# Regular half-linear second order differential equations

Archivum Mathematicum (2003)

- Volume: 039, Issue: 3, page 233-245
- ISSN: 0044-8753

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topDošlý, Ondřej, and Řezníčková, Jana. "Regular half-linear second order differential equations." Archivum Mathematicum 039.3 (2003): 233-245. <http://eudml.org/doc/249117>.

@article{Došlý2003,

abstract = {We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation \[ \left(r(t)\Phi (x^\{\prime \})\right)^\{\prime \}+c(t)\Phi (x)=0\,,\quad \Phi (x):=|x|^\{p-2\}x\,,\quad p>1 \qquad \mathrm \{\{(*)\}\}\]
and we show that if (*) is regular, a solution $x$ of this equation such that $x^\{\prime \}(t)\ne 0$ for large $t$ is principal if and only if \[ \int ^\infty \frac\{dt\}\{r(t)x^2(t)|x^\{\prime \}(t)|^\{p-2\}\}=\infty \,. \]
Conditions on the functions $r,c$ are given which guarantee that (*) is regular.},

author = {Došlý, Ondřej, Řezníčková, Jana},

journal = {Archivum Mathematicum},

keywords = {regular half-linear equation; principal solution; Picone’s identity; Riccati-type equation; principal solution; Picone's identity; Riccati-type equation},

language = {eng},

number = {3},

pages = {233-245},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Regular half-linear second order differential equations},

url = {http://eudml.org/doc/249117},

volume = {039},

year = {2003},

}

TY - JOUR

AU - Došlý, Ondřej

AU - Řezníčková, Jana

TI - Regular half-linear second order differential equations

JO - Archivum Mathematicum

PY - 2003

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 039

IS - 3

SP - 233

EP - 245

AB - We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation \[ \left(r(t)\Phi (x^{\prime })\right)^{\prime }+c(t)\Phi (x)=0\,,\quad \Phi (x):=|x|^{p-2}x\,,\quad p>1 \qquad \mathrm {{(*)}}\]
and we show that if (*) is regular, a solution $x$ of this equation such that $x^{\prime }(t)\ne 0$ for large $t$ is principal if and only if \[ \int ^\infty \frac{dt}{r(t)x^2(t)|x^{\prime }(t)|^{p-2}}=\infty \,. \]
Conditions on the functions $r,c$ are given which guarantee that (*) is regular.

LA - eng

KW - regular half-linear equation; principal solution; Picone’s identity; Riccati-type equation; principal solution; Picone's identity; Riccati-type equation

UR - http://eudml.org/doc/249117

ER -

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## Citations in EuDML Documents

top- Mariella Cecchi, Zuzana Došlá, Mauro Marini, Limit and integral properties of principal solutions for half-linear differential equations
- Ondřej Došlý, Zuzana Pátíková, Hille-Wintner type comparison kriteria for half-linear second order differential equations
- Zuzana Pátíková, Hartman-Wintner type criteria for half-linear second order differential equations
- Ondřej Došlý, Jaroslav Jaroš, A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations

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