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Stability and instability for dynamic equations on time scales. (English) Zbl 1093.34023

The authors examine stability properties of the trivial solution of a first-order system of dynamic equations

𝐱 Δ =f(t,𝐱)

on time scales. Here, the concepts of Lyapunov stability and LaSalle’s invariance principle are conveyed to the time scales case. The lack of an appropriate chain rule in the general setting is circumvented by sufficient Lyapunov conditions on V ˙(t,𝐱):=[V(𝐱(t))] Δ involving the solution of the system. The theorems in the last section are results from general theory of discrete dynamical systems.

MSC:
34D20Stability of ODE
39A12Discrete version of topics in analysis