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Select with a check-mark ☑ and solve either A-4.I
or A-4.II:
☐ **A-4.I.** Assume \(f:D\to\mathbb{R}^N\) is continuous and \(f\) is bounded by a
constant \(m\) on a subdomain \(D_0\subset D\). Let \(u(t)\) be a solution of \(u'=f(t,u)\) with
\((t,u(t))\in D_0\) on \(a0>f(a)\), then \(d(f,(a,b),0)=1\).
**(b)** If \(f(b)<00\), then \(d(f,(a,b),0)=0\).
Select with a check-mark \checkedchoice{} and solve either A-4.I
or A-4.II:
\choicebox{} \textbf{A-4.I.} Assume \(f:D\to\R^N\) is continuous and \(f\) is bounded by a
constant \(m\) on a subdomain \(D_0\subset D\). Let \(u(t)\) be a solution of \(u'=f(t,u)\) with
\((t,u(t))\in D_0\) on \(a<t<b\).
\begin{enumerate}
\item[(a)] Prove that \(u(t)\) satisfies a Lipschitz condition
\(\lvert u(t_1)-u(t_2)\rvert\le m\lvert t_1-t_2\rvert\).
\item[(b)] Prove that \(u(t)\) has one-sided limits at \(t=a\) and \(t=b\):
\(\lim_{t\to a+}u(t)\) and \(\lim_{t\to b-}u(t)\) exist and are finite.
\item[(c)] Explain the connection between (b) and the extension of solutions of initial
value problems to a maximal interval of existence.
\end{enumerate}
\choicebox{} \textbf{A-4.II.} Let \(f:[a,b]\to\R^1\) be continuous and assume
\(f(a)f(b)\ne0\). Verify the following properties of topological degree:
\begin{enumerate}
\item[(a)] If \(f(b)>0>f(a)\), then \(d(f,(a,b),0)=1\).
\item[(b)] If \(f(b)<0<f(a)\), then \(d(f,(a,b),0)=-1\).
\item[(c)] If \(f(a)f(b)>0\), then \(d(f,(a,b),0)=0\).
\end{enumerate}
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