Real Analysis Exchange

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

Dan Dobrovolschi

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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

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

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A48: Monotonic functions, generalizations 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 34D20: Stability

monotone function preimage of an open set differentiable function Darboux function Lyapunov function


Dobrovolschi, Dan. Strictly Monotone Functions on Preimages of Open Sets Leading to Lyapunov Functions. Real Anal. Exchange 37 (2011), no. 2, 291--304.

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  • S. K. Berberian, Fundamentals of Real Analysis, Universitext, Springer–Verlag, 1999.
  • A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Mathematics 659, Springer–Verlag, 1978.
  • J. L. Corne, and N. Rouche, Attractivity of Closed Sets Proved by Using a Family of Liapunov Functions, J. Differential Equations, 13 (1973), 231–246.
  • V. Ene, Monotonicity and local systems, Real Anal. Exchange, 17 (1991/92), 291–313.
  • M. Ghil, and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer, New York, 1987.
  • Ph. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964.
  • M. W. Hirsch, and S. Smale, Differential equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.
  • J. P. LaSalle, Stability Theory for Ordinary Differential Equations, J. Differential Equations, 4 (1968), 57–65.
  • E. N. Lorenz, Deterministic Nonperiodic Flow, J. Atmospheric Sci. 20 (1963), 130–141.
  • R. Ortega and L. A. Sanchez, Abstract Competitive Systems and Orbital Stability in $\mathbb{R}^3$, Proc. Amer. Math. Soc., 128 (2000), 2911–2919.
  • C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New York, 1982.
  • B. S. Thomson, Real Functions, Lecture Notes in Mathematics 1170, Springer, Berlin, 1985.
  • B. S. Thomson, J. B. Bruckner, and A. M. Bruckner, Elementary Real Analysis, Second Edition,, 2008.