Math 6410 • ODE

Let \(g:[0,\infty)\times\R^d\to\R^d\) be a continuously differentiable function and
\(p>1\) a constant such that
\[
  \abs{g(t,x)}\le \abs{x}^p\quad\text{for all }t\ge0\text{ and }x\in\R^d.
\]
Let \(A\) be a real matrix such that \(\operatorname{Re}\lambda<0\) for all eigenvalues
of \(A\).
\begin{enumerate}
\item[(a)] Show that there is \(\delta>0\) such that if \(\abs{x_0}\le\delta\), then
the initial value problem
\[
  \begin{aligned}
    \dot{x}&=Ax+g(t,x)\\
    x(0)&=x_0
  \end{aligned}
  \tag{1}
\]
has a bounded solution on \([0,\infty)\). [Provide the complete proof. Do not just
quote a theorem.]
\item[(b)] Is the zero solution of (1) asymptotically stable? Explain.

[Hint: let \(f(t)\), \(\varphi(t)\) be nonnegative continuous functions on the interval
\(J=(\alpha,\beta)\) containing \(t_0\). Let \(c_0\ge0\). Gronwall's Lemma says that if
\[
  f(t)\le c_0+\left|\int_{t_0}^{t}\varphi(s)f(s)\dd s\right|
\]
for all \(t\in J\) then
\[
  f(t)\le c_0\exp\left|\int_{t_0}^{t}\varphi(s)\dd s\right|
\]
for all \(t\in J\).]
\end{enumerate}