\chapter{Generalized solutions of ordinary differential equations}

Throughout this chapter $J$ will be some subinterval of $\R$. Starting point of our overview of generalized solution concepts
for ordinary differential equations is the initial value problem
$$
\ode
$$
where the right-hand side $a(t,x)=(a_1(t,x), \dots, a_n(t,x))$ is a map from $J \times \R^n$ to $\R^n$. 
Occasionally we will indicate dependence on the initial conditions by writing $\xi(\cdot;t,x)$ for a solution $\xi$ with $\xi(t)=x$.
 
Let us first consider the well-posedness: The left-hand side of the equation requires a possible solution $\xi$ 
to be at least absolutely continuous in order to ensure differentiability 
almost everywhere on $J$. 
Furthermore it is apparent that the right-hand side of the equation 
has to be in $L^1_{loc}(J)^n$. 
Nevertheless in general it is not sufficient to have $a \in L^1_{\rm loc}(J \times \R^n)^n$ 
in order to have the composition $s \mapsto a(s, f(s))$ in $L^1_{\rm loc}(J)^n$ 
for $f \in AC(J)^n$. 

In the first section we investigate the classical Caratheodory theory, 
which requires the right-hand side $a$ to be in the space 
$L^1_{\rm loc}(J, L^{\infty}(\R^n))^n$ with $a(\cdot, x) \in L^1_{\rm loc}(J)^n$ 
for all $x\in \R^n$ and $a(s,\cdot) \in C(J)^n$ for almost all $s \in J$ 
(referred to as \emph{Caratheodory conditions}), then the existence of 
a solution $\xi \in AC(J)^n$ can be proved.

Of course the continuity required in the space-variable of the 
right-hand side restricts the applicability of the Caratheodory theory. 
Nevertheless if we want to drop this requirement we cannot retain 
on the classical concept of a solution. 

A natural generalization of an ordinary differential equation is the differential inclusion
$$
\odie,
$$
where $A_{t,x}$ is a subset of $\R^n$ for each $(t,x) \in J \times \R^n$. The map
$A:(t,x) \mapsto A_{t,x}$ is called a \emph{set-valued map} (cf. Section \ref{set_valued_map_section}). 
As in the case of Caratheodory theory we have to impose certain conditions (Filippov conditions) on the set-valued map 
$A$ in order to obtain an existence result for the differential inclusion. These results will be presented in the second section.
 
The concept of a differential inclusion can then be utilized to obtain a generalized solution concept 
for ordinary differential equations, when the right-hand side fails continuity in the space variable, 
but is still in $L^1_{\rm loc}(J; L^{\infty}(\R^n))^n$. It is obtained by mapping the right-hand side $a$ to some $A$ 
in the space of set-valued functions satisfying the above-mentioned Filippov conditions
in a way that $A_{t,x} =\{a(t,x)\}$ whenever $a$ fulfills the classical Caratheodory conditions. 
Then the absolutely continuous solution of the differential inclusion is called a  Filippov generalized solution.

Another approach to generalize the classical solution concept for ordinary differential equations 
is the regularization of the right-hand side $a$ by means of convolution with a net of smooth mollifiers, such that
the resulting net of smooth functions $(A_{\varepsilon})_{\varepsilon}$ has the property
$\lim_{\varepsilon \rightarrow 0} \langle A_{\varepsilon}, \varphi \rangle = \langle a, \varphi\rangle$ for all 
$\varphi \in \mathcal{D}'(\R^n)$.
Then a net of solutions $(\xi_{\varepsilon})_{\varepsilon}$ can be obtained by solving the differential equation for fixed 
$\varepsilon \in ]0,1]$ using the classical existence results. If the net of solutions $(\xi_{\varepsilon})_{\varepsilon}$
possesses a subnet converging to some $\xi \in AC(J)^n$, i.e. there exists some $\tau \in \mathcal{T}$ such that
$\xi_{\tau(\varepsilon)}(s) \rightarrow \xi(s)$ uniformly on compact subsets of $J$, then the function $\xi$ 
is a generalized solution in the sense that it satisfies the classical ordinary differential equation 
whenever the initial right-hand side $a$ satisfies the Caratheodory conditions. 

It is remarkable that the latter approach is closely related to the former, since the generalized solution obtained by the
regularization procedure satisfies a certain differential inclusion. The concept of a generalized graph (cf. Chapter \ref{generalized_graph}) 
plays a crucial role in understanding this relation.
 
Another viewpoint of the regularization approach is that the net $(A_{\varepsilon})_{\varepsilon}$ 
can be considered as an element in the Colombeau algebra of generalized functions. 
We put $A := [(A_{\varepsilon})_{\varepsilon}]\in \cf{J\times \R^n}^n$. Note that in general $A=(A_1, \dots, A_n) \ne (\iota(a_1), \cdots, \iota(a_n))$, 
where $\iota: \mathcal{D}'(\R^n) \rightarrow \cf{\R^n}$ denotes the standard embedding, 
since we did not specify the net of smooth mollifier any further.
Occasionally it may be convenient to use mollifiers with non-vanishing moments 
(f.e. positive or compactly supported mollifiers), which is not possible when using $\iota$ to embed the right-hand side $a$.

If the ordinary differential equation
$$
\colode
$$
can be solved in the setting of Colombeau generalized functions by some 
$\xi := [(\xi_{\varepsilon})_{\varepsilon}] \in \cf{J}^n$, we call $\xi$ a Colombeau solution with inital condition 
$(\widetilde{t},\widetilde{x}) \in \widetilde{J} \times \widetilde{\R}^n$.

It generalizes the classical solution concept whenever there exists some $\tau \in \mathcal{T}$ 
such that $(\xi_{\tau(\varepsilon)})_{\varepsilon}$ converges to a function $\zeta \in AC(J)^n$ 
uniformly on compact sets of $J$. 
We say $\xi$ has an absolutely continuous \emph{sub-shadow} $\zeta$\index{sub-shadow}. This notion is justified
by the fact that $\zeta$ is an absolutely continuous shadow of $\xi$, 
if $(\xi_{\varepsilon})_{\varepsilon}$ converges to $\zeta$ uniformly on compact subsets of $J$.

$\zeta$ is a generalized solution in the sense that it satisfies the 
classical ordinary differential equation whenever the initial right-hand side fulfills the 
Caratheodory conditions.


\section{Solutions of ordinary differential equations according to Caratheodory} \label{caratheodory_solution_concept}


In this section we will review the classical Caratheodory theory for ordinary differential equations
\begin{equation} \label{ode}
\ode
\end{equation}
with non-smooth right-hand side $a: J \times \R^n \rightarrow \R^n$ or equivalently the corresponding integral equation
\begin{equation} \label{int_eq}
\xi(s) = x + \int_t^s a(\tau,\xi(\tau)) \m{\tau}.
\end{equation}

We follow closely the introduction from the book \cite{Filippov:88}. 
Throughout the entire section the right-hand side $a=(a_1,\ldots, a_n)$ satisfies the conditions
\begin{enumerate} 
\item $a(t,x)$ is Lebesgue measurable in $t$ for all fixed $x\in \mathbb{R}^n$,
\item $a(t,x)$ is continuous in $x$ for almost all $t \in J$, and
\item $\sup_{x\in\mathbb{R}^n} |a(t,x)| \le \beta(t)$ almost everywhere for some positive function $\beta \in L_{\rm loc}^1(J)$,
\end{enumerate}
referred to as \emph{Caratheodory conditions} (CC). \index{Caratheodory conditions}

Note that the first two Caratheodory conditions ensure Lebesgue measurability of the composition $s\mapsto a(s,f(s))$ 
for all $f\in {C}(J)^n$ (due to Lemma \ref{measurability_of_composition_lemma}), 
while the third condition is crucial in the existence proof. 


\begin{theorem}[Existence theorem for ordinary differential equations] \label{caratheodory_ode_existence}
Let $J$ be some subinterval of $\R$ and assume that $a=(a_1,\ldots, a_n)$ satisfies (CC) on $J\times\mathbb{R}^n$. 
Let $(t,x) \in J \times \mathbb{R}^n$, then there exists an absolutely continuous solution $\xi=(\xi_1, \ldots, \xi_n)$ to
the ordinary differential equation
$$
\ode
$$
\end{theorem}

\begin{proof}
For sake of completeness we include this classical proof, following Filippov (\cite[Ch. 1,Thm. 1]{Filippov:88}).
Without loss of generality we assume that $t$ lies in the interior of $J$. 

Note that the Caratheodory condition (i) and (ii) imply (due to Lemma \ref{measurability_of_composition_lemma}) 
that for any $f \in C(J)^n$ the composition $s \mapsto a(s,f(s))$ is in $L^1_{\rm loc}(J)$.
Let us consider a compact subset of $J$ in the forward direction, i.e. set $J^{+}:= J \cap [t,T^{+}]$ for some 
$T_{+} \in J$ with $T^{+}>t$.
For any $k\in\N$ we put $\lambda_k:=\frac{T^{+}-t}{k}$ in order to obtain a decomposition
of the interval $J^+$ in $k$ subintervals by $J_l^{+} := [t+(l-1) \lambda_k, t+ l\lambda_k]$ for $1\le l \le k$, such that
$$
J^{+} := \bigcup_{l=1}^k J^+_{l}.
$$ 
We construct a function $\xi_k \in AC(J^+)$ inductively by setting
$$
\xi_k(s) := x +  \int_{t}^{\max{(t,s-\lambda_k)}} a(\tau+ \lambda_k, \xi_k(\tau)) \m{\tau} 
$$
for all $s\in J^+$. Whenever $s\in J_l^+$ for $l\ge 2$, 
the integral expression on the right-hand side of the above definition depends only on $\xi_k$ restricted to $J^+_{l-1}$.
For $s \in  J_1^+$ we have $\xi_k(s) =x$.

Caratheodory condition (iii) implies 
\begin{eqnarray*} 
|\xi_k(s) - \xi_k(r)| = \left|\, \int_{\max{(t,r-\lambda_k)}}^{\max{(t,s-\lambda_k)}} a(\tau+ \lambda_k, \xi_k(\tau)) \m{\tau}\right| 
\le \int^{\max{(r,s)}}_{\min{(r,s)}} \beta(\tau) \m{\tau} \quad \text{ for } s,r \ge t
\end{eqnarray*} 
and $\xi_k(s) \in B_{\rho}(x)$ for all $s\in J^+$, where $\rho := \int^{T_{+}}_t \beta(\tau) \m{\tau}$.
Obviously $(\xi_k)_k$ is a equi-continuous and equi-bounded family in $C(J^+)^n$. The Theorem of Arzela-Ascoli yields an uniformly convergent subsequence 
$\xi_{k_j} \stackrel{j\rightarrow \infty}{\rightarrow} \xi$, where $\xi \in C(J^+)^n$ with $\xi(t)=x$.

Observe that
\begin{multline*}
|\xi_{k_j}(s-\lambda_{k_j}) - \xi(s)| \le |\xi_{k_j}(s-\lambda_{k_j}) - \xi_{k_j}(s)|+|\xi_{k_j}(s) - \xi(s)| \\
= \int_{s- \lambda_{k_j}}^{s} \beta(\tau) \m{\tau} + |\xi_{k_j}(s) - \xi(s)| \stackrel{j \rightarrow \infty}{\rightarrow} 0
\end{multline*}
holds. So we finally obtain by the theorem of dominated convergence that
\begin{multline*}
\xi(s) = x + \lim_{j\rightarrow \infty} \int_{t}^{\max{(t,s-\lambda_{k_j})}} a(\tau+ \lambda_{k_j}, \xi_{k_j}(\tau)) \m{\tau} 
=  
x + \lim_{j\rightarrow \infty} \int_{t}^{\max{(t,s)}} a(\tau, \xi_{k_j}(\tau- \lambda_{k_j})) \m{\tau}  \\=
x +  \int_{t}^{\max{(t,s)}} \lim_{j\rightarrow \infty} a(\tau, \xi_{k_j}(\tau- \lambda_{k_j})) \m{\tau} =
x +  \int_{t}^{\max{(t,s)}} a(\tau, \xi(\tau) ) \m{\tau}.
\end{multline*}
Equivalently $\xi$ satisfies the ordinary differential equation $\dot{\xi}(s)= a(s, \xi(s)),\ \xi(t)=x$ for almost all $s \in J^+$. The proof for backward problem on some $J^{-}:= J \cap [T^{-}, t]$ for some $T^{-} \in J$ with $T^{-}<t$ is analogously.

In the case where $J$ is an open interval or unbounded, it is straight-forward to decompose the domain in countable many compact subintervals. Then the local solutions for each subinterval are continuously glued together (by choosing appropriate initial conditions for the local solutions) to obtain a global solution.
\end{proof}


\begin{definition}\label{solution_set} \index{solution set}
Let $\Omega$ be a non-empty subset of $J \times \R^n$, then we define
$$
\Xi(\Omega) := \{ \xi \in AC(J)^n \mid  \exists (t,x)\in \Omega: \dot{\xi}(s) = a(s,\xi(s)) \text{ a.e., }\ \xi(t)=x \}
$$
denoted as (Caratheodory) solution set with initial conditions in $\Omega$. 
If we put $\Xi_{t,x} := \Xi(\{(t,x)\})$ it obviously holds that 
$$
\Xi(\Omega) = \bigcup_{(t,x) \in \Omega} \Xi_{t,x}.
$$

The solution set $\Xi(\Omega) $ is called forward (resp. backward) unique, if $\xi_1, \xi_2 \in \Xi(\Omega)$
with $\xi_1(t) = \xi_2(t)$ implies $\xi_1(s)=\xi_2(s)$ for all $s\ge t$ (resp. $s\le t$).

In the case where $J= \R$ and $\Omega'$ is a non-empty subset of $\R^n$, we refer to a solution set $\Xi(\R\times \Omega')$ as autonomous, if $\xi\in \Xi(\R \times \Omega')$
implies $\xi(\cdot+r) \in \Xi(\R \times \Omega')$ for any $r\in \R$. 
\end{definition}


\begin{theorem}[Properties of the solution set] \label{caratheodory_solution_set_properties}
Let $\Omega$ be a non-empty subset of $\R^n$, then the solution set $\Xi(J \times \Omega)$ has the following properties:
\begin{enumerate} 
\item For every $(t,x) \in J \times \Omega$, there exists $\xi \in \Xi(J \times \Omega)$ with $\xi(t)=x$.
\item If $t\in J$ and $\Omega$ is closed, then $\Xi(\{t\} \times \Omega)$ is a closed subset of $C(J)^n$. 
\item $(\xi)_{\xi \in \Xi(J \times \Omega)}$ is an equi-continuous family in $C(J)^n$.
\item If $\Omega \csub \R^n$, then $\Xi(\{t\} \times \Omega)$ is a compact subset of $C(J)^n$.
\end{enumerate}
If the right-hand side is time-independent, we put $J=\R$ and the solution set $\Xi(\R \times \Omega)$ is autonomous.
\end{theorem}

\begin{proof}
(i) By Theorem \ref{caratheodory_ode_existence} the set $\Xi(J \times \Omega)$ is a non-empty subset of 
$C(J)^n$, such that for every $(t,x) \in J \times \Omega$ there exists a $\xi \in \Xi(J \times \Omega)$ with $\xi(t)=x$. 

(ii) First we prove that $\Xi(\{t\} \times \Omega)$ is a closed set in $C(J)^n$ if $\Omega$ is closed:
Let $(\xi_k)_{k\in\N}$  be sequence in $\Xi(\{t\} \times \Omega)$ uniformly converging (on compact sets)
to some $\xi \in C(J)^n$. 
In particular $x_k=\xi_k(t) \rightarrow \xi(t)=:x \in \Omega$, since $\Omega$ is closed.  
It follows that  
\begin{multline*}
\xi(s) =\lim_{k\rightarrow\infty} \xi_k(s) =
 \lim_{k\rightarrow\infty} \left( x_k+\int_{t}^s a(\tau, \xi_k(\tau), w) \m{\tau} \right) \\
= x +  \int_{t}^s \lim_{k \rightarrow \infty} a(\tau, \xi_k(\tau)) \m{\tau} 
= x + \int_{t}^s a(\tau, \xi(\tau)) \m{\tau}, 
\end{multline*}
which implies $\xi \in \Xi(\{t\} \times \Omega)$ satisfying the initial condition $\xi(t)=x \in \Omega$, 
so $\Xi(\{t\} \times \Omega)$ is a closed subset of $C(J)^n$.

(iii) Using the integral equation (\ref{int_eq}) we have for all $s,t \in J$ that 
\begin{equation} \label{caratheodory_equi_continuous_estimate}
|\xi(s) -\xi(t)| \le  \int_{\min{(t,s)}}^{\max{(t,s)}} \beta(\tau) \m{\tau}
\end{equation}
holds uniformly for all $\xi \in \Xi(J \times \Omega)$, thus $\Xi(J \times \Omega)$ is an equi-continuous family of continuous
functions.

(iv) Since $K\csub \R^n$ the estimate
\begin{eqnarray*}
\sup_{\xi \in \Xi(\{t\} \times K)} |\xi(s)| \le \sup_{x \in K} |x| + 
\sup_{\xi \in \Xi(\{t\} \times K)} \left| \int_t^s a(\tau, \xi(\tau)) \m{\tau} \right| \le \sup_{x \in K} |x| + 
\int_{\min{(t,s)}}^{\max{(t,s)}} \beta(\tau) \m{\tau} < \infty
\end{eqnarray*}
holds for all $s \in J$, thus $(\xi )_{\xi \in \Xi(\{t\} \times K)}$ is an equi-bounded family. 

By the Theorem of Arzela-Ascoli it follows that $\{ \xi : \xi \in \Xi(\{t\} \times K) \}$ is pre-compact in 
$C(J)^n$. The closedness of $\{ \xi :\xi \in \Xi(\{t\} \times K) \}$ in $C(J)^n$ was already shown in (ii), thus
$\Xi(\{t\} \times K ) \csub C(J)^n$.

If the right-hand side of (\ref{ode}) is time-independent, it is  obvious that we can put $J=\R$. For any $\xi \in \Xi( \R \times \Omega)$
with $\xi(t) =x \in \Omega$ we can define $\widetilde{\xi} = \xi(\cdot - r)$ for some $r\in \R$.
Upon change of variable in the integral on the right-hand side of (\ref{int_eq}) it follows directly that 
$\widetilde{\xi}$ again satisfies the ordinary differential equation with modified initial condition 
$\widetilde{\xi}(t+r)=x \in \Omega$, thus $\widetilde{\xi} \in \Xi(\R \times \Omega)$.
\end{proof}


\begin{theorem}[Unique solution sets] \label{flow_map}
If there exists some $\alpha \in L^{1}_{\rm loc}(J)$ such that the right-hand side $a$ satisfies
the forward uniqueness condition
\begin{equation} \label{forward_uniqueness_condition} \index{forward uniqueness condition}
\langle x-y,  a(s,x)-a(s,y) \rangle  \le \alpha(s) |x-y|^2 
\end{equation}
resp. the backward uniqueness condition
\begin{equation} \label{backward_uniqueness_condition} \index{backward uniqueness condition}
 \langle x-y,  a(s,x)-a(s,y) \rangle  \ge - \alpha(s) |x-y|^2
\end{equation}
for almost all $s \in J$ and all $x,y\in \R^n$, then the solution set $\Xi(J \times \R^n)$ is forward (resp. backward) unique. 
\end{theorem}

\begin{proof}
We prove only the forward uniqueness (as the proof of backward uniqueness is analogous):
Let $\xi_1, \xi_2 \in \Xi(J \times \Omega)$ with $\xi_1(t)=\xi_2(t)=x$. 
Setting $\zeta(s) :=|\xi_1(s)- \xi_2(s)|^2$ we observe $\zeta(t)=  |\xi_1(t) -\xi_2(t)|^2 = 0$ and using that 
$\xi_1, \xi_2$ satisfy the ordinary differential equation, we derive that the estimate
\begin{eqnarray*}
\partial_s \zeta(s)  &=& 2 \langle \xi_1(s) -\xi_2(s), \dot{\xi_1}(s)-\dot{\xi_2}(s)  \rangle 
= 2 \langle \xi_1(s) -\xi_2(s) , a(s,\xi_1(s)) - a(s,\xi_2(s)) \rangle \\
&\le &  2 \alpha(s) \zeta(s)
\end{eqnarray*}
holds for almost all $s\in J$, thus Gronwall's inequality gives
\begin{eqnarray*}
\zeta(s) \le \zeta(t) \exp{\left(2\int_t^s \alpha(r) \m{r}\right)} = 0
\end{eqnarray*}
for all $s \ge t$. It follows that $\zeta(s) = |\xi_1(s)- \xi_2(s)|^2 \equiv 0$ for all $s\ge t$, 
so $\Xi(J\times \Omega)$ is forward unique. 
\end{proof}


\begin{theorem}[Forward unique flow map] \label{forward_flow_map} \index{flow!forward unique}
Let $\Xi(J\times \R^n)$ be a forward unique solution set. Put $J_t^+:=[t,\infty[ \cap J$, then the map 
defined by
\begin{equation*}
\begin{array}{rll}
 \chi_t: J_t^+ \times \R^n & \rightarrow & \R^n \\
 (s,x) &\mapsto& \xi(s;t,x) 
\end{array}
\end{equation*}
for some $\xi(\cdot;t,x) \in \Xi(J\times \R^n)$ with $\xi(t;t,x)=x$ is uniquely defined,
called \emph{forward flow map}.

It has the properties that
\begin{enumerate}
\item $\chi_t$ is a continuous map, 
\item $\chi_t(s, \cdot)$ is proper and onto for all $s\in J_{t}^+$ fixed, and
\item $\chi_r(s,\chi_t(r,x)) = \chi_t(s,x)$ for $s\ge r \ge t$.
\end{enumerate}

If the right-hand side of (\ref{ode}) is time-independent, we have the additional flow property
$\chi_t(s, x) =\chi_0(s-t, x)$ for all $s \ge t$, so property (iii) can be written in the well-known form
$$
\chi_{t_1}(s, \chi_{t_2}(r,x)) = \chi_{t_1+t_2}(s+ r ,x) 
$$
for $s\in J_{t_1}^+, r\in J_{t_2}^+$ with $t_1,t_2 \in J$.
\end{theorem}

\begin{proof}
First of all we note that $\chi_t$ is uniquely defined, as the solution set $\Xi(J \times \R^n)$ is forward unique.

(i) Considering the continuity of $\chi_t$, we first prove that $\chi_t(s,x)$ is separately continuous. 
The continuity in $s$ is obvious by the definition of $\chi_t$.

Continuity in $x$ we shall prove by contradiction: Assume $\chi_t(s,x)$ is not continuous in $x$ at fixed $s$. Then
there exists a sequence $(x_k)_{k\in\mathbb{N}}$ converging to $x$, $\gamma >0$ and $k_0\in \N$ such that
$|\chi_t(s,x) - \chi_t(s,x_k)|>\gamma$ for all $k\ge k_0$.
Choose $K\csub \R^n$ such that $x_k \in K$ for all $k\in\N$.

For each $x_k$ we can find a $\xi_k\in\Xi(\{t\} \times K)$ such that $\xi_k(t) = x_k$ and $\chi_t(s,x_k)=\xi_k(s)$ for all $s\in J_t^+$. 
Furthermore there exists a $\xi\in\Xi(\{t\} \times K)$ such that $\xi(t) = x$ and $\chi_t(s,x)=\xi(s)$  for all $s\in J_t^+$.
The compactness of $\Xi(\{t\} \times K))$ in $C(J)^n$ yields that, there exists a locally uniformly converging subsequence 
$\xi_{k_l} \stackrel{l \rightarrow \infty}{\rightarrow} \tilde{\xi} \in \Xi(\{t\} \times K))$. 
The forward uniqueness of $\Xi(J \times \R^n)$ gives that $\widetilde{\xi}(s) = \xi(s)$ for all $s\in J_t^+$, thus $\lim_{l\rightarrow \infty} |\chi_t(s,x)-\chi_t(s, x_{k_l})| = 0 < \gamma$ contradicting the assumption.

The equi-continuity of $\Xi(J \times \R^n)$ yields the estimate
\begin{multline*}
|\chi_t(s,x) - \chi_t(s',x')| \le |\chi_t(s,x) - \chi_t(s,x')| + |\chi_t(s,x') - \chi_t(s',x')| \le
|\chi_t(s,x) - \chi_t(s,x')| \\+ \sup_{\xi \in \Xi(J \times \R^n)} |\xi(s) - \xi(s')|,
\end{multline*}
which, as $\chi_t$ was already shown to be separately continuous, implies the joint continuity of $\chi_t$.

(ii)
There exists some $\xi \in \Xi_{t,x}$ with $\chi_t(s,x) = \xi(s)$, thus 
$$
 |\chi_t(s,x) -x| = |\xi(s)-x| = \left|\int_t^s a(\tau,\xi(\tau)) \m{\tau} \right|\le \int_t^s \beta(\tau) \m{\tau}
$$
holds for all $s \ge t$. For any compact set $K\csub \R^n$ we have that 
$\chi_t^{-1}(\{s\}\times K) \subseteq K':=\left\{x\in \R^n \mid \exists x_0 \in K : |x-x_0| \le  \int_t^s \beta(\tau) \m{\tau} \right\}$,
implying the properness of $x \mapsto \chi_t(s,x)$.

To show that $x\mapsto \chi_t(s,x)$ is onto for all $s\in J$ fixed, we have to show $\chi_t(s, \R^n)=\R^n$. 
We prove by contradiction: Assume there exists some $y\not\in \chi_t(s,\R^n)$, then by Theorem
\ref{caratheodory_solution_set_properties} (i) there exists a solution $\xi \in \Xi_{s,y}$ with $\xi(s)=y$. Putting $x:=\xi(t)$ we observe that $\xi \in \Xi_{t,x}$ and
forward-uniqueness of $\Xi(J\times\R^n)$ yields that $\xi(r) = \chi_t(r,x)$ for all $r\ge t$. In particular we have $y =\xi(s)=\chi_t(s,x) \in \chi_t(s,\R^n)$ which is a contradiction. 

(iii)
For fixed $x\in\mathbb{R}^n$ we have that $\chi_t(s,x) = \xi_1(s), s\in J_t^+$ for some $\xi_1 \in \Xi(J \times \R^n)$
with $\xi_1(t)=x$.
Fixing $r\in J_t^{+}$ we may choose $\xi_2 \in \Xi(J \times \R^n)$ with 
$\xi_2(r) = \chi_t(r,x)$ and $\xi_2(s) =\chi_r(s, \chi_t(r,x))$ for $s\ge r$. Since $\xi_1, \xi_2 \in \Xi(J \times \R^n)$ with $\xi_1(r) =  \chi_t(r,x)$ and $\xi_2(r) =  \chi_t(r,x)$ it follows by the forward uniqueness of $\Xi(J \times \R^n)$ that $\xi_1(s)=\xi_2(s)$ for all $s\ge r$, thus
$$
 \chi_r(s,\chi_t(r,x)) = \chi_t(s,x) 
$$
for $s\in J_t^{+} \cap J_r^{+}$ and $r\in J_t^{+}$. 

In the case where the right-hand side of \ref{ode} is time-independent, we put $J=\R$ and have that $\Xi(\R^{n+1})$ is autonomous
due to Theorem \ref{solution_set}. As $\chi_0(s, x) = \xi_1(s)$ for some $\xi_1 \in \Xi(\R^{n+1})$ with $\xi_1(0)=x$
and $\chi_t(s,x) = \xi_2(s)$ for some $\xi_2 \in \Xi(\R^{n+1})$ with $\xi_2(t)=x$. Due to the autonomy of the solution set 
it follows that $\widetilde{\xi_2}(s) := \xi_2(s-t)$ is again in $\Xi(\R^{n+1})$ . As $\widetilde{\xi_2}(t) = \xi_1(t) =x$ the forward uniqueness of $\Xi(\R^{n+1})$ immediately yields $\widetilde{\xi_2}(s) = \xi_1(s)$ for all $s \in J_t^+$, thus
$\chi_t(s,x) = \chi_0(s-t,x)$ for all $s\in J_t^+$. Using this relation and replacing $r$ by $r+t-t_2$ and $s$ by $s+r+t-t_2-t_1$ we can rewrite property (iii) in order to obtain
$$
\chi_{t_1}(s, \chi_{t_2}(r,x)) = \chi_{t_1+t_2}(s+ r ,x) 
$$
for all $r\in J_{t_1}^+, s \in J_{t_2}^+$ with $t_1, t_2 \in \R$.
\end{proof}


\begin{theorem}[Backward unique flow maps]\label{backward_flow_map} \index{flow!backward unique}
Let $\Xi(J\times \R^n)$ be a backward unique solution set. Put $J_t^-:=]-\infty,t] \cap J$, then the map 
defined by
\begin{equation*}
\begin{array}{rll}
 \Phi_t: J_t^+ \times \R^n & \rightarrow & \R^n \\
 (s,x) &\mapsto& \xi(s;t,x) 
\end{array}
\end{equation*}
for some $\xi(\cdot;t,x) \in \Xi(J\times \R^n)$ with $\xi(t;t,x)=x$ is uniquely defined,
called \emph{backward flow map}.

It has the properties that
\begin{enumerate}
\item $\Phi_t$ is a continuous map, 
\item $\Phi_t(s, \cdot)$ is proper for all $s\in J_{t}^-$ fixed, and
\item $\Phi_r(s,\Phi_t(r,x)) = \Phi_t(s,x)$ for $s\le r \le t$.
\end{enumerate}

If the right-hand side of (\ref{ode}) is time-independent, we have the additional property
$ \Phi_t(s, x) =\Phi_0(s-t, x)$ for all $s \le t$, so property (iii) can be written in the well-known form
$$
\Phi_{t_1}(s, \Phi_{t_2}(r,x)) = \Phi_{t_1+t_2}(s+ r ,x) 
$$
for $s\in J_{t_1}^-, r\in J_{t_2}^-$ with $t_1,t_2 \in \R$.
\end{theorem}

\begin{proof}
Analogous to the proof of Theorem \ref{forward_flow_map}.
\end{proof}


\begin{remark}[Discontinuous flow maps] \index{flow!discontinuous}
Due to Theorem \ref{caratheodory_ode_existence} it is of course possible (regardless of any uniqueness property of the solution set $\Xi(J\times \R^n)$) to define a "global" flow map $\Theta_t: J \times \R^n \rightarrow \R^n, \quad (s,x) \mapsto \xi(s)$ where $\xi \in \Xi(J\times \R^n)$ with $\xi(t) =x$. But such a map is not uniquely defined and
 $\Theta_t(s,\cdot)$ is in general neither continuous, nor onto $\R^n$ for fixed $s,t \in J$.
 Nevertheless at least $(\Theta_t(\cdot, x))_{x \in \R^n}$ is equi-continuous family in $C(J)^n$ and
$\Theta_t(s, \cdot)$ is a proper map, i.e. for all $K \csub \R^n$ there exists some $K' \csub \R^n$ such that
the inverse image $\Theta_t^{-1}(\{s\} \times K)$ is contained in $K'$. 
In the case where $\Xi(J \times \R^n)$ is forward (resp. backward) unique, we have that
 $$
 \Theta_t \mid_{J_t^+} = \chi_t \quad (\textrm{resp. } \Theta_t \mid_{J_t^-} = \Phi_t )
 $$
holds. In the case when the solution set is unique, the globally defined flow map $\Theta_t$ is uniquely defined by 
the forward flow on $J_t^+$ and the backward flow on $J_t^-$, as we will see.
\end{remark}


\begin{theorem}[Unique flow maps] \label{unique_flow_map} \index{flow!properties}
Let $\Xi(J \times \R^n)$ be a unique solution set, then it generates a unique, global (defined on the whole interval J) flow map by
$$
\Theta_t(s,x) := \left\{\begin{array}{ll}
\chi_t(s,x) & s \ge t \\
\Phi_t(s,x) & s < t
\end{array} \right.
$$ for $t\in J$ fixed. 
It satisfies the properties 
\begin{enumerate}
\item $\Theta_t$ is a continuous map,
\item $\Theta_t(s, \cdot)$ is proper for all $s \in J$ fixed, and
\item $\Theta_s(r,\Theta_t(s,x)) = \Theta_t(r,x)$ for $s,r,t \in J$ with the special case (setting $r=t$)\\
 $\Theta_s(t,\Theta_t(s,\cdot)) = {\rm id}_{\R^n}$ for $s,t \in J$.
\end{enumerate}
We can write property (iii) in terms of the forward and backward flow as
$$
\Phi_s(t, \cdot) = (\chi_t(s, \cdot))^{-1} 
$$
for all $s,t\in J$ with $s\ge t$. 

In the case where the right-hand side of (\ref{ode}) is time independent we again obtain
$\Theta_t(s,x) = \Theta_0(s-t,x)$ for all $s,t \in J$ and $x\in \R^n$.
\end{theorem}

\begin{proof}
The uniqueness of $\Theta_t$ follows immediately by combining the uniqueness results from Theorem \ref{forward_flow_map} and Theorem \ref{backward_flow_map}. It is globally defined since $J_t^{-} \bigcup J_t^+ = J$.

(i) The continuity of $\Theta_t$ is obvious by Definition, as $\chi_t(t,\cdot) = \Phi_t(t,\cdot) = {\rm id}_{\R^n}$, $\chi_t$ is continuous on $J_{t}^+ \times \R^n$ and $\Phi_t$ is continuous on $J_t^{-} \times \R^n$ due to Theorem \ref{forward_flow_map} and \ref{backward_flow_map}.

(ii) $\Theta_t(s, \cdot)$ is proper follows immediately from the properness of $\chi_t(s, \cdot)$ and $\Phi_t(s, \dot)$ 
on the respective domains as proved in Theorem \ref{forward_flow_map} and \ref{backward_flow_map}.

(iii) 
The case where $r\le s \le t$ or $t \le r \le s$ follows again directly from Theorem \ref{forward_flow_map} and \ref{backward_flow_map}.
The $r\le s \le t$, $t \le r \le s$, $r \le t \le s$ and $s \le t \le r$ need some additional work, but the proof is analogous
to the respective proofs in Theorem \ref{forward_flow_map} and \ref{backward_flow_map}. 
To illustrate the proving technique one more time, we consider the case $r\le s \le t$: We have $\Theta_s(r, \Theta_t(s,x)= \Phi_s(r, \chi_t(s,x)) = \xi_1(r)$ for some $\xi_1 \in \Xi(J\times\R^n)$ with $\xi_1(s) =\chi_t(s,x)$ and $\Theta_t(r,x) = \chi_t(r,x)=\xi_2(r)$ for some $\xi_1 \in \Xi(J\times\R^n)$ with $\xi_2(t) = x$. If $r=s$ we have $\xi_1(s) = \xi_2(s)$, thus uniqueness of $\Xi(J\times \R^n)$ implies that $\xi_1 \equiv \xi_2$ on the whole interval $J$.

The relation $\Theta_t(s,x) = \Theta_0(s-t, x)$ for all $s,t \in J$ and $x \in \R^n$ follows directly from
Theorem \ref{forward_flow_map} and \ref{backward_flow_map}.
\end{proof}


\begin{remark}[$C^1$ regularity of the flow] \label{ode_derivatives}
Let $a$ satisfy the Caratheodory condition on $J \times \mathbb{R}^{n}$ and assume that $a(s, \cdot) \in \Con^1(\R^n)^n$ for almost all $s\in J$.
Then a the solution set $\Xi(J)$ is unique and the
uniquely generated flow map $\Theta_t$ satisfies the properties 
\begin{eqnarray*}
(d_x \Theta_t) (s,x) &=& \exp{\left(\int_t^s d_x a(\tau, \Theta_t(\tau,x)) \m{\tau} \right)}  
\end{eqnarray*}
and 
\begin{eqnarray*}
\det(d_x \Theta_t) (s,x) &=& \exp{\left(\int_t^s \div{a}(\tau, \Theta_t(\tau,x)) \m{\tau} \right)}. 
\end{eqnarray*}
for all $(s,x) \in J \times \mathbb{R}^n$. For a proof we refer to \cite[Theorem 1.2.5.]{Hoermander:97}.
\end{remark}


\section{Differential inclusions and integral inequalities} \label{differential_inclusion_equation_section}


In this section we introduce a natural generalization of the ordinary differential equation, namely the \emph{differential inclusion} which is of the form
\begin{equation} \label{odie}
\odie,
\end{equation}
where $A_{t,x}$ is a non-empty, closed and convex subset of $\R^n$ for all $(t,x) \in J \times \R^n$. 

It turns out that the notion of a \emph{set-valued map} (cf. Section \ref{set_valued_map_section}) is suitable for the description of such 
generalized right-hand sides of the form $(t,x)\mapsto A_{t,x}$. 

As in the case of Caratheodory theory we have to impose certain conditions on the set-valued map $A$
in order to obtain an existence result for the differential inclusions.  

Throughout this section we assume that the set-valued map $A$ satisfies
\begin{enumerate} \label{filippov_conditions}
\item $t \mapsto A_{t,x}$ is Lebesgue measurable on $J$ for all fixed $x \in \mathbb{R}^n$,
\item $x \mapsto A_{t,x}$ is a upper semi-continuous for almost all $t \in J$,        
\item there exists a positive function $\beta \in L_{\rm loc}^1(J)$ such that $\sup_{x\in\mathbb{R}^n} |A_{t,x}| \le \beta(t)$ almost everywhere on $J$,
\end{enumerate}
referred to as \emph{Filippov conditions} (FC). \index{Filippov conditions}

It should be noted that $\sup_{x\in \R^n}|A_{t,x}| \le \beta(t)$ implies that $x\mapsto A_{t,x}$
is bounded for almost all $t\in J$. Due to Theorem \ref{set_valued_map_main_theorem} condition (ii)
is equivalent to have $x\mapsto  A_{t,x}$ being a closed set-valued map for almost all $t\in J$.  

We should think of $t \mapsto A_{t,x}$ as a set-valued map in $L^1_{\rm loc}(J; \mathcal{K}_0(\R^n))$.
This means that the Filippov conditions do not depend on the chosen representative of $t \mapsto A_{t,x}$ for $x\in \R^n$ fixed, so
without loss of generality we can assume that $x\mapsto A_{t,x}$ is upper semi-continuous (or equivalently - closed) for all $t\in J$.  

Equivalently we can formulate these conditions for the 
supporting $H$ (according to Theorem \ref{set_valued_map_main_theorem}) by
\begin{enumerate}
\item $t \mapsto H(t,x,w)$ is Lebesgue measurable on $J$ for all $x,w \in \R^n$,
\item $x \mapsto H(t,x,w)$ is upper semi-continuous for almost all $t \in J$ and $w \in \R^n$,        
\item there exists a positive function $\beta \in L_{\rm loc}^1(J)$ such that 
 $\sup_{x\in\mathbb{R}^n} |H(t,x,w/|w|)| \le \beta(t)$ for almost all $t\in J$ and all $w\in \R^n$.
\end{enumerate}

Furthermore the differential inclusion (\ref{odie}) is equivalent to the following integral inequality \index{integral inequality}
\begin{equation}
\langle \xi(s), w\rangle \le \langle x,w\rangle + \int_t^s H(\tau, \xi(\tau), w) \m{\tau} ,\quad \forall w\in \R^n.
\end{equation}
 


\begin{theorem}[Existence theorem for differential inclusions] \label{filippov_existence_theorem}
Let $A$ be set-valued map satisfying (FC). Let $(t,x) \in J \times \R^n$ be the initial value. Then there exists a solution
$\xi \in AC(J)^n$ of the differential inclusion
\begin{eqnarray*}
\dot{\xi}(s) \in A_{s, \xi(s)} \  a.e., \quad \xi(t)=x.
\end{eqnarray*} 
\end{theorem}

\begin{proof}
First of all we notice that for any $f\in C(J)^n$ it holds that the composition $t \mapsto A_{t, f(t)}$ is a Lebesgue measurable set-valued map from $J$ to $\R^n$.
By Definition \ref{set_valued_integral} and Corollary \ref{composition_single_valued_function} the measurability of the set-valued map $t \mapsto A_{t, f(t)}$ is equivalent to the measurability its supporting function
$t\mapsto H(t, f(t), w)$ for all $w\in\R^n$, which follows by Lemma \ref{measurability_of_composition_lemma} 
since $H$ satisfies the Filippov condition (i) and (ii).

We construct a sequence of continuous functions $(\xi_{k})_{k \in \N}$
which approximates the solution of the differential inclusion.

Put $\xi_0(s)=x$ for all $s\in J$, then using Theorem \ref{primitive_set-valued map} we iteratively construct 
a sequence of set-valued maps $(X_k)_{k \in \N}$ with $X_k: J \rightarrow \mathcal{K}_0(\R^n)$ by
$$
X_{k+1}(s) = \{x\} + \int_{t}^{s} A_{\tau,\xi_k(\tau)} \m{\tau}
$$
Furthermore Theorem  \ref{primitive_set-valued map} yields an absolutely continuous 
selection $\xi_{k+1}$ of $X_{k+1}$ with $\xi_{k+1}(t)=x$, so we obtain a family $(\xi_{k})_{k \in \N}$ of
absolutely continuous functions with the property
\begin{equation} \label{approx_solution}
\langle \xi_k(s) - \xi_k(r), w \rangle \le \int_r^s  H(\tau,\xi_k(\tau), w ) \m{\tau}
\end{equation}
for $s,r \in J$ and $w\in\R^n$. 

We observe that the (FC) (iii) yields the estimate
$$ 
| \xi_k(s) - \xi_k(r)| \le \int_{\min{(r,s)}}^{\max{(r,s)}}  \sup_{w\in\R^n}{H(\tau,\xi_{k}(\tau), w/|w| )} \m{\tau} 
\le \int_{\min{(r,s)}}^{\max{(r,s)}} \beta(\tau) \m{\tau} \quad \text{ for } s,r \in J.
$$
Thus $(\xi_k)_{k\in\N}$ is an equi-continuous family. The estimate
$$
|\xi_k(s)| \le |x| + \int_r^s  \sup_{w\in\R^n }H(\tau,\xi_k(\tau), w/|w| ) \m{\tau}
$$
yields the equi-boundedness of $(\xi_k)_{k\in\N}$.

By the Theorem of Arzela Ascoli there exists an uniformly convergent (on compact subsets of $J$) subsequence $\xi_{k_j} \stackrel{j\rightarrow \infty}{\rightarrow} \xi$, where $\xi \in C(J)^n$ with $\xi(t)=x$.

We have for all $s,r\in J$,
\begin{multline} \label{inclusion_equi_cont}
|\xi(s) - \xi(r)| \le |\xi(s) - \xi_{k_j}(s)|+ |\xi_{k_j}(s)-\xi_{k_j}(r)| +|\xi_{k_j}(r) - \xi(r)|  \\
\le |\xi(s) - \xi_{k_j}(s)| +\int_{\min{(r,s)}}^{\max{(r,s)}} \beta(\tau) \m{\tau} +|\xi_{k_j}(r) - \xi(r)| \stackrel{j \rightarrow \infty}{\rightarrow}\int_{\min{(r,s)}}^{\max{(r,s)}} \beta(\tau) \m{\tau} , 
\end{multline}
hence $\xi$ is absolutely continuous in $J$.

By the upper semi-continuity of $x \mapsto H(t,x,w)$, it follows that
$$
\limsup_{j \rightarrow \infty}  H(t, \xi_{k_j}(\tau), w )  \le 
 H(t, \xi(t),w ).  
$$
The theorem of dominated convergence \cite[Theorem 12.24]{HewStr:65} yields
$$
\langle \xi(s),w \rangle  \le \limsup_{j\rightarrow \infty} \langle \xi_{k_j}(s) ,w \rangle = \langle x, w \rangle  + \limsup_{j\rightarrow \infty} \int_t^s H(\tau, \xi_{k_j}(\tau), w )  \m{\tau} \le \langle x, w \rangle + \int_{t}^s H(\tau, \xi(\tau),w) \m{\tau} 
$$
Differentiation with respect to $s$ yields  
$$
\langle \dot{\xi}(s) , w \rangle \le H(s,\xi(s), w)
$$
for all $w\in \R^n$, thus $\dot{\xi}(s) \in A_{s, \xi(s)}$ for almost all $s\in J$ and $\xi(s)=x$.   
\end{proof}


\begin{definition}\label{odi_solution_set} \index{inclusion solution set}
Let $\Omega$ be a non-empty subset of $\R^n$. Analogous to the case of classical ordinary differential 
equations, we define the \emph{inclusion solution set} by
$$
\Xi_I(J \times \Omega) := \{ \xi \in AC(J)^n \mid  \exists (t,x)\in J\times \Omega : 
\dot{\xi}(s) \in A_{s,\xi(s)} \text{ a.e., }\ \xi(t)=x \}.
$$
If we put $\Xi_{t,x} := \Xi_I(\{(t,x)\})$ it obviously holds that 
$$
\Xi_I(J \times \Omega) = \bigcup_{(t,x) \in J \times \Omega} \Xi_{t,x}.
$$

The solution set $\Xi_I(J \times \Omega) $ is called \emph{forward (resp. backward) unique}, if $\xi_1, \xi_2 \in \Xi_I(J \times \Omega)$
with $\xi_1(t) = \xi_2(t)$ implies $\xi_1(s)=\xi_2(s)$ for all $s\ge t$ (resp. $s\le t$).

In the case where $J= \R$, we refer to a solution set $\Xi_I(\R\times \Omega)$ as \emph{autonomous}, if $\xi\in \Xi_I(\R \times \Omega)$
implies $\xi(\cdot+r) \in \Xi_I(\R \times \Omega)$ for any $r\in \R$. 
\end{definition}


\begin{proposition}[Properties of the inclusion solution set] \label{filippov_solution_set_prop} 
Let $\Omega$ be a non-empty subset of $\R^n$, then the inclusion solution set $\Xi_I(J \times \Omega)$ has the following properties:
\begin{enumerate} 
\item For every $(t,x) \in J \times \Omega$, there exists $\xi \in \Xi_I(J \times \Omega)$ with $\xi(t)=x$.
\item If $t\in J$ and $\Omega$ is closed, then $\Xi_I(\{t\} \times \Omega)$ is a closed subset of $C(J)^n$. 
\item $(\xi)_{\xi \in \Xi_I(J \times \Omega)}$ is an equi-continuous family in $C(J)^n$.
\item If $\Omega \csub \R^n$, then $\Xi_I(\{t\} \times \Omega)$ is a compact subset of $C(J)^n$.
\end{enumerate}
If the right-hand side of (\ref{odie}) is time-independent, we can put $J=\R$ and the (Filippov) solution set $\Xi_I (\R \times \Omega)$ is autonomous.
\end{proposition}

\begin{proof}
(i) By Theorem \ref{filippov_existence_theorem} the set $\Xi_I(J \times\Omega)$ is a non-empty subset of 
$C(J)^n$, such that for every $(t,x) \in \Omega$ there exists a $\xi \in \Xi_I(J \times\Omega)$ with $\xi(t)=x$. 

(ii) First we prove that $\Xi_I(\{t\} \times \Omega)$ is a closed set in $C(J)^n$ if $\Omega$ is closed:
Let $(\xi_k)_{k\in\N}$  be sequence in $\Xi_I(\{t\} \times \Omega)$ uniformly converging (on compact sets)
to some $\xi \in C(J)^n$. 
In particular $x_k=\xi_k(t) \rightarrow \xi(t)=:x \in \Omega$, since $\Omega$ is closed.  
It follows that  
\begin{multline*}
\langle \xi(s), w\rangle =\lim_{k\rightarrow\infty} \langle \xi_k(s), w\rangle = \lim_{k\rightarrow\infty} 
\left( \langle x_k, w\rangle + \int_{t}^s H(\tau, \xi_k(\tau), w) \m{\tau} \right) \\
\le \langle x,w \rangle + \int_{t}^s \limsup_{k \rightarrow \infty} H(\tau, \xi_k(\tau), w) \m{\tau} 
\le \langle x,w \rangle 
+ \int_{t}^s H(\tau, \xi(\tau), w) \m{\tau} 
\end{multline*}

which implies $\xi \in \Xi_I(\{t\} \times \Omega)$ satisfying the initial condition $\xi(t)=x \in \Omega$, 
so $\Xi_I(\{t\} \times \Omega)$ is a closed subset of $C(J)^n$.

(iii) For all $t\in J$, there exist some compact neighborhood $K_t \csub J$, such that by (\ref{inclusion_equi_cont})
\begin{equation} \label{filippov_equi_continuous_estimate}
|\xi(s) -\xi(r)| \le  \int_{\min{(r,s)}}^{\max{(r,s)}} \beta(\tau) \m{\tau}
\end{equation}
holds uniformly for all $\xi \in \Xi_I(J \times \Omega)$, thus $\Xi_I(J \times \Omega)$ is a equi-continuous family.

(iv) Since $K\csub \R^n$ the estimate
\begin{multline*}
\sup_{\xi \in \Xi_I(\{t\} \times K)} |\xi(s)| \le \sup_{x \in K} |x| + 
\sup_{\xi \in \Xi_I(\{t\} \times K)} \left| \int_t^s \sup_{w\in \R^n} H(\tau, \xi(\tau),w/|w|) \m{\tau} \right| \\
\le \sup_{x \in K} |x| + \int_{\min{(t,s)}}^{\max{(t,s)}} \beta(\tau) \m{\tau} < \infty
\end{multline*}
holds for all $s \in J$, thus $(\xi )_{\xi \in \Xi_I(\{t\} \times K)}$ is an equi-bounded family. 

By the Theorem of Arzela-Ascoli it follows that $\{ \xi : \xi \in \Xi_I(\{t\} \times K) \}$ is pre-compact in 
$C(J)^n$. The closedness of $\{ \xi :\xi \in \Xi_I(\{t\} \times K) \}$ in $C(J)^n$ was already shown in (ii), thus
$\Xi_I(\{t\} \times K ) \csub C(J)^n$.

If the rightside of (\ref{odie}) is time-independent, it is  obvious that we can put $J=\R$. For any $\xi \in \Xi_I( \R \times \Omega)$
with $\xi(t) =x \in \Omega$ we can define $\widetilde{\xi} = \xi(\cdot - r)$ for some $r\in \R$. 
By integral substitution on the right-hand side of (\ref{int_eq}) it follows directly 
that $\widetilde{\xi}$ again satisfies the ordinary differential equation with modified initial condition 
$\widetilde{\xi}(t+r)=x \in \Omega$, thus $\widetilde{\xi} \in \Xi_I(\R \times \Omega)$ .
\end{proof}


\begin{theorem}[Properties of the inclusion solution set] 
Let $A$ satisfy conditions (FC) and assume there exists some positive 
$\alpha(t)\in L^{1}_{\rm loc}(J)$ such that the supporting function $H$ of $A$ satisfies
\begin{eqnarray} \label{set_valued_function_forward_uniqueness_condition}
H(s,x,x-y)+H(s,y,y-x) \le \alpha(s) |x-y|^2 
\end{eqnarray}
resp. 
\begin{eqnarray} \label{set_valued_function_backward_uniqueness_condition}
 H(s,x,x-y)+H(s,y,y-x)\ge - \alpha(s) |x-y|^2)
\end{eqnarray} 
for all $x,y\in\R^n$ and almost all $s \in J$, then the solution set $\Xi_I(J\times \R^n)$ is a forward (resp. a backward) unique 
transport set. 
\end{theorem}

\begin{proof}
We prove only the forward uniqueness (the proof of backward uniqueness is done analogously):
Let $\xi_1, \xi_2 \in \Xi_I(J \times \R^n)$ with $\xi_1(t)=\xi_2(t)=x$. 
Setting $\zeta(s) :=|\xi_1(s)- \xi_2(s)|^2$ we observe $\zeta(t)=  |\xi_1(t) -\xi_2(t)|^2 = 0$ 
and using the definition of $\Xi_I(J\times \R^n)$ and the forward uniqueness 
condition (\ref{set_valued_function_forward_uniqueness_condition}), we obtain the estimate
\begin{eqnarray*}
\partial_s \zeta(s)  &=& 2 \langle \xi_1(s) -\xi_2(s), \dot{\xi_1}(s)-\dot{\xi_2}(s)  \rangle 
\le 2 ( H(s, \xi_1(s) , \xi_1(s) -\xi_2(s) ) + H(s, \xi_2(s) , \xi_2(s) -\xi_1(s) )\\
&\le &  2 \alpha(s) \zeta(s)
\end{eqnarray*}
for almost all $s\in J$, thus Gronwall's inequality gives
\begin{eqnarray*}
\zeta(s) \le \zeta(t) \exp{\left(2\int_t^s \alpha(\tau) \m{\tau}\right)} = 0.
\end{eqnarray*}
It follows that $\zeta(s) = |\xi_1(s)- \xi_2(s)|^2 \equiv 0$ for all $s\ge t$ if $\xi_1(t)=\xi_2(t)$, 
so $\Xi_I(J\times \R^n)$ is a forward unique transport set.
\end{proof}


\begin{theorem}[Forward unique (inclusion) flow map] \label{filippov_forward_flow_map} \index{inclusion flow!forward unique}
Let $\Xi_I(J\times \R^n)$ be a forward unique solution set. Put $J_t^+:=[t,\infty[ \cap J$, then the map 
defined by
\begin{equation*}
\begin{array}{rll}
 \chi_t: J_t^+ \times \R^n & \rightarrow & \R^n \\
 (s,x) &\mapsto& \xi(s;t,x) 
\end{array}
\end{equation*}
for some $\xi(\cdot;t,x) \in \Xi_I(J\times \R^n)$ with $\xi(t;t,x)=x$ is uniquely defined,
called \emph{(inclusion) forward flow map}.

It satisfies the following properties:
\begin{enumerate}
\item $\chi_t$ is a continuous map, 
\item $\chi_t(s, \cdot)$ is proper and onto for all $s\in J_{t}^+$ fixed, and
\item $\chi_r(s,\chi_t(r,x)) = \chi_t(s,x)$ for $s\ge r \ge t$.
\end{enumerate}

If the right-hand side of (\ref{odie}) is time-independent, we have the additional flow property
$\chi_t(s, x) =\chi_0(s-t, x)$ for all $s \ge t$, so property (iii) can be written in the well-known form
$$
\chi_{t_1}(s, \chi_{t_2}(r,x)) = \chi_{t_1+t_2}(s+ r ,x) 
$$
for $s\in J_{t_1}^+, r\in J_{t_2}^+$ with $t_1,t_2 \in \R$.
\end{theorem}

\begin{proof}
First of all we note that $\chi_t$ is uniquely defined, as the solution set $\Xi_I(J \times \R^n)$ is forward unique.

(i) Considering the continuity of $\chi_t$, we first prove that $\chi_t(s,x)$ is separately continuous. 
The continuity in $s$ is obvious by the definition of $\chi_t$.

Continuity in $x$ we shall prove by contradiction: Assume $\chi_t(s,x)$ is not continuous in $x$ at fixed $s$. Then
there exists a sequence $(x_k)_{k\in\mathbb{N}}$ converging to $x$, $\gamma >0$ and $k_0\in \N$ such that
$|\chi_t(s,x) - \chi_t(s,x_k)|>\gamma$ for all $k\ge k_0$.
Choose $K\csub \R^n$ such that $x_k \in K$ for all $k\in\N$.

For each $x_k$ we can find a $\xi_k\in\Xi_I(\{t\} \times K)$ such that $\xi_k(t) = x_k$ and $\chi_t(s,x_k)=\xi_k(s)$ for all $s\in J_t^+$. 
Furthermore there exists a $\xi\in\Xi_I(\{t\} \times K)$ such that $\xi(t) = x$ and $\chi_t(s,x)=\xi(s)$  for all $s\in J_t^+$.
The compactness of $\Xi_I(\{t\} \times K))$ in $C(J)^n$ yields that, there exists a locally uniformly converging subsequence 
$\xi_{k_l} \stackrel{l \rightarrow \infty}{\rightarrow} \tilde{\xi} \in \Xi_I(\{t\} \times K))$. 
The forward uniqueness of $\Xi_I(J \times \R^n)$ gives that $\widetilde{\xi}(s) = \xi(s)$ for all $s\in J_t^+$, 
thus $\lim_{l\rightarrow \infty} |\chi_t(s,x)-\chi_t(s, x_{k_l})| = 0 < \gamma$ contradicting the assumption.

The equi-continuity of $\Xi_I(J \times \R^n)$ yields the estimate
\begin{multline*}
|\chi_t(s,x) - \chi_t(s',x')| \le |\chi_t(s,x) - \chi_t(s,x')| + |\chi_t(s,x') - \chi_t(s',x')| \le
|\chi_t(s,x) - \chi_t(s,x')| \\+ \sup_{\xi \in \Xi_I(J \times \R^n)} |\xi(s) - \xi(s')|,
\end{multline*}
which, as $\chi_t$ was already shown to be separately continuous, implies the joint continuity of $\chi_t$.

(ii)
There exists some $\xi \in \Xi_{t,x}$ with $\chi_t(s,x) = \xi(s)$, thus 
$$
 |\chi_t(s,x) -x| = \sup_{w\in \R^n} \langle \chi_t(s,x) -x,w  \rangle  \le \int_t^s \sup_{w \in \R^n} \langle H(\tau,\xi(\tau),w/|w|)  \m{\tau} 
\le \int_t^s \beta(\tau) \m{\tau}
$$
holds for all $s \ge t$. For any compact set $K\csub \R^n$ we have that 
$\chi_t^{-1}(\{s\}\times K) \subseteq K':=\left\{x\in \R^n \mid \exists x_0 \in K : |x-x_0| \le  \int_t^s \beta(\tau) \m{\tau} \right\}$,
implying the properness of $x \mapsto \chi_t(s,x)$.

To show that $x\mapsto \chi_t(s,x)$ is onto for all $s\in J$ fixed, we have to show $\chi_t(s, \R^n)=\R^n$. 
We prove by contradiction: Assume there exists some $y\not\in \chi_t(s,\R^n)$, then by Theorem
\ref{filippov_solution_set_prop} (i) there exists a solution $\xi \in \Xi_{s,y}$ with $\xi(s)=y$. Putting $x:=\xi(t)$ we observe that $\xi \in \Xi_{t,x}$ and
forward-uniqueness of $\Xi_I(J\times\R^n)$ yields that $\xi(r) = \chi_t(r,x)$ for all $r\ge t$. In particular we have $y =\xi(s)=\chi_t(s,x) \in \chi_t(s,\R^n)$ which is a contradiction. 

(iii)
For fixed $x\in\mathbb{R}^n$ we have that $\chi_t(s,x) = \xi_1(s), s\in J_t^+$ for some $\xi_1 \in \Xi_I(J \times \R^n)$
with $\xi_1(t)=x$.
Fixing $r\in J_t^{+}$ we may choose $\xi_2 \in \Xi_I(J \times \R^n)$ with 
$\xi_2(r) = \chi_t(r,x)$ and $\xi_2(s) =\chi_r(s, \chi_t(r,x))$ for $s\ge r$. Since $\xi_1, \xi_2 \in \Xi_I(J \times \R^n)$ with $\xi_1(r) =  \chi_t(r,x)$ and $\xi_2(r) =  \chi_t(r,x)$ it follows by the forward uniqueness of $\Xi_I(J \times \R^n)$ that $\xi_1(s)=\xi_2(s)$ for all $s\ge r$, thus
$$
 \chi_r(s,\chi_t(r,x)) = \chi_t(s,x) 
$$
for $s\in J_t^{+} \cap J_r^{+}$ and $r\in J_t^{+}$. 

In the case where the right-hand side of \ref{odie} is time-independent, we put $J=\R$ and have that $\Xi_I(\R^{n+1})$ is autonomous
due to Theorem \ref{solution_set}. As $\chi_0(s, x) = \xi_1(s)$ for some $\xi_1 \in \Xi_I(\R^{n+1})$ with $\xi_1(0)=x$
and $\chi_t(s,x) = \xi_2(s)$ for some $\xi_2 \in \Xi_I(\R^{n+1})$ with $\xi_2(t)=x$. Due to the autonomy of the solution set 
it follows that $\widetilde{\xi_2}(s) := \xi_2(s-t)$ is again in $\Xi_I(\R^{n+1})$ . As $\widetilde{\xi_2}(t) = \xi_1(t) =x$ the forward uniqueness of $\Xi_I(\R^{n+1})$ immediately yields $\widetilde{\xi_2}(s) = \xi_1(s)$ for all $s \in J_t^+$, thus
$\chi_t(s,x) = \chi_0(s-t,x)$ for all $s\in J_t^+$. Using this relation and replacing $r$ by $r+t-t_2$ and $s$ by $s+r+t-t_2-t_1$ we can rewrite property (iii) in order to obtain
$$
\chi_{t_1}(s, \chi_{t_2}(r,x)) = \chi_{t_1+t_2}(s+ r ,x) 
$$
for all $r\in J_{t_1}^+, s \in J_{t_2}^+$ with $t_1, t_2 \in \R$.
\end{proof}


\begin{theorem}[Backward unique flow maps]\label{filippov_backward_flow_map} \index{inclusion flow!backward unique}
Let $\Xi_I(J\times \R^n)$ be a backward unique solution set. Put $J_t^-:=]-\infty,t] \cap J$, then the map 
defined by
\begin{equation*}
\begin{array}{rll}
 \Phi_t: J_t^+ \times \R^n & \rightarrow & \R^n \\
 (s,x) &\mapsto& \xi(s;t,x) 
\end{array}
\end{equation*}
for some $\xi(\cdot;t,x) \in \Xi_I(J\times \R^n)$ with $\xi(t;t,x)=x$ is uniquely defined,
called \emph{(inclusion) backward flow map}.

It has the properties that
\begin{enumerate}
\item $\Phi_t$ is a continuous map, 
\item $\Phi_t(s, \cdot)$ is proper for all $s\in J_{t}^-$ fixed, and
\item $\Phi_r(s,\Phi_t(r,x)) = \Phi_t(s,x)$ for $s\le r \le t$.
\end{enumerate}

If the right-hand side of \ref{odie} is time-independent, we have the additional property
$ \Phi_t(s, x) =\Phi_0(s-t, x)$ for all $s \le t$, so property (iii) can be written in the well-known form
$$
\Phi_{t_1}(s, \Phi_{t_2}(r,x)) = \Phi_{t_1+t_2}(s+ r ,x) 
$$
for $s\in J_{t_1}^-, r\in J_{t_2}^-$ with $t_1,t_2 \in \R$.
\end{theorem}

\begin{proof}
Analogous to the proof of Theorem \ref{filippov_forward_flow_map}.
\end{proof}


\section{Generalized solution concepts for ordinary differential equations} \label{section_generalized_solution_concepts_ode}


It is possible to generalize the classical solution concept for 
ordinary differential equations with discontinuous right-hand side $a\in L^1_{\rm loc}(J; L^{\infty}(\R^n))^n$,
by replacing $a$ with a suitable set-valued map $A$  satisfying (FC) 
and solving the resulting differential inclusion by means of Theorem \ref{filippov_existence_theorem}.

Generalization of the classical solution concept means that whenever a right-hand side 
$a$ satisfies the Caratheodory conditions the graph of the set-valued map $A$ 
and the graph of $a\in L^1_{\rm loc}(J; L^{\infty}(\R^n))^n$ are equal.
One way of mapping functions $a\in L^1_{\rm loc}(J; L^{\infty}(\R^n))^n$ into set-valued maps 
satisfying (FC) is by means of the essential convex hull. 

Another method to generalize the classical solution concept, is to regularize 
the right-hand side $a$ to obtain a net of smooth functions $(A_{\varepsilon})_{\varepsilon \in ]0,1]}$ 
with $\lim_{\varepsilon \rightarrow 0}\langle A_{\varepsilon}, \varphi \rangle = \langle a, \varphi \rangle$ 
for all $\varphi \in\mathcal{D}(\R^n)$.
Then a net of solutions $(\xi_{\varepsilon})_{\varepsilon}$ is obtained by solving
the family of ordinary differential equations
\begin{eqnarray} \label{ode_net}
\dot{\xi_{\varepsilon}}(s) = A_{\varepsilon}(s, \xi_{\varepsilon}(s)), \ \xi_{\varepsilon}(t_{\varepsilon}) = x_{\varepsilon},
\end{eqnarray}
where $(t_{\varepsilon}, x_\varepsilon)_{\varepsilon}$ is converging net with $\lim_{\varepsilon \rightarrow 0} (t_{\varepsilon}, x_{\varepsilon}) = (t,x)$.

Such nets of solutions $(\xi_{\varepsilon})_{\varepsilon}$ have convergent subnets $(\xi_{\tau(\varepsilon)})_{\varepsilon}$ with $\tau \in \mathcal{T}$ and $\xi_{\tau(\varepsilon)} \stackrel{\varepsilon \rightarrow 0}{\rightarrow} \xi \in C(J)^n$ uniformly on compact subsets of $J$, when the net of smooth functions $(A_{\varepsilon})_{\varepsilon}$ satisfies
certain boundedness assumptions. The generalized solution $\xi$ does in general not solve any differential equation, 
but it will turn out that $\xi$ solves a differential inclusion. So obviously these two solution concepts are closely related.


\subsection{The Filippov solution concept} \label{filippov_solution_concept}
If $a\in L^1_{\rm loc}(J; L^{\infty}(\R^n))^n$ we can consider its
essential convex hull $A_{t,x}$ for almost all $t\in J$ and all $x\in \R^n$.
We show that $A: (t,x) \mapsto A_{t,x}$ determines a set-valued map, satisfying 
the conditions (FC).
Furthermore the essential convex hull $A_{t,x}$ is equal to $\{a(t,x)\}$ for almost
all $t\in J$ and all $x\in \R^n$, if $a$ satisfies the Caratheodory conditions.

By Theorem \ref{filippov_existence_theorem} there exists a solution $\xi$ of differential
inclusion
$$
\odie,
$$
which we call \emph{Filippov (generalized) solution} of the ordinary \index{Filippov generalized solution}
differential equation $\dot{\xi} = a(s, \xi(s))$ when $(t,x) \mapsto A_{t,x}$ denotes the essential convex hull of $a$.

This type of generalized solution concept for ordinary differential equations with discontinuous right-hand side
was presented in \cite{Filippov:60}, \cite[Chapter 2]{AubinCellina:84}, and \cite[Chapter 1.4]{Hoermander:97}.


\begin{definition} \label{essential_convex_hull} \index{essential convex hull}
Let $a \in L^{1}_{\rm loc}(J; L^{\infty}(\R^n))^n$ and define the essential supporting function for $a$ by
$$
H_a(t,x, w) := \lim_{\delta \rightarrow 0} \esup_{y \in B_{\delta}(x)}{\langle a(t,y), w \rangle}. 
$$
which is defined for almost all $t \in J$ and all $x,w\in \R^n$.

The generated set-valued map $A: (t,x)\mapsto A_{t,x}$ obtained by
$$
A_{t,x} := \{ y\in \R^n \mid \langle y, w \rangle \le H_a(t,x,w), w \in \R^n \}.
$$
is called \emph{essential convex hull} of $a$.
\end{definition}


\begin{proposition}
The essential convex hull of any function $a \in L^{1}_{\rm loc}(J; L^{\infty}(\R^n))^n$ satisfies (FC).
\end{proposition}

\begin{proof}
By Definition \ref{essential_convex_hull} it is obvious that $w \mapsto H_a(t,x,w)$ 
is positively homogeneous and convex in $w$ for almost all $t\in J$ and all $x\in\R^n$. 

We put 
$$
h_{\delta}(t,x) := \esup_{y\in B_{\delta}(x)} \langle a(t,y),w \rangle 
$$
By \cite[Theorem 20.14]{HewStr:65} there exists a null set $N$, such that
$$
h_{\delta}(t,x) = \sup_{y\in B_{\delta}(x)/N} \langle a(t,y),w \rangle.
$$ 
Due to Lemma \ref{supremum_approximating_sequence} we can find a sequence 
$(x_k)_{k\in \N}$ in $B_{\delta}(x) / N$ such that
$$
h_{\delta}(t,x,w) = \limsup_{k \rightarrow \infty} \langle a(t,x_k),w \rangle,
$$
thus $(t,x) \mapsto h_{\delta}(t,x,w)$ is measurable as the pointwise 
limes superior of a family of measurable functions. 

Observe that $h_{\delta_1}(t,x,w) \le h_{\delta_2}(t,x,w)$ for $\delta_2 \ge \delta_1$,
so $(h_{\delta}(t,x,w))_{\delta \in ]0,1]}$ is a decreasing net, bounded for almost all $t\in J$.
For $y\in B_{\delta}(x)/N$ it holds that
$h_{\delta}(t,y,w) \ge \langle a(t,x),w \rangle$, thus $h_{\delta}(t,y,w)$ is bounded from below for almost all $t\in J$.
It follows that
$$
\lim_{\delta \rightarrow 0} h_{\delta}(t,x,w) = \inf_{\delta \in ]0,1]} h_{\delta}(t,x,w) 
$$
and since the pointwise infimum of a net of upper semi-continuous functions is again
upper semi-continuous (see \cite[Exercise 7.23 (a)]{HewStr:65}), we have that
$x \mapsto H_a(t,x,w)$ is upper semi-continuous for almost all $t\in J$ and all $w\in \R^n$.

We have
$$
H_a(t,x,w/|w|) \le h_1(t,x,w/|w|) \le \esup_{y \in \R^n } |a(t,y)| \in L^1_{\rm loc}(J)
$$
since $a \in  L^1_{\rm loc}(J; L^\infty(\R^n))^n$.
\end{proof}


\begin{theorem}[Approximation property of the essential convex hull] \label{filippov_approximation_property}
Let $a \in L^{1}_{\rm loc}(J; L^{\infty}(\R^n))^n$ and let $\rho$ be a positive mollifier in $\mathcal{D}(\R^n)$ with
$\int \rho(z) \m{z}=1$. Put $\rho_{\varepsilon}(x) = \varepsilon^{-n} \rho(x/\varepsilon)$.

Then the approximating net $a_{\varepsilon}(s,x) = a(s,\cdot) \ast \rho_{\varepsilon}$ has the property, that if $(\xi_{\varepsilon})_{\varepsilon}$ is a net 
of smooth functions converging to some $\xi \in C(J)^n$ uniformly on compact sets, then it holds
that 
$$
\limsup_{\varepsilon \rightarrow 0} 
\langle a_{\varepsilon}(s,\xi_{\varepsilon}(s)), w \rangle \le  H_a(s,\xi(s),w)
$$
for almost all $s\in J$.
\end{theorem}

\begin{proof}
Let $R > 0$ be such that $\supp{\rho} \subseteq B_{R}(0)$.  
Then we observe that 
$$
\sup_{\mu \in ]0, \varepsilon]} \langle a_{\mu}(s,\xi_{\mu}(s)), w \rangle 
= \sup_{\mu \in ]0, \varepsilon]} \int_{\R^n} \langle a(s,y) ,w \rangle \rho_{\mu}(\xi_{\mu}(s)-y) \m{y} 
\le \sup_{\mu \in ]0, \varepsilon]} \sup_{y \in B_{R \cdot \mu}(\xi_{\mu}(s))} \langle a(s,y) ,w \rangle 
 < \infty.
$$
for almost all $s\in J$.

Let $K\csub J$ fixed.

Then for all $\varepsilon > 0$ there exists 
$\mu'(\varepsilon)\in ]0,\varepsilon/(2R)]$  such 
that $|\xi_{\mu}(s)-\xi(s)|< \varepsilon/2$ for all $s\in K$ and 
$\mu \in]0,\mu'(\varepsilon)]$.
It follows that 
$$B_{R \cdot \mu }(\xi_{\mu}(s)) \subseteq 
B_{R \cdot \mu }(\xi(s)) + B_{\varepsilon/2}(\xi(s)) =
B_{R \cdot \mu+\varepsilon/2 }(\xi(s)) \subseteq B_{\varepsilon}(\xi(s)) 
$$
for all $s\in K$ and $\mu \in ]0, \mu'(\varepsilon)]$, implying 
$$
\sup_{\mu \in ]0, \mu'(\varepsilon)]} 
\sup_{y \in B_{R \cdot \mu}(\xi_{\mu}(s))} \langle a(s,y) ,w \rangle
\le 
\sup_{y \in B_{\varepsilon}(\xi(s))} \langle a(s,y) ,w \rangle. 
$$
Observe that 
$
\left(\sup_{\mu \in ]0, \varepsilon]} \langle a_{\mu}(s,\xi_{\mu}(s)), w \rangle \right)_{\varepsilon}
$
is a decreasing, bounded net for almost all $s\in J$, thus converging.
We conclude that any subnet of the net 
$(\sup_{\mu \in ]0,\varepsilon]} \langle a_{\mu}(s, \xi_{\mu}(s)),w \rangle)_{\varepsilon}$, 
has the same limit, thus
$$
\lim_{\varepsilon \rightarrow 0} \sup_{\mu \in ]0,\varepsilon]} \langle a_{\mu}(s, \xi_{\mu}(s)),w \rangle
=
\lim_{\varepsilon \rightarrow 0} \sup_{\mu \in ]0,\mu'(\varepsilon)]} \langle a_{\mu}(s, \xi_{\mu}(s)),w \rangle.
$$
Finally we obtain
\begin{multline*}
\limsup_{\varepsilon \rightarrow 0} \langle a_{\varepsilon}(s, \xi_{\varepsilon}(s)),w \rangle =
\lim_{\varepsilon \rightarrow 0} \sup_{\mu \in ]0,\varepsilon]} \langle a_{\mu}(s, \xi_{\mu}(s)),w \rangle
=
\lim_{\varepsilon \rightarrow 0} \sup_{\mu \in ]0,\mu'(\varepsilon)]} \langle a_{\mu}(s, \xi_{\mu}(s)),w \rangle
\\\le
\lim_{\varepsilon \rightarrow 0} \sup_{y \in B_{\varepsilon}(\xi(s))} \langle a(s,y) ,w \rangle 
=
H_a(s,\xi(s),w)
\end{multline*}
for almost all $s\in K$ and all $w\in \R^n$.
\end{proof}


\begin{remark}
Theorem \ref{filippov_approximation_property} shows that a Filippov generalized solution for an ordinary differential equation with
discontinuous right-hand side  $a\in L^{1}_{\rm loc}(J; L^{\infty}(\R^n))^n$ can be obtained by regularizing 
the right-hand side by convolution with a positive, smooth and compactly supported mollifier (instead of applying the existence theorem \ref{filippov_existence_theorem} for differential inclusions).
Such an approach for constructing Filippov generalized solutions is pursued in \cite[Chapter 1.4]{Hoermander:97}.

Putting $a_{\varepsilon}(s,x) = a(s,\cdot) \ast \rho_{\varepsilon}$ we have that any
net of regular functions $(\xi_{\varepsilon})_{\varepsilon}$ solving
$$
\dot{\xi_{\varepsilon}} = a_{\varepsilon}(s,\xi_{\varepsilon}(s)),\quad \xi_{\varepsilon}(t) = x
$$
has the property that 
\begin{multline*}
\limsup_{\varepsilon \rightarrow 0} \langle \xi_{\varepsilon}(s), w\rangle 
= \langle x,w\rangle + \limsup_{\varepsilon \rightarrow 0} \int_{t}^s \langle a_{\varepsilon}(\tau,\xi_{\varepsilon}(\tau)), w \rangle \m{\tau}  \\
\le
 \langle x,w\rangle + \int_{t}^s \limsup_{\varepsilon \rightarrow 0} \langle a_{\varepsilon}(\tau,\xi_{\varepsilon}(\tau)), w \rangle \m{\tau} 
\le  \langle x,w\rangle +\int_t^s H_a(\tau,\limsup_{\varepsilon \rightarrow 0}  \xi_{\varepsilon}(\tau),w) \m{\tau}
\end{multline*}
for all $w\in \R^n$, which implies that any subnet $(\xi_{\tau(\varepsilon)})_{\varepsilon}$ (with $\tau\in\mathcal{T}$)
converging uniformly on compact subsets to some $\xi \in C(J)^n$, satisfies the differential inclusion
$$
\dot{\xi}(s) \in A_{s,\xi(s)}, \quad \xi(t)=x,
$$
where $A$ denotes the essential convex hull. 
\end{remark}


\begin{theorem} \label{filippov_forward_uniqueness}
Let $a \in L^{1}_{\rm loc}(J; L^{\infty}(\R^n))^n$ satisfy the forward uniqueness condition
\begin{equation} 
\langle x-y,  a(s,x) - a(s,y) \rangle  \le \alpha(s) |x-y|^2 
\end{equation}
resp. the backward uniqueness condition
\begin{equation}
 \langle x-y,  a(s,x) - a(s,y) \rangle  \ge - \alpha(s) |x-y|^2,
\end{equation}
for some positive $\alpha \in L^1_{\rm loc}(J)$,
then the supporting function of the essential convex hull $H_f$
satisfies the forward uniqueness condition (\ref{set_valued_function_forward_uniqueness_condition})
resp. backward uniqueness condition (\ref{set_valued_function_backward_uniqueness_condition}).
\end{theorem}

\begin{proof}
We prove only the case where $a$ satisfies the forward-uniqueness condition.
\begin{multline*}
\langle a(s,z_1), x-y \rangle - \langle a(s,z_2), x-y \rangle
=
\langle a(s,z_1), z_1-z_2 \rangle + \langle a(s,z_2), z_2-z_1 \rangle +
2 \beta(s) (|z_1 - x|+|z_2 - y|)
\\
\le 
|z_1-z_2|^2 \alpha(s) +  4 \delta \beta(s) 
\end{multline*}
for all $z_1 \in B_{\delta}(x), z_2 \in B_{\delta}(y)$ and almost all $s\in J$. It follows that 
\begin{multline*}
\esup_{z\in B_{\delta}(x)} \langle a(s,z), x-y \rangle + \esup_{z\in B_{\delta}(y)} \langle a(s,z), y-x \rangle
\le \sup_{(z_1,z_2)\in B_{\delta}(x) \times  B_{\delta}(y)} |z_1-z_2|^2 \alpha(s) +  4 \delta \beta(s) \\
= |x-y|^2 \alpha(s) + 2 \delta \alpha(s) +  4 \delta \beta(s), 
\end{multline*}
thus 
\begin{multline*}
 H_a(s,x, x-y) + H_a(s,y, y-x) = 
\lim_{\delta \rightarrow 0}
(\esup_{z\in B_{\delta}(x)} \langle a(s,z), x-y \rangle + \esup_{z\in B_{\delta}(y)} \langle a(s,z), y-x \rangle)
\\ \le |x-y|^2 \alpha(s)
\end{multline*}
holds for almost all $s\in J$.
\end{proof}


\subsection{Colombeau solutions for ordinary differential equations}


Finally we consider generalized solutions for ordinary differential equation
in the setting of Colombeau generalized functions. 

Let $(\widetilde{t},\widetilde{x}) \in \widetilde{J \times \R^n}$ be the initial condition, then
we study the ordinary differential equation
\begin{eqnarray*}
\dot{\xi}(s) &=& A(s,\xi(s)), \quad \xi(\widetilde{t})=\widetilde{x},
\end{eqnarray*}
where $A:=[(A_{\varepsilon})_{\varepsilon}] \in \colmap{J\times \R^n}{\R^n}$ such that there exists some representative satisfying the bound
\begin{eqnarray} \label{colombeau_existence_bound}
\sup_{x\in\mathbb{R}^n}|A_{\varepsilon}(t,x)| \le  \beta(t) ,\ \ \varepsilon \in\, ]0,1], \ \text{almost everywhere in}\ t\in J
\end{eqnarray}
for some positive function $\beta \in L^1_{\rm loc}(J)$.
It will turn out that the generalized solution $\xi:=[(\xi_{\varepsilon})_{\varepsilon}]$ then allows 
for picking a subnet $(\xi_{\tau(\varepsilon)})_{\varepsilon}$ converging uniformly on compact subsets of 
$J$ to some $\zeta \in AC(J)^n$ with $\zeta(t)=x$. 


\begin{theorem}[Existence] \label{colombeau_existence_theorem} \index{Colombeau generalized solution}
Let $A \in \colmap{J \times \R^n}{\R^n}$ be Colombeau map of natural type 
with a representative $(A_{\varepsilon})_{\varepsilon}$ satisfying (\ref{colombeau_existence_bound}).

If $(\widetilde{t},\widetilde{x}) \in \widetilde{J \times \R^n}$, then there exists a c-bounded solution $\xi \in \G(J)^n$ to the initial value problem
\begin{eqnarray*}
\dot{\xi}(s) &=& A(s,\xi(s)), \quad \xi(\widetilde{t})=\widetilde{x}.
\end{eqnarray*}
Furthermore, there exists some $(t,x) \in \overline{J}\times \Omega$, $\zeta \in AC(J)^n$ such that for any representative $(\xi_{\varepsilon})_{\varepsilon}$ of $\xi$, $(\widetilde{t}_\varepsilon, \widetilde{x}_{\varepsilon})_{\varepsilon}$ of $(\widetilde{t},\widetilde{x})$ there exists subnets $(\widetilde{t}_{\tau(\varepsilon)}$, $\widetilde{x}_{\tau(\varepsilon)})_{\varepsilon}$,$(\xi_{\tau(\varepsilon)})_{\varepsilon}$ with $\tau \in \mathcal{T}$ and
$\lim_{\varepsilon \rightarrow 0} (\widetilde{t}_{\tau(\varepsilon)}, \widetilde{x}_{\tau(\varepsilon)}) =(t,x)$
 and $\xi_{\tau(\varepsilon)} \stackrel{\varepsilon \rightarrow 0}{\rightarrow} \zeta$ uniformly on compact sets of $J$.

It satisfies the integral inequality
\begin{equation} \label{colombeau_solution_pseudo_shadow} 
\langle \zeta(s),w \rangle \le \langle x, w\rangle + \int_t^s H_A(\tau, \zeta(\tau), w) \m{\tau}
\end{equation}
for all $w\in \R^n$, where $(x,w) \mapsto H_A(t,x,w)$ denotes the supporting function of 
the generalized graph $\ggraph{A(t,\cdot)}$.
\end{theorem}

\begin{proof}
Let $(A_{\varepsilon})_{\varepsilon}$ be a representative of $A$ and let $(t_{\varepsilon}, x_{\varepsilon})_{\varepsilon}$ be a representative of $(\widetilde{t},\widetilde{x})$.
By classical existence and uniqueness we obtain a solution $\xi_{\varepsilon}$ for each $\varepsilon \in ]0,1]$ such that
\begin{eqnarray*}
\xi_{\varepsilon}(s) &=& x_{\varepsilon} + \int_{t_{\varepsilon}}^s A_{\varepsilon}(\tau,\xi_{\varepsilon}(\tau)) \m{\tau} 
\end{eqnarray*}
holds.  (\ref{colombeau_existence_bound}) yields $|\xi_{\varepsilon}(s)| \le |x_{\varepsilon}| + \int_{t_\varepsilon}^s \beta(\tau) \m{\tau}$ for all
$s\in J$, hence $(\xi_{\varepsilon})_{\varepsilon}$ is bounded on compact subsets  of $J$. 

Since $(t_{\varepsilon}, x_{\varepsilon})_{\varepsilon}$ is bounded we have that for all $s\in J$ there exists some $C>0$ such that $|\xi_{\varepsilon}(s)| \le C$ for all $\varepsilon \in ]0,1]$, thus $(\xi_{\varepsilon})_{\varepsilon}$ is an equi-bounded family. The estimate
\begin{eqnarray*}
|\xi_{\varepsilon}(s) - \xi_{\varepsilon}(r)| \le  \int_{\min{(r,s)}}^{\max{(r,s)}} \beta(\tau) \m{\tau} \quad s,r \in J, \varepsilon \in ]0,1].
\end{eqnarray*}
yields that the family $(\xi_{\varepsilon})_{\varepsilon}$  is  equi-continuous.

The theorem of Arzela-Ascoli yields a subnet $(\xi_{\tau(\varepsilon)})_{\varepsilon}$ with $\tau \in \mathcal{T}$ converging to some $\zeta \in C(J)^n$ uniformly on compact sets.
Without loss of generality we can assume that $(t_{\tau(\varepsilon)},x_{\tau(\varepsilon)})_{\varepsilon}$ 
converges to some $(t,x) \in \overline{J} \times \R^n$ with $\zeta(t)=x$.

We have for all $s,r\in J$,
\begin{multline*}
|\zeta(s) - \zeta(r)| \le |\zeta(s) - \xi_{ \tau(\varepsilon) }(s)|+ |\xi_{ \tau(\varepsilon) }(s)-\xi_{ \tau(\varepsilon) }(r)| +|\xi_{ \tau(\varepsilon) }(r) - \zeta(r)|  \\
\le |\zeta(s) - \xi_{ \tau(\varepsilon)  }(s)| +\int_{s}^{r} \beta(\tau) \m{\tau} +|\xi_{\tau(\varepsilon)} (r) - \zeta(r)| \stackrel{\varepsilon \rightarrow 0}{\rightarrow}\int_{s}^{r} \beta(\tau) \m{\tau} , 
\end{multline*}
hence $\zeta$ is absolutely continuous on $J$.

Note that the equi-continuity of $(\xi_{\varepsilon})_{\varepsilon}$ implies its c-boundedness. Then
the moderateness of $(\xi_{\varepsilon})_{\varepsilon}$ follows by
the same argument as in the proof of \cite{GKOS:01}[Proposition 1.2.8], since
the derivative $\dot{\xi_{\varepsilon}}(s)$ may be written as the composition of the moderate net $A_{\varepsilon}$
and the c-bounded net $(s, \xi_{\varepsilon}(s))$.

By Theorem \ref{colombeau_supporting_function_proposition} we obtain that
$A$ induces a supporting function satisfying (FC). By the theorem of dominated convergence
\cite[Theorem 12.24]{HewStr:65} and Theorem
\ref{supporting_function_approximation_property} it follows that 
\begin{multline*}
\langle \zeta(s), w \rangle = \lim_{j \rightarrow \infty} \langle \xi_{\tau(\varepsilon)}, w \rangle \le 
 \langle x,w\rangle + \limsup_{j \rightarrow \infty} \int_t^s A_{\tau(\varepsilon)}(\tau, \xi_{\tau(\varepsilon)}(\tau)) \m{\tau} \\
\le 
\langle x,w\rangle + \int_t^s \limsup_{j \rightarrow \infty}  A_{\tau(\varepsilon)}(\tau, \xi_{\tau(\varepsilon)}(\tau)) \m{\tau} \\
\le 
\langle x,w\rangle + \int_t^s H_{A}(\tau, \zeta(\tau), w ) \m{\tau} 
\end{multline*}
for all $w\in\R^n$.
\end{proof}


\begin{remark} \label{colombeau_solution_remark}
By Proposition \ref{generalized_graph_equicontinuous_proposition} the equi-continuity of 
$(\xi_{\varepsilon})_{\varepsilon}$ yields that
$$
\ggraph{\xi} = \bigcup_{s\in J} \{s\} \times \cp{(\xi_{\varepsilon}(s))_{\varepsilon}}.
$$
Then $\zeta \in AC(J)^n$ as in (\ref{colombeau_solution_pseudo_shadow})
is a selection of the set-valued map $s \mapsto \ggraph{\xi}_s$, i.e. $\zeta(s) \in \ggraph{\xi}_s$ for all $s\in J$.

If $(r,z) \in \ggraph{\xi}$ we can find some $\tau \in \mathcal{T}$ such that 
$\lim_{\varepsilon \rightarrow 0} \xi_{\tau(\varepsilon)}(r) = z$. 

Then the Theorem of Arzela-Ascoli yields that it is possible to
choose a subnet of $(\xi_{\tau(\varepsilon)})_{\varepsilon}$ 
converging uniformly on compact sets to some $\zeta \in AC(J)^n$ with $\zeta(r)=z$ and $\zeta(t)=x$.

This shows that the $\ggraph{\xi}$ is the union of $\graph{\zeta}$ of all possible $\zeta\in AC(J)^n$ as in
Theorem \ref{colombeau_existence_theorem}. We can reformulate
property (\ref{colombeau_solution_pseudo_shadow}) to obtain
\begin{equation} \label{hs_graph}
\ggraph{\xi}_s \subseteq \{x\}+ \int_t^s \ch{\ggraph{A} \circ ( \{\tau\} \times \ggraph{\xi}_{\tau} )} \m{\tau}
\end{equation}
where "$\circ$" denotes the composition of set-valued maps (cf. Theorem \ref{set_valued_composition_theorem})
and the integral is according to Definition \ref{set_valued_integral}.
\end{remark}


\begin{example}[Hurd-Sattinger characteristic flow] \label{hs_flow_example} \index{Hurd-Sattinger!characteristic flow}
Consider the ordinary differential equation: 
$$
\dot{\xi}(s) = \Theta(-\xi(s)) ,\quad \xi(t)=x,
$$
where $\Theta = [(\Theta_{\varepsilon})_{\varepsilon}]\in \cf{\R}$  is defined by $\Theta_{\varepsilon}(x) =  \rho_{\varepsilon} \ast H(-\cdot)$
with $\rho_{\varepsilon} = \frac{1}{\varepsilon}\rho (\cdot /  \varepsilon)$  
for some $\rho \in \mathcal{S}(\mathbb{R})$ with $\int \rho(x) \m{x}=1$. 

We put $g(x):= \int_x^{\infty} \rho(z) \m{z}$ and denote $\overline{\ran{g}} = [\alpha_{-},\alpha_{+}]$ for some $\alpha_{-},\alpha_{+} \in \R$.
Note that $\lim_{x \rightarrow -\infty}g(x) = 1$ and $\lim_{x \rightarrow \infty}g(x) = 0$ implies $\alpha_{-}\le 0$ and 
$\alpha_{+}\ge 1$. It holds that $\alpha_{-}=0$ and $\alpha_{+}=1$ if $\rho \ge 0$.

In our example we allow $\rho$ to have non-vanishing moments, 
so $\Theta$ is not necessarily the embedding of the Heaviside function.

Nevertheless $(\Theta_{\varepsilon})_{\varepsilon}$ converges to the Heaviside function in $L^1_{\rm loc}(\R)$
and since $\sup_{x\in\R} |\Theta_{\varepsilon}(-x)| < \sup_{x\in\R} |g(x/\varepsilon)| \le \alpha_+$ it satisfies
(\ref{colombeau_existence_bound}). 

Due to Theorem \ref{colombeau_existence_theorem} there exists a c-bounded Colombeau solution $\xi = [(\xi_{\varepsilon})_{\varepsilon}]$, such that
there exists a converging subnet with $\lim_{\varepsilon \rightarrow 0} \xi_{\tau(\varepsilon)} \rightarrow \zeta$
uniformly on compact set for some $\zeta \in \AC(\R)^n$. 

As in Example \ref{heaviside_ggraph_example} we obtain that the generalized graph of $\Theta$ is
$$
\ggraph{\Theta(-\cdot)} := \left(\{x\in\mathbb{R}\mid x<0\} \times \{1\}\right) \cup \left(\{0\} \times \overline{\ran{g}} \right) \cup \left(\{x\in\mathbb{R}\mid x>0\} \times \{0\}\right).
$$
Using its support function (Example \ref{supportfunction_heaviside_example}) the property 
(\ref{colombeau_solution_pseudo_shadow}) of $\zeta$ translates to 
\begin{eqnarray*}
x + \int_t^s 1_{\zeta(\tau) < 0}(\tau) \m{\tau} \le \zeta(s) \le x + \int_t^s 1_{\zeta(\tau) < 0}(\tau) \m{\tau}.
\end{eqnarray*}
Then it is straight-forward to derive that 
\begin{equation} \label{hs_shadow}
\zeta(s;t,x) \in \left\{
\begin{array}{ll}
\{\min{(x+ s-t,0)}\} & x < 0 \\
\{0\} & x = 0,  s \ge  t \\  
\left[s-t,0\right] & x = 0, s \le  t \\  
\{x\} & x > 0.
\end{array}\right.
\end{equation}
holds, which implies that $\zeta$ is forward unique. It follows that 
$$
\lim_{\varepsilon \rightarrow }\sup_{s\in K\cap[t,+\infty[} |\xi_{\varepsilon}(s)-\zeta(s)|=0
$$
for all $K \csub \R$. 

Note that the support function of $\ggraph{\Theta(-\cdot)}$ does not satisfy the forward uniqueness condition 
(\ref{set_valued_function_forward_uniqueness_condition}) if $\alpha_{+}>1$ or $\alpha_{-}<0$, nevertheless 
$\zeta$ is forward unique due to (\ref{hs_shadow}) regardless of the values $\alpha_{-},\alpha_{+}$.

By Remark \ref{colombeau_solution_remark} we obtain that the Colombeau solution $\xi$ of the Hurd-Sattinger equation
has the property that 
$$
\ggraph{\xi}_s \subseteq \{x\} + \int_t^s \ch{\ggraph{\Theta} \circ (\{\tau\} \times \ggraph{\xi}_\tau)} \m{\tau} =
\{x\} + \int_t^s 1_{\ggraph{\xi}_\tau<0} (\tau)  \m{\tau} 
$$
which in (\ref{hs_shadow}) implies
$$
\ggraph{\xi}_s \subseteq \left\{
\begin{array}{ll}
\{\min{(x+ s-t,0)}\} & x < 0 \\
\{0\} & x = 0,  s \ge  t \\  
\left[s-t,0\right] & x = 0, s \le  t \\  
\{x\} & x > 0.
\end{array}\right.
$$
\end{example}


\begin{theorem}[Forward uniqueness of the distributional shadow] \label{colombeau_uniqueness_theorem} \index{distributional shadow!forward uniqueness}
Let $A  \in \G(\overline{\Omega_T})^n$ with representative $(A_{\varepsilon})_{\varepsilon}$ 
satisfying (\ref{colombeau_existence_bound}) and in addition the forward uniqueness condition
\begin{eqnarray} \label{colombeau_forwarduniqueness_condition}
\langle A_{\varepsilon}(s,x)-A_{\varepsilon'}(s,y), x-y \rangle \le \alpha(s) |x-y|^2 + \gamma_{\varepsilon, \varepsilon'}(s)  
\end{eqnarray}
resp. the backward uniqueness condition
\begin{eqnarray} \label{colombeau_backwarduniqueness_condition}
\langle A_{\varepsilon}(s,x)-A_{\varepsilon'}(s,y), x-y \rangle \ge -\alpha(s) |x-y|^2 - \gamma_{\varepsilon, \varepsilon'}(s)  
\end{eqnarray}
holds for some positive functions $\alpha,\gamma_{\varepsilon, \varepsilon'}\in L^1([0,T])$ such that 
$\int_0^T \gamma_{\varepsilon, \varepsilon'}(\tau) \m{\tau} =O(\varepsilon+\varepsilon')$ as 
$\varepsilon,\varepsilon' \rightarrow 0$. 

Then the supporting function $H_A$ of the generalized graph $\ggraph{A(s,\cdot)}$ satisfies the forward uniqueness property
$$
H_A(s,x,x-y)+H_A(s,y,y-x) \le \alpha(s) |x-y|^2 
$$
resp.
$$ 
H_A(s,x,x-y)+H_A(s,y,y-x)\ge - \alpha(s) |x-y|^2).
$$
\end{theorem}

\begin{proof}
By the proof of Theorem \ref{generalized_graph_supporting_function_proposition} we can write
$$
H_A(s,x,x-y) = \lim_{\varepsilon \rightarrow 0} 
\langle A_{\tau_1(\varepsilon)}(x_{\varepsilon}), x-y \rangle
$$
and 
$$
H_A(s,y,y-x) = \lim_{\varepsilon \rightarrow 0} \langle 
A_{\tau_2(\varepsilon)}(y_{\varepsilon}), y-x \rangle
$$
for fixed $x,y \in \R^n$ and
for some $\tau_1, \tau_2 \in\mathcal{T}$ and nets 
$(x_{\varepsilon})_{\varepsilon},(y_{\varepsilon})_{\varepsilon}$
with $x_{\varepsilon} \rightarrow x$, $y_{\varepsilon} \rightarrow y$. 
For almost all $s\in J$ and all $x,y \in \R^n$ we have
\begin{multline*}
H_A(s,x,x-y)+H_A(s,y,y-x) =  \lim_{\varepsilon \rightarrow 0}  
(\langle A_{\tau_1(\varepsilon)}(s,x_{\varepsilon}), x-y \rangle + \langle A_{\tau_2(\varepsilon)}(s,y_{\varepsilon}), y-x \rangle) 
\\
\le
 \lim_{\varepsilon \rightarrow 0}  
(\langle A_{\tau_1(\varepsilon)}(s,x_{\varepsilon}), x_{\varepsilon} - y_{\varepsilon} \rangle 
- \langle A_{\tau_2(\varepsilon)}(s,y_{\varepsilon}),x_{\varepsilon} - y_{\varepsilon} \rangle 
+\beta(s)  ( |x- x_{\varepsilon}|+ |y- y_{\varepsilon}|) ) 
\\
\le
  \lim_{\varepsilon \rightarrow 0} (\alpha(s) |x_{\varepsilon} - y_{\varepsilon}|^2 +
\beta(s)  (|x- x_{\varepsilon}|+ |y- y_{\varepsilon}|)  
+ \gamma_{\tau_1(\varepsilon), \tau_2(\varepsilon)}(t) )  
\le
\alpha(t) |x - y|^2.
\end{multline*}
The proof for the backward uniqueness condition is analogous.
\end{proof}


\begin{remark}
By means of Theorems \ref{colombeau_uniqueness_theorem} it immediately follows that the 'sub-shadow' $\zeta$
of the Colombeau solution $\xi = [(\xi_{\varepsilon})_{\varepsilon}] \in  \cf{J}^n$ (as obtained by Theorem \ref{colombeau_existence_theorem}), is forward unique resp. backward unique if
the right-hand side $A:=[(A_{\varepsilon})_{\varepsilon}] \in \colmap{J \times \R^n}{\R^n}$
has the property that the supporting function of $\ggraph{A(t,\cdot)}$ satisfies the forward uniqueness condition \ref{colombeau_forwarduniqueness_condition}
resp. the backward uniqueness condition \ref{colombeau_backwarduniqueness_condition}.
\end{remark}


\begin{example}\label{colombeau_solution_uniqueness_example}
Let $\widetilde{a}\in {L^1(\R, L^{\infty}(\mathbb{R}^n))}^n$ 
satisfy the classical forward uniqueness condition 
\begin{eqnarray} \label{limit_solution_forward_uniqueness_example}
\langle \widetilde{a}(t,x) - \widetilde{a}(t,y), x-y  \rangle \le 
\widetilde{\alpha}(t) |x-y|^2 \ \ \text{for almost all}\ (t,x), (t,y) \in \R^{n+1}
\end{eqnarray} 
for some positive $\widetilde{\alpha} \in L^1(\R)$.

Assume $\rho \in\mathcal{D}(\mathbb{R}^{n+1})$ positive, $\int_{\mathbb{R}^n} \rho(z) \m{z} = 1$ and define $\rho_{\varepsilon}(t,x):=\varepsilon^{-n-1} \rho(t /\varepsilon,x/\varepsilon)$.  
Consider the Colombeau function $A:=[(A_{\varepsilon})_{\varepsilon}] \in \colmap{\R^{n+1}}{\R^n}$ defined by
\begin{eqnarray*} 
A_{\varepsilon} := \widetilde{a} \ast \rho_{\varepsilon},
\end{eqnarray*}
then $[(A_{\varepsilon})_{\varepsilon}]$ satisfies the condition (\ref{colombeau_existence_bound}) for the existence 
theorem and the forward uniqueness condition (\ref{colombeau_forwarduniqueness_condition}).
\end{example}

\begin{proof}
 Since $\widetilde{a} \in L^{1}(\mathbb{R}, L^{\infty}(\mathbb{R}^n))^n$ it holds that for almost all $t \in \mathbb{R}$, we have
 $\widetilde{\beta}(t):= \sup_{z\in \mathbb{R}^n} |\widetilde{a}(t,z)|$ where $\widetilde{\beta}$ is some positive function in $L^1(\mathbb{R})$. 
 First, observe that
 \begin{multline} \label{colombeau_example_1}
  \sup_{z\in \mathbb{R}^n}  |A_{\varepsilon}(t, z)|   =   
  \sup_{z \in \mathbb{R}^n} \left|\int_{\mathbb{R}}\int_{\mathbb{R}^n}   a(t-\varepsilon \tau, z-\varepsilon y) \rho(\tau,y) \m{y}  \m{\tau}\right|  \\
  \le  \int_{\mathbb{R}} \widetilde{\beta}(t-\varepsilon \tau)   
\left(  \int_{\mathbb{R}^n} \rho(\tau,y) \m{y} \right) \m{\tau} 
 \le \beta(t) \ \ \text{for almost all}\  t\in\R
 \end{multline}
 where $\beta(t):= \sup_{\varepsilon\in]0,1]}\int_{\mathbb{R}} \widetilde{\beta}(t-\varepsilon s) \left(  \int_{\mathbb{R}^n} \rho(\tau,y) \m{y} \right) \m{\tau}$ is a positive function in $L^1(\R)$ and set $\varphi(s) := \int_{\mathbb{R}^n} \rho(s,y) \m{y} $. 
Hence $A_{\varepsilon}(t,x)$ satisfies (\ref{colombeau_existence_bound}).
 Using (\ref{limit_solution_forward_uniqueness_example}) 
and writing $A_{\varepsilon}$ explicitly as convolution integral, a 
 straight-forward calculation and estimation gives
 \begin{multline} \label{expression_colombeau_forwarduniqueness}
 \langle A_{\varepsilon}(t,x) - A_{\varepsilon'}(t,y), x-y\rangle \\ \le 2 \widetilde{\alpha}(t)  |x-y|^2  + 2 C_1 \widetilde{\alpha}(t) (\varepsilon+\varepsilon')^2 + 2 C_2  \sup_{z\in\mathbb{R}^n}{|\widetilde{a}(t,z)|} (\varepsilon+\varepsilon') + r_{\varepsilon, \varepsilon'}(t) \frac{1}{2}(1+|x-y|^2)
 \end{multline}
 where $C_1, C_2 $ are positive constants and 
$r_{\varepsilon,\varepsilon'}(t) := \lambda_{\varepsilon}(t)-\lambda_{\varepsilon'}(t)$ with
 \begin{eqnarray*}
 \lambda_{\varepsilon}(t) = \frac{1}{\varepsilon|x-y|} \int_{\mathbb{R}^n} \int_{\mathbb{R}} \langle \widetilde{a}(s, x-\varepsilon z) - \widetilde{a}(t,x-\varepsilon z) , x-y \rangle \rho(t-s/\varepsilon,z)  \m{s} \m{z} .
 \end{eqnarray*} 
 One easily verifies that
 \begin{eqnarray} \label{expression_lambda_estimate}
 \int_0^T \lambda_{\varepsilon}(t) \m{t} = O(\varepsilon),
 \end{eqnarray}
 hence $\int_{0}^T r_{\varepsilon,\varepsilon'}(\tau) \m{\tau} 
\le \int_{0}^T \lambda_{\varepsilon}(\tau) \m{\tau}+
\int_{0}^T \lambda_{\varepsilon'}(\tau) \m{\tau} = O(\varepsilon+\varepsilon')$. 
  
  The constants $C_1$ and $C_2$ are derived 
  from the estimates
  \begin{eqnarray*}
  \int \int |\varepsilon z-\varepsilon' z'|^2 \phi(z) \phi(z') \m{z} \m{z'} 
  &\le& \int \int ((\varepsilon z)^2 -2 \varepsilon \varepsilon' \langle z',  z \rangle +(\varepsilon' z')^2)  \phi(z) \phi(z') \m{z} \m{z'} \\
  &\le&
  \int \int ((\varepsilon z)^2 + \varepsilon \varepsilon' ((z')^2+ (z)^2) +(\varepsilon' z')^2)  \phi(z) \phi(z') \m{z} \m{z'}\\
  &=&
   2(\varepsilon +\varepsilon')^2  \int z^2 \rho(z) \m{z} =:  2(\varepsilon +\varepsilon')^2  C_1
  \end{eqnarray*}
  and 
  \begin{eqnarray*}
  \int \int |\varepsilon z-\varepsilon' z'| \phi(z) \phi(z') \m{z} \m{z'} 
  &\le& \int \int (|\varepsilon z| +|\varepsilon' z'|)  \phi(z) \phi(z') \m{z} \m{z'} \\
  &\le& (\varepsilon +\varepsilon')  \int |z| \rho(z) \m{z}  =:  (\varepsilon +\varepsilon')  C_2.
  \end{eqnarray*}
  Finally we can put (\ref{expression_colombeau_forwarduniqueness}) and (\ref{expression_lambda_estimate}) 
together to obtain
  \begin{multline*}
  \langle A_{\varepsilon}(t,x) - A_{\varepsilon'}(t,y), x-y\rangle \\ \le
  (2 \widetilde{\alpha}(t)+ \frac{1}{2}\gamma_{\varepsilon, \varepsilon'}(t)) |x-y|^2  + 2 C_1 \widetilde{\alpha}(t) (\varepsilon+\varepsilon')^2 + 2 C_2  \sup_{z\in\mathbb{R}^n}{|\widetilde{a}(t,z)|} (\varepsilon+\varepsilon') + \frac{1}{2} \gamma_{\varepsilon, \varepsilon'}(t) 
  \end{multline*}
 Setting
 \begin{eqnarray*}
 \alpha(t) &:=& (2 \widetilde{\alpha}(t)+ \frac{1}{2} \sup_{\varepsilon, \varepsilon' \in ]0,1]}r_{\varepsilon, \varepsilon'}(t))\\
 \gamma_{\varepsilon, \varepsilon'}(t)&:=& 2 C_1 \widetilde{\alpha}(t) (\varepsilon+\varepsilon')^2 + 2 C_2  \sup_{z\in\mathbb{R}^n}{|\widetilde{a}(t,z)|} (\varepsilon+\varepsilon') + \frac{1}{2} r_{\varepsilon, \varepsilon'}(t). 
 \end{eqnarray*}
 we obtain that $A_{\varepsilon}(t,x)$ satisfies (\ref{colombeau_forwarduniqueness_condition}). 
So the requirements for Theorem 
 \ref{colombeau_existence_theorem} and \ref{colombeau_uniqueness_theorem} are fulfilled.
\end{proof}
{"config": "arxiv", "file": "0806.1451/ode.tex"}