First and second order linear dynamic equations on time scales. (English) Zbl 0993.39010

The authors investigate first and second order dynamic equations on time scales. Recall that a time scale \(\mathbb T\) is any closed subset of the reals \(\mathbb R\) and for a function \(f:\mathbb T\to \mathbb R\) one can establish the calculus which reduces to the differential and integral calculus if \(\mathbb T=\mathbb R\) and to the calculus of finite differences if \(\mathbb T=\mathbb Z\) – the set of integers. For more details concerning the time scales calculus see S. Hilger [Result. Math. 18, No. 1/2, 18-56 (1990; Zbl 0722.39001)] or the recent book of M. Bohner and A. Peterson [Dynamic equations on time scales, Birkhäuser, Basel (2001; Zbl 0978.39001)].
A particular attention is devoted to explicitly solvable first and second order dynamic equations. It is shown that in the theory of time scales dynamic equations the generalized time scales elementary functions (exponential, trigonometric, hyperbolic) play essentially the same role as their classical counterparts in the theory of differential equations. The order reduction formula and the variation of parameters method are important tools in this investigation. General results are illustrated by a number of examples.


39A12 Discrete version of topics in analysis
39A10 Additive difference equations
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