Math 6410 • ODE

Let \(f \in C^1(U,\mathbb{R}^n)\) for \(U \subset \mathbb{R}^n\) and
\(x_0 \in U\). Given the Banach space
\(X=C([0,T],\mathbb{R}^n)\) with norm
\(\|x\|=\max_{0\le t\le T}|x(t)|\), let
\[
K(x)(t)=x_0+\int_0^t f(x(s))\,ds
\]
for \(x \in X\). Define
\(V=\{x\in X\mid \|x-x_0\|\le \epsilon\}\) for fixed \(\epsilon>0\) and suppose
\(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\)
with \(V\) a closed subset of \(X\).
\begin{enumerate}

\item
Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant
\(L_0\), and taking \(x,y\in V\), show that
\[
|K(x(t))-K(y(t))|\le L_0t\|x-y\|.
\]
Hence, show that
\[
\|K(x)-K(y)\|\le L_0T\|x-y\|,\quad x,y\in V.
\]


\item
Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the
integral equation has a unique continuous solution \(x(t)\) for all
\(t\in[0,T]\) and sufficiently small \(T\). Hence establish existence and
uniqueness of the initial value problem
\[
\frac{dx}{dt}=f(x),\quad x(0)=x_0.
\]


\item
Show that the solution of the equation
\[
\dot{x}=x^2,\quad x(0)=x_0
\]
blows up in finite time. Does this contradict the result of part (b)?

\end{enumerate}