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

Basic information
Original title:A note on asymptotics and nonoscillation of linear $q$-difference equations
Author:Pavel Řehák
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Citation:ŘEHÁK, Pavel. A note on asymptotics and nonoscillation of linear $q$-difference equations. Electronic Journal of Qualitative Theory of Differential Equations, Szeged: Bolyai Institute, University of Szeged, 2012, neuveden, 4.5.2012, p. 1-12. ISSN 1417-3875.Export BibTeX
@article{981582,
author = {Řehák, Pavel},
article_location = {Szeged},
article_number = {4.5.2012},
keywords = {q-difference equation; asymptotic behavior; nonoscillation},
language = {eng},
issn = {1417-3875},
journal = {Electronic Journal of Qualitative Theory of Differential Equations},
note = {E. J. Qualitative Theory of Diff. Equ., Proc. 9'th Coll. Qualitative Theory of Diff. Equ., No. 12 (2011), pp. 1-12.},
title = {A note on asymptotics and nonoscillation of linear $q$-difference equations},
url = {http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=3&paramtipus_ertek=publication&param_ertek=1073},
volume = {neuveden},
year = {2012}
}
Original language:English
Field:General mathematics
WWW:link to a new windowhttp://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=3&paramtipus_ertek=publication&param_ertek=1073
Type:Article in Periodical
Keywords:q-difference equation; asymptotic behavior; nonoscillation

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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