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"
}