\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\DeclareMathOperator{\Ker}{{\rm Ker}}
\DeclareMathOperator{\Image}{{\rm Image}}
\def\Fbb{\mathbb{F}}
\begin{document}
\doclabel{Math 617: Homework 11}
\docdate{Due: April 20, 2012}
Let $a,b\in\Reals$ with $a<b$. Let
$A\in C[a,b]\otimes \Fbb^{n\times n}$, let $f\in C[a,b]\otimes \Fbb^n$, and let $x_0\in\Fbb^n$.  We would like to show that there exist a unique solution in $C^1[a,b]\otimes\Fbb^n$ of
$$
x'=Ax+f
$$
with the initial condition $x(a)=x_0$.  This homework guides you through
a (likely) unfamiliar approach to showing this is true.

The usual approach to showing solutions of differential equations 
exist is done using a contraction mapping argument, but this only
ensures existence of solutions on a small time interval.  There is
then an argument to show that if the right-hand side is nice enough,
you can extend the solutions to a large time interval.  For linear
equations, like the one studied here, one can always extend.  The 
method of proof demonstrated here exploits linearity to obtain
an existence on the whole 
time interval without a continuation argument.

\begin{aproblems}

\aproblem  Suppose $u\in C^1[a,b]$, $\alpha\in C^0[a,b]$, and suppose
\[
u'\le \alpha u
\]
on $[a,b]$. Note that $u$ is {\bf real} valued, not $\Fbb^n$-valued.

Let $v(x)=\exp(\int_a^x \alpha(s)\;ds)$.  Show that $v'=\alpha v$, $v(a)=1$ and that
\[
u\le u(a) v
\]
on $[a,b]$.  Hint: Observe that $v \neq 0$ and consider $u/v$.

Moral:  The solution of $w'=\alpha w$ with $w(a)=u(a)$ dominates $u$.

\aproblem  Suppose $x\in C^1([a,b])\otimes \Fbb^n$ and 
$$
x'=Ax
$$
on $[a,b]$ and $x(a)=0$.  Show that $x\equiv 0$.  Hint:
Let $z=||x||^2$ and show that $z$ satisfies a differential
inequality of the form $z'\le \alpha z$ on $[a,b]$.  Then
apply Problem 1.

\aproblem Given $v\in L^2[a,b]\otimes\Fbb^n$ let 
$$
Tv(x) = \int_a^b K(x,y) v(y)\; dy
$$
where $K(x,y)=A(y)$ if $y<x$ and $K(x,y)=0$ otherwise.
Show that
\begin{subproblems}
\item If $v\in L^2[a,b]\otimes \Fbb^n$, $Tv\in C[a,b]\otimes\Fbb^n$.
\item $T$ is a compact map from  $L^2[a,b]\otimes\Fbb^n$ to itself.  
\item If $v\in C[a,b]\otimes \Fbb^n$ and $u=Tv$, then 
$u\in C^1[a,b]\otimes\Fbb^n$ and $u'=Av$.
\end{subproblems}

\aproblem Show that if $u\in L^2[a,b]\otimes\Fbb^n$ and
$Tu=u$ then $u=0$.  Hint: Be sure to first show that $u\in C^1[a,b]\otimes F^n$.

\aproblem Problem 2 from the previous assignment holds even when
$K$ is not self adjoint; you'll show this in future work.  Assuming
this for the moment, show that given $g\in L^2[a,b]\otimes \Fbb^n $ 
there exists a unique solution $u\in L^2[a,b]\otimes\Fbb^n$ of
$$
u-Tu = g.
$$

\aproblem Finish the job.

\end{aproblems}
\end{document}