Bounds for the periods of periodic solutions of ordinary differential equations (Q1699449)

Consider in \(\mathbb{R}^n\) the nonautonomous differential system \[ {dx\over dt}= X(t,x)\tag{\(*\)} \] under the assumption that \(X\) is continuously differentiable and that has a periodic solution with period \(T>0\) in some region \(\Omega\subset \mathbb{R}^n\). Additionally, the author supposes that there exists \[ \lambda:=\sup_{\substack{ t\in\mathbb{R}_+\\ x\in\Omega}}\;{\|{d\over dt}\,X(t, x(t))|_{(*)}\|\over \| X(t,x)\|}. \] Then he proves \(T\geq {2\pi\over\lambda}\).