Publication detail
Critical oscillation constant for Euler-type dynamic equations on time scales
VÍTOVEC, J.
Original Title
Critical oscillation constant for Euler-type dynamic equations on time scales
Czech Title
Kritická oscilační konstanta pro Eulerův typ dynamické rovnice na časových škálách
English Title
Critical oscillation constant for Euler-type dynamic equations on time scales
Type
journal article
Language
en
Original Abstract
In this paper we study the second-order dynamic equation on the time scale $\T$ of the form $$(r(t)y^{\Delta })^\Delta + \frac{\gamma q(t)}{t\sigma(t)}y^{\sigma}=0,$$ where $r$, $q$ are positive rd-continuous periodic functions with $\inf\{r(t),\, t\in\T\}>0$ and $\gamma$ is an arbitrary real constant. This equation corresponds to Euler-type differential (resp. Euler-type difference) equation for continuous (resp. discrete) case. Our aim is to prove that this equation is conditionally oscillatory, i.e., there exists a constant $\Gamma>0$ such that studied equation is oscillatory for $\gamma>\Gamma$ and non-oscillatory for $\gamma<\Gamma$.
Czech abstract
V tomto článku studujeme dynamickou rovnici druhého řádu na časové škále $\T$ tvaru $$(r(t)y^{\Delta })^\Delta + \frac{\gamma q(t)}{t\sigma(t)}y^{\sigma}=0,$$ kde $r$, $q$ jsou kladné rd-spojité periodické funkce splňující $\inf\{r(t),\, t\in\T\}>0$ a $\gamma$ je libovolná reálná konstanta. Tato rovnice odpovídá Eulerovu typu diferenciální (resp. Eulerovu typu diferenční) rovnice pro spojitý (resp. diskrétní) případ. Našim cílem je dokázat, že that tato rovnice je podmíněně oscilatorická, tj. existuje konstanta $\Gamma>0$ taková, že studovaná rovnice je oscilatorická pro $\gamma>\Gamma$ a neoscilatorická pro $\gamma<\Gamma$.
English abstract
In this paper we study the second-order dynamic equation on the time scale $\T$ of the form $$(r(t)y^{\Delta })^\Delta + \frac{\gamma q(t)}{t\sigma(t)}y^{\sigma}=0,$$ where $r$, $q$ are positive rd-continuous periodic functions with $\inf\{r(t),\, t\in\T\}>0$ and $\gamma$ is an arbitrary real constant. This equation corresponds to Euler-type differential (resp. Euler-type difference) equation for continuous (resp. discrete) case. Our aim is to prove that this equation is conditionally oscillatory, i.e., there exists a constant $\Gamma>0$ such that studied equation is oscillatory for $\gamma>\Gamma$ and non-oscillatory for $\gamma<\Gamma$.
Keywords
Time scale; Dynamic equation; Non(oscillation) criteria; Periodic coefficient
RIV year
2014
Released
09.07.2014
Pages from
838
Pages to
848
Pages count
11
URL
BibTex
@article{BUT108316,
author="Jiří {Vítovec}",
title="Critical oscillation constant for Euler-type dynamic equations on time scales",
annote="In this paper we study the second-order dynamic equation on the time scale $\T$ of the form $$(r(t)y^{\Delta })^\Delta + \frac{\gamma q(t)}{t\sigma(t)}y^{\sigma}=0,$$ where $r$, $q$ are positive rd-continuous periodic functions with $\inf\{r(t),\, t\in\T\}>0$ and $\gamma$ is an arbitrary real constant. This equation corresponds to Euler-type differential (resp. Euler-type difference) equation for continuous (resp. discrete) case. Our aim is to prove that this equation is conditionally oscillatory, i.e., there exists a constant $\Gamma>0$ such that studied equation is oscillatory for $\gamma>\Gamma$ and non-oscillatory for $\gamma<\Gamma$.",
chapter="108316",
doi="10.1016/j.amc.2014.06.066",
howpublished="online",
number="7",
volume="243",
year="2014",
month="july",
pages="838--848",
type="journal article"
}