Math 6410 • ODE

Consider the problem
\[
\begin{aligned}
  \frac{dx}{dt} &= f(x,t),\\
  x(t_0) &= x_0
\end{aligned}
\tag{1.1}
\]
where \(f\in C(D,\R^n)\) and \(D\subset\R^n\times\R\) is open. Suppose that \(f\) is
Lipschitz continuous in \(x\) for \((x,t)\in D\).
\begin{enumerate}
\item Prove the existence of a local (in \(t\)) solution to (1.1) using successive
approximations.
\item Prove the uniqueness of this solution and show that it can be extended until it
meets \(\partial D\).
\item Prove that the solution is continuous in \((x_0,t_0)\).
\end{enumerate}