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

0376946 - MÚ 2013 RIV HU eng J - Journal Article
Řehák, Pavel
A note on asymptotics and nonoscillation of linear q-difference equations.
Electronic Journal of Qualitative Theory of Differential Equations. Roč. 12, May 04 (2012), s. 1-12. ISSN 1417-3875. E-ISSN 1417-3875.
[Colloquium on the Qualitative Theory of Differential Equations /9./. Szeged, 28.06.2011-01.07.2011]
Institutional research plan: CEZ:AV0Z10190503
Keywords : q-difference equation * oscillation * asymptotic behavior
Subject RIV: BA - General Mathematics
Impact factor: 0.740, year: 2012
http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=3&paramtipus_ertek=publication&param_ertek=1073

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.
Permanent Link: http://hdl.handle.net/11104/0209225