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The strict stability of dynamic systems on time scales. (English) Zbl 0996.34043

First, the author considers existence and uniqueness theorems for initial value problems \[ x^\Delta=f(t,x), \quad x(t_0)=x_0, \tag{1} \] on an arbitrary time scale \({\mathbb{T}}\) (a nonempty closed subset of \({\mathbb{R}}\)) with rd-continuous \(f\). This includes as special cases the differential (\({\mathbb{T}}={\mathbb{R}}\)) and difference (\({\mathbb{T}}={\mathbb{Z}}\)) equations. Peano-type existence result and Perron-type existence and uniqueness result are proven by using the continuous induction principle and Hilger’s restriction function \(f^{t]}\) [S. Hilger, Result. Math. 18, No. 1/2, 18-56 (1990; Zbl 0722.39001)]. In the second part, the author defines several notions of strict (uniform, attractive, asymptotic) stability for equation (1) and proves sufficient conditions for the strict (uniform, attractive, asymptotic) stability of the trivial “solution” to (1) in terms of the Dini derivative \(D^+ V_\eta^\Delta(t,x)\). Furthermore, another system of the same form as (1) is considered for a strict stability comparison result. However, the author neither assumes that the trivial “solution” is indeed a solution to equation (1), nor gives condition(s) under which it exists. Moreover, the author does not apparently pay attention to some details such as that \({\mathbb{T}}\) should be unbounded above or the symbols \(V_\eta(t,x)\), \(S_\rho\) should be introduced. The paper will be useful for researchers interested in time scales (measure chains) and/or in stability criteria for differential and difference equations.

MSC:

34D20 Stability of solutions to ordinary differential equations
39A11 Stability of difference equations (MSC2000)
34E13 Multiple scale methods for ordinary differential equations

Citations:

Zbl 0722.39001
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