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\noindent
\Large Math 555\hfill Winter 2006

\noindent
{\bf Homework Set 1\hfill Due Friday January 9}
\bigskip

\begin{enumerate}
\item
Osgood's Uniqueness Theorem:
Suppose that $\map{\phi}{\R_+}{\R}$ is a continuous
increasing positive function on $\R_+$ for which
$
\int_0^1 \frac{du}{\phi(u)}=\infty.
$
Show that if $\map{f}{\R\times\R^n}{\R^n}$ is continuous and
$
|f(t,x)-f(t,y)|\le \phi(|x-y|),
$
then the IVP $x'=f(t,x),\ x(0)=x_0$ has at most one
solution.

\item
Let $\mathcal{D}\subset\mathbb{R}^2$ be open and nonempty, and
suppose that $f:\mathcal{D}\to \mathbb{R}$ is continuous.
Let $(t_0,x_0)\in \mathcal{D}$.
Show that if there exits $\beta>0$ and convex set $\Omega\subset\mathbb{R}$
with $[t_0,t_0+\beta]\times\Omega\subset\mathcal{D}$
such that the initial value problem
\[
IVP:\quad x'=f(t,x),\quad x(t_0)=x_0
\]
has two distinct solutions $x_1(t)$ and $x_2(t)$
with $x_i(t)\in\Omega$ for all $t\in [t_0,t_0+\beta]$, i=1,2,
then there are infinitely many
solutions to the initials value problem IVT on $I$.

\item
Let $n=1,\ \F=\R$. Let $f(u)$ be a positive continuous function on 
$[u_0,\infty)$. Consider the IVP $u'=f(u),\ u(0)=u_0.$.
\begin{enumerate}
\item
Use the inverse function theorem to give a {\it rigorous} justification
of the method of {\it separation of variables} to solve this problem
by proving that the equation 
$$
\int_{u_0}^u \frac{dv}{f(v)}=t
$$
determines a $\cC^1$ function $u(t)$ that is the unique solution
of the IVP for $t\ge 0$.
\item
Show that the solution of this IVP exists for all time $t\ge 0$
if and only if 
$\int_{u_0}^\infty \frac{dv}{f(v)}=\infty$.
\end{enumerate}

\item
Let $\map{V}{\R^n}{\R}$ and $\map{f}{\R^N}{\R^N}$ be $\cC^1$.
Suppose that 
$$
\nabla V(x)\cdot f(x)\le 0\quad\mbox{for all}\ x\in\R^n\ .
$$
\begin{enumerate}
\item
Interpret this inequality geometrically.
\item
Further, suppose that there is an $\a > 0$ such that
$$
V(x)\ge \a |x|^2\quad\mbox{for all}\ x\in\R^n\ .
$$
Show that for any $x_0\in\R^n$, the solution of the IVP
$x'=f(x),\ x(0)=x_0$ can be extended to all of $[0,\infty)$.

\noindent
Remark: $V(x)$ is called a {\it Liapunov function} for $f(x)$.
\end{enumerate}


\end{enumerate}
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