A note on asymptotics and nonoscillation of linear $q$-difference equations
| Autoři | |
|---|---|
| Rok publikování | 2012 |
| Druh | Článek v odborném periodiku |
| Časopis / Zdroj | Electronic Journal of Qualitative Theory of Differential Equations |
| Fakulta / Pracoviště MU | |
| Citace | |
| www | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=3¶mtipus_ertek=publication¶m_ertek=1073 |
| Obor | Obecná matematika |
| Klíčová slova | q-difference equation; asymptotic behavior; nonoscillation |
| Popis | We study the linear second order $q$-difference equation $ y(q^2t)+a(t)y(qt)+b(t)y(t)=0 $ on the $q$-uniform lattice $\{q^k:k\in\N_0\}$ with $q>1$, where $b(t)\ne0$. We establish various conditions guaranteeing the existence of solutions satisfying certain estimates resp. (non)oscillation of all solutions resp. $q$-regular boundedness of solutions resp. $q$-regular variation of solutions. Such results may provide quite precise information about their asymptotic behavior. Some of our results generalize existing Kneser type criteria and asymptotic formulas, which were stated for the equation $D_q^2y(qt)+p(t)y(qt)=0$, $D_q$ being the Jackson derivative. In the proofs however we use an original approach. |
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