## Real Analysis Exchange

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

Dan Dobrovolschi

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 37, Number 2 (2011), 291-304.

Dates
First available in Project Euclid: 15 April 2013

https://projecteuclid.org/euclid.rae/1366030625

Mathematical Reviews number (MathSciNet)
MR3080592

Zentralblatt MATH identifier
1278.26005

#### Citation

Dobrovolschi, Dan. Strictly Monotone Functions on Preimages of Open Sets Leading to Lyapunov Functions. Real Anal. Exchange 37 (2011), no. 2, 291--304. https://projecteuclid.org/euclid.rae/1366030625

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