q-Karamata functions and second order q-difference equations
In this paper we introduce and study $q$-rapidly varying functions on the lattice $q^{N_0}:=\{q^k:k\in N_0\}$, $q>1$, 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 function...
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| Format: | Article |
| Language: | English |
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University of Szeged
2011-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=626 |
| _version_ | 1797830696072380416 |
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| author | Pavel Řehák J. Vítovec |
| author_facet | Pavel Řehák J. Vítovec |
| author_sort | Pavel Řehák |
| collection | DOAJ |
| description | In this paper we introduce and study $q$-rapidly varying functions on the lattice $q^{N_0}:=\{q^k:k\in N_0\}$, $q>1$, 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 $q^{N_0}$ 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. |
| first_indexed | 2024-04-09T13:40:13Z |
| format | Article |
| id | doaj.art-273cc6f84fbd49c49ce43f71a9f86e50 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:40:13Z |
| publishDate | 2011-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-273cc6f84fbd49c49ce43f71a9f86e502023-05-09T07:53:01ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752011-04-0120112412010.14232/ejqtde.2011.1.24626q-Karamata functions and second order q-difference equationsPavel Řehák0J. Vítovec1Academy of Sciences of the Czech Republic, Brno, Czech RepublicBrno University of Technology, Brno, Czech RepublicIn this paper we introduce and study $q$-rapidly varying functions on the lattice $q^{N_0}:=\{q^k:k\in N_0\}$, $q>1$, 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 $q^{N_0}$ 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.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=626regularly varying functionsrapidly varying functions$q$-difference equationsasymptotic behavior |
| spellingShingle | Pavel Řehák J. Vítovec q-Karamata functions and second order q-difference equations Electronic Journal of Qualitative Theory of Differential Equations regularly varying functions rapidly varying functions $q$-difference equations asymptotic behavior |
| title | q-Karamata functions and second order q-difference equations |
| title_full | q-Karamata functions and second order q-difference equations |
| title_fullStr | q-Karamata functions and second order q-difference equations |
| title_full_unstemmed | q-Karamata functions and second order q-difference equations |
| title_short | q-Karamata functions and second order q-difference equations |
| title_sort | q karamata functions and second order q difference equations |
| topic | regularly varying functions rapidly varying functions $q$-difference equations asymptotic behavior |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=626 |
| work_keys_str_mv | AT pavelrehak qkaramatafunctionsandsecondorderqdifferenceequations AT jvitovec qkaramatafunctionsandsecondorderqdifferenceequations |