\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 64, pp. 1--18.\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 2004 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2004/64\hfil A limit set trichotomy for order-preserving systems]
{A limit set trichotomy for order-preserving systems on time scales} 

\author[Christian P\"otzsche \& Stefan Siegmund \hfil EJDE-2004/64\hfilneg]
{Christian P\"otzsche \& Stefan Siegmund}  % in alphabetical order

\address{Christian P\"otzsche \hfill\break
University of Augsburg\\
Department of Mathematics\\
86135 Augsburg\\
Germany}
\email{poetzsche@math.uni-augsburg.de}
\urladdr{http://www.math.uni-augsburg.de/$\sim$poetzsch} 
\thanks{}

\address{Stefan Siegmund \hfill\break
J. W. Goethe University\\
Department of Mathematics\\
60325 Frankfurt\\
Germany}
\email{siegmund@math.uni-frankfurt.de}
\urladdr{http://www.math.uni-frankfurt.de/$\sim$siegmund}

\date{}
\thanks{Submitted April 15, 2004. Published April 27, 2004.}
\thanks{The first author is supported by the ``Graduiertenkolleg: Nichtlineare
Probleme in Analysis, \hfill\break\indent
Geometrie und Physik'' (GRK~283) financed by
the DFG and the State of Bavaria.
The second\hfill\break\indent 
author is an Emmy Noether Fellow supported by the DFG}

\subjclass[2000]{37C65. 37B55, 92D25}
\keywords{Limit set trichotomy, 2-parameter semiflow, dynamic equation, 
\hfill\break\indent time scale}


\begin{abstract}
  In this paper we derive a limit set trichotomy for abstract
  order-preserving 2-parameter semiflows in normal cones of
  strongly ordered Banach spaces. Additionally, to provide an
  example, M\"uller's theorem is generalized to dynamic equations
  on arbitrary time scales and applied to a model from population
  dynamics.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newcommand{\ang}[1]{\langle#1\rangle}
\newcommand{\norm}[1]{\|#1\|}
\newcommand{\set}[1]{\{#1\}}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{defn}[thm]{Definition}
\newtheorem{example}[thm]{Example}
\newtheorem{rem}[thm]{Remark}
\renewcommand{\labelenumi}{(\alph{enumi})}

\section{Introduction}

In certain relevant situations it happens that a dynamical system preserves
a (partial) order relation on its state space. These systems are called
\emph{order-preserving} or \emph{monotone} and the ground for their
qualitative theory was laid by Krasnoselskii in his two books
\cite{kra64,kra68}. Meanwhile many others made further important
contributions for different types of such dynamical systems like
(semi-)flows of ordinary differential equation \cite{hir82,hir84,hir85,smi86},
functional differential equations \cite{smi87,ab90}, semilinear parabolic
equations \cite{hir88,tak91}, ordinary difference equations \cite{hir85,tak90},
\cite{kn93,kr92}, \cite{pt91,hp93}, random dynamical systems \cite{ac98} or
general skew-product flows \cite{chu01}; compare also the monographs
\cite{smi96} and \cite{chu02} for numerous examples and applications.

The essential property of order-preserving dynamical systems is that
they possess a surprisingly simple asymptotic behavior. In fact Krause
et al.~\cite{kn93,kr92} proved a so-called \emph{limit set trichotomy}
(cf.~also \cite{nes99} for nonautonomous difference equations or
\cite{ac01} for random dynamical systems), describing the only three
asymptotic scenarios of such systems under a certain kind of concavity.

In this paper we prove such a limit set trichotomy for a general model of
nonexpansive dynamical processes, namely \emph{$2$-parameter semiflows}
in normal cones on time scales. They include the solution operators of
dynamic equations on time scales (cf.~\cite{hil90,bp01}) and in particular
of nonautonomous difference and differential equations. Beyond the
unification aspect, dynamic equations on time scales are predestinated
to describe the interaction of biological species with hibernation
periods. The crucial point is that we provide sufficient criteria for
the nonexpansiveness of such solution operators in terms of concavity
and cooperativity conditions on the right-hand sides of the
corresponding equations.

On this occasion we generalize the classical theorem of M\"uller
(cf., e.g., \cite{mue26}) to dynamic equations in real Banach spaces.
Thereby we closely follow the arguments of \cite{wal01}, who considers
finite-dimensional ordinary differential equations and orderings with respect to
arbitrary cones. However, although our state spaces are allowed to be
infinite-dimensional, we have to make the assumption that cones have
nonempty interior. The use of arbitrary order cones instead of $\mathbb{R}_+^d$
even in finite dimensions has the advantage that certain equations are
cooperative (see Definition~\ref{defcoop}) with respect to an ordering different
from the component-wise.

\section{Preliminaries} \label{sec2}

Let $\mathbb{T}$ be an arbitrary \emph{time scale}, i.e., a canonically ordered
closed subset of the real axis $\mathbb{R}$. Since we are interested in the
asymptotic behavior of systems on such sets $\mathbb{T}$, it is reasonable to
assume that $\mathbb{T}$ is unbounded above in the whole paper. Moreover, $\mathbb{T}$ is called
\emph{homogeneous}, if $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}=h\mathbb{Z}$, $h>0$. A \emph{$\mathbb{T}$-interval}
is the intersection of a real interval with the set $\mathbb{T}$, for $a,b\in\mathbb{R}$
we write $[a,b]_\mathbb{T}:=[a,b]\cap\mathbb{T}$ and (half-)open intervals
are defined analogously. $(X,d)$ denotes a metric space from now on.

\begin{defn}\label{sflow} \rm
    A mapping $\varphi:\set{(t,\tau)\in\mathbb{T}^2:\,\tau\leq t}\times X\to X$ is
    denoted as a \emph{$2$-parameter semiflow on $X$}, if the mappings
    $\varphi(t,\tau,\cdot)=\varphi(t,\tau) : X \to X$, $\tau\leq t$,
    satisfy the following properties:
    \begin{itemize}
        \item[(i)] $\varphi(\tau,\tau)x=x$ for all $\tau\in\mathbb{T}$, $x\in X$,

        \item[(ii)] $\varphi(t,s)\varphi(s,\tau)=\varphi(t,\tau)$ for all
        $\tau,s,t\in\mathbb{T}$, $\tau\leq s\leq t$,

        \item[(iii)] $\varphi(\,\cdot,\cdot)x:\set{(t,\tau)\in\mathbb{T}^2:\,\tau\leq t}
        \to X$ is continuous for all $x\in X$.
    \end{itemize}
\end{defn}

\begin{rem} \rm % rmk2.2
\emph{(1)}\; Sometimes $2$-parameter semiflows are also called
    \emph{(evolutionary) processes} (cf., e.g., \cite[p.~100, Definition~1.1]{hv93}).

\emph{(2)}\; To provide some concepts from classical ($1$-parameter)
    semiflows, we denote a point $x_0\in X$ as an \emph{equilibrium}
    of $\varphi$, if $\varphi(t,\tau)x_0=x_0$ for all $\tau\le t$ holds.
    Moreover, for $\tau\in\mathbb{T}$ and $x\in X$, the \emph{orbit emanating
    from $(\tau,x)$} is
    $$
        \gamma_\tau^+(x):=\set{\varphi(t,\tau)x\in X:\,\tau\le t}
    $$
    and the \emph{$\omega$-limit set of $(\tau,x)$} is given by
    $$
        \omega_\tau^+(x)
        :=
        \bigcap_{\tau\le t}\mathop{\rm closure}\set{\varphi(s,\tau)x\in X:\,t\le s}.
    $$
    Equivalently, $\omega_\tau^+(x)$ consists of all the points
    $x^\ast\in X$ such that there exists a sequence $t_n\to\infty$
    in $\mathbb{T}$ with $x^\ast=\lim_{n\to\infty}\varphi(t_n,\tau)x$. A
    subset $U\subset X$ is denoted as \emph{forward invariant},
    if $\varphi(t,\tau)U\subset U$ holds for $\tau\leq t$.
    
\end{rem}

\begin{example}\label{fex} \rm
\emph{(1)}\; For homogeneous time scales $\mathbb{T}$, any strongly continuous
    discrete $(\mathbb{T}:=h\mathbb{Z},\,h>0)$ or continuous $(\mathbb{T}:=\mathbb{R})$ $1$-parameter
    semiflow $\set{\phi_t}_{t\geq 0}$ evidently generates a $2$-parameter
    semiflow $\varphi$ via $\varphi(t,\tau):=\phi_{t-\tau}$.

\emph{(2)}\; Let $X$ be some Banach space $V$ and $f:\mathbb{T}\times V\to V$.
    Then the standard examples for $2$-parameter-semiflows are the solution
    operators $\varphi(t,\tau,\cdot):V\to V$, $\tau\leq t$, of nonautonomous
    difference equations $v(t+1)=f(t,v(t))$, $t\in\mathbb{T} := \mathbb{Z}$, or of
    nonautonomous ordinary differential equations $\dot{v}(t)=f(t,v(t))$,
    $t\in\mathbb{T} := \mathbb{R}$ in $V$, provided that in the ODE case, $f$ is e.g.~measurable
    in $t$, (locally) Lipschitzian in $v$, and satisfies a certain growth
    condition to exclude finite escape times. The general situation, where
    $\mathbb{T}$ is an arbitrary closed subset of the reals, occurs in the context
    of dynamic equations $v^\Delta=f(t,v)$ on time scales
    (see\ Section \ref{sec5}).

\emph{(3)}\; Let $r\geq 0$ be real, $X:=C([-r,0],\mathbb{R}^d)$ the space of
    continuous functions endowed with the sup-norm, and $f:\mathbb{R}\times X\to\mathbb{R}^d$
    be continuous and (locally) Lipschitzian in the second argument. Then,
    if no finite escape times appear, the solution
    $v(\cdot,\tau,v^0):[\tau,\infty)\to\mathbb{R}^d$ of the retarded functional
    differential equation
    \begin{gather*}
        \dot{v}(t)=f(t,v_t),\\
        v_t(\theta):=v(t+\theta) \quad\mbox{for all }\theta\in[-r,0]
    \end{gather*}
    satisfying the initial condition $v_\tau=v^0$ for $\tau\in\mathbb{R}$,
    $v^0\in X$, defines a $2$-parameter semiflow on $X$ with $\mathbb{T}=\mathbb{R}$
    via $\varphi(t,\tau)v^0:=v_t(\cdot,\tau,v^0)$ (cf.~\cite{hv93}).

\emph{(4)}\; Criteria for more abstract nonautonomous evolutionary
    equations to generate a $2$-parameter semiflow can be found in
    \cite{am96} and the references therein.
\end{example}

We need some further terminology. A self-mapping $\Phi:X\to X$ will be called
\emph{nonexpansive} (on $(X,d)$), if $d(\Phi x,\Phi\bar{x})\leq d(x,\bar{x})$
for all $x,\bar{x}\in X$, and $\Phi$ will be called \emph{contractive}, if
$d(\Phi x,\Phi\bar{x})<d(x,\bar{x})$ for all $x,\bar{x}\in X$, $x\neq\bar{x}$.
If $P$ is a nonempty set, then a family of parameter-dependent self-mappings
$\Phi(p):X\to X$, $p\in P$, is called \emph{uniformly contractive}, if there
exists a continuous function $c:X\times X\to\mathbb{R}_+$, such that the following two
conditions are fulfilled (cf.~\cite{nes99}):
\begin{itemize}
    \item[(i)] $c(x,\bar{x})<d(x,\bar{x})$ for all $x,\bar{x}\in X$,
    $x\neq\bar{x}$,

    \item[(ii)] $d(\Phi(p)x,\Phi(p)\bar{x})\leq c(x,\bar{x})$ for all
    $p\in P$, $x,\bar{x}\in X$.
\end{itemize}

Assume from now on that the metric space $X$ is a cone $V_+$ in a real Banach
space $(V,\norm{\cdot})$. Recall that a \emph{cone} is a nonempty closed convex
set $V_+\subset V$ such that $\alpha V_+\subset V_+$ for $\alpha\geq 0$ and
$V_+\cap(-V_+)=\set{0}$. Moreover, define $V_+^\ast:=V_+\setminus\set{0}$.
Any cone defines a partial order relation on $V$ via $u\leq v$, if $v-u\in V_+$,
which is preserved under addition and scalar multiplication with nonnegative
reals. Furthermore, we write $u<v$ when $u\leq v$ and $u\neq v$. If $V_+$ has
nonempty interior $\mathop{\rm int} V_+$, we say that $V$ is \emph{strongly ordered} and
write $u\ll v$, if $v-u\in\mathop{\rm int} V_+$. A cone $V_+$ is called \emph{normal}, if
there exists a real number $M\geq 0$ such that $\norm{u}\leq M\norm{v}$ for
all $u,v\in V_+$ with $u\leq v$. In fact, without loss of generality, one
can assume the norm $\norm{\cdot}$ to be monotone, i.e., $\norm{u}\leq\norm{v}$,
if $u\leq v$; otherwise an equivalent norm on $V$ can be found for which $M=1$
(cf.~\cite{sch71}). Finally we define the \emph{order interval}
$[u,v]:=\set{w\in V:\,u\leq w\leq v}$ for $u,v\in V$, $u\leq v$.
Explicit examples of normal cones and strongly ordered Banach spaces can be
found in, e.g., \cite[pp.~219ff]{dei85}.

Although forthcoming results on the boundedness of orbits are stated in
the norm topology on $V_+$, our contractivity condition for $2$-parameter
semiflows will be formulated in a different metric topology:

\begin{defn}\label{partdef} \rm
    \begin{itemize}
        \item[(i)] The equivalence classes under the equivalence relation
        defined by $u \sim v$, if there exists $\alpha > 0$ such that
        $\alpha^{-1}u\leq v\leq \alpha u$ on the cone $V_+$ are called
        the \emph{parts} of $V_+$.

        \item[(ii)] Let $C$ be a part of $V_+$. Then $p:C\times C\to\mathbb{R}_+$,
        \begin{displaymath}
            p(u,v) :=
            \inf\set{\log \alpha \,: \alpha^{-1} u\leq v\leq \alpha u}
            \quad\mbox{for all } u,v\in C,
        \end{displaymath}
        defines a metric on $C$ called the \emph{part metric} of $C$.
    \end{itemize}
\end{defn}

\begin{rem}\label{rem-partmetric} \rm
\emph{(1)}\; $u$ and $v$ lie in the same part, if and only if $p(u,v)<\infty$.

\emph{(2)}\; Clearly $\mathop{\rm int} V_+$ is a part and the closure of every part
    is also a convex cone in the Banach space $V$. For a proof of the fact
    that $p$ is a metric on $C$ and for other properties of the part metric
    we refer to \cite{bb69} or \cite[pp.~83--86]{chu02}.

\emph{(3)}\; If the cone $V_+$ is normal, then $\mathop{\rm int} V_+$ is a complete
    metric space with respect to the part metric $p$ (cf.~\cite{tho63}).
\end{rem}

Norm distance and the part metric are related by the following inequality:

\begin{lem}\label{partprop}
    \begin{enumerate}
        \item If $V_+$ is normal with monotone norm, then
        \begin{displaymath}
            \norm{v-\bar{v}}
            \leq
            \big(2e^{p(v,\bar{v})}-e^{-p(v,\bar{v})}-1)
            \min\set{\norm{v},\norm{\bar{v}}} \quad\mbox{for all } v,\bar{v}\in V_+^\ast,
        \end{displaymath}

        \item $p|_{\mathop{\rm int} V_+\times\mathop{\rm int} V_+}$ is continuous in the
        norm topology on $\mathop{\rm int} V_+\times\mathop{\rm int} V_+$.
    \end{enumerate}
\end{lem}

\begin{proof}
    See \cite[Lemma~2.3]{kn93}
    for (a), while assertion (b) can be found in \cite[Proof of Theorem~2]{nes99}.
\end{proof}

\section{A Limit Set Trichotomy}\label{sec3}

The following theorem is a clear manifestation of the general experience
that contractivity drastically simplifies the possible long-term behavior
of a dynamical system. It is the main result in the abstract part of this
paper.

In the autonomous discrete time case a limit set trichotomy was discovered
(and so named) by Krause and Ranft~\cite{kr92} and generalized in \cite{kn93}
to infinite-dimensional autonomous difference equations; in addition,
\cite{nes99} considers such nonautonomous systems.

\begin{thm}[Limit Set Trichotomy]\label{lst}
    Let $V_+\subset V$ be a normal cone, $\mathop{\rm int} V_+\neq\emptyset$ and
    assume that $\varphi$ is a $2$-parameter semiflow on $V_+$ with
    the following properties:
    \begin{itemize}
        \item[(i)] There exists a real $T>0$ such that for all
        $t,\tau\in\mathbb{T}$ satisfying $T\leq t-\tau$, one has
        $\varphi(t,\tau)V_+^\ast\subset\mathop{\rm int} V_+$ and that the mapping
        $\varphi(t,\tau)|_{\mathop{\rm int} V_+}$ is nonexpansive,

        \item[(ii)] for all $(\tau,v)\in\mathbb{T}\times V_+$ every bounded orbit
        $\gamma_\tau^+(v)$ is relatively compact in the norm topology.
    \end{itemize}
    Then for every $\tau\in\mathbb{T}$ the following trichotomy holds, i.e.,
    precisely one of the following three cases applies:
    \begin{enumerate}
        \item For all $v\in V_+^\ast$ the orbits $\gamma_\tau^+(v)$
        are unbounded in norm,

        \item for all $v\in V_+$ the orbits $\gamma_\tau^+(v)$
        are bounded in norm and for all $v\in V_+^\ast$
        we have $\lim_{t\to\infty}\norm{\varphi(t,\tau)v}=0$,

        \item for all $v\in V_+$ the orbits $\gamma_\tau^+(v)$ are
        bounded in norm, the $\omega$-limit sets $\omega_\tau^+(v)$
        are nonempty and for all $v\in V_+^\ast$ they have
        a nontrivial accumulation point.
    \end{enumerate}
    If, moreover, $\omega_\tau^+(v)\subset\mathop{\rm int} V_+\cup\set{0}$ for all
    $v\in V_+^\ast$ and the mappings $\varphi(t,\tau)|_{\mathop{\rm int} V_+}$ are
    uniformly contractive for all $t,\tau\in\mathbb{T}$ with $T\leq t-\tau$,
    then in case (c) we have
    \begin{equation}
        \lim_{t\to\infty}\big[\varphi(t,\tau)v_1-\varphi(t,\tau)v_2\big]
        = 0 \quad\mbox{for all }v_1,v_2\in V_+^\ast.
        \label{star2}
    \end{equation}
\end{thm}

\begin{rem} \rm %rmk3.2
\emph{(1)}\; Condition \eqref{star2} implies that all $\omega$-limit sets
    $\omega_\tau^+(v)$, $v\in V_+^\ast$, are identical, and it excludes the
    existence of two different equilibria of $\varphi$. In fact, if $\varphi$
    possesses an equilibrium $v_0\in V_+^\ast$, then \eqref{star2} guarantees
    $\omega_\tau^+(v)=\set{v_0}$ for all $v\in V_+^\ast$. In the ``autonomous''
    situation of a homogeneous time scale $\mathbb{T}$ and a $2$-parameter semiflow
    induced by a $1$-parameter semiflow (cf.\ Example \ref{fex}(1)), the
    assumption $\omega_\tau^+(v)\subset\mathop{\rm int} V_+\cup\set{0}$ becomes superfluous.
    This yields by the invariance of $\omega_\tau^+(v)$ and hypothesis (ii),
    i.e., $\omega_\tau^+(v)=\varphi(t,\tau)\omega_\tau^+(v)\subset\mathop{\rm int} V_+$ for
    $T\leq t-\tau$.

\emph{(2)}\; One can show a stronger limit set trichotomy, if $\varphi$ is
    induced by a discrete $1$-parameter semiflow (cf.~\cite[Theorem~3.1]{kn93}).
    More results in the finite-dimensional situation $V_+=\mathbb{R}_+^d$ can be found
    in \cite[Theorems~1,~2]{kr92} and related topics are contained in
    \cite[Theorem~1.1]{tak90} or \cite[Theorem~4.1]{pt91}. Furthermore,
    \cite[Lemma 4]{nes99} provides sufficient conditions for the right-hand
    side of nonautonomous difference equations to generate a uniformly
    contractive $2$-parameter semigroup. On general time scales, stronger limit
    set trichotomies can be found in \cite{poe04} under the assumption that
    $\varphi$ is uniformly ascending.

\emph{(3)}\; We also briefly comment the situation when $\varphi$ comes
    from an ordinary differential equation $(\mathbb{T}=\mathbb{R})$. For autonomous
    cooperative systems in $\mathbb{R}^2$, a prototype result has been given by
    \cite[Theorem~2.3]{hir82}. If $\varphi$ comes from a time-periodic equation,
    \cite[Theorem~3.1]{smi86} proved a ``limit set dichotomy'' under certain
    assumptions on the Floquet multipliers. Similar results are given by
    \cite[Theorems~3,~4]{kr92}; \cite[Theorem~6.8]{hir88} considers general
    continuous $1$-parameter semiflows, and \cite[Theorem~3.1]{chu01} proves
    a limit set trichotomy for order-preserving skew-product flows.

\emph{(4)}\; Finally, the case of random dynamical systems is
    considered in \cite[Theorem~4.2]{ac01} and
    \cite[pp.~123--124, Theorem~4.4.1]{chu02}.
    
\end{rem}

\begin{proof}
    Let $\tau\in\mathbb{T}$ be arbitrary, but fixed. If (a) holds, then obviously
    (b) and (c) cannot hold. If (a) does not hold, then there exists a
    $v_1\in V_+^\ast$ such that the orbit $\gamma_\tau^+(v_1)$ is bounded,
    i.e., $\|\varphi(t,\tau)v_1\|\leq M$ for some $M\geq 0$ and all $t\geq\tau$.
    Now we show that in this case every orbit $\gamma_\tau^+(v)$, $v\in V_+$,
    is bounded in norm. Let the vector $v_2\in V_+$ be arbitrary. Then either
    (i) $\gamma_\tau^+(v_2)$ is bounded or (ii) there exists a $t'\in\mathbb{T}$ with
    $T\leq t'-\tau$ such that $\varphi(t',\tau)v_2\not=0$. In case (ii) it
    follows from assumption~(i) that
    $\varphi(t,\tau)v_1,\varphi(t,\tau)v_2\in\mathop{\rm int} V_+$ for $t\geq t'$.
    The Remarks~\ref{rem-partmetric} (1) and (2) imply
    $K:=p(\varphi(t',\tau)v_1,\varphi(t',\tau)v_2)<\infty$. Using the
    $2$-parameter semiflow property of $\varphi$ (cf.~Definition~\ref{sflow}(ii))
    together with the fact that the mappings $\varphi(t,t')$, $t\geq t'+T$, are
    nonexpansive, we obtain
    \begin{displaymath}
        p(\varphi(t,\tau)v_1,\varphi(t,\tau)v_2)\leq K
        \quad
        \text{for } t\geq t'+T
        \,.
    \end{displaymath}
    Consequently, Lemma~\ref{partprop}(a) provides the estimate
    \begin{displaymath}
        \norm{\varphi(t,\tau)v_2}
        \leq
        \norm{\varphi(t,\tau)v_2-\varphi(t,\tau)v_1}+\norm{\varphi(t,\tau)v_1}
        \leq
        2e^K M
    \end{displaymath}
    for all $t\geq t'+T$, proving that $\gamma_\tau^+(v_2)$ is bounded.

    Now we show that either (b) or (c) holds. By assumption (ii) the orbits
    $\gamma_\tau^+(v)$, $v\in V_+$, are relatively compact and
    therefore $\omega_\tau^+(v) \not= \emptyset$, moreover,
    the relation $\omega_\tau^+(v) = \set{0}$ is equivalent to
    $\lim_{t\to\infty}\varphi(t,\tau)v = 0$. We show that
    \begin{displaymath}
        \omega_\tau^+(v_1) = \set{0}
        \text{ for a single } v_1\in V_+^\ast
        \quad\implies\quad
        \omega_\tau^+(v) = \set{0}
        \text{ for any } v\in V_+^\ast
        \,.
    \end{displaymath}
    To this end, we assume that there exist $v_1,v_2,v_2^\ast\in V_+^\ast$
    with $\omega_\tau^+(v_1)=\set{0}$ and
    $v_2^\ast\in\omega_\tau^+(v_2)\setminus\set{0}$.
    Then there exists a sequence $t_n\to\infty$ in $\mathbb{T}$ with
    \begin{displaymath}
        \lim_{n\to\infty}\varphi(t_n,\tau)v_1 = 0
        \quad\text{and} \quad
        \lim_{n\to\infty}\varphi(t_n,\tau)v_2 = v_2^\ast
        \,,
    \end{displaymath}
    where we assume without lost of generaliry that $t_0 = \tau$ and $t_{n+1}-t_n\geq T$,
    which implies by assumption (i) that $\varphi(t_n,\tau)v_i\in\mathop{\rm int} V_+$
    for $i=1,2$ and $n\in\mathbb{N}$.
    Using the 2-parameter semiflow property and the fact that the
    mappings $\varphi(t_{n+1},t_n)$, $n\in\mathbb{N}$, are nonexpansive, we get
    \begin{displaymath}
        p(\varphi(t_n,\tau)v_1,\varphi(t_n,\tau)v_2)
        \leq
        \dots
        \leq
        p(\varphi(t_1,\tau)v_1,\varphi(t_1,\tau)v_2)
        \leq
        p(v_1,v_2)
    \end{displaymath}
    for $n\in\mathbb{N}$. Choosing $N\in\mathbb{N}$ such that $\|\varphi(t_n,\tau)v_1\|\leq
    \|\varphi(t_n,\tau)v_2\|$ for $n\geq N$, Lemma~\ref{partprop}(a)
    implies the contradiction
    \begin{align*}
        \|\varphi(t_n,\tau)v_2\|
        & \leq
        \|\varphi(t_n,\tau)v_2-\varphi(t_n,\tau)v_1\| + \|\varphi(t_n,\tau)v_1\|
        \\
        & \leq
        3 e^{p(v_1,v_2)} \|\varphi(t_n,\tau)v_1\|
        \to 0
        \quad\text{for } n\to\infty
        \,,
    \end{align*}
    proving that either (b) or (c) is true.

    It remains to show \eqref{star2} under the additional assumptions
    that the mappings $\varphi(t,s)$, $T\leq t-s$, are uniformly
    contractive, and that $\omega_\tau^+(v)\in\mathop{\rm int} V_+\cup\set{0}$
    for all $v\in V_+^\ast$. Assume that \eqref{star2} does not hold.
    Then there exists $v_1, v_2\in V_+^\ast$, an $\varepsilon > 0$ and a
    sequence $t_n\to\infty$ in $\mathbb{T}$ with
    \begin{equation}
        \norm{\varphi(t_n,\tau)v_1-\varphi(t_n,\tau)v_2}
        \geq
        \varepsilon \quad\mbox{for all } n\in\mathbb{N},
        \label{star3}
    \end{equation}
    where we assume without lost of generaliry $t_1\geq\tau+T$ and $t_{n+1}-t_n\geq T$, which
    implies that $\varphi(t_n,\tau)v_i \not= 0$ for $i=1,2$ and $n\in\mathbb{N}$. Since
    the orbits $\gamma_\tau^+(v_1)$ and $\gamma_\tau^+(v_2)$ are relatively
    compact there exists a subsequence of $(t_n)_{n\in\mathbb{N}}$, which we denote
    by $(t_n)_{n\in\mathbb{N}}$ again, such that the limits
    \begin{displaymath}
        v_1^\ast := \lim_{n\to\infty}\varphi(t_n,\tau)v_1
        \quad\text{and} \quad
        v_2^\ast := \lim_{n\to\infty}\varphi(t_n,\tau)v_2
    \end{displaymath}
    exist. By assumption $v_1^\ast,v_2^\ast\in\mathop{\rm int} V_+\cup\set{0}$
    and by \eqref{star3} $v_1^\ast\neq v_2^\ast$. We can also rule
    out that $v_1^\ast = 0$ and $v_2^\ast\in\mathop{\rm int} V_+$, since in this
    case, choosing $N\in\mathbb{N}$ such that $\|\varphi(t_n,\tau)v_1\|\leq
    \|\varphi(t_n,\tau)v_2\|$ for $n\geq N$, Lemma~\ref{partprop}(a)
    would imply
    \begin{align*}
        \norm{\varphi(t_n,\tau)v_2}
        & \leq
        \norm{\varphi(t_n,\tau)v_2-\varphi(t_n,\tau)v_1}+\norm{\varphi(t_n,\tau)v_1}
        \\
        & \leq
        3 e^{p(\varphi(t_1,\tau)v_1,\varphi(t_1,\tau)v_2)} \|\varphi(t_n,\tau)v_1\|
        \to 0
        \quad\text{for } n\to\infty
        \,,
    \end{align*}
    contradicting $v_1^\ast \not= v_2^\ast$. Hence we have
    $v_1^\ast,v_2^\ast\in\mathop{\rm int} V_+$. The $2$-parameter semiflow
    property and the fact that the mappings $\varphi(t_{n+1},t_n)$,
    $n\in\mathbb{N}$, are uniformly contractive, imply the estimates
    \begin{gather*}
         p(\varphi(t_n,\tau)v_1,\varphi(t_n,\tau)v_2)
        \leq
        c(\varphi(t_n,\tau)v_1,\varphi(t_n,\tau)v_2)
         < \dots
        \\
         <
        p(\varphi(t_1,\tau)v_1,\varphi(t_1,\tau)v_2)
        \leq
        c(\varphi(t_1,\tau)v_1,\varphi(t_1,\tau)v_2)
        \,.
    \end{gather*}
    Since monotone and bounded sequences converge and the mappings $p$ and
    $c$ are continuous in $(v_1^\ast,v_2^\ast)$ by Lemma~\ref{partprop}(b)
    and assumption, respectively, we get
    \begin{displaymath}
        p(v_1^\ast,v_2^\ast) = c(v_1^\ast,v_2^\ast)
    \end{displaymath}
    in contradiction to $c(v_1^\ast,v_2^\ast) < p(v_1^\ast,v_2^\ast)$,
    thus proving that \eqref{star2} holds.
\end{proof}


\section{Subhomogeneous Order-Preserving Semiflows}

The results of Section \ref{sec3} are only helpful, if one has
verifiable conditions which guarantee that a $2$-parameter semiflow
is nonexpansive with respect to the part-metric. Sufficient conditions
will be given by the following notions.
\begin{defn} \rm %def4.1
    Let $U\subset V_+$. A $2$-parameter semiflow $\varphi$
    on $V_+$ is said to be
    \begin{itemize}
        \item[(i)] \emph{order-preserving on $U$} if
        \begin{displaymath}
            u,v\in U,\,u\leq v
            \quad \implies \quad
            \varphi(t,\tau)u\leq \varphi(t,\tau)v  \quad\mbox{for all }\tau\leq t;
        \end{displaymath}

        \item[(ii)] \emph{strictly order-preserving on $U$}, if it is
        order-preserving on $U$ and
        \begin{displaymath}
            u,v\in U,\,u < v
            \quad\implies\quad
            \varphi(t,\tau)u <\varphi(t,\tau)v \quad\mbox{for all }\tau\leq t;
        \end{displaymath}

        \item[(iii)] \emph{strongly order-preserving on $U$}, if
        $\mathop{\rm int} V_+\neq\emptyset$, if $\varphi$ is order-preserving on $U$ and
        \begin{displaymath}
            u,v\in U,\,u \ll v
            \quad\implies\quad
            \varphi(t,\tau)u \ll\varphi(t,\tau)v \quad\mbox{for all }\tau\leq t.
        \end{displaymath}
    \end{itemize}
\end{defn}

We now introduce a class of order-preserving $2$-parameter semiflows
which possess a certain concavity property we call \emph{subhomogeneity}
(sometimes also named sublinearity). Subhomogeneity means concavity for
the particular case in which one of the reference points is $0$, hence
asks less and is thus more general than classical concavity. The autonomous
version of this property plays an important role in many studies and
applications, see \cite{kra64,kra68}, \cite{kn93,kr92}, \cite{smi86,tak90}
and the references therein.

\begin{defn} \rm %def4.2
    A $2$-parameter semiflow $\varphi$, which is order-preserving
    on $V_+$, is said to be
    \begin{itemize}
        \item[(i)] \emph{subhomogeneous}, if for any $v\in V_+$ and for any
        $\alpha\in (0,1)$ we have
        \begin{equation}\label{sublin}
            \alpha\varphi(t,\tau)v
            \leq
            \varphi(t,\tau)\alpha v \quad\mbox{for all }\tau<t;
        \end{equation}

        \item[(ii)] \emph{strictly subhomogeneous}, if we have in addition
        for any $v\in\mathop{\rm int} V_+$ the strict inequality
        \begin{equation}\label{strictsublin}
            \alpha\varphi(t,\tau)v
            <
            \varphi(t,\tau)\alpha v \quad\mbox{for all }\tau<t.
        \end{equation}
    \end{itemize}
\end{defn}

\begin{rem} \rm %rmk4.3
    Inequality \eqref{sublin} holds automatically for $t = \tau$ and for
    $\alpha\in\set{0,1}$; it can be equivalently rewritten as follows:
    For any $v\in V_+$ and for any $\alpha > 1$ we have
    \begin{equation}\label{no33}
        \varphi(t,\tau)\alpha v
        \leq
        \alpha \varphi(t,\tau)v \quad\mbox{for all }\tau<t
    \end{equation}
    and $\varphi(t,\tau)\alpha v<\alpha\varphi(t,\tau)v$ instead of
    \eqref{strictsublin}, respectively.
\end{rem}

\begin{lem}\label{sub}
    Let $\varphi$ be a subhomogeneous $2$-parameter semiflow on $V_+$,
    which is order-preserving on $V_+$. Then
    \begin{enumerate}
        \item $\varphi$ preserves the equivalence relation from
        Definition \ref{partdef}(i) and is nonexpansive under
        the part metric on every part $C$ of $V_+$.

        \item If, moreover, $\varphi$ is strictly subhomogeneous on a part
        $C$ of $V_+$, it is contractive under the part metric on $C$.
    \end{enumerate}
\end{lem}

\begin{rem} \rm % rmk4.5
    It is easy to see that a contractive $2$-parameter semiflow possesses
    at most one equilibrium in $C$.
\end{rem}

\begin{proof}
    (a) It follows from \eqref{sublin} and \eqref{no33} that,
    if for $v,\bar{v}\in C$ and some $\alpha\geq 1$ the estimate
    $\alpha^{-1}v\leq\bar{v}\leq\alpha \bar{v}$ implies
    \begin{displaymath}
        \alpha^{-1}\varphi(t,\tau)v\leq\varphi(t,\tau)\bar{v}
        \leq
        \alpha\varphi(t,\tau)v \quad\mbox{for all }\tau\leq t
    \end{displaymath}
    and hence by the definition of the part metric
    \begin{displaymath}
        p(\varphi(t,\tau)v,\varphi(t,\tau)\bar{v})
        \leq
        p(v,\bar{v}) \quad\mbox{for all }\tau\leq t.
    \end{displaymath}
    (b) Analogously, under the assumption \eqref{strictsublin}, it follows
    \begin{displaymath}
        \alpha^{-1}\varphi(t,\tau)v<\varphi(t,\tau)\bar{v}
        <
        \alpha\varphi(t,\tau)v \quad\mbox{for all }t>\tau
    \end{displaymath}
    and this leads to the contractivity of $\varphi$.
\end{proof}

\section{Order-Preserving Dynamic Equations}\label{sec5}

Supplementing our explanations from Section \ref{sec2}, we need
some further terminology. $I_V$ denotes the identity map on $V$.
The \emph{dual cone} $V_+'$ of $V_+$ is the set of all linear and
continuous mappings $v':V\to\mathbb{R}$ such that $\ang{v,v'}\geq 0$ for
all $v\in V_+$, where $\ang{v,v'}:=v'(v)$ is the duality mapping.
If $V$ is a Hilbert space, then $V_+'$ can be identified with a
subset of $V$ through the Riesz representation theorem
(cf.~\cite[p.~104, Theorem~2.1]{lan93}). The elements of
$V_+^\star:=V_+'\setminus\set{0}$ are called \emph{supporting forms}
and we define $\mathcal{L}(V_+):=\set{T\in\mathcal{L}(V):\,T(V_+)\subset V_+}$. Any
such operator $T\in\mathcal{L}(V_+)$ is called \emph{positive}, and
\emph{strictly positive}, if $Tv=0$ implies $v=0$ for any $v\in V_+$.

First of all, we can characterize the (interior) points of a cone in
terms of linear functionals.

\begin{lem}\label{lem51}
    For any $v\in V$ the following holds:
    \begin{enumerate}
        \item $v\in V_+$ $\Leftrightarrow$ $\ang{v,v'}\geq 0$ for all
        $v'\in V_+^\star$,

        \item $v\in\mathop{\rm int} V_+$ $\Leftrightarrow$ $\ang{v,v'}>0$ for all
        $v'\in V_+^\star$.
    \end{enumerate}
\end{lem}

\begin{proof}
    See~\cite[p.~221, Proposition 19.3]{dei85}.
\end{proof}

At this point we introduce some further notation concerning the
calculus on time scales (see also~\cite{bp01}).
Remember that $\mathbb{T}$ is a closed subset
of $\mathbb{R}$ which is assumed to be unbounded above.
$\sigma(t):=\inf\set{s\in\mathbb{T}:\,t<s}$ defines the
\emph{forward jump operator} $\sigma:\mathbb{T}\to\mathbb{T}$ and
$\mu(t):=\sigma(t)-t$ the \emph{graininess} of $\mathbb{T}$. A point
$t\in\mathbb{T}$ is called \emph{right-dense}, if $\sigma(t)=t$ and otherwise
\emph{right-scattered}. Analogously, in case $\sup\set{s\in\mathbb{T}:\,s<t}=t$,
the point $t\in\mathbb{T}$ is said to be \emph{left-dense}. It is worth to
mention that all the results of this paper remain true with obvious
modifications, if the time scale $\mathbb{T}$ is replaced by a general
measure chain (cf.~\cite{hil90}).

Now we will discuss dynamic equations in real Banach spaces of the form
\begin{equation}\label{system}
v^\Delta=f(t,v),
\end{equation}
where the \emph{right-hand side} $f:\mathbb{T}\times V\to V$ satisfies the
following assumptions:
\begin{itemize}
    \item[(H0)] $V$ is a strongly ordered Banach space with cone $V_+$,
    i.e., $\mathop{\rm int} V_+\neq\emptyset$,

    \item[(H1)] $f:\mathbb{T}\times V\to V$ is rd-continuously differentiable
    with respect to the second variable, i.e., the partial derivative
    $D_2f:\mathbb{T}\times V\to V$ is assumed to exist and, furthermore,
    is rd-continuous,

    \item[(H2)] for any $\tau\in\mathbb{T}$ and $v\in V$ the solution
    $t\mapsto\varphi(t,\tau,v)$ of \eqref{system} starting at time
    $\tau$ in $v$ exists for all $\tau\leq t$.
\end{itemize}
Under the condition (H1) the solutions of \eqref{system} exist and
are unique locally in forward time (cf.~\cite[p.~324, Theorem~8.20]{bp01}),
with continuous partial derivatives $D_3\varphi(t,\tau,v)\in\mathcal{L}(V)$,
$\tau\leq t$, $v\in V$ (cf.~\cite[pp.~47--48, Satz~1.2.22]{poe02})
and by the absence of finite escape times, obviously $\varphi$ defines a
$2$-parameter semiflow on $V$ (cf.~\cite[p.~42, Korollar 1.2.19]{poe02}),
where all the results from the previous sections apply to $\varphi$ under
certain assumptions on $f$.

\begin{lem}[Characterization of Forward Invariance]\label{lem52}
    The following three statements are equivalent.
    \begin{enumerate}
        \item The cone $V_+$ is forward invariant for \eqref{system}.

        \item For every right-dense $t_0\in\mathbb{T}$, any $v\in\partial V_+$,
        $v'\in V_+^\star$ such that $\ang{v,v'}=0$ satisfy
        $\ang{f(t_0,v),v'}\geq 0$, and, for every right-scattered $t_0\in\mathbb{T}$
        any $v\in V_+$ satisfies $v+\mu(t_0)f(t_0,v)\in V_+$.

        \item For every $t_0\in\mathbb{T}$ it holds
        \begin{equation}
            \lim_{h\searrow\mu(t_0)}
            \frac{\mathop{\rm dist}(v+hf(t_0,v),V_+)}{h}=0,
            \label{nagumo}
        \end{equation}
        if $v\in\partial V_+$ and $t_0$ is right-dense,
        or, $v\in V_+$ and $t_0$ is right-scattered.
    \end{enumerate}
\end{lem}

\begin{rem} \rm  % rmk5.3
    The condition \eqref{nagumo} provides a descriptive geometric
    interpretation: In a right-scattered point $t_0\in\mathbb{T}$ it simply means
    that $v+\mu(t)f(t,v)\in V_+$, if $v\in V_+$. In a right-dense point
    $t_0\in\mathbb{T}$, and at a boundary point $\nu(t_0)\in\partial V_+$ with a
    tangent, $f(t_0,\nu(t_0))$ and hence the vector $\nu^\Delta(t_0)$ have to be
    directed to the interior of $V_+$, i.e., this vector does not point into the
    outer half space. Both conditions force the solutions to remain in $V_+$.
\end{rem}

\begin{proof}
    Let $t_0\in\mathbb{T}$ be arbitrary. We proceed in four steps:

    \textit{(I)} In case of a right-dense $t_0$ the equivalence of (b) and (c)
    is shown in~\cite[p.~51, Example 4.1]{dei77}. In a right-scattered
    $t_0$ the relation \eqref{nagumo} obviously holds, if and only if
    $v+\mu(t_0)f(t_0,v)\in V_+$, since $V_+$ is closed.

    \textit{(II)} We show that the forward invariance of $V_+$ implies (b).
    Thereto let $t_0\in\mathbb{T}$, $v\in V_+$ be arbitrary and let
    $\nu$ be the solution of \eqref{system} with $\nu(t_0)=v$. In a
    right-scattered $t_0$ the invariance of $V_+$ implies
    \begin{displaymath}
        v+\mu(t_0)f(t_0,v)
        =
        \nu(t_0)+\mu(t_0)f(t_0,\nu(t_0))
        =
        \nu(\sigma(t_0))\in V_+
        \,.
    \end{displaymath}
    On the other hand, if $t_0$ is right-dense and $v\in\partial V_+$,
    choose $v'\in V_+^\star$ such that $\ang{v,v'}=0$ (cf.~Lemma~\ref{lem51}).
    Then the assumption $\ang{f(t_0,v),v'}<0$ would imply the existence of a
    right-sided $\mathbb{T}$-neighborhood $N$ of $t_0$ with $\ang{\nu(t),v'}<0$
    for $t\in N$ and hence the contradiction $\nu(t)\not\in V_+$ (cf.\
    Lemma~\ref{lem51}(a)).

    \textit{(III)} In the remaining two steps we show the forward invariance
    of $V_+$ under the condition (b). For the present, we strengthen (b) to the
    hypothesis that for any right-dense point $t_0$, every $v\in\partial V_+$
    and every $v'\in V_+^\star$ such that $\ang{v,v'}=0$, one has
    $\ang{f(t_0,v),v'}>0$. Thus, let $\nu$ denote a solution of
    \eqref{system} starting at $\tau\in\mathbb{T}$ in $\nu(\tau)\in\mathop{\rm int} V_+$. If
    the claim were false, then there exists a finite $t^\ast\in\mathbb{T}$ given by
    \[
        t^\ast:=\sup T,\quad
        T:=\set{t\geq\tau:\nu(s)\in V_+\text{ for all }s\in[\tau,t]_{\mathbb{T}}}.
    \]
    Since the cone $V_+$ is closed we have $\nu(t^\ast)\in V_+$. The point
    $t^\ast$ is right-dense, because otherwise $\nu(\sigma(t^\ast))=\nu(t^\ast)+
    \mu(t^\ast)f(t^\ast,\nu(t^\ast))\in V_+$ would yield the contradiction
    $t^\ast<\sigma(t^\ast)\in T$. Moreover, $\nu(t^\ast)\in\partial V_+$,
    because the assumption $\nu(t^\ast)\in\mathop{\rm int} V_+$ would imply the existence
    of a neighborhood $U\subset V_+$ of $\nu(t^\ast)$ and a $\mathbb{T}$-neighborhood
    $N$ of $t^\ast$ with $\nu(t)\in U$ for $t\in N$, since $\nu$
    is continuous as the solution of \eqref{system}. This again contradicts
    the definition of $t^\ast$. Now by Lemma~\ref{lem51}(b) there exists a
    supporting form $v'\in V_+^\star$ such that $\ang{\nu(t^\ast),v'}=0$ and
    by definition $t^\ast$ is the time when the solution $\nu$ leaves $V_+$,
    which by Lemma~\ref{lem51}(a) gives us $\ang{\nu(t),v'}\leq 0$ for $t$
    from a right-sided neighborhood of $t^\ast$. One finds
    $\ang{\frac{\nu(t)-\nu(t^\ast)}{t-t^\ast},v'}\leq 0$ and in the
    limit $t\searrow t^\ast$, we therefore obtain the contradiction
    \begin{displaymath}
        0
        \geq
        \ang{\nu^\Delta(t^\ast),v'}
        =
        \ang{f(t^\ast,\nu(t^\ast)),v'}
        >
        0
    \end{displaymath}
    with a view to the above (strengthened) hypothesis.

    \textit{(IV)} The verification of the assumption under the general
    hypothesis yields as follows: For arbitrary reals $\varepsilon>0$ and some
    fixed $e\in\mathop{\rm int} V_+$ one can apply the above step (III) to the
    solution $\nu_\varepsilon$ of the dynamic equation
    \begin{displaymath}
        v^\Delta=f(t,v)+\varepsilon e
    \end{displaymath}
    and consequently, for any $\tau\in\mathbb{T}$ and $v_0\in V_+$ one
    obtains $\nu_\varepsilon(t)\in V_+$ for $\tau\leq t$, provided that
    $\nu_\varepsilon(\tau)\in V_+$. Now let $\nu$ denote the solution of
    \eqref{system} satisfying $\nu(\tau)=\nu_\varepsilon(\tau)$. One should
    bear in mind that $\nu_\varepsilon(t)$ is continuous in $(\varepsilon,t)$
    (cf.~\cite[p.~39, Satz~1.2.17]{poe02}) and
    consequently uniformly continuous on the set
    $K\times[0,\varepsilon_0]$, where
    $K\subset[\tau,\infty)_{\mathbb{T}}$ is a compact $\mathbb{T}$-interval
    and $\varepsilon_0>0$ arbitrary. By a standard argument, the solutions
    $\nu_\varepsilon$ converge to $\nu$ uniformly on $K$ as $\varepsilon\searrow 0$.
\end{proof}

\begin{cor}\label{inv}
    Let $V_+\subset V$ be a normal cone and assume that
    \begin{itemize}
        \item[(H3)] in any left-dense $t_0\in\mathbb{T}$ there exists a
        left-sided $\mathbb{T}$-neighborhood $N_0(t_0)$ of $t_0$ such that
        $f(s,0)\in V_+$ for all $s\in N_0(t_0)$.
    \end{itemize}
    Then $V_+^\ast$ is forward invariant for \eqref{system},
    if and only if every right-dense $t_0\in\mathbb{T}$, any
    $v\in\partial V_+$, $v'\in V_+^\star$ such that
    $\ang{v,v'}=0$ satisfy $\ang{f(t_0,v),v'}\geq 0$, and,
    for every right-scattered $t_0\in\mathbb{T}$ any $v\in V_+^\ast$
    satisfies $v+\mu(t_0)f(t_0,v)\in V_+^\ast$.
\end{cor}

\begin{proof}
    We have to show two directions:

    $(\Rightarrow)$ If $V_+^\ast$ is a forward invariant set, then the
    assertion can be shown analogously to step (II) in the proof of
    Lemma~\ref{lem52}.

    $(\Leftarrow)$ Using the induction principle
    (cf.~\cite[p.~4, Theorem~1.7]{bp01}) we deduce the statement
    \begin{displaymath}
        \mathcal{A}(t): v\neq 0
        \quad\implies\quad
        \varphi(t,\tau)v\neq 0 \quad\mbox{for all }\tau\leq t.
    \end{displaymath}
    Above all, choose $v\in V_+^\ast$ arbitrarily.
    \begin{itemize}
        \item $\mathcal{A}(\tau)$ obviously holds since $\varphi(\tau,\tau)v=v$.

        \item Let $t\geq\tau$ be right-scattered and $\mathcal{A}(t)$ be true.
        Then by the $2$-parameter semiflow property and the assumption
        one immediately gets
        \begin{displaymath}
            \varphi(\sigma(t),\tau)v
            =
            \varphi(t,\tau)v+\mu(t)f(t,\varphi(t,\tau)v)
            \neq
            0
        \end{displaymath}
        i.e., $\mathcal{A}(\sigma(t))$ holds.

        \item Let $t\geq\tau$ be right-dense and assume that $\mathcal{A}(t)$ is
        valid. Then $\varphi(t,\tau)v\neq 0$ implies that $\varphi(s,\tau)v\neq 0$
        in a right-sided $\mathbb{T}$-neighborhood $N$ of $t$. Hence $\mathcal{A}(t)$ yields
        $\varphi(s,\tau)v\neq 0$ for $s\in N$.

        \item Let $t\geq\tau$ be left-dense and $\mathcal{A}(s)$ be true for $s<t$.
        We want to show $\mathcal{A}(t)$ and proceed indirectly, i.e., assume that
        we have $\varphi(t,\tau)v_0=0$ for some $v_0\in V_+^\ast$. Since
        (H3) holds, we get from Lemma~\ref{lem52} that
        \begin{align*}
            0
            & \leq
            \varphi(s,\tau)v_0
            =
            -\int_{s}^{t}f(\rho,\varphi(\rho,\tau)v_0)\,\Delta\rho
            \\
            & \leq
            \int_{s}^{t}
            \big[f(\rho,0)-f(\rho,\varphi(\rho,\tau)v_0)\big]\,\Delta\rho
            \quad\mbox{for all }s\in N_0(t)
        \end{align*}
        and Hypothesis (H1) implies that
        $C(\rho):=\sup_{h\in[0,1]}\norm{D_2f(\rho,h\varphi(\rho,\tau)v_0)}$
        exists as an rd-continuous function in $\rho\in N_0(t)$. By
        assumption the cone $V_+$ is normal and therefore
        \begin{align*}
            \norm{\varphi(s,\tau)v_0}
            & \leq
            \int_{s}^{t}
            \norm{f(\rho,0)-f(\rho,\varphi(\rho,\tau)v_0)}\,\Delta\rho
            \\
            & \leq
            -\int_{t}^{s}
            C(\rho)\norm{\varphi(\rho,\tau)v_0}\,\Delta\rho
            \quad\mbox{for all } s\in N_0(t).
        \end{align*}
        Due to the limit relation $\lim_{\rho\nearrow t}\mu(t)=0$
        one can choose a left-sided $\mathbb{T}$-neigh\-bor\-hood
        $N\subset N_0(t)\cap[\tau,t]_\mathbb{T}$ such that we have
        $C(\rho)\mu(\rho)<1$ for $\rho\in N\setminus\set{t}$. Thus
        $-C(\rho)$ is positively regressive on $N\setminus\set{t}$
        and from the Gronwall lemma (cf.~\cite[p.~256, Theorem~6.4]{bp01})
        we obtain $\varphi(s,\tau)v_0=0$ for $s\in N$. This contradicts
        $\mathcal{A}(s)$.
    \end{itemize}
    Hence the proof of Corollary~\ref{inv} is complete.
\end{proof}

Before stating the next result we refer to
\cite[p.~54, Definition~1.3.5]{poe02} for the definition of the
\emph{transition operator} $\Phi_A(t,\tau)\in\mathcal{L}(V)$ of a linear
dynamic equation
\begin{equation}
    v^\Delta=A(t)v
    \label{linsys}
\end{equation}
in the nonregressive case. Now the forward invariance of $V_+$
with respect to \eqref{linsys} is a necessary and sufficient condition
for the positivity of $\Phi_A(t,\tau)$.

\begin{cor}\label{cor53}
    Let $A:\mathbb{T}\to\mathcal{L}(V)$ be rd-continuous and $t,\tau\in\mathbb{T}$. Then the
    following statements are equivalent:
    \begin{enumerate}
        \item $\Phi_A(t,\tau)\in\mathcal{L}(V_+)$ for $\tau\leq t$.

        \item For every right-dense $t\geq\tau$, $v\in\partial V_+$
        and $v'\in V_+^\star$ satisfying $\ang{v,v'}=0$, the
        inequality $\ang{A(t)v,v'}\geq 0$ holds, and, moreover,
        for every right-scat\-tered $t\geq\tau$, $v\in V_+$ the inclusion
        $v+\mu(t)A(t)v\in V_+$ holds.

        \item For every $t\geq\tau$ it holds
        \begin{displaymath}
            \lim_{h\searrow\mu(t)}
            \frac{\mathop{\rm dist}(v+hA(t)v,V_+)}{h}=0,
        \end{displaymath}
        if $v\in\partial V_+$ and $t$ is right-dense, or,
        $v\in V_+$ and $t$ is right-scattered.
    \end{enumerate}
\end{cor}

\begin{proof}
    Evidently Lemma~\ref{lem52} applies to \eqref{linsys} and therefore
    $V_+$ is forward invariant with respect to \eqref{linsys}, which, in turn,
    yields $\Phi_A(t,\tau)V_+\subset V_+$ for $\tau\leq t$.
\end{proof}

Adopting terminology introduced in \cite{hir82}, we denote a nonvoid
subset $U\subset V$ as \emph{$V_+$-convex}, if for any $u,v\in U$ such
that $u\leq v$, the whole line segment between $u$ and $v$ is contained
in $U$, i.e., $u+h(v-u)\in U$ for $h\in[0,1]$. Evidently the cone $V_+$
itself is $V_+$-convex. With all the above preliminaries at hand, we can
proceed to an appropriate definition of cooperativity.

\begin{defn}\label{defcoop} \rm
    Let $U\subset V$ be $V_+$-convex. A dynamic equation of the form
    \eqref{system} is called
    \begin{itemize}
        \item[(i)] \emph{$V_+$-cooperative on $U$}, if for all right-dense
        $t\in\mathbb{T}$, $u\in U$, $v\in\partial V_+$ and $v'\in V_+^\star$ such
        that $\ang{v,v'}=0$, the inequality $\ang{v',D_2f(t,u)v}\geq 0$
        holds and, moreover, if for every right-scattered $t\in\mathbb{T}$, $u\in U$,
        $v\in V_+$ the inclusion $v+\mu(t)D_2f(t,u)v\in V_+$ holds,

        \item[(ii)] \emph{strictly $V_+$-cooperative on $U$}, if
        \eqref{system} is $V_+$-cooperative on $U$, satisfies (H3),
        and if for every right-scattered $t\in\mathbb{T}$, $u\in U$, $v\in V_+$
        the implication $v+\mu(t)D_2f(t,u)v=0$ $\Rightarrow$ $v=0$ holds.
    \end{itemize}
\end{defn}

\begin{rem}\label{vrem} \rm
    Fix $\tau\in\mathbb{T}$ and $u\in U$ arbitrarily.

    \textit{(1)} Since the partial derivative
    $D_3\varphi(\cdot,\tau,u):[\tau,\infty)_{\mathbb{T}}\to\mathcal{L}(V)$
    solves the variational equation
    \begin{equation}
        X^\Delta=D_2f(t,\varphi(t,\tau,u))X
        \label{var}
    \end{equation}
    to the initial condition $X(\tau)=I_V$ on $[\tau,\infty)_{\mathbb{T}}$
    (cf.~\cite[pp.~47--48, Satz~1.2.22]{poe02}), by Corollary~\ref{cor53}
    the dynamic equation \eqref{system} is $V_+$-cooperative on the set $U$,
    if and only if $D_3\varphi(t,\tau,u)\in\mathcal{L}(V_+)$ holds for $\tau\leq t$ and $u\in U$.

    \textit{(2)} Assume that $V_+$ is normal and that \eqref{system}
    satisfies (H3). By using Corollary~\ref{inv} instead of Lemma~\ref{lem52}
    in the proof of Corollary~\ref{cor53}, it is not
    difficult to see that \eqref{system} is strictly $V_+$-cooperative
    on $U$, if and only if $D_3\varphi(t,\tau,u)$ is strictly positive for
    $\tau\leq t$ and $u\in U$.
\end{rem}

\begin{example}\label{char-cooperative} \rm
    Let $V_+=\mathbb{R}_+^d$ be the nonnegative orthant in the Banach space $V=\mathbb{R}^d$.
    Then $\mathcal{L}(V_+)$ is (isomorphic to) the set $\mathbb{R}_+^{d\times d}$ of
    nonnegative matrices. A mapping $f:\mathbb{T}\times\mathbb{R}^d\to\mathbb{R}^d$ is
    $\mathbb{R}_+^d$-cooperative, if in each right-dense point $t$ the
    off-diagonal elements of $D_2f(t,u)\in\mathbb{R}^{d\times d}$ are nonnegative
    and, if in each right-scattered point $t$ the matrix
    $I_{\mathbb{R}^d}+\mu(t)D_2f(t,u)$ is nonnegative for every $u\in U$.
    In the case of ordinary differential equations, where $\mathbb{T}=\mathbb{R}$
    consists of right-dense points, this definition coincides with the
    one from \cite{hir82}.
\end{example}

\begin{rem}[Euler discretization of cooperative ODEs] \rm
    Consider an $\mathbb{R}_+^d$-coop\-era\-tive ordinary differential equation
    $\dot{v} = f(t,v)$. Then according to Example \ref{char-cooperative} its
    Euler discretization $\nu(t_{n+1})=\nu(t_n)+(t_{n+1}-t_n)f(t_n,\nu(t_n))$
    on a discrete time scale $\mathbb{T}=\set{t_n}_{n\in\mathbb{N}_0}$ with $t_{n+1}>t_n$,
    is also $\mathbb{R}_+^d$-cooperative if the matrix
    $I_{\mathbb{R}^d}+(t_{n+1}-t_n)D_2f(t_n,u)$ is nonnegative. Since the off-diagonal
    elements of $D_2f(t,u)$ are nonnegative, this is true, if the diagonal
    entries $a_{ii}(t_n,u)$, $i=1,\dots,d$, of the matrix $D_2f(t_n,u)$
    satisfy the condition $a_{ii}(t_n,u)\geq\frac{-1}{t_{n+1}-t_n}$ with
    the stepsize $\mu(t_n) = t_{n+1}-t_n$ of the Euler discretization.
\end{rem}

\begin{thm}[M\"uller's Theorem]\label{muller}
    Let $U\subset V$ be $V_+$-convex.
    \begin{enumerate}
        \item If \eqref{system} is $V_+$-cooperative on $U$, then $\varphi$ is
        order-preserving on $U$.

        \item Conversely, if $\varphi$ is order-preserving on $U$, then the
        dynamic equation \eqref{system} is $V_+$-cooperative on $U$.
    \end{enumerate}
\end{thm}

\begin{proof}
    (a) Choose $u,v\in U$ with $u\leq v$ and because of the $V_+$-convexity
    of $U$ one has $u+h(v-u)\in U$ for $h\in[0,1]$. Then the mean value
    theorem (cf.~\cite[p.~341, Theorem~4.2]{lan93}) yields
    \begin{displaymath}
        \varphi(t,\tau,v)-\varphi(t,\tau,u)
        =
        \int_0^1D_3\varphi(t,\tau,u+h(v-u))(v-u)\,dh
        \quad
        \text{for all }\tau\le t
    \end{displaymath}
    and since $D_3\varphi(t,\tau,w)\in\mathcal{L}(V_+)$, $w\in U$, we obtain
    $D_3\varphi(t,\tau,w)(v-u)\in V_+$. Now convexity of the integral
    implies the claim $\varphi(t,\tau,v)\leq\varphi(t,\tau,u)$.

    (b) Using the fact that $D_3\varphi(t,\tau,v)$ solves the dynamic
    equation \eqref{var} in $\mathcal{L}(V_+)$, we get the assertion (b) with
    a view to Corollary~\ref{cor53}.
\end{proof}

The next part of this section is dedicated to sufficient conditions for
strictly and strongly order-preserving mappings. Since we have not assumed
regressivity of $f$ (cf.~\cite[pp.~321--322, Definition~8.14(ii)]{bp01})
the mapping $\varphi(t,\tau):V\to V$, $\tau\leq t$, needs not to be a
homeomorphism. Hence the arguments
of~\cite[pp.~32--33, Proof of Proposition 1.1]{smi96} do not apply directly.

\begin{cor}\label{cor1}
    Let $U\subset V$ be $V_+$-convex. If for a $V_+$-cooperative
    system \eqref{system} on $U$ one of the following conditions
    \begin{itemize}
        \item[(i)] $I_V+\mu(t)f(t,\cdot):V\to V$ in one-to-one
        on $U$ for any $t\in\mathbb{T}$,

        \item[(ii)] $I_V+\mu(t)f(t,\cdot):V\to V$ is strictly
        order-preserving on $U$ for any $t\in\mathbb{T}$
    \end{itemize}
    holds, then $\varphi$ is strictly order-preserving on $U$.
\end{cor}

\begin{proof}
    We proceed in two steps:

    \textit{(I)} To show that (i) implies (ii), fix arbitrary
    $t\in\mathbb{T}$, $u\in U$ and abbreviate $F(u):=u+\mu(t)f(t,u)$.
    Observing the fact
    \begin{equation}
        \varphi(\sigma(t),t)u=u+\mu(t)f(t,u)=F(u),
        \label{star}
    \end{equation}
    it is evident that $F$ is strictly order-preserving on $U$.

    \textit{(II)} We apply the induction principle
    (cf.~\cite[p.~4, Theorem~1.7]{bp01}) to the statement
    \begin{displaymath}
        \mathcal{A}(t): u<v
        \quad\implies\quad
        \varphi(t,\tau)u<\varphi(t,\tau)v \quad\mbox{for all }\tau\leq t.
    \end{displaymath}
    First of all, choose $u,v\in U$, $u<v$ arbitrarily.
    \begin{itemize}
        \item $\mathcal{A}(\tau)$ is clearly satisfied since $\varphi(\tau,\tau)u=u$.

        \item Let $t\geq\tau$ be right-scattered and $\mathcal{A}(t)$ be true.
        Then by $2$-parameter semiflow property and (ii) one obtains
        \begin{align*}
            \varphi(\sigma(t),\tau)u
            & \stackrel{\eqref{star}}{=}
            \varphi(t,\tau)u+\mu(t)f(t,\varphi(t,\tau)u)<\\
            & <
            \varphi(t,\tau)v+\mu(t)f(t,\varphi(t,\tau)v)
            \stackrel{\eqref{star}}{=}
            \varphi(\sigma(t),\tau)v
            \,,
        \end{align*}
        i.e., $\mathcal{A}(\sigma(t))$ holds.

        \item Let $t\geq\tau$ be right-dense and assume that $\mathcal{A}(t)$ is
        valid. Then there exists a $\mathbb{T}$-neighborhood $N$ of $t$ such
        that $\varphi(s,t):V\to V$ is a homeomorphism for $s\in N$
        and in particular one-to-one. Hence $\mathcal{A}(t)$ yields
        $\varphi(s,\tau)u<\varphi(s,\tau)v$ for $s\in N$.

        \item Let $t\geq\tau$ be left-dense and $\mathcal{A}(s)$ be true for $s<t$.
        Similar to the above we get that $\varphi(t,s):V\to V$ is a
        homeomorphism for $s$ in some $\mathbb{T}$-neighborhood of $t$, which gives
        us $\varphi(t,\tau)u<\varphi(t,\tau)v$, i.e., $\mathcal{A}(t)$ is valid.
    \end{itemize}
    Thus the proof is complete.
\end{proof}

\begin{cor}\label{cor2}
    Let $U\subset V$ be $V_+$-convex. If for a $V_+$-cooperative
    system \eqref{system} on $U$ one of the following conditions
    \begin{itemize}
        \item[(i)] $I_V+\mu(t)D_2f(t,u)\in\mathcal{L}(V)$ is onto for any
        $u\in U$ and any $t\in\mathbb{T}$,

        \item[(ii)] $I_V+\mu(t)f(t,\cdot):V\to V$ is strongly
        order-preserving on $U$ for any $t\in\mathbb{T}$
    \end{itemize}
    holds, then $\varphi$ is strongly order-preserving on $U$.
\end{cor}

\begin{rem} \rm %rmk5.13
    In the case of ordinary differential equations, where $\mathbb{T}=\mathbb{R}$ consists
    of right-dense points, we have $\mu(t)\equiv 0$ on $\mathbb{T}$, and both
    conditions (i) and (ii) in Corollary~\ref{cor1} and \ref{cor2} are
    dispensable. Therefore, solutions of $V_+$-cooperative ODEs are always
    strictly and strongly order-preserving.
\end{rem}

\begin{proof}
    We proceed in two steps again:

    \textit{(I)} To show that (i) implies (ii) fix arbitrary $t\in\mathbb{T}$,
    $u,v\in U$ with $u\ll v$ and use the notation from the proof of
    Corollary~\ref{cor1}. Then Theorem~\ref{muller}(a) yields that $F$
    maps the order-interval $[u,v]$ into the order-interval $[F(u),F(v)]$.
    Now we prove that the latter set has nonempty interior, which guarantees
    $F(u)\ll F(v)$. To do so, pick some $w\in\mathop{\rm int}[u,v]$ arbitrarily. Using the
    hypothesis (i) we see that $F$ must be locally open in a neighborhood of $w$
    by the Surjective Mapping Theorem (cf.~\cite[p.~397, Theorem~3.5]{lan93}).
    Consequently, we obtain the inclusion $F(w)\in\mathop{\rm int}[F(u),F(w)]$ and $F$ is
    strongly order-preserving.

    \textit{(II)} We apply the induction principle
    (cf.~\cite[p.~4, Theorem~1.7]{bp01}) to the statement
    \begin{displaymath}
        \mathcal{A}(t): u\ll v
        \quad\implies\quad
        \varphi(t,\tau)u\ll\varphi(t,\tau)v
        \quad\text{for all $\tau\leq t$.}
    \end{displaymath}
    First of all, choose $u,v\in U$, $u\ll v$ arbitrarily.
    \begin{itemize}
        \item $\mathcal{A}(\tau)$ is clearly satisfied since $\varphi(\tau,\tau)u=u$.

        \item The implication $\mathcal{A}(t)\Rightarrow\mathcal{A}(\sigma(t))$ for
        right-scattered $t\geq\tau$ results as in the corresponding part of
        the proof of Corollary~\ref{cor1} with the relation $<$ replaced by
        $\ll$.

        \item Let $t\geq\tau$ be right-dense and assume that $\mathcal{A}(t)$
        is valid. Then there exists a $\mathbb{T}$-neighborhood $N$ of $t$
        such that $\varphi(s,t):V\to V$ is a homeomorphism for $s\in N$.
        Since $[\varphi(t,\tau)u,\varphi(t,\tau)v]$ has nonempty interior
        by the induction hypothesis $\mathcal{A}(t)$, also
        $\mathop{\rm int}[\varphi(s,\tau)u,\varphi(s,\tau)v]\neq\emptyset$ holds for
        $s\in N$, which is equivalent to $\varphi(s,\tau)u\ll\varphi(s,\tau)v$.

        \item Let $t\geq\tau$ be left-dense and $\mathcal{A}(s)$ be true for
        $s<t$. Similar to the above we get that $\varphi(t,s):V\to V$
        is a homeomorphism for $s$ in some $\mathbb{T}$-neigh\-borhood of $t$,
        which, in turn, yields $\varphi(t,\tau)u\ll\varphi(t,\tau)v$,
        i.e., $\mathcal{A}(t)$ is valid.
    \end{itemize}
    Thus the proof is complete.
\end{proof}

So far, Theorem~\ref{muller} provides a criterion that the solution operator
$\varphi$ of \eqref{system} is order-preserving. In order to apply Theorem
\ref{lst}, and in reference to Lemma~\ref{sub}, we need additional conditions
for the subhomogeneity of $\varphi$.
\begin{lem}\label{hlem}
    Let \eqref{system} be $V_+$-cooperative on $V_+$. Then
    \begin{enumerate}
        \item $\varphi$ is subhomogeneous, if and only if
        \begin{equation}
            D_3\varphi(t,\tau,v)v
            \leq
            \varphi(t,\tau,v) \quad\mbox{for all }\tau\leq t,\,v\in V_+;
            \label{cond}
        \end{equation}

        \item $\varphi$ is strictly subhomogeneous, if
        \begin{displaymath}
            D_3\varphi(t,\tau,v)v
            <
            \varphi(t,\tau,v) \quad\mbox{for all }\tau<t,\,v\in V_+^\ast.
        \end{displaymath}
    \end{enumerate}
\end{lem}

\begin{proof}
    (a) Let $\tau\leq t$ be fixed in $\mathbb{T}$. Consider for $v'\in V_+^\star$ and
    $v\in V_+$ the function $\phi_{v',v}:(0,\infty)\to\mathbb{R}$,
    $\phi_{v',v}(\alpha):=\frac{1}{\alpha}\ang{\varphi(t,\tau)\alpha v,v'}$.
    We show that $\varphi$ is subhomogeneous, if and only if $\phi_{v',v}$ is
    decreasing for all $v'\in V_+^\star$, $v\in V_+$:\\
    $(\Rightarrow)$ If $\varphi$ is subhomogeneous, then for arbitrary
    $0<\alpha\leq\beta$ there holds the inequality
    $\frac{\alpha}{\beta}\varphi(t,\tau)\beta v
    \leq\varphi(t,\tau)\alpha v$, i.e., we have
    $\frac{1}{\beta}\varphi(t,\tau)\beta v\leq
    \frac{1}{\alpha}\varphi(t,\tau)\alpha v$. By Lemma~\ref{lem51}(a) this
    implies that $\phi_{v',v}$ is decreasing.\\
    $(\Leftarrow)$ Conversely, let $\phi_{v',v}$ be decreasing in
    $0<\alpha<1$. Then $\phi_{v',v}(1)\leq\phi_{v',v}(\alpha)$, and
    since $v'\in V_+^\star$ was arbitrary, we readily obtain
    $\varphi(t,\tau)v\leq\frac{1}{\alpha}\varphi(t,\tau)\alpha v$ from
    Lemma~\ref{lem51}(a).\\
    By assumption on $f$, the function $\phi_{v',v}$ is differentiable
    and the chain rule implies
    \begin{displaymath}
        \phi_{v',v}'(\alpha)
        =
        \frac{
        \ang{\alpha D_3\varphi(t,\tau,\alpha v)v-\varphi(t,\tau,\alpha v),v'}}
             {\alpha^2} \quad\mbox{for all }\alpha>0.
    \end{displaymath}
    Thus the subhomogeneity of the mapping $\varphi$
    is equivalent to the property that
    $\ang{\alpha D_3\varphi(t,\tau,\alpha v)v-\varphi(t,\tau,\alpha v),v'}\leq 0$,
    i.e., by Lemma~\ref{lem51}(a) to the condition \eqref{cond}.

    (b) Now let $\tau<t$ be arbitrary points in $\mathbb{T}$. Along the same lines
    as in (a), one shows that $\varphi$ is strictly subhomogeneous, if and
    only if the mapping $\phi_{v',v}$ is strictly decreasing. This property,
    in turn, is necessary for $\phi_{v',v}'(\alpha)<0$, $0<\alpha$, and by
    Lemma~\ref{lem51}(b) we obtain the assertion.
\end{proof}

\begin{thm}\label{subt}
    Let \eqref{system} be $V_+$-cooperative on $V_+$. Then
    \begin{enumerate}
        \item $\varphi$ is subhomogeneous, if
        \begin{equation}
            D_2f(t,v)v
            \leq
            f(t,v) \quad\mbox{for all }t\in\mathbb{T},\,v\in V_+;
            \label{s1}
        \end{equation}

        \item $\varphi$ is strictly subhomogeneous, if, moreover,
        \eqref{system} is strictly $V_+$-cooperative on $V_+$ and
        \begin{displaymath}
            D_2f(t,v)v
            <
            f(t,v) \quad\mbox{for all }t\in\mathbb{T},\,v\in V_+^\ast.
        \end{displaymath}
    \end{enumerate}
\end{thm}
\begin{proof}
    Let $\tau\leq t$ and $u\in V_+$ be fixed.

    (a) We are going to show that the
    mapping $\Lambda:[\tau,\infty)_\mathbb{T}\to V$,
    $\Lambda(t):=\varphi(t,\tau,u)-D_3\varphi(t,\tau,u)u$ has values in
    the cone $V_+$. Thereto consider
    \begin{align*}
            \Lambda^\Delta(t)
            & \stackrel{\eqref{system}}{=}
            f(t,\varphi(t,\tau,u))-
            D_2f(t,\varphi(t,\tau,u))D_3\varphi(t,\tau,u)u=\\
            & =
            D_2f(t,\varphi(t,\tau,u))\Lambda(t)+l(t)
    \end{align*}
    with
    $l(t):=f(t,\varphi(t,\tau,u))-D_2f(t,\varphi(t,\tau,u))\varphi(t,\tau,u)$.
    Since $l:[\tau,\infty)_\mathbb{T}\to V$ is rd-continuous and since
    $D_3\varphi(\cdot,\tau,u)$ solves \eqref{var} with respect to the
    initial condition $X(\tau)=I_V$, the variation of constants
    formula (cf.~\cite[p.~56, Satz~1.3.11]{poe02}) yields
    \begin{displaymath}
        \Lambda(t)=\int_{\tau}^{t}\Psi_u(t,\sigma(s))l(s)\,\Delta s,
    \end{displaymath}
    where $\Psi_u(t,\tau)\in\mathcal{L}(V)$ is the transition operator of \eqref{var}.
    By assumption, \eqref{system} is $V_+$-cooperative on $V_+$ and
    similarly to Remark \ref{vrem}(1) one sees the inclusion
    $\Psi_u(t,\tau)\in\mathcal{L}(V_+)$
    for $\tau\leq t$. Furthermore, \eqref{s1} implies $l(t)\in V_+$ and
    by the convexity of the Cauchy-integral on $\mathbb{T}$ it follows
    $\Lambda(t)\in V_+$ for $\tau\leq t$. Now Lemma~\ref{hlem}(a)
    leads to the assertion.

    (b) Proceed like in the proof of (a). Here Remark \ref{vrem}(2)
    yields that $\Psi_u(t,\tau)$ is strictly positive and the assertion
    follows from Lemma~\ref{hlem}(b).
\end{proof}

\section{Application: Symbiotic Interaction}

In the following last section we demonstrate the importance of the
limit set trichotomy from Theorem~\ref{lst} in an application from
biology within the calculus on time scales. Thereto we restrict our
considerations to time scales of the form
\begin{displaymath}
    \mathbb{T}=\bigcup_{n\in\mathbb{N}_0}[\tau_n,t_n],
\end{displaymath}
where $(\tau_n)_{n\in\mathbb{N}_0}$, $(t_n)_{n\in\mathbb{N}_0}$ are real sequences
with $\lim_{n\to\infty}\tau_n=\lim_{n\to\infty}t_n=\infty$ and
$\tau_n\leq t_n<\tau_{n+1}$ for all $n\in\mathbb{N}_0$. Hence we have a
continuous ODE dynamical behavior of \eqref{system} on the intervals
$[\tau_n,t_n]$, $n\in\mathbb{N}_0$, while the dynamic on the ``gaps''
$(t_n,\tau_{n+1})$ is discrete, i.e., difference equation-like.
For technical reasons we additionally assume that the differences
$\tau_{n+1}-\tau_n$, $n\in\mathbb{N}_0$, are bounded above by some real
$T\geq 0$.

Consider a symbiotic interaction between $d\geq 2$, e.g., insect
populations, i.e., an interaction that results in a benefit between
the populations. The life span of each population is given by the
interval $[\tau_n,t_n]$, $n\in\mathbb{N}_0$, which can be interpreted
as a summer period. Suppose that just before the populations die out,
eggs are laid at time $t=t_n$ and hatch after
the winter period $(t_n,\tau_{n+1})$ at time $t=\tau_{n+1}$.
During the winter, a certain amount of eggs dies, but to prevent
each species from dying out, an exterior influence adds additional
eggs. If $v_i(t_n)\geq 0$, $n\in\mathbb{N}_0$, denotes the biomass of the
$i$th, $i=1,\dots,d$, population at time $t=t_n$, we model this
behavior over the winter periods with the equations
\begin{equation}
    v_i(\tau_{n+1})=q_i(t_n)v_i(t_n)+p_i(t_n) 
    \quad\mbox{for all }i=1,\dots,d
    \label{dis}
\end{equation}
and $n\in\mathbb{N}_0$, where $q_i(t_n)\in[0,1[$ describes the natural decay
in the winter and $p_i(t_n)>0$ the external ``seed''. The equation \eqref{dis}
guarantees that we have the inclusion $v(\tau_{n+1})\in\mathop{\rm int}\mathbb{R}_+^d$ after each
winter --- independent of $v(t_n)\in\mathbb{R}_+^d$. For the continuous growth we lean
on \cite{kr92} and consider the ODEs
\begin{equation}
    \dot{v}_i=v_iF_i(v,t) \quad\mbox{for all }i=1,\dots,d,
    \label{con}
\end{equation}
on the intervals $[\tau_n,t_n]$, $n\in\mathbb{N}_0$, where the mappings
$F_i:\mathbb{R}^d\times\mathbb{T}\to\mathbb{R}$ are continuously
differentiable in each state space variable $v_1,\dots,v_d$. Obviously the boundary
$\partial\mathbb{R}_+^d$ is forward invariant with respect to \eqref{con} and therefore any
solution of \eqref{con} cannot leave the standard cone $\mathbb{R}_+^d$ for
times $t\in[\tau_n,t_n]$, $n\in\mathbb{N}_0$. Combining both situations, we
arrive at a dynamic equation \eqref{system} with right-hand side
$f=(f_1,\dots,f_d)$ and
\begin{displaymath}
    f_i(t,v)
    :=
    \begin{cases}
        v_iF_i(v,t) &\mbox{for } t\in[\tau_n,t_n)\\
        \frac{q_i(t)-1}{\mu(t)}v_i+\frac{p_i(t)}{\mu(t)} 
   &\mbox{for } t=t_n\,.
    \end{cases}
\end{displaymath}
If we assume that the ODE \eqref{con} has no finite escape times,
then the mapping $f$ satisfies the assumptions (H1)--(H2).
In addition, the standard cone $\mathbb{R}_+^d$ is forward-invariant
with respect to \eqref{system}.

As a canonical state space for \eqref{system} we consider the cone
$V_+=\mathbb{R}_+^d$, which evidently satisfies the assumption (H0), and
is $\mathbb{R}_+^d$-convex, since the nonnegative orthant is convex. Under
the assumption
\begin{itemize}
    \item[$(C)$] $D_jF_i(u,t)\geq 0$ for all $u\in\mathbb{R}_+^d$, $i\neq j$,
    and $t\in\bigcup_{n\in\mathbb{N}_0}[\tau_n,t_n)$,
\end{itemize}
the system \eqref{system} is $\mathbb{R}_+^d$-cooperative on $\mathbb{R}_+^d$ and we
obtain from Theorem~\ref{muller}(a) that its solution $\varphi$ is
order-preserving. On the other hand, if we suppose
\begin{itemize}
    \item[$(S)$] $\sum_{j=1}^{d}v_jD_jF_i(v,t)\leq 0$ for all $v\in\mathbb{R}_+^d$,
    $i=1,\dots,d$ and $t\in\bigcup_{n\in\mathbb{N}_0}[\tau_n,t_n)$,
\end{itemize}
then using Theorem~\ref{subt}(a) one can show that $\varphi$ is also
subhomogeneous. So, due to Lemma~\ref{sub}(a), $\varphi(t,\tau)$,
$\tau\leq t$, must be nonexpansive with respect to the part metric on $\mathbb{R}_+^d$.
Finally, since each $\mathbb{T}$-interval of length greater or equal than $T$
contains a right-scattered point, we have $\varphi(t,\tau,v)\in\mathop{\rm int}\mathbb{R}_+^d$
for $T\leq t-\tau$ and $v\in\mathbb{R}_+^d$.
Therefore the assumptions of Theorem~\ref{lst} are satisfied and our
limit set trichotomy applies. In particular, if $p_i$ is bounded away
from zero, we can exclude case (b) of Theorem~\ref{lst} and all
solutions of the general nonautonomous dynamic equation \eqref{system}
are either unbounded, or bounded with nonempty $\omega$-limit sets.

\begin{example}[Kolmogorov systems] \rm
    A particularly relevant special case of the symbiotic interaction
    discussed above, are so-called \emph{Kolmogorov systems} which have
    the following biological interpretation (cf.~\cite{fs95}): Think of a
    hierarchy of species $v_1,\dots,v_d$, where $v_i(t)$ is the biomass
    of the $i$th species. In this hierarchy, $v_i$ interacts only with
    $v_{i-1}$ and $v_{i+1}$. Such a hierarchy may occur in steep
    mountain side or in a lake, where each population dominates a specific
    altitude or depth, respectively, but is obliged to cooperate with other
    populations in the (narrow) overlap of their zones of dominance. So we
    only modify the law for the continuous growth and consider the system
    of ODEs
    \begin{equation}
 \begin{gathered}
            \dot{v}_1 = v_1F_1(v_1,v_2,t)\\
            \dot{v}_i = v_iF_i(v_{i-1},v_i,v_{i+1},t)
       \quad\mbox{for all }i=2,\dots,d-1\\
            \dot{v}_d = v_dF_d(v_{d-1},v_d,t)
        \end{gathered}
        \label{last}
    \end{equation}
    to describe the behavior on the intervals $[\tau_n,t_n]$, where
    the mappings $F_1,\dots,F_d$ are continuously differentiable in their
    state space variables. Furthermore, the conditions $(C)$ and $(S)$
    reduce to
    \begin{itemize}
        \item $D_2F_1(v_1,v_2,t),D_1F_i(v_1,v_2,v_3,t),D_3F_i(v_1,v_2,v_3,t),
        D_1F_d(v_1,v_2,t)\geq 0$ for all $v_1,v_2,v_3\in\mathbb{R}_+$, $i=2,\dots,d-1$,
        and $t\in\bigcup_{n\in\mathbb{N}_0}[\tau_n,t_n)$,
        \item $\sum_{j=1}^{2}v_jD_jF_1(v_1,v_2,t)\leq 0$ and
        $\sum_{j=1}^{3}v_jD_jF_i(v_1,v_2,v_3,t)\leq 0$, as well as
        $\sum_{j=1}^{2}v_jD_jF_d(v_1,v_2,t)\leq 0$ for all
        $v_1,v_2,v_3\in\mathbb{R}_+$, $i=2,\dots,d-1$, and
        $t\in\bigcup_{n\in\mathbb{N}_0}[\tau_n,t_n)$,
    \end{itemize}
    respectively. They guarantee that the right-hand side of \eqref{system}
    is $\mathbb{R}_+^d$-cooperative and generates a subhomogeneous $2$-parameter
    semiflow. Consequently our limit set trichotomy from Theorem~\ref{lst}
    applies. Explicit biological systems modelled by \eqref{last} can be
    found in \cite{fs95}.
    
\end{example}

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\end{document}