Math 6410 • ODE

Let \(A\) be a real matrix such that \(\operatorname{Re}\lambda<0\) for all eigenvalues of \(A\).
\begin{enumerate}

\item
Show that there is \(\delta>0\) such that if \(|x_0|\le\delta\), then the initial value problem

\[
\dot{x}=Ax+e^{-t}|x|x
\]

(1)

\[
x(0)=x_0
\]


has a bounded solution on \([0,\infty)\). [Provide the complete proof. Do not just quote a
theorem.]


\item
Is the zero solution of (1) asymptotically stable? Explain.
[Hint: let \(f(t)\), \(\phi(t)\) be nonnegative continuous functions on the
interval \(J=(\alpha,\beta)\) containing \(t_0\). Let \(c_0\ge0\). Gronwall's
Lemma says that if
\[
f(t)\le c_0+\left|\int_{t_0}^t \phi(s)f(s)\,ds\right|
\]
for all \(t\in J\), then
\[
f(t)\le c_0\exp\left|\int_{t_0}^t \phi(s)\,ds\right|
\]
for all \(t\in J\).]

\end{enumerate}