A note on asymptotics and nonoscillation of linear $q$-difference equations

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Authors

ŘEHÁK Pavel

Year of publication 2012
Type Article in Periodical
Magazine / Source Electronic Journal of Qualitative Theory of Differential Equations
MU Faculty or unit

Faculty of Education

Citation
Web http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=3&paramtipus_ertek=publication&param_ertek=1073
Field General mathematics
Keywords q-difference equation; asymptotic behavior; nonoscillation
Description 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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