A note on continuous dependence of solutions of dynamic equations on time scales

On continuous dependence of solutions of dynamic equations

“…[23,33] or [13]) -see Section 4. Note that some estimation related to this result can be also found in [1,Appendix]. The above Lemma allows us to compare the presented approach with some earlier results.…”

“…The first result on the continuous dependence of solutions on the domain T for (1) is proved by Adamec [1]. In fact, he deals with the case of two dynamic equations on different time scales which are ''close'' to each other with respect to the Hausdorff metric h. As mentioned before this kind of convergence is not the general one.…”

“…If we restrict our attention to the case of bounded time scales T n & T, the uniform convergence on T n (as in [1]) can be useful. If this not the case we can extend time scales T n by considering S n ¼ T n [ T, but this means that the graininess function for the last time scale is different than for T n .…”

“…This approach will be described in our Section 2.2. Basic ideas in this direction can be found in [1,12,17,19,24,27,30,32], but (almost) all of that papers are devoted to study the case of compact time scales and the convergence (if any) is taken in the sense of the Hausdorff distance [1,19,21,27].…”

“…In this case we need to have a nonempty intersection of domains for both problems and to compare the solutions on this common set -cf. [1], where the problem was announced and described. It is satisfactory solved on compact domains by extending all the problems to the case of real intervals (cf.…”