Stability for first order delay dynamic equations on time scales

On Different Types of Stability for Linear Delay Dynamic Equations

“…Thus, by Remark 5.1, the condition (8) or equivalently the condition (39) are satisfied, see [39,40,47] and [18,23,26,33,36,37,48,49]. For the asymptotic stability of dynamic equations, we can refer to [46], which assumes (39). So, this can make an impression that (8) is necessary for asymptotic stability.…”

“…By using Corollary 7.5, similar idea can be applied to show equivalence of stability of R) and (48) holds. This discussion also brings an exponential estimate for the solutions of two-term equations (46). For instance, if the solution y of (47) satisfies Let {t n } n∈N 0 be an increasing unbounded sequence of reals and consider on the time scale T := {t n } n∈N 0 , the equation…”

“…For example, extend the results of [32,37,38] and the present paper. (P2) In addition to equations with concentrated delays, consider equations with distributed delays, i.e., equations of the form [30,46]). (P4) In [17] it is shown that under certain conditions nonoscillation of a dynamic equation is monotonic, i.e., nonoscillation of (1) .…”

“…However, we cannot apply Theorem 3.6 to provide uniform exponential stability for the trivial solution. The delay is not strict since α(sinh(n)) = sinh(n) for n ∈ N, which shows that the result in [46] (see also [30]) does not apply. The graininess being unbounded implies that the results in [16] fail for this equation.…”

Global stability for nonlinear dynamic equations with variable coefficients

“…Another pair of authors dealing with first-order linear delay dynamic equations is Wu and Zhou [12], with the simpler equation…”