Math 6410 • ODE

\begin{enumerate}
\item[(a)] Find the first three successive approximations \(u_1(t)\), \(u_2(t)\), \(u_3(t)\)
for the initial value problem (IVP)
\[
  \dot{x}=x^2,\qquad x(0)=1.
\]
Use mathematical induction to show that for all \(n\ge 1\),
\(u_n(t)=1+t+\cdots+t^n+O(t^{n+1})\) as \(t\to0\).
\item[(b)] Solve the IVP in part (a) and show that the function
\(x(t)=1/(1-t)\) is a solution to the IVP on the interval \((-\infty,1)\).
Compare with the approximation of part (a).
\item[(c)] Let \(f\in C^1(U,\R^n)\) for \(U\subset\R^n\) and \(x_0\in U\). Given the
Banach space \(X=C([0,T],\R^n)\) with norm
\(\norm{x}=\max_{0\le t\le T}\abs{x(t)}\), let
\[
  K(x)(t)=x_0+\int_0^t f(x(s))\dd s
\]
for \(x\in X\). Define
\(V=\{x\in X\mid\norm{x-x_0}\le\epsilon\}\) 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 \(X\). Using the fact that \(f\) is locally Lipschitz in \(U\)
with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that
\[
  \abs{K(x(t))-K(y(t))}\le L_0t\norm{x-y}.
\]
Hence, show that
\[
  \norm{K(x)-K(y)}\le L_0T\norm{x-y}\qquad x,y\in V.
\]
\end{enumerate}