Math 6410 • ODE

Let \(p(t)\), \(q(t)\) and \(r(t)\) be continuous functions defined for \(t\in\R\) and
\(t_0,x_0,x_1\in\R\). Consider the initial value problem
\[
  \begin{cases}
    \ddot{x}+p(t)\dot{x}+q(t)x=r(t)\\
    x(t_0)=x_0,\qquad \dot{x}(t_0)=x_1.
  \end{cases}
  \tag{1}
\]
\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)] Carefully reformulate (1) as a fixed point problem. Prove that the fixed
point problem is equivalent to the local existence problem for (1).
\item[(c)] Using the Contraction Mapping Theorem, show that there is a \(\delta>0\) and
a unique function \(x(t)\in C^2([t_0,t_0+\delta])\) that satisfies your fixed point
problem, and hence (1), for \(t_0\le t\le t_0+\delta\).
\end{enumerate}