×

zbMATH — the first resource for mathematics

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.
Reviewer: Pavel Rehak (Brno)

MSC:
39A12 Discrete version of topics in analysis
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.P.; Bohner, M., Quadratic functionals for second order matrix equations on time scales, Nonlinear anal., 33, 675-692, (1998) · Zbl 0938.49001
[2] Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 1-2, 3-22, (1999) · Zbl 0927.39003
[3] C.D. Ahlbrandt, J. Ridenhour. Putzer algorithms for matrix exponentials, matrix powers, and matrix logarithms, 2000, in preparation. · Zbl 1032.39005
[4] Bézivin, J.P., Sur LES équations fonctionnelles aux q-différences, Aequationes math., 43, 159-176, (1993) · Zbl 0757.39002
[5] Bohner, M.; Došlý, O., Disconjugacy and transformations for symplectic systems, Rocky mountain J. math., 27, 3, 707-743, (1997) · Zbl 0894.39005
[6] Bohner, M.; Eloe, P.W., Higher order dynamic equations on measure chains: Wronskians, disconjugacy, and interpolating families of functions, J. math. anal. appl., 246, 639-656, (2000) · Zbl 0957.34033
[7] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. · Zbl 0978.39001
[8] M. Bohner, A. Peterson, First and second order linear dynamic equations on measure chains, J. Differ. Equations Appl. 2001, to appear. · Zbl 0993.39010
[9] F.B. Christiansen, T.M. Fenchel, Theories of Populations in Biological Communities, Lecture Notes in Ecological Studies, vol. 20, Springer, Berlin, 1977. · Zbl 0354.92025
[10] O. Došlý, R. Hilscher, Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales, J. Differ. Equations Appl. 2001, to appear.
[11] S. Hilger, Ein Maß kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universität Würzburg, 1988. · Zbl 0695.34001
[12] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[13] S. Hilger, Special functions, Laplace and Fourier transform on measure chains, Dynam. Systems Appl. 8(3-4) (1999) 471-488 (Special Issue on “Discrete and Continuous Hamiltonian Systems”, edited by R.P. Agarwal and M. Bohner). · Zbl 0943.39006
[14] R. Hilscher, Reid roundabout theorem for symplectic dynamic systems on time scales, Appl. Math. Optim. 2001, to appear. · Zbl 0990.39017
[15] S. Keller, Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen. Ph.D. thesis, Universität Augsburg, 1999.
[16] Trijtzinsky, W.J., Analytic theory of linear q-difference equations, Acta math., 61, 1-38, (1933) · Zbl 0007.21103
[17] Zhang, C., Sur la sommabilité des séries entières solutions d’équations aux q-différences, I., C.R. acad. sci. Paris Sér. I math., 327, 349-352, (1998) · Zbl 0913.39002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.