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q-Karamata functions and second order q-difference equations
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SYSNO ASEP 0374109 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title q-Karamata functions and second order q-difference equations Author(s) Řehák, Pavel (MU-W) RID, SAI, ORCID
Vítovec, J. (CZ)Source Title Electronic Journal of Qualitative Theory of Differential Equations. - : University of Szeged - ISSN 1417-3875
-, č. 24 (2011), s. 1-20Number of pages 20 s. Language eng - English Country HU - Hungary Keywords regularly varying functions ; rapidly varying functions ; q-difference equations Subject RIV BA - General Mathematics CEZ AV0Z10190503 - MU-W (2005-2011) UT WOS 000289152400001 EID SCOPUS 79955571543 Annotation In this paper we introduce and study q-rapidly varying functions on the lattice q(N0) := {q(k) : k is an element of N(0)}, q > 1, which naturally extend the recently established concept of q-regularly varying functions. These types of functions together form the class of the so-called q-Karamata functions. The theory of q-Karamata functions is then applied to half-linear q-difference equations to get information about asymptotic behavior of nonoscillatory solutions. The obtained results can be seen as q-versions of the existing ones in the linear and half-linear differential equation case. However two important aspects need to be emphasized. First, a new method of the proof is presented. This method is designed just for the q-calculus case and turns out to be an elegant and powerful tool also for the examination of the asymptotic behavior to many other q-difference equations, which then may serve to predict how their (trickily detectable) continuous counterparts look like. Second, our results show that q(N0) is a very natural setting for the theory of q-rapidly and q-regularly varying functions and its applications, and reveal some interesting phenomena, which are not known from the continuous theory. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2012
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