Gauge functions and limit sets for nonautonomous ordinary differential equations

Abstract:

A gauge function V for the differential equation $({\text {S}})x’ = f(x,t)$ is a scalar-valued function sufficiently smooth for $dV(\phi (t),t)/dt$ to exist almost everywhere for solutions $x = \phi (t)$, ${t_0} \leqq t < \tau _\phi ^ +$, of (S). Let (S) have gauge function V that satisfies the following conditions: (1) ${\lim _{t \to + \infty }}V(x,t) \equiv \lambda (x)$ exists; (2) V is continuous in x uniformly with respect to t; (3) the upper, right derivate of V with respect to (S) is nonpositive. Then, if a solution $x = \phi (t)$ of (S) has an $\omega$-limit point P, there is a unique constant $c(\phi )$ such that $\lambda (p) = c(\phi )$. An application to second order, linear equations is given.