Half-linear dynamic equations on time scales: IVP and
oscillatory properties

J 2002

Half-linear dynamic equations on time scales: IVP and oscillatory properties

ŘEHÁK, Pavel

Základní údaje

Originální název

Half-linear dynamic equations on time scales: IVP and oscillatory properties

Autoři

ŘEHÁK, Pavel (203 Česká republika, garant)

Vydání

Nonlinear Functional Analysis and Applications, Kyungnam University, Masan, Kyungnam University Press, 2002, 1229-1595

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Korejská republika

Utajení

není předmětem státního či obchodního tajemství

Kód RIV

RIV/00216224:14410/02:00008101

Organizační jednotka

Pedagogická fakulta

Klíčová slova anglicky

Half-linear dynamic equation; time scale; measure chain; Roundabout theorem; Picone identity; Sturmian theory; Riccati technique; variational principle

Štítky

Half-linear dynamic equation, measure chain, Picone identity, Riccati technique, Roundabout theorem, Sturmian theory, time scale, variational principle

Změněno: 26. 10. 2006 10:38, prof. Mgr. Pavel Řehák, Ph.D.

Anotace

V originále

In this paper we show how the basic results of oscillation theory of the Sturm--Liouville linear differential equation $$ (r(t)y')'+p(t)y=0 $$ can be extended to the half-linear dynamic equation $$ (r(t)\Phi(y^\Delta))^\Delta+p(t)\Phi(y^\sigma)=0 \tag{HL$^\Delta$E} $$ on an arbitrary time scale, where $\Phi(x)=|x|^{\alpha-1}\sgn x$ with $\alpha>1$. In particular, the generalization of the so called Roundabout theorem is proved for equation (HL$^\triangle$E), which provides powerful tools for the investigation of oscillatory properties of this equation, namely the Riccati technique and variational principle. As an application we present Sturmian theory, oscillation and nonoscillation criteria for (HL$^\Delta$E). The questions concerning the existence and uniqueness of a solution of initial value problem are also discussed.

Návaznosti

GA201/01/0079, projekt VaV

Název: Kvalitativní teorie řešení diferenčních rovnic

Investor: Grantová agentura ČR, Kvalitativní teorie řešení diferenčních rovnic

GP201/01/P041, projekt VaV

Název: Kvalitativní teorie pololineárních diferenciálních a diferenčních rovnic

Investor: Grantová agentura ČR, Kvalitativní teorie pololineárních diferenciálních a diferenčních rovnic

Citovat

ŘEHÁK, Pavel. Half-linear dynamic equations on time scales: IVP and oscillatory properties. Nonlinear Functional Analysis and Applications. Kyungnam University, Masan: Kyungnam University Press, 2002, roč. 7, č. 3, s. 361-404. ISSN 1229-1595.

@article{487585,
   author = {Řehák, Pavel},
   article_location = {Kyungnam University, Masan},
   article_number = {3},
   keywords = {Half-linear dynamic equation; time scale; measure chain; Roundabout theorem; Picone identity; Sturmian theory; Riccati technique; variational principle},
   language = {eng},
   issn = {1229-1595},
   journal = {Nonlinear Functional Analysis and Applications},
   title = {Half-linear dynamic equations on time scales: IVP and oscillatory properties},
   volume = {7},
   year = {2002}
}

TY  - JOUR
ID  - 487585
AU  - Řehák, Pavel
PY  - 2002
TI  - Half-linear dynamic equations on time scales: IVP and oscillatory properties
JF  - Nonlinear Functional Analysis and Applications
VL  - 7
IS  - 3
SP  - 361
EP  - 361
PB  - Kyungnam University Press
SN  - 12291595
KW  - Half-linear dynamic equation
KW  - time scale
KW  - measure chain
KW  - Roundabout theorem
KW  - Picone identity
KW  - Sturmian theory
KW  - Riccati technique
KW  - variational principle
N2  - In this paper we show how the basic results of oscillation theory of the Sturm--Liouville linear differential equation $$ (r(t)y')'+p(t)y=0 $$ can be extended to the half-linear dynamic equation $$ (r(t)\Phi(y^\Delta))^\Delta+p(t)\Phi(y^\sigma)=0 \tag{HL$^\Delta$E} $$ on an arbitrary time scale, where $\Phi(x)=|x|^{\alpha-1}\sgn x$ with $\alpha>1$. In particular, the generalization of the so called Roundabout theorem is proved for equation (HL$^\triangle$E), which provides powerful tools for the investigation of oscillatory properties of this equation, namely the Riccati technique and variational principle. As an application we present Sturmian theory, oscillation and nonoscillation criteria for (HL$^\Delta$E). The questions concerning the existence and uniqueness of a solution of initial value problem are also discussed.
ER  -

ŘEHÁK, Pavel. Half-linear dynamic equations on time scales: IVP and oscillatory properties. \textit{Nonlinear Functional Analysis and Applications}. Kyungnam University, Masan: Kyungnam University Press, 2002, roč.~7, č.~3, s.~361-404. ISSN~1229-1595.

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