Strictly monotone functions on preimages of open sets leading to Lyapunov functions. (English) Zbl 1278.26005

Dobrovolschi, Dan

Real Anal. Exch. 37(2011-2012), No. 2, 291-304 (2012).

The work deals with monotonicity of real functions of one variable defined on open sets in preimages of prescribed sets. There are two main results, the first one stating that if \(f\) is Darboux and right continuous on an open interval \(I\), strictly decreasing at each \(x \in f^{-1}(D)\) for an arbitrary \(D \subseteq R\), then \(f\) is strictly decreasing on \(f^{-1}(D)\). The second result claims that if \(f\) has a negative derivative on the preimage of an open set \(D\), then it is strictly decreasing on this preimage. Both results hold also for increasing functions. Examples for autonomous ODE and some examples are also presented.

Reviewer: Vladimír Janiš (Banská Bystrica)

MSC:

26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A48	Monotonic functions, generalizations
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
34D20	Stability of solutions to ordinary differential equations

Keywords:

monotone function; open sets; preimage; differentiable function; Darboux function; Lyapunov function

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