K}(u_{0})$ and a
sequence $(v_{k})$ in $\mathbb{R}^N$ convergent to 0 such that
$(v_{k}^i)_{k \in \mathbb{N}^{*}}$ is strictly
decreasing for all $i=1,\dots ,N$ and
$$\min_{i\in \{1,\dots ,N\}}(A^i(u+v_{k})-A^iu)>0\eqno (2.3)$$
for each $k\in \mathbb{N}^{*}$ and each $u \in V$.
\paragraph{Remark 2.2}
1) In terms of production operator
($A^iu$ is the production of the $i$-th input of
the product $u$),
the uniform ascent property at $u_{0}$ means
that in a neighborhood of $u_{0}$ it is possible to
increase the level of production by means of small uniform augmentations
around $u_{0}$.
\\
2) The notion of uniform ascent plays
a crucial part in this work.
In our opinion, this concept is new, but it was inspired
from the progression notion carried out in \cite{d3}.
\\
3) The uniform ascent property may be connected to the submarkovian property namely,
$$A(u+C)-Au \geq 0 \eqno (2.4)$$
for all $u\in \mathbb{R}^N$ and all $C\in \mathbb{R}^{+}$.
Notice that a submarkovian derivor in $\mathbb{R}^N$ is accretive
in ($\mathbb{R}^N,\|\cdot \|_{\infty}$);
the verification of this claim is left to the reader.
\\
4) The following dual notion of
uniform ascent at $u_{0}$ provides again the results of Section 3:
There are $V \in V_{K}(u_{0})$
and a strictly increasing
sequence $(v_{k})$ in $\mathbb{R}^N$ convergent to 0 such that
$(v_{k}^i)_{k \in \mathbb{N}^{*}}$ is strictly
increasing for all $i=1,\dots ,N$ and
$$\sup_{i\in \{1,\dots ,N\}}(A^i(u+v_{k})-A^iu)<0$$
for each
$k\in \mathbb{N}^{*}$ and each $u \in V$.
\\
5) In the case $A=I-P$ with an increasing operator $P$,
(see Remark 2.1.d)), Definition 2.2 means that the required sequence
$(v_{k})$ satisfies
$$P^i(u+v_{k})-P^i(u)0$
are single-valued contractions. But
in our case $(I+\lambda A)^{-1}$ is a priori multi valued. Nevertheless
it is possible to define
suitable selectors $J_{\lambda}$ of $(I+\lambda A)^{-1}$
as claimed in the following lemma.
\paragraph{Lemma 2.4}
{\sl Let $A$ be a moderate continuous derivor on $\mathbb{R}^N$.
Let $u$ in $\mathbb{R}^N$, $\lambda \in \mathbb{R}^{+}$
and $v$ a solution of
$$ \begin{gathered} v\leq u\cr
Av \leq 0.\end{gathered}\eqno (2.7)$$
Then the system
$$\begin{gathered}w\geq v\cr
(I+\lambda A)w \geq u\end{gathered}\eqno (2.8)$$
has a smallest solution denoted by
$J_{\lambda,v}u$.
Moreover we have
$$J_{\lambda,v}u\in (I+\lambda A)^{-1}(u).\eqno (2.9)$$
}
\paragraph{Proof.}
According to (ii) in definition 2.1, Systems (2.7) and (2.8)
have solutions. Let $v$ be a solution of (2.7).
Since $B=I+\lambda A$ is a continuous derivor,
the existence of the smallest solution
$J_{\lambda,v}u$ of (2.8) is guaranteed
by the Theorem 2.4.
It remains to prove
$$(I+\lambda A)J_{\lambda,v}u=u.\eqno (2.10)$$
Since the constraints are optimal in (2.8),
we have for each $i\in \{1,\dots ,N\}$,
$((I+\lambda A)J_{\lambda,v}u)^i=u^i$
or $(J_{\lambda,v}u)^i=v^i$.
Thus we have
to prove $((I+\lambda A)J_{\lambda,v}u)^i=u^i$
when $(J_{\lambda,v}u)^i=v^i$.
So assume $(J_{\lambda,v}u)^i=v^i$
for some $i\in \{1,\dots ,N\}$,
Relation (2.1) and $J_{\lambda,v}u\geq v$ yield
$$(AJ_{\lambda,v}u)^i\leq (Av)^i.\eqno (2.11)$$
Now (2.11) and (2.7) provide
$$
(J_{\lambda,v}u)^i+\lambda (AJ_{\lambda,v}u)^i
=v^i+\lambda (AJ_{\lambda,v}u)^i
\leq u^i+\lambda (AJ_{\lambda,v}u)^i
\leq u^i.
$$
Therefore, $((I+\lambda A)J_{\lambda,v}u)^i\leq u^i$. But
from (2.8) we have $((I+\lambda A)J_{\lambda,v}u)^i\geq u^i$.
Finally (2.10) is proved. \hfill$\square$\medskip
In the same way, let $v$ be a solution of
$$\begin{gathered} v\geq u\cr
Av \geq 0.\end{gathered}\eqno (2.12)$$
Then the system
$$\begin{gathered} w\leq v\cr
(I+\lambda A)w \leq u\end{gathered}\eqno (2.13)$$
has a largest solution $w={\tilde J}_{\lambda,v}u$.
Moreover ${\tilde J}_{\lambda,v}u$ satisfies again (2.9).
Set $J_{\lambda}u=J_{\lambda,v}u$ (resp. $J_{\lambda}u={\tilde
J}_{\lambda,v}u$) for an arbitrary $v$
satisfying (2.7) (resp. (2.12)).
Let us notice that $J_{\lambda}$ is defined on
$D_{v}=\{u \in \mathbb{R}^N, u\geq v\}$
(resp. $D_{v}=\{u \in \mathbb{R}^N, u\leq v\}$).
The family of selectors
$(J_{\lambda})_{\lambda\geq 0}$ of $(I+\lambda A)^{-1}$
is said to be the {\bf resolvent} of $A$.
\paragraph{Definition 2.4}
For $u$ given, the notation ${\overline u}$ (resp. $ {\hat u}$ ) stands
for the largest solution of (2.7) (resp. the smallest
solution of (2.12)). \smallskip
Thanks to Theorem 2.4, such extremal elements ${\overline u}$ and
$ {\hat u}$ exist.
Furthermore we have clearly
$$u\leq v \Longrightarrow ({\overline u}\leq {\overline v}
\quad\mbox{and}\quad {\hat u}\leq {\hat v})\eqno (2.14) $$
and
$${\overline {\overline u}}={\overline u}\quad\mbox{and}\quad
{\hat {\hat u}}={\hat u}.\eqno (2.15)$$
The resolvent operators satisfy the following properties.
\paragraph{Proposition 2.4}
{\sl For a given $u\in \mathbb{R}^N$,
let $v,v'\in \mathbb{R}^N$ satisfying (2.7) and $w,w'
\in \mathbb{R}^N$ satisfying (2.12). Then \begin{enumerate}
\item[(a)] The map $J_{\lambda}$ is single-valued and increasing
on $D_{v}$.
\item[(b)] We have
$$v\leq J_{\lambda}u \leq w\eqno (2.16)$$
In particular
$${\overline u}\leq J_{\lambda}u \leq {\hat u}\eqno (2.17)$$
\item[(c)] If $Au\geq 0$ (resp. $Au\leq 0$), then
$AJ_{\lambda}u\geq 0$ (resp. $AJ_{\lambda}u\leq 0$)
for each $\lambda \geq 0$.
\item[(d)] If $Au\geq 0$, then $\lambda \to J_{\lambda}u$ is decreasing
on $\mathbb{R}^N$ (and increasing if $Au\leq 0$).
\item[(e)] We have $J_{\lambda,v}u\leq {\tilde J}_{\lambda,w}u$.
In particular
$J_{\lambda,{\overline u}}u\leq {\tilde J}_{\lambda,{\hat u}}u$.
\item[(f)] $J_{\lambda,{\overline u}}{\overline u}={\overline u}$
and ${\tilde J}_{\lambda,{\hat u}}{\hat u}={\hat u}$.
\item[(g)] If $v\leq v'$ and $w\leq w'$, then
$J_{\lambda,v}u\leq J_{\lambda,v'}u$ and
${\tilde J}_{\lambda,w}u\leq {\tilde J}_{\lambda,w'}u$.
\end{enumerate}
} %end proposition
\paragraph{Proof.}
We prove only results (a),(b),(c),(d) in the case
$J_{\lambda}=J_{\lambda,v}$.\\
(a) Let $u\geq w$. Then $J_{\lambda,v}u$ satisfies
$$\displaylines{J_{\lambda,v}u\geq v\cr
(I+\lambda A)J_{\lambda,v}u \geq u \geq w.}$$
Hence we get (a) from minimality of $J_{\lambda,v}w$ for the previous
system. \\
(b) Inequality $J_{\lambda,v}u\geq v$
is required in the definition of $J_{\lambda,v}$.
Since $w$ satisfies $$\displaylines{w\geq v\cr
(I+\lambda A)w \geq w\geq u,}$$
we get (b) from minimality of $J_{\lambda,v}u$ in the previous system.
\\
(c) Let $Au\geq 0$ and $\lambda \geq 0$. We have
$$\displaylines{(I+\lambda A)u\geq u\cr
u \geq {v}}$$ so $J_{\lambda,v}u \leq u$.
Hence $u=J_{\lambda,v}u +\lambda AJ_{\lambda,v}u \leq u+\lambda
AJ_{\lambda,v}u$ and so $AJ_{\lambda}u \geq 0$.
\\
(d) Let $0\leq \lambda \leq \mu$. Then
$u=(I+\lambda A)J_{\lambda,v}u \leq (I+\mu A)J_{\lambda,v}u$.
Since we have $J_{\lambda,v}u \geq v$,
from minimality of $J_{\mu}u$ for these two constraints,
it comes $J_{\mu,v}u \leq J_{\lambda,v}u$.
The proof is similar when $Au\leq 0$.
\\
(e) Since $Av\leq 0$ and $Aw\geq 0$, from (c) it follows
$AJ_{\lambda,v}u\geq 0$ and $A{\tilde
J}_{\lambda,w}u\leq 0$. Hence (e) results from
$J_{\lambda,v}u+\lambda AJ_{\lambda,v}u=u={\tilde
J}_{\lambda,w}u+\lambda A{\tilde J}_{\lambda,w}u$.
\\
Properties (f) and (g) result immediately from the definitions.
\hfill$\square$
\subsection*{Solution of (1.1)}
We recall that a (local) \textbf{strong solution} of (1.1)
is a continuous function $u$ defined on
$[0,\theta) \subset [0,T],\theta >0$ such that
$u(t)=u_{0}+\int_{0}^{t}(-Au(\tau)+f(\tau))d\tau$
for $t\in [0,\theta)$.
In the sequel we only look for (local) strong solutions of (1.1).
A {\bf maximal} (resp. {\bf minimal}) {\bf solution } of (1.1)
is the strong solution $u=S_{A,f}^{\text{max}}(t)u_{0}$
(resp. $u=S_{A,f}^{\text{min}}(t)u_{0}$) of (1.1) defined as follows:
\\
(i) The interval of definition $[0,\theta)$
of $S_{A,f}^{\text{max}}(.)u_{0}$ (resp. $S_{A,f}^{\text{min}}(.)u_{0}$) is maximal
on $[0,T]$, i.e. there is no solution $v \not= u$,
such that $v=u$ on $[0,\theta]$.
\\
(ii) For each solution $v$ of (1.1) on $[0,T_{1})\subset [0,T]$, we have
$v(t)\leq S_{A,f}^{\text{max}}(t)u_{0}$
(resp. $v(t)\geq S_{A,f}^{\text{min}}(t)u_{0}$) on $[0,\inf (\theta, T_{1}))$.
\section{Main results}
For the following results, we assume the hypothesis \textrm{H1T} defined
in Section 2.
\paragraph{Theorem 3.1}
{\sl The problem $CP(A,f,u_{0})$ has a unique local solution denoted by
$S_{A,f}(t)u_{0}$ (or $S_{A}(t)u_{0}$ if $f=0$) and
defined on a maximal interval $[0,T_{\max}) \subset [0,T]$.
Moreover
if $u_{0}\leq u_{1}$ in $\mathbb{R}^N$ and if $f\leq g$
in $L^{1}([0,T],\mathbb{R}^N)$ then
$S_{A,f}(t)u_{0}\leq S_{A,g}(t)u_{1}$
on the common domain of existence of these two solutions.} \medskip
The next result concerns the autonomous case, for which
we have global solutions.
\paragraph{Theorem 3.2}
{\sl Assume that $f\equiv 0$. Then
$S_{A}(.)u_{0}$ is defined on the whole interval $[0,T]$
and
$$S_{A}(t)u_{0}=\lim_{n\to +\infty}J_{{t/n},n}(u_{0}),\eqno (3.1)$$
for $t\in [0,T]$, where $J_{\lambda}=J_{\lambda,{\overline u}_{0}}$
is as defined in Section 2.}
This is an exponential Crandall-Liggett's type formula, but
here $(I+{\lambda}A)^{-1}$ is a priori multi-valued.
In the non-autonomous case $f \not\equiv 0$, it is possible to exhibit
a formula as (3.1) which gives the solution of (1.1) as a limit of a
discrete scheme. But such a formula is more complicated than (3.1)
and thus, is not of a particular interest.
When $f \in L^{\infty}([0,T], \mathbb{R}^N)$, from Theorem 3.1 and
Theorem 3.2, we can deduce that $CP(A,f)$
has solution on $[0,T]$ if $A$ is a strong continuous derivor
(see def. 2.1). Unfortunately, we do not know what happens in the
general case $f \in L^{1}([0,T], \mathbb{R}^N)$
without extra assumptions.
\section{Proofs}
The proof of Theorem 3.1 follows immediately from the three lemmas
below.
\paragraph{Lemma 4.1}
{\sl Let $A$ be a continuous derivor. Let $V$ be an element of
$V_{K}(u_{0})$. Then the operator B defined by
$$B(v):=\inf_{w\in V}[A(w+v)-A(w)]\eqno (4.1)$$
is a continuous derivor.}
\paragraph{Proof.}
1.) Let us show that $B$ is a derivor on $\mathbb{R}^N$.
If $u\leq v$ and $u^i=v^i$
for some $i\in \{1,\dots ,N\}$,
we have $u+w\leq v+w$ and $(u+w)^i=(v+w)^i$
for each $w\in V$. Since A is a derivor, it follows
$A^i(u+w)-A^iw\geq A^i(v+w)-A^iw$.
Thus
$$\inf_{w\in V}(A^i(u+w)-A^iw)\geq \inf_{w\in V}(A^i(v+w)-A^iw).$$
So $B^iu\geq B^iv$ for $u\leq v$ and $u^i=v^i$.
\\
2.) At this stage we will show that $B$ is continuous on $\mathbb{R}^N$.
According to (4.1), $B$ is clearly upper semi-continuous
(see \cite[pp. 132-137]{c1}).
So it is enough to prove that for each $i\in \{1,\dots ,N\}$, $B^i$
is lower semi-continuous on $\mathbb{R}^N$. Fix $i\in \{1,\dots ,N\}$.
For each $u\in \mathbb{R}^N$, thanks to the
compactness of $V$, there exists $\chi(u)$ (which depends on $i$) in $V$
satisfying
$$B^i u=A^i( u+{\chi(u)})-A^i({\chi(u)})\eqno (4.2)$$
We have to prove now that $B^i$ is lower semi-continuous,
that is $(B^i)^{-1}(]-\infty,\alpha])$ is closed for all
$\alpha \in \mathbb{R}$. In this goal, consider $\alpha \in \mathbb{R}$
and a sequence $(u_{k})_{k \in \mathbb{N}^{*}}$
of elements of $\mathbb{R}^N$ such that $\lim u_{k}=u_{\infty}$
and $B^i(u_{k})\leq \alpha$. It suffices to prove
$B^i(u_{\infty})\leq \alpha$.
By contradiction, let us suppose $B^i(u_{\infty})> \alpha$.
Without loss of generality, thanks to the compactness of $V$,
we can suppose
$$\lim_{k\to +\infty}{\chi(u_{k})}=v_{\infty}\in V.$$
Equation (4.1) yields
$$\alpha **0$
such that $v^i(\tau)\leq u_{n}^i(\tau)$
for $\tau \in [t_{0},t_{0}+\eta]\subset [0,\tilde T \wedge T_{n})$.
Finally $E$ is open in $[0,\tilde T \wedge T_{n})$ and thus
$E=[0,\tilde T \wedge T_{n})$.
\\
2) We have $u_{n+1}\leq u_{n}$ on $[0,T_{n+1} \wedge T_{n})$.
Indeed the proof is the same as 1) if we replace $v$ by $u_{n+1}$
and $-Au_{n}(\tau)+Av(\tau)$ by $-Au_{n}(\tau)+Au_{n+1}(\tau)$.
\\
3) We have $\tilde T \wedge T_{n} \geq \tilde T \wedge T_{1}$.
Indeed, from parts 1) and 2), for each $n\in \mathbb{N^{*}}$
we have
$$v\leq u_{n+1} \leq u_{n} \leq u_{1}\eqno (4.7)$$
on the common interval of existence of these solutions.
Then the extension principle of solutions
implies $\tilde T \wedge T_{n+1}\geq \tilde T \wedge T_{n}$
since a bounded solution is extendable.
\\
4) The sequence $(u_{n})$ converges uniformly to $u_{\infty}$
on each compact sub-interval of $[0,\tilde T \wedge T_{1})$ thanks to
(4.7) and the Lebesgue's Dominated Convergence Theorem. Furthermore
$u_{\infty}$ is solution of $CP(A,f,u_{0},\tilde T \wedge T_{1})$
on $[0,\tilde T \wedge T_{1})$.
Moreover, clearly $u_{\infty}$ is the maximal solution
of $CP(A,f,u_{0},\tilde T \wedge T_{1})$ (see Section 2).
Let $F$ be the set of $S\in [0,T]$ such that
$u_{\infty}$ is extendable into a continuous function on $[0,S)$
which is the maximal solution of $CP(A,f,u_{0},S)$.
One has $\tilde T \wedge T_{1}\in F$.
By considering $S_{\infty}=\sup F$,
we obtain a maximal extension of $u_{\infty}$
as a local solution of $CP(A,f,u_{0},T)$
which is by construction the maximal solution of
$CP(A,f,u_{0},T)$. \hfill$\square$\smallskip
The next lemma makes use of the ascent assumption.
\paragraph{Lemma 4.3}
{\sl With the notation in Lemma 4.2, if \textrm{H1T} holds, we have
$$S_{A,f}^{\text{min}}(t)u_{0}=S_{A,f}^{\text{max}}(t)u_{0}$$ on $[0,T^{1}\wedge T^{2})
=[0,T^{1})$.}
\paragraph{Proof.}
Thanks to Lemma 4.2(a), (1.1) has a maximal solution
$S_{A,f}^{\text{max}}(t)u_{0}$ defined on a sub-interval $[0,T^{1})$ of $[0,T]$
and a minimal solution $u(t)=S_{A,f}^{\text{min}}(t)u_{0}$
defined on a sub-interval $[0,T^{2})$ of $[0,T]$.
Set $T_{3}=T^{1}\wedge T^{2}$ and
$$w(t):=S_{A,f}^{\text{max}}(t)u_{0}-S_{A,f}^{\text{min}}(t)u_{0}\eqno (4.8)$$
for $t\in [0,T_{3})$.
We have to prove $w=0$ on $[0,T_{3})$, that is $E=[0,T_{3})$
where $E=\{ t\in [0,T_{3}), w(\tau)=0,\forall \tau \in [0,t]\}$.
Since $E=w^{-1}(0)$ is closed in $[0,T_{3})$ ($w$ being continuous), it
just remains to show that $E$ is open to the right.
Let $t_{0} \in E, t_{0}< T_{3}$. We have to prove that there exists $h>0$
such that $w=0$ on $[t_{0},t_{0}+h]$. Eventually, by changing
$w$ into $w(t_{0}+.)$ and $f$ into $f(t_{0}+.)$, we will suppose
$t_{0}=0$.
Let $V \in V_{K}(u_{0})$ and $B$ as in (4.1),
in view of the continuity of $u$ at 0, there exists $T_{4}\in ]0,T_{3}[$
such that, for each $t\in [0,T_{4}]$, $u(t)\in V$,
hence $w$ satisfies a.e.:
$$\begin{gathered} w'(t)=-(A(u(t)+w(t))-Au(t))\leq -Bw(t) \cr
w(0)=0,\end{gathered}\eqno (4.9)$$
a.e. $t\in [0,T_{4}]$.
By using Lemma 4.2 (b) with $B$ instead of $A$,
we have $$w(t)\leq S_{B}^{\text{max}}(t)(0)\eqno (4.10)$$ for each
$t\in [0,T_{4}\wedge T_{5}]$,
where $[0,T_{5}]$ is the maximal interval of existence
of $S_{B}^{\text{max}}(t)(0)$.
The function $x(t)=S_{B}^{\text{max}}(t)(0)$ satisfies
$$\begin{gathered} x'(t)= -Bx(t)\cr
x(0)=0.\end{gathered}\eqno (4.11)$$
Let $(v_{k})_{k\in \mathbb{N^{*}}}$ be a sequence
which defines a uniform ascent at the point $u_{0}$
for the operator $A$ on the set$ V$ (see section 2).
$$B^i(v_{k})=A^i(v_{k}+{\hat v_{k}}(i))-A^i({\hat v_{k}}(i))>0
\eqno (4.12)$$
for $k\in \mathbb{N^{*}}$ and $i \in \{1,\dots ,N\}$
where ${\hat v_{k}}(i)$ is a vector minimizing
$ v \to A^i(v_{k}+ v)-A^i(v)$ on $V$.
Let $k\in \mathbb{N}$ be fixed, then due to Lemma 4.2(b)
there exists $s_{k}>0$ such that $s_{k}\leq T_{4}\wedge T_{5}$
and
$$S_{B}^{\text{max}}(t)(0)\leq S_{B}^{\text{max}}(t)(v_{k})\eqno (4.13)$$
for each $t \in [0,s_{k}]$.
Equation (4.12) and the continuity of $B$
give the existence of $t_{k}>0$ and $t_{k}\leq s_{k}$
such that:
$$B(S_{B}^{\text{max}}(t)(v_{k}))\geq 0$$ for $t\in [0,t_{k}]$.
Thus $t \to S_{B}^{\text{max}}(t)(v_{k})$ is decreasing on $[0,t_{k}]$.
Consequently, from (4.10) and (4.13), it results
$$w(t)\leq S_{B}^{\text{max}}(t)(v_{k})\leq S_{B}^{\text{max}}(0)(v_{k})=v_{k}\eqno (4.14)$$
for each $t\in [0,t_{k}]$.
In particular, we have $w(t_{k})\leq v_{k} $.
If we put $y(t)=w(t_{k}+t)$, we get
$$\begin{gathered} y'(t)\leq -By(t) \cr
y(0)=w(t_{k})\leq v_{k}\end{gathered}\eqno (4.15)$$
for a.e. $t\in [0,t_{k}]$.
Hence, according to (4.14) and (4.15), one has
$$w(t_{k}+t)\leq S_{B}^{\text{max}}(t)(v_{k})\leq v_{k}$$
for $t\in [0,t_{k}]$.
So $w(t)\leq v_{k}$ for $t\in [0,2t_{k}\wedge T_{4}]$.
Whence by induction, we get
$$0\leq w(t)\leq v_{k}\eqno (4.16)$$ for $t\in [0,T_{4}]$.
Since (4.16)
is valid for each $k\in \mathbb{N^{*}}$ and
$\lim v_{k}=0$, it follows $w(t)=0$
for each $t\in [0,T_{4}]$. Hence
for $h=T_{4}>0$, we have $[0,h] \subset E$ which completes the proof.
\hfill$\square$
\subsection*{Proof of Theorem 3.2}
In this subsection, we assume that $A$ satisfies \textrm{H1T}, and
$f \equiv 0$ on $[0,T]$. First, let us recall some basic facts
about the discretization (1.1) in the Theory of Nonlinear Semigroups.
It is known \cite{c5} that a strong solution of (1.1) is a mild
solution, i.e. a continuous function which is a uniform limit of Euler's
implicit discrete schemes. Such discrete schemes are defined as follows.
Let $\epsilon >0$ be fixed. Then an $\epsilon$-discretization on $[0,T]$
of $\dot u +Au=0$ on $[0,T]$ consists of a partition $0=t_{0}\leq t_{1}\leq \dots \leq
t_{n}$ of the interval $[0,t_{n}]$ and a finite sequence
$(f_{1},f_{2},\dots ,f_{n})$ in $\mathbb{R}^N$ such that
\\
(a) $t_{i}-t_{i-1}<\epsilon$ for $i=1,\dots ,n$
and $T-\epsilon0$
there is an $\epsilon$-approximate solution $v$
of $CP(A,0,u_{0})$ on $[0,T]$ such that $\| v(t)-u(t) \| \leq \epsilon$
for $t$ in the domain of $v$.
Now, for $n\in \mathbb{N^{*}}$, let
$J=J_{T/n, {\overline u_{0}}}$, and
define the function $u_{n}$ by
$u_{n}(0)=u_{0}$ and $u_{n}(t)=J^i(u_{0})$
for $(i-1)T/n 0$ there exists
$N_{\epsilon}\in \mathbb{N}$ and $\eta_{\epsilon}>0$
such that ($n\geq N_{\epsilon}$ and
$\vert t-s \vert \leq \eta_{\epsilon}$)
implies $\| u_{n}(t)-u_{n}(s) \|_{\infty}\leq \epsilon$.
Indeed, relations (4.17) lead to
$$
u_{n}(t_{j}^{n})-u_{n}(t_{i}^{n})=
-\int_{t_{i}^{n}}^{t_{j}^{n}}Au_{n}(t)dt.\eqno(4.21)
$$
Using (4.18), Relation (4.21) yields
$$\| u_{n}(t)-u_{n}(s) \| \leq M(\vert t-s \vert +2\frac{T}{n}),
\eqno (4.22)$$
where $M=\sup_{{\overline u}_{0}\leq w \leq {\hat u_{0}}} \| Aw \|$.
Consequently (see \cite[p. 260]{d4}) the sequence
$(u_{n})$ is relatively compact in the Banach space
${\cal B}([0,T],\mathbb{R}^N, \| \hskip 0.2cm \|_{\infty})$ of bounded
functions on $[0,T]$ with values in $\mathbb{R}^N$. So
there exists a subsequence $(u_{n_{k}})$ converging to a continuous
function $u_{\infty}$ which is a mild solution of $CP(A,0,u_{0})$.
Then, passing to the limit in (4.21)
(or from \cite[p. 314]{c5}), we see that $u_{\infty}$ is a strong
(even a classical) solution of $CP(A,0,u_{0})$ on $[0,T]$.
From Theorem 3.1, it results
$$u_{\infty}=S_{A}(.)u_{0}\eqno (4.23)$$
on $[0,T]$. Thus (4.20) follows from (4.23) and (4.18) on $[0,T]$.
Then, taking $T=t$, (4.23) yields
$$S_{A}(t)u_{0}=\lim_{n\to +\infty}J_{t/n}^{n}
(u_{0}),
$$
where $J_{t/n}=J_{{t/n},{\overline u_{0}}}$.
The proof is complete.
\section{An example in $\mathbb{R}^{2}$}
Let $A_{0}$ be the operator defined on $\mathbb{R}^{2}$ by
$$A_{0}\begin{pmatrix} x\cr y\end{pmatrix}
:=\begin{pmatrix} x+(x-2y)^{1/3}\cr
y+(2y-x)^{1/5}\end{pmatrix} \eqno (5.1)$$
\paragraph{Lemma 5.1}
{\sl The operator $A_{0}$ satisfies \textrm{H1T} and
\textrm{H2T} for all $T>0$.} \medskip
The proof is left to the reader. In particular, the relation
$$A_{0}\begin{pmatrix} x\cr y\end{pmatrix}
+\begin{pmatrix} 2t\cr t\end{pmatrix}
=A_{0}\begin{pmatrix} x\cr y\end{pmatrix}
+\begin{pmatrix} 2t\cr t\end{pmatrix}\eqno (5.2)$$
for $t\in \mathbb{R}^{+}$, provides uniform ascents at each point.
The sublinearity at infinity implies \textrm{H2T}.
Therefore we can apply the results of Section 3 to the operator $A_{0}$
for any $T>0$. Hence $CP(A_{0},f,u_{0},+\infty)$ has a unique global
solution, on $[0,+\infty[$. Now, our task is to prove that
no condition of Nagumo-Osgood-Kamke and no accretivity condition
(even in a generalized sense) can be applied
to obtain the uniqueness of solutions of $CP(A_{0},f,u_{0})$.
\subsection*{Generalized accretivity conditions}
Let $\|\cdot \|_{p}$, $p\in [1,+\infty]$, be
the classical $l_{p}$-norm in $\mathbb{R}^{2}$.
As usual (see \cite{c5,d1}), we set
$$[u,v]=\lim_{\lambda \downarrow 0}
\frac{\| u+\lambda v \|-\| u \|}{\lambda}\eqno (5.3)
$$
for $u,v\in \mathbb{R}^{2}$.
For $p\in [1,+\infty]$, the notation $[u,v]_{p}, p\in [1,+\infty]$ means
$[u,v]$, with $\| \cdot \|_{p}$ instead of $\| \cdot \|$ in (5.3).
In the sequel, $\phi$ stands for a continuous function
$\phi:\mathbb{R}\to \mathbb{R}^{+}$ satisfying
the following condition ${\cal U}$:
For each $T_{0}$, the function
$x\equiv 0$ is the unique positive solution on $[0,T_{0}]$ of
$$\displaylines{\dot x(t)=\phi(x(t))\cr
x(0)=0.
}$$
\paragraph{Definition 5.2}
We will say that an operator $B$
defined on $\mathbb{R}^{2}$ is {\bf $\phi$-accretive }
in $(\mathbb{R}^{2},\| \cdot \|)$ if
$$-[u-v,Bu-Bv]\leq \phi(\| u-v \|)\eqno (5.4)$$
for all $u,v\in \mathbb{R}^{2}$.
We will say that $B$ satisfies a
{\bf $\phi$-Osgood condition} if
$$\| Bu-Bv \| \leq \phi(\| u-v \|)$$ for all $u,v\in \mathbb{R}^{2}$.
\paragraph{Remark}
a) The condition $B+\omega I$ is accretive $(\omega \geq 0)$
means $B$ is $\phi$-accretive with $\phi(x)=\omega x$.
General studies of $\phi$-accretive conditions can be found in
\cite{c4,k2}.
\\
b) A $\phi$-{\bf Osgood condition} is a particular case of
$\phi$-accretivity.
\paragraph{Lemma 5.2}
{\sl Let $p\in [1,+\infty]$. Then, there is no $\phi$,
such that $A_{0}$ is $\phi$-accretive in
($\mathbb{R}^{2},\|\cdot \|_{p}$).
Moreover, there are no $\phi$ and no norm $\|\cdot \|$
such that $A_{0}$ satisfies a $\phi$-Osgood condition in
($\mathbb{R}^{2},\|\cdot\|$).}
\paragraph{Proof.}
a) Suppose first $p=+\infty$. By contradiction, suppose that $A_{0}$ is
$\phi$-accretive in ($\mathbb{R}^{2},\| \cdot \|_{\infty}$)
for some $\phi$. Let $x\in [0,1[$. A direct computation
yields $A_{0}\begin{pmatrix}0\cr 0\end{pmatrix}
=\begin{pmatrix}0\cr 0\end{pmatrix}$
and $$[\begin{pmatrix}x\cr x-\frac 12 x^{2}\end{pmatrix},
A\begin{pmatrix} x\cr x-\frac 12 x^{2}\end{pmatrix}]_{\infty}
=x^{1/3}(x^{2/3}+(-1+x)^{1/3}).\eqno (5.5)
$$
So, thanks to the $\phi$-accretivity, (5.5) implies
$$\frac 12 x^{1/3}\leq \phi(x),\eqno (5.6)$$
for $x\geq 0$ sufficiently small.
Set
$$z(t)=H^{-1}(t)\,,\quad H(\sigma)
=\int_{0}^{\sigma}\frac{d\xi}{\phi(\xi)}.\eqno (5.7)$$
From (5.6), $H$ is defined for $\sigma\geq 0$
sufficiently small and $$z(t)>0\eqno (5.8)$$ on some interval
$]0,T_{0}]$ with $T_{0}>0$.
By using (5.7), a straightforward computation gives
$z'(t)=\phi (z(t))$ and $z(0)=0$.
Then ${\cal U}$ provides
$$z\equiv 0 \quad\mbox{on } [0,T_{0}].\eqno (5.9)
$$
Hence there is a contradiction between (5.8) and (5.9).
\noindent b) Suppose now $p\in [1,+\infty[$.
By contradiction again, suppose that $A_{0}$ is $\phi$-accretive in
($\mathbb{R}^{2},\| \cdot \|_{p}$).
In this case, for $x\in [0,1]$, by setting
$$u=\begin{pmatrix} x\cr \frac 12 x-\frac 12 x^{2} \end{pmatrix}
$$
a direct computation gives
$$[u, A_{0}u]_{p}
=\left(1+\frac{x^{p-\frac 13}-(\frac{x-x^2}{2})^{p-1}x^{2/5}}
{\| u\|_{p}^{p}}\right)\| u \|_{p}.\eqno (5.10)$$
According to (5.10), the reader can check that the
$\phi$-accretivity property implies
$\phi(\| u \|_{p})\geq -[u,A_{0}u]_{p}\geq \frac{1}{2^{p}+1}x^{2/5}$
for $x\in [0,1]$ sufficiently small. Then we can deduce
that for some $x_{0}\in ]0,1]$, there is $C>0$ (for instance
$C=\frac{e^{-1/5}}{2(2^{p}+1)}$), such that
$$C\| u \|_{p}^{2/5}\leq \phi(\| u \|_{p})
$$ for all $x\in [0,x_{0}]$.
Finally, there exists $\xi_{0}> 0$ such that
$\phi(\xi)\geq C\xi^{2/5}$ for $\xi \in [0,\xi_{0}]$.
Now, as in step a), using the function $H$ defined in (5.7),
we can easily derive a contradiction.
\noindent c) Let $\| \cdot \|$ be a norm in $\mathbb{R}^{2}$
and suppose that $A_{0}$ satisfies a $\phi$-Osgood condition in
$(\mathbb{R}^{2}, \| \cdot \|)$.
Then, by taking $u=\begin{pmatrix} 0\cr x\end{pmatrix}$, in the
$\phi$-Osgood property
we obtain $\phi(\xi)\geq c{\xi^{1/5}}$ for a constant $c>0$,
$\xi_{1}>0$ and all $\xi \in [0,\xi_{1}]$.
So we can conclude as before and the lemma is proved.
\hfill$\square$
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1.eps}
\end{center}
\caption{Flow relative to $A_{0}$}
\end{figure}
\section{Asymptotic behavior}
Figure 5.1 motivates the following remarks about asymptotic behavior
of solutions of (1.1). Hypothesis \textbf{H3} stands for following three
conditions\begin{itemize}
\item $f\equiv 0$
\item The assumption \textrm{H2T} holds for all $T>0$
\item $A$ is a continuous derivor on $ \mathbb{R}^N$.
\end{itemize}
We do not assume the uniqueness of solutions
of $CP(A,0,u_{0},+\infty)$.
We set $A^{+}=\{u;Au\geq 0\}$ and $A^{-}=\{u;Au\leq 0\}$.
\paragraph{Definition 6.1}
A derivor $A$ is \textit{absorbent} if $ u_{0}\in A^{+}$
(resp. $ u_{0}\in A^{-}$) implies $u(t)\in A^{+}$
(resp. $ u(t)\in A^{-}$) for all $t\geq 0$ and
each solution $u(.)$ of the autonomous problem
$CP(A,0,u_{0},+\infty)$. We say that A is $u_{\infty}$-absorbent
if B defined by $Bu=Au-Au_{\infty}$ is absorbent.
\paragraph{Proposition 6.2} {\sl
Assume \textrm{H3}. Let $u_{0},v_{0},w_{0}$ be in $\mathbb{R}^N$
such that $Av_{0}\leq 0$, $Aw_{0}\geq 0$, $v_{0}\leq
w_{0}$ and $v_{0} \leq u_{0} \leq w_{0}$.
Suppose $A$ is a continuous $u_{\infty}$-absorbent derivor on
$\mathbb{R}^N$ such that the equation $Av=0$
has a unique solution $u_{\infty}$ in $[v_{0},w_{0}]$.
Then every solution $u$ of $CP(A,0,u_{0},+\infty)$ satisfies
$$\lim_{t\to +\infty}u(t)=u_{\infty}
$$} %\end proposition
\paragraph{Proof.}
It is sufficient to prove the result for $S_{A}^{\text{max}}(t)w_{0}$
and $S_{A}^{\text{min}}(t)v_{0}$ since from Lemma 4.2
such extremal solutions exist and satisfy
$S_{A}^{\text{min}}(t)v_{0}\leq u(t) \leq S_{A}^{\text{max}}(t)w_{0}$,
$t \in [0,+\infty [$. If $w(t)=S_{A}^{\text{max}}(t)w_{0}$
we have $$w(t)-w_{0}=-\int_{0}^{t}Aw(x)dx. \eqno (5.11)$$
Consequently $t\to w(t)$ is decreasing because
from the absorbent property
$w'(t)=-Aw(t)\leq 0$.
In an analogous way $v(t)=S_{A}^{\text{min}}(t)u$ is increasing
because $v'(t)=-Av(t)\leq 0$ for each $t\in [0,+\infty [$.
Hence we get
$$v_{0}\leq v(t)
\leq w(t)\leq w_{0}.$$
Then $l_{1}=\lim_{t\to +\infty}w(t)$
and $l_{2}=\lim_{t\to +\infty}v(t)$ exist in $\mathbb{R}^N$.
Hence, according to (5.11), $\int_{0}^{+\infty}Av(\tau)d\tau$ and
$\int_{0}^{+\infty}Aw(\tau)d\tau$ converge.
Since $\lim_{t\to +\infty}Aw(t)=Al_{1}$ and
$\lim_{t\to +\infty}Av(t)=Al_{2}$, we have necessarily
$Al_{1}=Al_{2}=0$. So by hypothesis
$\lim_{t\to +\infty}w(t)=\lim_{t\to +\infty}v(t)=u_{\infty}$.
\paragraph{Corollary 6.3}
{\sl For the operator $A_{0}$ introduced in (5.1), we have
$$\lim_{t\to \infty}S_{A_{0}}(t)(u_{0})=
\begin{pmatrix} 0\cr 0\end{pmatrix}.$$}
\paragraph{Proof.}
We can show that $A_{0}u=\begin{pmatrix} 0\cr 0\end{pmatrix}$ holds if
and only if $u=\begin{pmatrix} 0\cr 0\end{pmatrix}$. Moreover $A_{0}$
is absorbent.
Indeed, with the notation of Lemma 4.4, let $u_{0}\in A_{0}^{+}$
(resp. $A_{0}^{-}$) and
$u_{n}(t)=J_{T/n}^i(u_{0})$
for $(i-1)T/ n **