Asymptotic Discontinuity of Smooth Solutions of Nonlinear $q$-Difference Equations
We investigate the asymptotic behavior of solutions of the simplest nonlinear q-difference equations having the form x(qt+ 1) = f(x(t)), q> 1, t∈ R +. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), t∈ R +. It is shown that, for “not very...
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| Datum: | 2000 |
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| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2000
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4566 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We investigate the asymptotic behavior of solutions of the simplest nonlinear q-difference equations having the form x(qt+ 1) = f(x(t)), q> 1, t∈ R +. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), t∈ R +. It is shown that, for “not very large” q> 1, the solutions of the q-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter qfor which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \) as T→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation. |
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