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 \(g(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) such that \(g(x)=O(\norm {x}^{\beta })\) for some \(\beta >1\) as \(x\to 0\). Consider the initial value problem for \(x_0\in \mathbb {R}^n\), \begin{equation} \begin {cases} \dfrac {dx}{dt}=Ax+g(x),\\ x(0)=x_0. \end {cases} \tag {2} \end{equation} Show that if \(\norm {x_0}\) is small enough, there is a solution to (2) which exists for all time.
[Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge 0\) is a constant that satisfy \(u(t)\le c_0+\int _0^t g(s)u(s)\dd s\) for all \(t\ge 0\). Then Gronwall’s Inequality implies \[ u(t)\le c_0\exp \left (\int _0^t g(s)\dd s\right ) \] for all \(t\ge 0\).]