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

Basic information

Original name

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

Authors

ŘEHÁK, Pavel

Edition

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

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

Republic of Korea

Confidentiality degree

is not subject to a state or trade secret

RIV identification code

RIV/00216224:14410/02:00008101

Organization unit

Faculty of Education

Keywords in English

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

Abstract

In the original language

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.

Links

GA201/01/0079, research and development project

Name: Kvalitativní teorie řešení diferenčních rovnic

Investor: Czech Science Foundation, Qualitative theory of solutions of difference equations

GP201/01/P041, research and development project

Name: Kvalitativní teorie pololineárních diferenciálních a diferenčních rovnic

Investor: Czech Science Foundation, Qualitative theory of half-linear differential and difference equations

Cite

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, vol. 7, No 3, p. 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, vol.~7, No~3, p.~361-404. ISSN~1229-1595.

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