Math 6410 • ODE

Let \(D\subset\R\times\R^n\) be an open set, \((t_0,x_0)\in D\) a point and
\(f(t,x)\in C(D,\R^n)\) be a map.
\begin{enumerate}
\item[(a)] Let \(X\) be a Banach Space with norm \(\norm{\bullet}\). Let \(C\subset X\) be
a closed subset. State the Contraction Mapping Theorem (otherwise known as the
Banach Fixed Point Theorem) for a map \(K:C\to X\).
\item[(b)] Define what it means for a map \(f(t,x)\in C(D,\R^n)\) to be locally
Lipschitz with respect to \(x\)
\item[(c)] Carefully reformulate the Initial Value Problems as a fixed point problem.
Prove that the fixed point problem is equivalent to the local existence problem for
the (IVP)
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(t,x),\\
    x(t_0)=x_0.
  \end{cases}
  \tag{IVP}
\]
\item[(d)] Assume \(f(t,x)\in C(D,\R^n)\) is locally Lipschitz with respect to \(x\).
Using the Contraction Mapping Theorem, show that there is a \(\epsilon>0\) and a
unique function \(y\in C^1([t_0,t_0+\epsilon],\R^n)\) that satisfies the initial
value problem (IVP) for \(t_0\le t\le t_0+\epsilon\).
\end{enumerate}