Strictly Monotone Functions on Preimages of Open Sets Leading to Lyapunov Functions

Abstract

We consider a real function \(f\) of a real variable such that, for every point \(x\) of the preimage \(f^{-1}(D)\) of a set \(D \subseteq \mathbb{R}\), \(f\) is strictly monotone at \(x\), and give sufficient conditions of strict monotonicity of \(f\) on \(f^{-1}(D)\). In particular, we prove that a differentiable function \(f\) on an open interval, whose derivative is strictly negative on \(f^{-1}(D)\), where \(D \subseteq \mathbb{R}\) is an open set, is strictly decreasing on \(f^{-1}(D)\). The latter result has applications in stability theory of differential equations on \(\mathbb{R}^N\). The first application provides Lyapunov functions \(V\) for preimages under \(V\) of closed sets. The second application is a generalization of the Lyapunov stability theorem, in which the role of the asymptotically equilibrium point is played by \(V^{-1}(-\infty, c_0]\), where \(V\) is a Lyapunov function for \(V^{-1}(-\infty, c_0]\), and all sublevel sets of \(V\) are assumed to be compact. Moreover, due to compactness, all solutions of the differential equation are global to the right. The second application is also a generalization of a boundedness result from Geophysical Fluid Dynamics; in particular, it proves rigorously that all trajectories of the famous Lorenz system eventually enter a compact set.