\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 142, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/142\hfil Continuous evolution of equations]
{Continuous evolution of equations and inclusions involving set-valued
contraction mappings with applications to generalized fractal transforms}

\author[H. Kunze, D. de La Torre, F. Mendivil, E. R. Vrscay\hfil EJDE-2014/142\hfilneg]
{Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay}  % in alphabetical order

\address{Herb Kunze \newline
Department of Mathematics and Statistics,
University of Guelph, Guelph, Ontario, Canada}
\email{hkunze@uoguelph.ca}

\address{Davide La Torre \newline
Department of Economics, Management, and Quantitative Methods,
University of Milan, Milan, Italy. \newline
Department of Applied Mathematics and Sciences,
Khalifa University, Abu Dhabi, UAE}
\email{davide.latorre@unimi.it, davide.latorre@kustar.ac.ae}

\address{Franklin Mendivil \newline
Department of Mathematics and Statistics, Acadia University,
Wolfville, Nova Scotia, Canada}
\email{franklin.mendivil@acadiau.ca}

\address{Edward R. Vrscay \newline
Department of Applied Mathematics, Faculty of Mathematics,
University of Waterloo, Waterloo, Ontario,  Canada}
\email{ervrscay@uwaterloo.ca}

\thanks{Submitted July 16, 2013. Published June 18, 2014.}
\subjclass[2000]{34A60, 28A80}
\keywords{Set-valued evolution inclusions, set-valued evolution equations,
\hfill\break\indent contractive set-valued functions, fixed points}

\begin{abstract}
 Let $T$ be a set-valued contraction mapping on a general Banach space
 $\mathcal{B}$. In the first part of this paper we introduce the evolution
 inclusion $\dot x + x \in Tx$ and study the convergence of solutions
 to this inclusion toward fixed points of $T$.
 Two cases are examined:  (i) $T$ has a fixed point $\bar y \in \mathcal{B}$
 in the usual sense, i.e., $\bar y = T \bar y$ and (ii) $T$ has a fixed
 point in the sense of inclusions, i.e., $\bar y \in T \bar y$.
 In the second part we extend this analysis to the case of set-valued
 evolution equations taking the form $\dot x + x = Tx$.
 We also provide some applications to generalized fractal transforms.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In \cite{BoVr}, it was shown that given a a Banach space $\mathcal{B}$
and a contraction mapping $T : \mathcal{B} \to \mathcal{B}$, the 
initial value problem
\begin{equation} \label{evol1}
\dot{y}(t) = Ty(t) - y(t),  \quad  y(0) = y_0 \in \mathcal{B},
\end{equation}
admits a unique solution $y : [0, \infty) \to \mathcal{B}$
which converges exponentially rapidly to the
unique fixed point $\bar y \in \mathcal{B}$ of $T$.
In other words, $\bar y = T \bar y$ is the unique globally asymptotically stable
solution of \eqref{evol1}.
The main purpose in introducing \eqref{evol1} was
to produce a {\em continuous evolution} toward $\bar y$,
as opposed to the usual discrete sequence of iterates
$y_n = T^n y_0$ that converges to $\bar y$, independent of $y_0$.
The original motivation for such an evolution
arose from a desire to perform continuous (in time)
nonlocal, fractal-like operations on images, in which
$\mathcal{B}$ denotes a Banach space of functions defined on a compact set
$X \subset {\bf R}^n$.

Nevertheless, the continuous evolution method
of \eqref{evol1} can also
be applied in other settings where discrete
iteration has normally been considered, for example,
complex analytic dynamics, including
(i) the iteration of rational maps and (ii) Newton's method (and its
generalizations) in the complex plane.

In this paper, we wish to consider an extension of the evolution equation
in \eqref{evol1} to the case of inclusions.
Set-valued differential inclusions appear to be the most
natural way to capture and explain the level of uncertainty, the
absence of controls and the variety of available dynamics that arise
in many applied disciplines, including physics, mechanics and engineering.
The importance of inclusions, which has been realized in the
control theory literature, motivated our introduction of the notion of
an Iterated Multifunction System (IMS) as a natural extension
of the classical notion of an
Iterated Function System (IFS) \cite{KuLaVr,Kunze}.

To produce such an extension, first rewrite \eqref{evol1} in the form
\begin{equation} \label{evol1p}
\dot{y}(t) + y(t) = T(y(t)) ,  \quad  y(0) = y_0 \in \mathcal{B}.
\end{equation}
Now let $T:\mathcal{B}\rightrightarrows \mathcal{B}$ be a set-valued contraction mapping
and consider the following evolution inclusion,
\begin{equation} \label{evolincl}
\dot{y}(t) + y(t) \in  T(y(t)) ,  \quad  y(0) = y_0 \in \mathcal{B}.
\end{equation}
The complexity of this inclusion is much greater than
\eqref{evol1p}.
One might think that the way to proceed
is by considering regular contractive selections.
In general, however, selections with this property are difficult to
establish.  It is even difficult to guarantee the existence
of Lipschitz selections.
As such, we consider here only cases in which the existence of continuous
selections is guaranteed.

Later in the paper we extend this approach to the case of set-valued equations
\begin{equation} \label{evol_set}
\dot{y}(t) + y(t) = T(y(t)) ,  \quad  y(0) = y_0 \in \mathcal{B},
\end{equation}
in which $y:[0,\infty)\rightrightarrows \mathcal{B}$ is a set-valued
solution and the derivative of $y$ w.r.t time is constructed by means
of the Minkowski sum.

We end this section with a few remarks regarding
\eqref{evol1} that will be helpful in our analysis.
In \cite{BoVr}, the solution of \eqref{evol1} was easily accomplished
by applying classical techniques to \eqref{evol1p}, namely,
Duhamel's formula which leads to the equation
\begin{equation} \label{evol1p1}
y(t) = y_0 e^{-t} + e^{-t} \int_0 ^t e^s (Ty)(s) \, ds .
\end{equation}
In the special case $y_0 = \bar y$, we have $y(t)=\bar y = T \bar y$
so that \eqref{evol1p1} leads to the trivial equation,
\begin{equation} \label{evol1p2}
\bar y  = \bar y e^{-t} + e^{-t} \int_0 ^t e^s (T \bar y )(s) \, ds .
\end{equation}
Subtraction, followed by Minkowski's integral inequality, etc., leads
to the desired result,
\begin{equation}
\| y(t) - \bar y \|  \leq  \| y_0 - \bar y \|  e^{(c_T - 1)t},
\end{equation}
where $c_T \in [0,1)$ is the contractivity factor of $T$.
One may also examine this problem in terms of semigroups, first
by writing \eqref{evol1} as
\begin{equation} \label{evolution}
\dot x(t) = Tx(t)- x(t) = (T-I)x(t)\,.
\end{equation}
The existence of a unique classical solution to \eqref{evolution}
can be established using the following result.

\begin{theorem}[\cite{BoVr}] \label{thm1.1}
Let $\mathcal{B}$ be a real Banach space and $T : \mathcal{B} \to \mathcal{B}$ 
a contraction map on $\mathcal{B}$ with fixed point function $\bar x$. 
Let us suppose that $T-I$ is a closed operator and that
the resolvent set of $T-I$ is nonempty. Then for any initial value 
$x_0\in \mathcal{B}$, the unique solution $x(t)$ to \eqref{evolution} 
converges exponentially rapidly to $\bar x$ as $t\to +\infty$.
\end{theorem}

The paper is organized as follows. 
In Section \ref{Sec2} we provide a brief overview of the method of Iterated 
Function Systems with Mappings which will be useful in the sequel. 
In Section \ref{Sec3} we present an extension to set-valued inclusions and 
we provide some results related to the convergence of fixed points. 
In Section \ref{Sec4} a different approach, involving set-valued equations 
in Banach spaces, is considered.

\section{Iterated Function Systems with Mappings and their continuous evolution}
\label{Sec2}

In general, the action of a generalized fractal transform (GFT) $T:S\to S$ 
on an element $u$ of the complete metric space $(S,d)$ can be summarized 
in the following steps. It first produces a set of $N$ spatially-contracted 
copies of $u$ and then modifies the values of these copies by means of a 
suitable range-mapping. Finally, it recombines these modified copies by 
means of an appropriate operator in order to define the element $v \in S$, 
$v = Tu$.  Under appropriate conditions, the fractal transform $T$ 
is a contraction on $(S,d)$ which, from Banach's Fixed Point Theorem,
guarantees the existence of a unique fixed point $\bar u = T \bar u$.

A special case of GFTs which operate on functions are
Iterated Function Systems with Mappings (IFSM), as formulated in
\cite{FoVr}.
We consider the case in which $u:[0,1]\to \mathbb{R}$ and thus the IFSM acts 
on the Banach space
\begin{equation}
\mathcal{B}=\{u:[0,1]\to \mathbb{R}, u\in L^2[0,1]\}.
\end{equation}
The ingredients of an $N$-map IFSM on $\mathcal{B}$ are
\begin{enumerate}
\item A set of $N$ contractive mappings $w=\{w_1, w_2, \ldots, w_N\}$, 
$w_i(x):[0,1]\to [0,1]$, such that $[0,1] = \cup_{i=1}^N w_i ([0,1])$.  In most
practical situations, the $w_i$ are assumed to be affine, i.e.,
\begin{equation}\label{affine_ifs_maps}
w_i(x)=s_i x+ a_i, \quad 0\le s_i<1, ~ 0\le a_i\le 1, ~ 0\le  s_i+a_i\le 1, ~~ i=1,2,\ldots, N;
\end{equation}

\item A set of associated functions\textemdash the so-called 
{\em greyscale maps}\textemdash \\
$\phi=\{\phi_1, \phi_2, \ldots,
    \phi_N\}$, $\phi_i:\mathbb{R}\to\mathbb{R}$, assumed to be Lipschitz.
Once again, in most practical situations, affine maps are employed, i.e.,
\begin{equation}\label{affine_phi_maps}
\phi_i(t)=\alpha_i t+\beta_i,
\end{equation}

\end{enumerate}
Associated with the $N$-map IFSM $(w,\phi)$ is the GFT operator $T$, 
the action of which on a
function $u \in \mathcal{B}$ is given by
\begin{equation}
\label{deterministic_ft}
(Tu)(x)=\sum_{i=1}^N {'} \, \phi_i(u(w_i^{-1}(x))),
\end{equation}
where the prime denotes that the sum operates on all those terms for 
which $w_i^{-1}(x)$ is defined.

\begin{theorem}\cite{FoVr} 
$T:\mathcal{B}\to \mathcal{B}$ and for any $u,v\in \mathcal{B}$ we have
\begin{equation}
d_2(Tu,Tv)\le C d_2(u,v) \, ,
\end{equation}
where
\begin{equation} \label{lipschitz_ifsm}
C=\sum_{i=1}^N s_i^{1/2} \alpha_i.
\end{equation}
\end{theorem}

When $C<1$, then $T$ is contractive on $X$, implying the existence of a unique 
fixed point $\bar u \in \mathcal{B}$ such that
$\bar{u}= T\bar{u}$. Also, from Banach's Fixed Point Theorem, the sequence
\begin{equation} \label{iteration}
u_{n+1} = T u_n
\end{equation}
converges to $\bar u$ for any initial value $u_0$.
\smallskip

\noindent\textbf{Example:}  
The 3-map IFSM on $[0,1]$ defined as follows,
\begin{equation} \label{ifsm_example}
\begin{gathered}
w_1(x) = 0.5 x \, , \quad  \phi_1(t) = 0.6 t + 0.2  \, , \\
w_2(x) = 0.4 x  + 0.3 \, , \quad  \phi_2(t) = 0.25 t + 0.25  \, , \\
w_3(x) = 0.6 x  + 0.4 \, , \quad  \phi_3(t) = 0.4 t + 0.6   \, .
\end{gathered}
\end{equation}
A quick calculation shows that the Lipschitz constant in \eqref{lipschitz_ifsm} is
$C \approx 0.8922 < 1$, which implies that the fractal transform $T$ associated
 with this IFSM is contractive.  In Figure \ref{fig1} are plotted the functions
$u_1(x), u_2(x)$ and $u_3(x)$ obtained from the iteration process
in \eqref{iteration} with seed function $u_0(x)=0$.
An approximation to the attractor
function $\bar u (x)$ is also plotted.
These plots were computed over a grid of 2001 equipartition points over $[0,1]$.
The approximation to
$\bar u$ required 43 iterations of $T$ to achieve convergence to one part
in $10^{-5}$. (The computation time was only 0.003 seconds.)

Note from this figure that the iterate $u_1(x)$ is a
piecewise constant function.  This is a consequence of the
use of the seed function $u_0(x)=0$ in the iteration process
involving an IFSM with affine greyscale maps.  In general, from
 \eqref{affine_phi_maps} and \eqref{deterministic_ft},
if $u_0(x)=0$, then
\begin{equation}
u_1(x) = \sum_{i=1}^N \beta_i \, I_{X_i}(x), \quad x \in [0,1],
\end{equation}
where $X_i = w_i([0,1])$ and
$I_S(x)$ denotes the indicator function of a set $S \subseteq [0,1]$.
In this example,
\begin{equation}
u_1(x) = 0.2 \, I_{[0,0.5]}(x) + 0.25 \, I_{[0.3,0.7]}(x) +
                        0.6 \, I_{[0.4,1.0]}(x) \, .
\end{equation}

\begin{figure}[ht] % two in a row
\centering
\makebox[\textwidth]{
 \subfigure[$u_1(x)$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig1a.png}}
 \subfigure[$u_2(x)$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig1b.png}}}
\makebox[\textwidth]{
 \subfigure[$u_3(x)$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig1c.png}}
 \subfigure[$\bar u (x)$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig1d.png}}}
\caption{Iterates $u_1$, $u_2$ and $u_3$ produced
by the iteration scheme $u_{n+1} = Tu_n$
for the IFSM in the Example in the text, starting with the seed
function $u_0(x) = 0$.  The attractor function $\bar u (x)$ of the
IFSM is also shown.}
\label{fig1}
\end{figure}

We now consider the continuous evolution associated with this
fractal transform $T$, as defined by \eqref{evol1}.
It is convenient to rewrite the evolution equation in the
 form
\begin{equation} \label{continuous_example}
\frac{\partial u}{\partial t} = T u - u , \quad u(x,0) = u_0(x) \, ,
\end{equation}
where the solution $u(x,t)$ is now expressed as a function
of the spatial variable $x \in [0,1]$ and the time variable
$t \geq 0$.  From the discussion in the Introduction, all solutions $u(x,t)$ approach
the attractor function $\bar u (x)$ of $T$ as $t \to \infty$.

In Figure \ref{fig2} are plotted approximations to the solution $u(x,t)$
to \eqref{continuous_example}, with initial condition $u_0(x) = 0$,
at times $t =  0.1, 0.5, 1.0$ and $2.0$.  The solutions were
computed over the 2001 equally-spaced gridpoints used in the previous
example by means of a simple forward Euler time-difference
scheme with step size $h=0.01$.
(Discretization of the time derivative of \eqref{evol1} is
discussed in \cite{BoVr} as well as in Section 3.1 of this paper.)
The vertical scale of Figure \ref{fig2} has been expanded
somewhat from that of Figure \ref{fig1} in order to accentuate the
differences between the graphs.
Note that the solution $u(x,1)$ at time $t=1$ is more ``fractal-like''
and not identical
to the piecewise constant solution $u_1(x)$ obtained
from the discrete iteration process
of \eqref{iteration} and shown in Figure \ref{fig1}.
In other words, solutions to the continuous
evolution equation do not necessarily interpolate the iterates
of the discrete evolution process.  This feature was discussed
in \cite{BoVr}.

The solutions $u(x,1)$ and $u(x,2)$ in Figure \ref{fig2} demonstrate
an evolution of $u(x,t)$ toward the attractor function $\bar u (x)$ of $T$ plotted in Figure \ref{fig1}.
Further evidence of numerical convergence is shown
in Figure \ref{fig3}, where the difference functions
$\Delta(x,t) = \bar u (x) - u(x,t)$
are plotted for $t=5,10$ and 15.  At $t=20$, the differences between
$u(x,20)$ and $\bar u (x)$ are within 2 parts in $10^{-3}$ over all 2001 gridpoints.
(The computation time is only 0.002 seconds.)

\begin{figure}[ht] % two in a row
\setcounter{subfigure}{0}
      \centering
\makebox[\textwidth]{
 \subfigure[$t=0.1$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig2a.png}}
 \subfigure[$t=0.2$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig2b.png}}
  }
\makebox[\textwidth]{
 \subfigure[$t=1$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig2c.png}}
 \subfigure[$t=2$]{
      \includegraphics[width=0.45\textwidth,height=0.3\textwidth]{fig2d.png}}
  }
\caption{Some solutions $u(x,t)$ to the continuous evolution equation
\eqref{continuous_example} where $T$ is the fractal transform
associated with the 3-map IFSM in \eqref{ifsm_example}.
Initial condition:  $u(x,0)=0$.}
\label{fig2}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.6\textwidth,height=0.3\textwidth]{fig3.png}
\caption{A plot of the difference functions $\Delta (x,t) = \bar u (x) - u(x,t)$ for
$t=5$ (top), $t=10$ (middle) and $t=15$ (bottom) for $0 \leq x \leq 1$, demonstrating the
numerical convergence of $u(x,t)$  to $\bar u (x)$.}
\label{fig3}
\end{figure}


\section{Set-valued inclusions involving set-valued contraction mappings}
\label{Sec3}

We now consider the following evolution inclusion,
\begin{equation} \label{inclusion}
\dot{y}(t)+y(t) \in  T(y(t)), \ \ y(0)=y_0 \in \mathcal{B} ,
\end{equation}
where $T:\mathcal{B}\to \mathcal{B}$ is a set-valued contraction mapping, as defined
below.

\begin{definition} \label{def3.1} \rm
Let $(Z,d)$ be a metric space and $T:Z\rightrightarrows Z$ be a set-valued mapping. 
We say that $T$ is a contraction if
there exists a $c_T \in [0,1)$ such that the following property holds
\begin{equation}
d_H(T(z_1),T(z_2))\le c_T d(z_1,z_2)
\end{equation}
for all $z_1,z_2\in Z$, where
\[
d_{H}(T(x_1),T(x_2)) = \max  \Big\{  \sup_{a_1\in Tx_1}\inf_{a_2\in Tx_2}
\| a_1-a_2\|,  \sup_{a_2\in Tx_2}\inf_{a_1\in Tx_1} \| a_1-a_2\| \Big\}
\]
is the standard Hausdorff distance between sets.
\end{definition}

For compact and convex sets $A,B \subset \mathcal{B}$,  it is easy to see that
\[
      d_H(A,B) = \sup_{p \in \mathcal{B}^*, \| p \| = 1 } 
| \operatorname{supp}(p, A) - \operatorname{supp}(p,B)|,
\]
where
\[
    \operatorname{supp}(\cdot, A) : \mathcal{B}^* \to \mathbb{R}   \text{ given by }
 p \mapsto  \sup_{a \in A}  p(a)
\]
is the \emph{support function}  of the convex set $A$ (see \cite[Ch. 13]{Ro} 
for details on support functions).


\begin{theorem}[\cite{CoNa}] \label{Covitz}
Let $\mathcal{B}$ be a  Banach space and 
$T:\mathcal{B} \rightrightarrows \mathcal{B}$ be a set-valued contraction mapping 
taking compact and convex values.
Then there exists at least one solution $\bar z\in \mathcal{B}$ to the fixed 
point inclusion $z\in T(z)$.
\end{theorem}

As mentioned earlier, the complexity of \eqref{inclusion}
is much greater than that of its counterpart in \eqref{evol1p}.
The existence of contractive selections is, in general, difficult to establish.
It is even difficult to guarantee the existence of Lipschitz selections.
For a finite dimensional Banach space
the following result holds.

\begin{theorem} \cite{Aubin}
 Consider a Lipschitz set-valued mapping $F$ from a metric space to nonempty 
closed convex subsets of $\mathbb{R}^n$. Then $F$ has a Lipschitz selection $f$.
\end{theorem}

The extension of this result to infinite dimensional Banach spaces is quite 
complex and it holds if extra conditions are satisfied.
The existence of a continuous selection is guaranteed by the Michael selection 
theorem (see \cite{Mi}).

\begin{theorem}[\cite{Mi}] \label{thm3.3}
A multivalued mapping $T : \mathcal{B}_1\rightrightarrows \mathcal{B}_2$ 
admits a continuous single-valued selection, provided that the following conditions
are satisfied:
\begin{itemize}
\item $\mathcal{B}_1$ is a paracompact space,
\item  $\mathcal{B}_2$ is a Banach space,
\item $T$ is a lower semicontinuous mapping,
\item For every $x\in X$, $T(x)$ is a nonempty convex subset of $\mathcal{B}_2$, and
\item For every $x \in X$, $T(x)$ is a closed subset of $\mathcal{B}_2$.
\end{itemize}
\end{theorem}

Since all metric (and thus Banach) spaces are paracompact, we do not need to consider paracompactness in what follows.

We now list some results which guarantee the convergence of trajectories 
towards fixed points. We assume that the existence of a continuous selection 
can be guaranteed, for example by assuming that $\mathcal{B}$ is a Banach space 
and $T$ takes nonempty closed and convex values.
We consider two possible cases for the contractive set-valued mapping
$T : \mathcal{B} \rightrightarrows \mathcal{B}$.
\smallskip

\noindent\textbf{Case 1:}  
A fixed point $\bar y \in \mathcal{B}$ satisfying the equation
$\bar y = T \bar y$ exists.
Note that a contractive
set-valued mapping $T : \mathcal{B} \rightrightarrows \mathcal{B}$ cannot have
two distinct fixed points
$\bar y_1$ and $\bar y_2$ since this would imply that
\begin{equation}
\|\bar y_1- \bar y_2\| = d_H(T(\bar y_1),T(\bar y_2))\le c_T
 \|\bar y_1- \bar y_2\|  \, .
\end{equation}
This, in turn, implies that $c_T \geq 1$, contradicting the
hypothesis of the contractivity of $T$.

\begin{proposition}\label{firstprop}
Let $\mathcal{B}$ be a real Banach space, 
$T:\mathcal{B}\rightrightarrows \mathcal{B}$ be a set-valued contraction mapping
taking nonempty closed and convex values, and $v:\mathcal{B}\to \mathcal{B}$, 
$v(x)\in T(x)$ for all $x\in \mathcal{B}$,
be a continuous selection. Let $y(t):[0,+\infty)\to \mathcal{B}$ be a continuous 
solution to the evolution equation
\begin{equation} \label{Setvaluedeq}
\dot{y}(t)+y(t) = v(y(t)), \quad y(0)=y_0 \in \mathcal{B}
\end{equation}
Suppose that there exists a $\bar y \in \mathcal{B}$ such that $\bar y=T\bar  y$.
Then $\lim_{t\to +\infty} \|y(t) - \bar y\|=0$.
\end{proposition}

\begin{proof}
We proceed in a manner somewhat parallel to
the approach employed in \cite{BoVr}.
Let $y:[0,+\infty)\to \mathcal{B}$ be a solution to the equation
\begin{equation}
\dot{y}(t)+y(t)=v(y(t)) \, .
\end{equation}
Multiplying both sides by $e^t$, we obtain
\begin{equation}
e^t \dot{y}(t)+ e^t y(t) = e^t v(y(t));
\end{equation}
that is,
\begin{equation}
(e^t y(t))' = e^t v(y(t)) .
\end{equation}
This implies that
\begin{equation} \label{yt}
y(t) = y_0 e^{-t} + \int_0^t e^{s-t}v(y(s)) \, ds.
\end{equation}
In the case $y_0 = \bar y$, the above equation becomes
\begin{equation} \label{ybart}
\bar y = \bar y e^{-t} + \int_0^t e^{s-t} \, \bar y \, ds \, .
\end{equation}
Subtracting \eqref{ybart} from \eqref{yt} yields
\begin{equation}
y(t) - \bar y = e^{-t}  (y_0 - \bar y) +
          \int_0^t e^{s-t}(v(y(s)) - \bar y) \, ds \, ,
\end{equation}
from which it follows that
\begin{equation}
\|y(t) - \bar y\| \le  e^{-t} \| y_0 - \bar y\| +
                 \int_0^t e^{s-t} \|v(y(s)) - \bar y\| \, ds \, .
\end{equation}
For each fixed $s \in [0,t]$, we take the
the supremum with respect to $v \in  T(y(s))$ to obtain
\begin{equation}
\begin{aligned}
\|y(t) - \bar y\|
& \le   e^{-t} \| y_0 - \bar v\| + \int_0^t e^{s-t} \sup_{v\in T(y(s))}
 \|v - \bar y\| \, ds   \\
& \leq  e^{-t} \|y_0 - \bar y\| + \int_0^t e^{s-t} \, d_H(T(y(s)),T(\bar y)) \, ds  \\
& \le  e^{-t} \|y_0 - \bar y\| + c_T \int_0^t e^{s-t} \|y(s)-\bar y\| \, ds \, .
\end{aligned}
\end{equation}
Gronwall's lemma then implies that for $t \geq 0$,
\begin{equation}
\|y(t)-\bar y\| \le \|y_0 - \bar y\| e^{(c_T -1)t} \, .
\end{equation}
Since $c_T \in [0,1)$, it follows that $\| y(t) - \bar y \| \to 0$
as $t \to \infty$, once again exponentially rapidly.
\end{proof}

\noindent
\textbf{Case 2:}  No points $\bar y \in \mathcal{B}$ satisfying the
fixed point equation $\bar y = T \bar y$ exist.

\begin{proposition}
Let $\mathcal{B}$ be a real Banach space, 
$T:\mathcal{B}\rightrightarrows \mathcal{B}$ be a set-valued contraction mapping 
taking nonempty closed and convex values, and $v:\mathcal{B}\to \mathcal{B}$, 
$v(x)\in T(x)$ for all $x\in \mathcal{B}$, be a continuous selection.
Let $\bar y\in T\bar y$ be one of its fixed points -- in the sense
of inclusions -- and suppose that
\begin{equation}
\sup_{s\in T(x)} \sup_{l\in T(\bar y)} \|s-l\| \le c \, \|x - \bar y \|
\end{equation}
for all $x\in \mathcal{B}$ and for some $c\in [0,1)$. Furthermore, let $y(t):[0,+\infty)\to \mathcal{B}$ be a solution to the evolution equation
\begin{equation}
\dot{y}(t)+y(t) = v(y(t)), \ \ y(0)=y_0 \in \mathcal{B}.
\end{equation}
Then $\lim_{t\to +\infty} \|y(t)-\bar y\|=0$.
\end{proposition}

\begin{proof}
The first part of the proof is identical to that of Proposition
\ref{firstprop} and is therefore omitted.
The only difference lies in the following development,
\begin{equation}
\begin{aligned}
\|y(t) - \bar y\| & \le    e^{-t} \| y_0 - \bar y\| +
               \int_0^t e^{s-t} \|v(y(s)) - \bar y\| \, ds   \\
& \leq
e^{-t} \| y_0 - \bar y\| +
     \int_0^t e^{s-t} \sup_{v\in T(y(s))} \sup_{l\in T\bar y}\|v - l\| \, ds
                  \\
& \leq
e^{-t} \| y_0 - \bar y\| + c \int_0^t e^{s-t}  \|y(s) - \bar y\| \, ds .
\end{aligned}
\end{equation}
Once again, Gronwall's lemma can be used to establish the desired result.
\end{proof}

\begin{lemma} \label{lem3.7}
Let $\mathcal{B}$ be a real Banach space, 
$T:\mathcal{B}\rightrightarrows \mathcal{B}$ be a set-valued contraction mapping
taking nonempty closed and convex values and let $\bar y\in \mathcal{B}$ 
one of its fixed points. Construct the single-valued map
\begin{equation}
\psi(x) =  \Pi_{T(x)} \bar y \, ,
\end{equation}
where $\Pi_{T(x)} \bar y$ is the projection of $\bar y$ onto $T(x)$.
Then $\psi$ is locally contractive at $\bar y$.
\end{lemma}

\begin{proof}
By computing, we have:
\begin{equation}
\|\psi(x)-\psi(\bar y)\| = \|\psi(x)-\bar y\|
 = d(T(x),\bar y) \le d_H(T(x),T(\bar y)) \le c \|x-\bar y\|.
\end{equation}
\end{proof}	

\begin{proposition} \label{prop3.8}
Let $\mathcal{B}$ be a real Banach space and 
$T:\mathcal{B}\rightrightarrows \mathcal{B}$ be a set-valued contraction mapping
taking nonempty closed and convex values. Let $\bar y\in T\bar y$ be one of its 
fixed points --  in the sense of inclusions.  Now consider the selection
$\psi:\mathcal{B}\to \mathcal{B}$ defined as follows,
\begin{equation}
\psi(x) =  \Pi_{T(x)} \bar y \, ,
\end{equation}
where $\Pi_{T(x)} \bar y$ denotes the projection of the element $\bar y$
onto the set $T(x)$. Suppose that the differential equation,
\begin{equation} \label{evol_scal}
\dot{y}(t)+y(t) = \psi(y(t)), \quad y(0)=y_0 \in \mathcal{B} ,
\end{equation}
admits a solution, $\forall t\ge 0$. Then $\lim_{t\to +\infty} \|y(t)-\bar y\|=0$.
\end{proposition}

\begin{proof}
Once again, the first part of the proof is identical to that of Proposition
\ref{firstprop} and is omitted.
We proceed with the following development,
\begin{equation}
\begin{aligned}
\|y(t) - \bar y\| & \le   e^{-t} \| y_0 - \bar y\| + \int_0^t e^{s-t} \|\psi(y(s)) - \bar y\| \, ds  \\
& \leq
e^{-t} \| y_0 - \bar y\| +
   \int_0^t e^{s-t} \inf_{v\in T(y(s))} \|v - \bar y\| \, ds  \\
& \leq
e^{-t} \| y_0 - \bar y\| +
    \int_0^t e^{s-t} \sup_{l\in T(\bar y)}\inf_{v\in T(y(s))} \|v - l\| \, ds
     \\
& \leq
e^{-t} \| y_0 - \bar y\| +
      \int_0^t e^{s-t} \, d_H(T(y(s)),T(\bar y)) \, ds  \\
& \leq  e^{-t} \| y_0 - \bar y\| + c_T \int_0^t e^{s-t} \, \|y(s) - \bar y\| \, ds .
\end{aligned}
\end{equation}
The proof then follows from Gronwall's lemma.
\end{proof}

Since a set-valued contraction mapping has, in general, more than one fixed point
it looks reasonable to use a particular $\bar y$ in the construction 
of \eqref{evol_scal}.
The following result states a global convergence result.

\begin{proposition} \label{prop3.9}
Let $\mathcal{B}$ be a real Banach space, 
$T:\mathcal{B}\rightrightarrows \mathcal{B}$ be a set-valued contraction mapping
taking nonempty closed and convex values. Let $\bar y\in T\bar y$ be one of its 
fixed points -- in the sense
of inclusions -- and suppose that
\begin{equation}
d_H(Tx,\bar y)\le c \|x-\bar y\|
\end{equation}
for all $x\in \mathcal{B}$ and for some $c\in [0,1)$.
Let $y(t):[0,+\infty)\to \mathcal{B}$ be a continuous solution to the evolution
inclusion
\begin{equation}
\dot{y}(t)+y(t) \in T(y(t)), \ \ y(0)=y_0 \in \mathcal{B}.
\end{equation}
Then $\lim_{t\to +\infty} \|y(t)-\bar y\|=0$.
\end{proposition}

\begin{proof}
By taking any $p$ in the dual of $\mathcal{B}$, we obtain
\begin{equation}
p(y(t))'+p(y(t)) \le \operatorname{supp} (p,T(y(t))).
\end{equation}
Standard arguments imply that the function $t\to \operatorname{supp} (p,T(y(t)))$
is measurable,  and so we have
\begin{equation}
p(y(t)) \le e^{-t} \int_0^t e^s \operatorname{supp} (p,T(y(s)))ds + e^{-t} p(y(0)).
\end{equation}
Then
\begin{equation}
\begin{aligned}
p(y(t))-p(\bar y)
& \leq  e^{-t} \int_0^t e^s (\operatorname{supp} (p,T(y(s))) - p(\bar y)) ds + e^{-t} (p(y(0))-p(\bar y))  \\
& \leq  e^{-t} \int_0^t e^s (\operatorname{supp} (p,T(y(s))) - \operatorname{supp} (p,\bar y)) ds + e^{-t} (p(y(0))-p(\bar y))  \\
& \leq  e^{-t} \int_0^t e^s d_H(T(y(s)),\bar y) ds + e^{-t} (p(y(0))-p(\bar y))  \\
& \leq  e^{-t} \int_0^t e^s \|y(s)-\bar y\| ds + e^{-t} (p(y(0))-p(\bar y)) .
\end{aligned}
\end{equation}
Gronwall's lemma can once again be used to establish the desired result.
\end{proof}

\subsection{Discretization}

As studied in \cite{BoVr}, employing the simple forward Euler scheme
with time step $h > 0$
for the derivative in \eqref{evol1} leads to the discrete dynamical
system,
\begin{equation} \label{dds1}
y_{n+1} = y_n + (Ty_n - y_n) h, \quad n=0,1,2, \cdots .
\end{equation}
Here, $y_0$ is the initial condition.
In the special case $h=1$, \eqref{dds1} becomes the usual iteration
procedure,
\begin{equation}
y_{n+1} = Ty_n  ,
\end{equation}
which necessarily converges to the fixed point $\bar y = T \bar y$.

The continuous evolution inclusion,
\begin{equation}
\dot{y}(t)+y(t)\in T(y(t)),
\end{equation}
can also be discretized according to classical numerical schemes,
including the above-mentioned Euler scheme with time step $h>0$, yielding
the discrete dynamical inclusion,
\begin{equation} \label{dds2}
y_{n+1}-y_n\in (T(y_n) - y_n)h ,
\end{equation}
where, once again, $y_0 \in \mathcal{B}$ denotes the initial condition.
When $h=1$, \eqref{dds2} becomes
\begin{equation}
y_{n+1}\in T(y_n).
\end{equation}
Let us suppose that $\bar y$ is a fixed point of $T$ and construct the
map $\psi(x) =  \Pi_{T(x)} \bar y$.
In the special case that for each $n \geq 1$,
we choose the particular point,
\begin{equation}
y_{n+1}=\Pi_{T(y_n)}\bar y\in T(y_n) ,
\end{equation}
that is, the projection $\Pi$ of $\bar y$ onto $T(y_n)$, then
we have a converging discrete iteration process (see also \cite{KuLaVr}).

\subsection{An application to fractal transforms}

The idea of using set-valued analysis in fractal theory is recent 
(see \cite{KuLaMeVr} and references therein). 
In this paragraph we introduce a set-valued extension of the classical IFSM 
operator which satisfies the hypotheses of the main results presented in the 
previous section.
Let $X=[0,1]$ and $\mathcal{B}=L^p([0,1])$ be the Banach space of all 
$p$-integrable functions
and consider the set-valued map $T:L^p([0,1])\rightrightarrows L^p([0,1])$ 
defined as
\begin{equation} \label{eq:IFSMM}
\begin{aligned}
(Tf)(x)
=\Big\{&g\in L^p([0,1]): g(x)=\sum_{i=1}^n \alpha_i f(w_i^{-1}(x))
 + \beta_i, \alpha_i\in  [\gamma_1,\gamma_2],\\
& \beta_i\in [\theta_1,\theta_2]  \Big\}\subset L^p([0,1])
\end{aligned}
\end{equation}
where $w_i:[0,1]\to [0,1]$ is a set of $N$ contractive affine maps,
$w_i(x)=s_i x + a_i$, $i=1\dots N$.
The following results prove that $T$ takes convex and closed values and
that it is a contraction.

\begin{proposition} \label{prop3.10}
$T:L^p([0,1])\rightrightarrows L^p([0,1])$ takes compact and convex values.
\end{proposition}

\begin{proof}
It is easy to see that for all $f\in L^p([0,1])$ we have that $Tf$ is 
a compact subset of $L^p([0,1])$, this follows from the fact
that $Tf$ is homeomorphic to the compact set 
$[\gamma_1,\gamma_2]\times [\theta_1,\theta_2]$. To prove $Tf$ is convex, 
let $\xi_1,\xi_2\in Tf$
and $\lambda\in [0,1]$. We have
\begin{gather}
\xi_1(x) = \sum_{i=1}^N \alpha_i^* f(w_i^{-1}(x))+ \beta_i^*,\\
\xi_2(x) = \sum_{i=1}^N \alpha_i^{**} f(w_i^{-1}(x))+ \beta_i^{**}
\end{gather}
and then
\begin{equation}
\lambda \xi_1(x) + (1-\lambda) \xi_2(x)
= \sum_{i=1}^N (\lambda \alpha_i^* + (1-\lambda)  \alpha_i^{**})
f(w_i^{-1}(x))+ \lambda \beta_i^*+ (1-\lambda)  \beta_i^{**}
\end{equation}
which proves the thesis.
\end{proof}

\begin{proposition} \label{prop:IFSMcontractive}
$T:L^p([0,1])\rightrightarrows L^p([0,1])$ is a contractive multifunction 
if $\sum_{i=1}^N s_i^{1/p} \max\{|\gamma_1|,|\gamma_2|\}<1$.
\end{proposition}

\begin{proof}
By means of some straightforward calculations,
\begin{align*}
&\max_{\alpha_i^{**}\in [\gamma_1,\gamma_2],~ \beta_i^{**}\in [\theta_1,\theta_2]}
 \, \,
\min_{\alpha_i^{*}\in [\gamma_1,\gamma_2], ~ \beta_i^{*}\in [\theta_1,\theta_2]}
\Big\| \sum_{i=1}^N  \alpha_i^* f_1(w_i^{-1}(x))+ \beta_i^* \\
&\quad - \sum_{i=1}^N  \alpha_i^{**} f_2(w_i^{-1}(x))- \beta_i^{**} \Big\|_p   \\
& \leq \max_{\alpha_i^{**}\in [\gamma_1,\gamma_2],~ \beta_i^{**}\in 
[\theta_1,\theta_2]} \, \,
\Big\| \sum_{i=1}^N  \alpha_i^{**} f_1(w_i^{-1}(x)) 
- \sum_{i=1}^N  \alpha_i^{**} f_2(w_i^{-1}(x))\Big\|_p   \\
& = \max_{\alpha_i^{**}\in [\gamma_1,\gamma_2]}
\big\| \sum_{i=1}^N  \alpha_i^{**} f_1(w_i^{-1}(x)) 
 - \sum_{i=1}^N  \alpha_i^{**} f_2(w_i^{-1}(x)) \Big\|_p  \\
& \leq \Big( \sum_{i=1}^N  s_i^{1/p} \max\{|\gamma_1|,|\gamma_2|\}\Big)
\|f_1-f_2\|_p \, ,
\end{align*}
which implies that $T$ is contractive multifunction.
\end{proof}

The results presented in the previous section can be applied to this 
set-valued mapping.

\section{Alternative approach:  Set-valued ODEs in Banach spaces}
\label{Sec4}

A completely different alternative approach to generalizing the results 
of \cite{BoVr} is to define a set-valued evolution equation.  This can be 
done in two equivalent ways, either directly as a set-valued ODE or as an 
ODE in a Banach space via the natural embedding of the space of compact 
and convex sets as the positive cone in a Banach space \cite{Ra}.  
In this section we discuss this approach.

Let $\mathcal{B}$ be a Banach space and 
${\mathcal{K}_{cc}} = \{ A \subset \mathcal{B} : \emptyset \ne A \text{ is compact and convex} \}$.
  Recall that ${\mathcal{K}_{cc}}$ is complete under the Hausdorff metric.
 Let $\mathcal{S}^* = \{ p \in \mathcal{B}^* : \| p \| = 1 \}$ be the collection 
of all continuous linear functionals on $\mathcal{B}$ of unit norm.   
We  assume that $T:\mathcal{B} \rightrightarrows \mathcal{B}$ is a contractive 
set-valued function with $T(x) \in {\mathcal{K}_{cc}}$ for all $x$.  We also assume that $T$ 
satisfies the  convexity condition
\begin{equation} \label{eq:Tconvex}
       \lambda T(x) + (1- \lambda) T(y) \subseteq  T( \lambda x + (1-\lambda) y ).
\end{equation}
These conditions on $T$ imply that if $A \in {\mathcal{K}_{cc}}$ then 
$T(A) = \{ T(a) : a \in A \} \in {\mathcal{K}_{cc}}$ as well.
 In particular, this means that $T$ induces a mapping $T:{\mathcal{K}_{cc}} \to {\mathcal{K}_{cc}}$ with the 
same contractivity factor as $T$ and so there is some $\bar{y} \in {\mathcal{K}_{cc}}$ with
 $T(\bar{y}) = \bar{y}$.

\begin{definition} \label{def:derivative} \rm
 For a set-valued mapping $T : \mathbb{R} \to {\mathcal{K}_{cc}}$, we say that $A \in {\mathcal{K}_{cc}}$ 
is the \emph{derivative of $T$ at $t_0$} if
\begin{equation} \label{eq:deriv}
          \lim_{h \to 0^+}   \frac{d_H( T(t_0-h) + A h , T(t_0))}{h} = 0,  
\quad
           \lim_{h \to 0^+}   \frac{d_H( T(t_0 + h), T(t_0) + A h)}{h} = 0.
\end{equation}
In this case, we write $A = T'(t_0)$.
\end{definition}

This is a natural extension of the notion of derivative for a function 
$f:\mathbb{R}^p \to \mathbb{R}^q$ but where we now use the Hausdorff metric to measure distance.  
This definition of derivative is also very naturally related to the embedding 
from \cite{Ra}.

The idea behind this embedding is that ${\mathcal{K}_{cc}}$ has a natural addition 
(Minkowski addition) and a natural scalar multiplication (at least for 
$\lambda \ge 0$).
 In addition, $\lambda(A+B) = \lambda A + \lambda B$ for all $\lambda$ and 
$\lambda_1 A + \lambda_2 A = (\lambda_1 + \lambda_2) A$ as
long as $\lambda_1$ and $\lambda_2$ have the same sign.
Furthermore, the Hausdorff distance on ${\mathcal{K}_{cc}}$ defines a natural ``norm'' 
by $\| A \| = d_H(A,\{ 0 \})$, since $\{ 0 \}$ is the additive identity for this 
semigroup.
The semigroup structure, along with the norm, is used to define the Banach 
space $\mathbb{X}$ in which ${\mathcal{K}_{cc}}$ is embedded as a closed cone.   The space $\mathbb{X}$ can 
be chosen so that
the closed cone has nonempty interior and we do so.
Since the norm on $\mathbb{X}$ is given by an extension of the Hausdorff distance 
between elements of ${\mathcal{K}_{cc}}$, the
definition of derivative in \eqref{eq:deriv} agrees with the derivative as 
defined in $\mathbb{X}$, at least for elements of the cone associated with ${\mathcal{K}_{cc}}$.

Using  definition \ref{def:derivative}, we interpret \eqref{evol1p}  directly 
as a set-valued evolution equation with initial condition $y(0) = y_0 \in {\mathcal{K}_{cc}}$.

By using support functions we can convert the set-valued ODE \eqref{evol1p} 
into a collection of scalar ODEs.
 That is, for each $p \in \mathcal{S}^*$ (each linear functional $p$ of norm one), we have
\begin{equation} \label{eq:supportODE1}
\begin{gathered}
       \operatorname{supp}(p,y'(t)) + \operatorname{supp}(p,y(t)) 
= \operatorname{supp}(p,T(y(t)), \\
  \operatorname{supp}(p,y(0)) = \operatorname{supp}(p,y_0) \quad
\text{ for } y_0 \in {\mathcal{K}_{cc}}.
\end{gathered}
\end{equation}
Now for $A,B \in {\mathcal{K}_{cc}}$, we have  
$d_H(A,B) = \sup\{ |\operatorname{supp}(p,A) - \operatorname{supp}(p,B)| 
:p \in \mathcal{S}^* \}$ and thus if $A = T'(t_0)$,
then we have for all $p \in \mathcal{B}^*$,
\begin{equation} \label{eq:supportderiv}
     \operatorname{supp}(p,A) = \operatorname{supp}(p, T'(t_0)) 
= \frac{d}{dt}\left( \operatorname{supp}( p, T(t))\right|_{t = t_0}.
\end{equation}
Thus, if we let $y_p(t) = \operatorname{supp}(p,y(t))$, 
we have the collection of ODEs (one for each $p \in \mathcal{S}^*$):
\begin{equation} \label{eq:supportODE2}
          y_p'(t) + y_p(t) = \operatorname{supp}(p, T(y(t))), \quad  
 y_p(0) = \operatorname{supp}(p,y_0).
\end{equation}
Notice that since $T$ is contractive, 
$x \mapsto \operatorname{supp}(p,T(x))$ is Lipschitz for all 
$p \in \mathcal{S}^*$ and thus we have existence and uniqueness of solutions 
for \eqref{eq:supportODE2} for all $p \in \mathcal{S}^*$.
However, this by itself is not enough to have existence and uniqueness 
of solutions to the set-valued ODE.
 In order for $y_p(t)$  (as $p$ ranges over $\mathcal{S}^*$) to be a collection of 
support functions for some compact and convex set $y(t)$, these functions 
must satisfy some additional conditions.   
These conditions are not obviously true just from being solutions 
to \eqref{eq:supportODE2}.

However, looking at \eqref{evol1p} instead as an ODE in the embedded Banach 
space $\mathbb{X}$ we do have existence and uniqueness of solutions in $\mathbb{X}$,
and as long as these solutions remain in the cone associated with ${\mathcal{K}_{cc}}$ we 
can interpret $y(t)$ as an element of ${\mathcal{K}_{cc}}$ and thus $y_p(t)$ is a collection of
support functions for $y(t) \in {\mathcal{K}_{cc}}$.    Our next result shows that the cone 
${\mathcal{K}_{cc}}$ in $\mathbb{X}$ is positively invariant and thus $y(t) \in {\mathcal{K}_{cc}}$ for all $t \ge 0$.

\begin{lemma}  \label{lemma:invariantcone}
Let $\mathbb{X}$ be a Banach space, $K \subset \mathbb{X}$ be closed and convex and 
$T:\mathbb{X} \to \mathbb{X}$ be Lipschitz and satisfy $T(K) \subseteq K$.  
Then the solution $y(t)$ to
\begin{equation}  \label{eq:odeprop}
      y' + y = T(y), \quad  y(0) = y_0 \in K
\end{equation}
satisfies $y(t) \in K$ for all $t \ge 0$.
\end{lemma}

\begin{proof}
We know that the solution to \eqref{eq:odeprop} can be written as
\begin{equation} \label{eq:odesoln}
    y(t) = y_0 e^{-t} + e^{-t} \int_0^t e^{-s} T(y(s)) \ ds.
\end{equation}
Choose $p \in \mathcal{S}^*$ and let $u = \operatorname{supp}(p,K)$.  Then
\begin{align*}
     p(y(t)) &= e^{-t} p(y_0) + e^{-t} \int_0^t e^{-s} p(T(y(s))) \, ds \\
             &\leq e^{-t} u + e^{-t} \int_0^t e^{-s} u \ ds \\
             &= e^{-t} u + u ( 1 - e^{-t}) = u.
\end{align*}
That is, $p(y(t)) \le u := \operatorname{supp}(p,K)$ for all $t \ge 0$.  
Since this is true for all $p \in \mathcal{S}^*$, this means that $y(t) \in K$ for all 
$t \ge 0$.
\end{proof}

In summary, we can either view \eqref{evol1p} as a set-valued evolution equation 
or alternatively  as a family \eqref{eq:supportODE2} of scalar ODEs, one for 
each ``direction'' $p \in \mathcal{S}^*$.

With this framework in place, we have the following result. 
All the hard work is in setting the framework.

\begin{proposition} \label{prop:setvalued1}
Let $T:\mathcal{B} \to {\mathcal{K}_{cc}}$ be contractive and satisfy the convexity 
condition \eqref{eq:Tconvex}.
Then the solution $y:[0,\infty) \to {\mathcal{K}_{cc}}$ of the set-valued evolution equation
\begin{equation} \label{eq:setvaluedevolution}
        y'(t) + y(t) = T(y(t)), \quad  y(0) = y_0 \in {\mathcal{K}_{cc}},
\end{equation}
satisfies  $d_H(y(t), \bar{y}) \to 0$ as $t \to \infty$.
\end{proposition}

The idea of this proof is the same as the proof of \cite[Theorem 1]{BoVr},
and is omitted.

\subsection{Weak* compact and convex sets}

For some applications (in particular to spaces of probability measures) 
it is important to be able to use weak* compact sets rather than norm compact sets.
Since the embedding from \cite{Ra} also works in this situation, the same 
framework will work.
To this end, in the case where $\mathcal{B}$ is separable and the dual of a 
Banach space, we also define
 ${\mathcal{K}_{wcc}} = \{ A \subset \mathcal{B} : \emptyset \ne A 
\text{ is bounded, weak$^*$ compact and convex} \}$.

Since $\mathcal{B}$ is separable, the weak* topology is metrizable but only 
on norm bounded subsets; this is the reason for the boundedness restriction 
in the definition of ${\mathcal{K}_{wcc}}$.
We need $T$ to preserve boundedness, so we assume that
\begin{equation} \label{eq:Tbounded}
   \text{there is some $\kappa > 0$ so that whenever }  \| A \| \le \kappa 
\text{ then } \| T(A) \| \le \kappa \text{ as well.}
\end{equation}
With this, we use as our ``base'' space 
$Y = \{ x \in \mathcal{B} : \| x \| \le \kappa \}$ and then
$\Omega = \{ A \subseteq Y :  \emptyset \ne A \text{ is weak* compact and convex} \}$
 replaces ${\mathcal{K}_{wcc}}$.
 Since $Y \subset \mathcal{B}$ is bounded, the weak* topology on $Y$ is metrizable 
and this in turn induces the Hausdorff metric on $\Omega$.   Further, $\Omega$ 
is complete because $Y$ is complete.
Note that we also have 
$\Omega = \{ A \subset \mathcal{B} : A \in {\mathcal{K}_{wcc}}, \| A \| \le \kappa \}$.
As a subspace of $\mathbb{X}$, $\Omega$ is the cone ${\mathcal{K}_{wcc}}$ intersected with the ball 
of radius $\kappa$ centered at $0$.  In particular, $\Omega$ is weak* compact 
and convex.

Given all of this setup, we obtain the same result as before.

\begin{proposition} \label{prop:setvalued2}
Let $T:\mathcal{B} \to {\mathcal{K}_{wcc}}$ be contractive and satisfy the convexity condition 
\eqref{eq:Tconvex} and the boundedness condition \eqref{eq:Tbounded}.
Let $y_0 \in {\mathcal{K}_{wcc}}$ be such that $\| y_0 \| \le \kappa$.   Then the solution 
$y:[0,\infty) \to {\mathcal{K}_{cc}}$ of the set-valued evolution equation
\begin{equation} \label{eq:setvaluedevolution1}
        y'(t) + y(t) = T(y(t)), \quad  y(0) = y_0,
\end{equation}
satisfies  $d_H(y(t), \bar{y}) \to 0$ as $t \to \infty$.
\end{proposition}

The collection of ODEs  in \eqref{eq:supportODE2}, one for each $p \in \mathcal{S}$, 
gives a practical method for obtaining a finite dimensional polyhedral 
approximation to the set-valued solution of the evolution equation 
\eqref{eq:setvaluedevolution1}.  This is especially simple in the case that 
$\mathcal{B}$ is a Hilbert space.

The strategy is to choose finitely many $p_i \in \mathcal{S}, i = 1, 2, \ldots, M$ 
and solve the corresponding scalar ODEs \eqref{eq:supportODE2} for each $p_i$;
 call this solution $y_i(t)$.  Then define the polyhedron  $P(t)$ to be the 
convex hull, $\overline{co}(y_i(t)p_i)$, of the points $y_i(t) p_i \in \mathcal{B}$.
 Since $y_0$ is compact, it is certainly possible to choose $p_i$ so that 
$\overline{co}(p_i)$ closely approximates $y_0$ in the Hausdorff distance.
 In principle because $y(t)$ is a differentiable function, it is also possible 
to choose the $p_i$ in such a way that $\overline{co}(y_i(t) p_i)$ is a close
 approximation to $y(t)$ for all $t$ in some finite time interval $[0,\tau]$.
However, in practice it is difficult to know \emph{a priori} which $p_i$ to 
choose since $y(t)$ is changing.  In the example below (using \eqref{eq:Tmultibeta} 
and \eqref{eq:intervalbeta}), the first few elements of a Haar basis for 
$L^2([0,1])$ adapted to the IFS $\{ w_i \}$ would work well.

\subsection{An application to fractal transforms}

Let $X=[0,1]$ and $\mathcal{B} = L^p([0,1])$ be the Banach space of all 
$p$-integrable functions and ${\mathcal{K}_{cc}}$ be the space of all nonempty, compact 
and convex subsets of $\mathcal{B}$.
Let $w_i:X \to X$, $i = 1, 2, \ldots, N$, be affine with the contraction 
factor of $w_i$ being $s_i$ and let $\alpha_i \in \mathbb{R}$.  
Then for any $f \in L^p([0,1])$ the function
$\sum_i \alpha_i f \circ w^{-1}$ is in $L^p([0,1])$ as well.
Next choose a fixed  $\beta \in {\mathcal{K}_{cc}}$ (so that $\beta \subset L^p([0,1])$ 
is compact and convex) and consider the set-valued map 
$T:L^p([0,1])\rightrightarrows L^p([0,1])$ defined as
\begin{equation} \label{eq:Tmultibeta}
       (Tf)=  \beta + \sum_i  \alpha_i f \circ w_i^{-1}.
\end{equation}
Notice that by definition $T(f) \in {\mathcal{K}_{cc}}$ for each $f \in \mathcal{B}$, 
since $T(f)$ is a translate of $\beta$.

For $\lambda \in [0,1]$ we have
\begin{align*}
     (\lambda T(f) + (1-\lambda) T(g)) 
&= \lambda \Big( \beta + \sum_i \alpha_i f \circ w_i^{-1} \Big) 
 + (1 - \lambda) \Big(  \beta + \sum_i \alpha_i g \circ w_i^{-1} \Big) \\
&= \lambda \beta + (1-\lambda) \beta  + \sum_i \alpha_i (\lambda f 
 + (1-\lambda) g) \circ w_i^{-1} \\
&= \beta+  \sum_i \alpha_i (\lambda f + (1 - \lambda) g)  \circ w_i^{-1}  \\
&= T(\lambda f + (1-\lambda) g).
\end{align*}
Note that $\lambda \beta + (1-\lambda) \beta = \beta$ since $\beta \in {\mathcal{K}_{cc}}$.   
Thus $T$ satisfies the convexity condition \eqref{eq:Tconvex} and so $T$ 
induces a mapping $T:{\mathcal{K}_{cc}} \to {\mathcal{K}_{cc}}$.

\begin{proposition} \label{prop4.5}
   $T:L^p([0,1])\rightrightarrows L^p([0,1])$ as defined in 
\eqref{eq:Tmultibeta} is a contractive multifunction if 
$ \sum_{i=1}^n s_i^{1/p} |\alpha_i| <1$.
\end{proposition}

\begin{proof}
This proof is very similar to the proof of proposition \ref{prop:IFSMcontractive}.
With $s_i$ the  contraction factor of $w_i$ and $\gamma_1,\gamma_2$ selections 
of $\beta$,
we compute
\begin{align*}
 &\max_{\gamma_1 \in \beta} \min_{\gamma_2 \in \beta}  \| \gamma_1 
 + \sum_i \alpha_i  f\circ w_i^{-1} - \gamma_2 
 - \sum_i \alpha_i g \circ w_i^{-1} \|_p \\
&\le \| \sum_i \alpha_i ( f \circ w_i^{-1} - g \circ w_i^{-1} ) \|_p \\
&\le \sum_i |\alpha_i| \| f \circ w_i^{-1} - g \circ w_i^{-1} \|_p \\
&\le \Big( \sum_i s_i^{1/p} | \alpha_i|\Big) \| f - g \|_p.
\end{align*}
Note that the inequality in the first line comes from choosing 
$\gamma_2 = \gamma_1$.  Thus we have our result.
\end{proof}

In particular, the induced mapping $T:{\mathcal{K}_{cc}} \to {\mathcal{K}_{cc}}$ has the same Lipschitz 
constant of $\sum_i s_i^{1/p} | \alpha_i|$ as $T$.  Thus we have a unique 
fixed set $A \in {\mathcal{K}_{cc}}$ and the solution $y(t)$ to the set-valued evolution 
equation \eqref{evol1p} will converge to $A$.
As in section 5 of \cite{LM}, we can write the fixed set $A$ of $T$ in ${\mathcal{K}_{cc}}$ as
\begin{equation} \label{eq:fixedset}
    A =  \beta +  \sum_{n=1}^{\infty} \sum_{\sigma \in \{1,2,\ldots, N \}^n}  
\alpha_{\sigma_1} \alpha_{\sigma_2} \cdots \alpha_{\sigma_n} \beta 
\circ (w_{\sigma_1}^{-1} \circ w_{\sigma_2}^{-1} \circ \cdots 
\circ w_{\sigma_n}^{-1}),
\end{equation}
where we use the notation  
$\beta \circ \phi = \{ g \circ \phi : g \in \beta \}$.  
Each set in the sum \eqref{eq:fixedset} is in ${\mathcal{K}_{cc}}$ and the limit for the 
infinite sum  is taken in the Hausdorff metric on ${\mathcal{K}_{cc}}$.

For a more definite example, we can take
\begin{equation} \label{eq:intervalbeta}
    \beta = \cup_{c \in [\beta_1, \beta_2]} \{  g(x) 
= c \text{ for all } x \in [0,1] \} \subset L^p([0,1]).
\end{equation}
Then the operator \eqref{eq:Tmultibeta} is related in a simple way to that 
in \eqref{eq:IFSMM} in that we vary the $\beta$ values but not the $\alpha$s.
Using this choice of $\beta$,   for each $g \in \beta$, the IFSM (see \cite{FoVr})
\begin{equation} \label{eq:IFSMTg}
    T_g(f) = g + \sum_i \alpha_i f \circ w_i^{-1}
\end{equation}
has a unique fixed point $f_g$ and the fixed set $A$ of $T$ 
(given by \eqref{eq:fixedset}) contains all of these fixed points.
Each of these operators $T_g$ could be thought of as a ``selection'' 
of the operator $T$.
 In a sense, the set-valued evolution will converge to a set which contains 
all the fixed points $f_g$ for all the ``selections'' $T_g$ of $T$.
Of course, $A$ contains much more than just the collection 
$\{ f_g : g \in \beta \}$, just like the attractor of a standard IFS contains
 much more than the collection of the fixed points of the individual $w_i$.
Even though this $\beta$ is one-dimensional, the limiting set $A$ is 
infinite dimensional.  To see this, we just notice that $T(\beta)$  
consists of functions which are piecewise constant on the $N$ sets 
$w_i([0,1])$, $T^2(\beta)$ consists of functions which are piecewise constant 
on the $N^2$ sets $w_i \circ w_j([0,1])$, etc..

\subsection*{Acknowledgements}
We gratefully acknowledge the partial support of this research provided
by Discovery Grants from the
Natural Sciences and Engineering Research Council of Canada (HK,FM,ERV).

\begin{thebibliography}{00}

\bibitem{Aubin}
J-P.~Aubin, H.~Frankowska; 
\emph{Set-valued analysis}, Birk\"hauser, Boston (1990).

\bibitem{BoVr}
J.~L.~Bona, E.~R.~Vrscay;
Continuous evolution of functions and mesures toward fixed points of 
contraction mappings, in {\em Fractals in Engineering: 
New Trends in Theory and Applications} (J. Levy-Vehel and E. Lutton eds.), 
Springer Verlag, London, 237-253 (2005).

\bibitem{CoNa}
H.~Covitz, S.~B.~Nadler;
 Multi-valued contraction mappings in generalized metric spaces, 
Israel Journal of Mathematics 8, 5--11 (1970).

\bibitem{FoVr}
B.~Forte, E.~R.~Vrscay, Theory of generalized fractal transforms, 
in {\em Fractal Image Coding and Analysis} (Y. Fisher ed.), 
NATO ASI Series F, Volume 159, Springer Verlag, New York (1998).

\bibitem{KuLaVr}
H.~Kunze, D.~La Torre, E.~R.~Vrscay;
 Contractive multifunctions, fixed point inclusions and iterated multifunction 
systems, Journal of Mathematical Analysis and Applications, 330 (1), 159--173 (2007).

\bibitem{KuLaMeVr}
H.~Kunze, D.~La Torre, F.~Mendivil, E.R.~Vrscay; 
\emph{Fractal-based Methods in Analysis}, Springer, New York (2012).

\bibitem{Kunze}
H.~Kunze H., D.~La Torre, E.~R.~Vrscay;
From iterated function systems to iterated multifunction systems, 
Communications on Applied Nonlinear Analysis, 15 (4), 1--13 (2008).

\bibitem{LM}
D. La Torre, F. Mendivil;
Iterated function systems on multifunctions and inverse problems, 
Journal of Mathematical Analysis and Applications 340 (2), 1469--1479 (2008).

\bibitem{Mi}
E.~A.~Michael; Continuous selections, 
Annals of Mathematics 63 (2), 361--382 (1956).

\bibitem{Ra}
H.~R\aa dstr\"om;
 An embedding theorem for spaces of convex sets, 
Proceedings of the American Mathematical Society 8 (1), 165--169 (1952).

\bibitem{Ro}
R.~Rockafellar; \emph{Convex Analysis}, Princeton University Press (1970).

\end{thebibliography}

\end{document}