Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\) satisfy
\(\Re\lambda_i<0\) for all \(i=1,\ldots,n\). Let \(f(t,x)\) be a \(C^2\) function such that
\(\lvert f(t,x)\rvert=o(\lvert x\rvert)\) as \(x\to0\), uniformly in \(t\). Consider the
initial value problem for \(x_0\in\R^n\),
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax+f(t,x),\\
    x(0)=x_0.
  \end{cases}
  \tag{3}
\end{equation}
\begin{enumerate}
\item[(a)] Show that if \(\lvert x_0\rvert\) is small enough, there is a bounded solution
to (3) that exists for all time.

[Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge0\) is a constant
that satisfy \(u(t)\le c_0+\int_0^t g(s)u(s)\dd s\) for all \(t\ge0\). Then Gronwall's
Inequality implies
\[
  u(t)\le c_0\exp\left(\int_0^t g(s)\dd s\right)
\]
for all \(t\ge0\).]
\item[(b)] Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\)
to be \textit{Lyapunov Stable} (some authors call this notion \textit{stable} .)
Show that \(z\) is Lyapunov stable for this equation.
\end{enumerate}