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Dynamic equations on time scales: A survey. (English) Zbl 1020.39008

The authors present a survey of some basic results concerning dynamic equations on time scales. The study of such objects goes back to S. Hilger [Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Diss. (1988; Zbl 0695.34001)], who created the calculus of time scales (or, more generally, the calculus of measure chains) in order to unify continuous and discrete analysis. This calculus enables an investigation of dynamic equations that cover as special cases differential equations, difference equations, as well as many other equations, where the domain of an unknown function is a closed subset of reals.

The authors give an introduction to the time scale calculus and present some basic properties of elementary functions on time scales, such as exponential, hyperbolic and trigonometric functions. Those are used to solve certain linear dynamic equations of first and second order. They give further basic results (as a variation of constants) for higher order linear equations, and also several examples and applications are considered. Finally, they mention results on the positivity of quadratic functionals and the solvability of Riccati dynamic equations which correspond to self-adjoint equations and, more generally, symplectic dynamic systems. The paper contains an extensive list of related publications.

MSC:
39A12Discrete version of topics in analysis
39-02Research monographs (functional equations)
37J10Symplectic mappings, fixed points (dynamical systems)