Math 6410 • ODE

Consider the initial value problem
\[
  x' = f(t,x), \qquad x(t_0)=x_0
\]
on the set \(X:\lvert t-t_0\rvert\le a\), \(\lvert x-x_0\rvert\le b\). Below, outline the
proof means a sequence of statements and lemmas and very brief details. In the outlines,
you may freely use the statement and details of proof from the Picard-Lindel\"of
existence and uniqueness theorem, which says that there exists a unique solution \(x(t)\)
defined on the interval
\(\lvert t-t_0\rvert\le\alpha=\min(a,b/m)\),
\(m=\max\{\lvert f(t,x)\rvert:(t,x)\in X\}\), provided \(f\) satisfies some special
conditions. Select with a check-mark \checkedchoice{} and solve either A-1.I or A-1.II:

\choicebox{} \textbf{A-1.I.} Let \(f(t,x)\) be measurable in \(t\) for each fixed \(x\),
continuous in \(x\) for fixed \(t\) a.e. and for some
\(m\in L^1(t_0-a,t_0+a)\), \(\lvert f(t,x)\rvert\le m(t)\) whenever \((t,x)\in X\). Outline
the proof of Carath\'eodory's existence theorem: There exists an absolutely continuous
function \(x(t)\) defined on an interval \(\lvert t-t_0\rvert\le h\) with \(h\le a\) such that
\(x'(t)=f(t,x(t))\) a.e., and \(x(t_0)=x_0\).

\choicebox{} \textbf{A-1.II.} Assume that \(f\) is continuous on \(X\). Outline the proof of
the Peano existence theorem: There exists for some \(h>0\) at least one solution \(x(t)\)
defined on the interval \(\lvert t-t_0\rvert<h\).