| Title:
|
Second order linear $q$-difference equations: nonoscillation and asymptotics (English) |
| Author:
|
Řehák, Pavel |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
1107-1134 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper can be understood as a completion of the $q$-Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$-difference equations. The $q$-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $q^{\mathbb {N}_0}:=\{q^k\colon k\in \mathbb {N}_0\}$ with $q>1$. In addition to recalling the existing concepts of $q$-regular variation and $q$-rapid variation we introduce $q$-regularly bounded functions and prove many related properties. The $q$-Karamata theory is then applied to describe (in an exhaustive way) the asymptotic behavior as $t\to \infty $ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p\colon \smash {q^{\mathbb {N}_0}}\to \mathbb {R}$. We also present the existing and some new criteria of Kneser type which are related to our subject. A comparison of our results with their continuous counterparts is made. It reveals interesting differences between the continuous case and the $q$-case and validates the fact that $q$-calculus is a natural setting for the Karamata like theory and provides a powerful tool in qualitative theory of dynamic equations. (English) |
| Keyword:
|
regularly varying functions |
| Keyword:
|
$q$-difference equations |
| Keyword:
|
asymptotic behavior |
| Keyword:
|
oscillation |
| MSC:
|
26A12 |
| MSC:
|
39A12 |
| MSC:
|
39A13 |
| MSC:
|
39A21 |
| idZBL:
|
Zbl 1249.26002 |
| idMR:
|
MR2886260 |
| DOI:
|
10.1007/s10587-011-0051-9 |
| . |
| Date available:
|
2011-12-16T15:53:17Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141810 |
| . |
| Reference:
|
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