q-Karamata functions and second order q-difference equations

J 2011

q-Karamata functions and second order q-difference equations

ŘEHÁK, Pavel a Jiří VÍTOVEC

Základní údaje

Originální název

q-Karamata functions and second order q-difference equations

Autoři

ŘEHÁK, Pavel (203 Česká republika, garant, domácí) a Jiří VÍTOVEC (203 Česká republika, domácí)

Vydání

Electronic Journal of Qualitative Theory of Differential Equations, Szeged, 2011, 1417-3875

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Maďarsko

Utajení

není předmětem státního či obchodního tajemství

Impakt faktor

Impact factor: 0.557

Kód RIV

RIV/00216224:14410/11:00050560

Organizační jednotka

Pedagogická fakulta

UT WoS

000289152400001

Klíčová slova anglicky

regularly varying functions; rapidly varying functions; q-difference equations; asymptotic behavior

Změněno: 25. 2. 2015 14:17, Dana Nesnídalová

Anotace

V originále

In this paper we introduce and study $q$-rapidly varying functions on the lattice $\qN:=\{q^k:k\in\N_0\}$, $q>1$, 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 $\qN$ 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.

Návaznosti

GAP201/10/1032, projekt VaV

Název: Diferenční rovnice a dynamické rovnice na ,,time scales'' III (Akronym: Difrov)

Investor: Grantová agentura ČR, Diferenční rovnice a dynamické rovnice na time scales III

Citovat

ŘEHÁK, Pavel a Jiří VÍTOVEC. q-Karamata functions and second order q-difference equations. Electronic Journal of Qualitative Theory of Differential Equations. Szeged, 2011, roč. 2011, č. 24, s. 1-20. ISSN 1417-3875.

@article{975039,
   author = {Řehák, Pavel and Vítovec, Jiří},
   article_location = {Szeged},
   article_number = {24},
   keywords = {regularly varying functions; rapidly varying functions; q-difference equations; asymptotic behavior},
   language = {eng},
   issn = {1417-3875},
   journal = {Electronic Journal of Qualitative Theory of Differential Equations},
   title = {q-Karamata functions and second order q-difference equations},
   volume = {2011},
   year = {2011}
}

TY  - JOUR
ID  - 975039
AU  - Řehák, Pavel - Vítovec, Jiří
PY  - 2011
TI  - q-Karamata functions and second order q-difference equations
JF  - Electronic Journal of Qualitative Theory of Differential Equations
VL  - 2011
IS  - 24
SP  - 1-20
EP  - 1-20
SN  - 14173875
KW  - regularly varying functions
KW  - rapidly varying functions
KW  - q-difference equations
KW  - asymptotic behavior
N2  - In this paper we introduce and study $q$-rapidly varying functions on the lattice $\qN:=\{q^k:k\in\N_0\}$, $q>1$, 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 $\qN$ 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.
ER  -

ŘEHÁK, Pavel a Jiří VÍTOVEC. q-Karamata functions and second order q-difference equations. \textit{Electronic Journal of Qualitative Theory of Differential Equations}. Szeged, 2011, roč.~2011, č.~24, s.~1-20. ISSN~1417-3875.

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