# Limit and integral properties of principal solutions for half-linear differential equations

Mariella Cecchi; Zuzana Došlá; Mauro Marini

Archivum Mathematicum (2007)

- Volume: 043, Issue: 1, page 75-86
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topCecchi, Mariella, Došlá, Zuzana, and Marini, Mauro. "Limit and integral properties of principal solutions for half-linear differential equations." Archivum Mathematicum 043.1 (2007): 75-86. <http://eudml.org/doc/250159>.

@article{Cecchi2007,

abstract = {Some asymptotic properties of principal solutions of the half-linear differential equation \[ (a(t)\Phi (x^\{\prime \}))^\{\prime \}+b(t)\Phi (x)=0\,, \qquad \mathrm \{(*)\}\]$\Phi (u)=|u|^\{p-2\}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.},

author = {Cecchi, Mariella, Došlá, Zuzana, Marini, Mauro},

journal = {Archivum Mathematicum},

keywords = {half-linear equation; principal solution; limit characterization; integral characterization; half-linear equation; principal solution; limit characterization; integral characterization},

language = {eng},

number = {1},

pages = {75-86},

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

title = {Limit and integral properties of principal solutions for half-linear differential equations},

url = {http://eudml.org/doc/250159},

volume = {043},

year = {2007},

}

TY - JOUR

AU - Cecchi, Mariella

AU - Došlá, Zuzana

AU - Marini, Mauro

TI - Limit and integral properties of principal solutions for half-linear differential equations

JO - Archivum Mathematicum

PY - 2007

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

VL - 043

IS - 1

SP - 75

EP - 86

AB - Some asymptotic properties of principal solutions of the half-linear differential equation \[ (a(t)\Phi (x^{\prime }))^{\prime }+b(t)\Phi (x)=0\,, \qquad \mathrm {(*)}\]$\Phi (u)=|u|^{p-2}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.

LA - eng

KW - half-linear equation; principal solution; limit characterization; integral characterization; half-linear equation; principal solution; limit characterization; integral characterization

UR - http://eudml.org/doc/250159

ER -

## References

top- Agarwal R. P., Grace S. R., O’Regan D., Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Kluwer Acad. Publ., Dordrecht, The Netherlands, 2002. Zbl1073.34002MR2091751
- Cecchi M., Došlá Z., Marini M., On nonoscillatory solutions of differential equations with $p$-Laplacian, Adv. Math. Sci. Appl. 11 1 (2001), 419–436. Zbl0996.34039MR1842385
- Cecchi M., Došlá Z., Marini M., Half-linear equations and characteristic properties of the principal solution, J. Differential Equ. 208, 2005, 494-507; Corrigendum, J. Differential Equations 221 (2006), 272–274. MR2109564
- Cecchi M., Došlá Z., Marini M., Half-linear differential equations with oscillating coefficient, Differential Integral Equations 18 11 (2005), 1243–1256. Zbl1212.34144MR2174819
- Cecchi M., Došlá Z., Marini M., Vrkoč I., Integral conditions for nonoscillation of second order nonlinear differential equations, Nonlinear Anal. 64 (2006), 1278–1289. Zbl1114.34031MR2200492
- Došlá Z., Vrkoč I., On extension of the Fubini theorem and its application to the second order differential equations, Nonlinear Anal. 57 (2004), 531–548. MR2062993
- Došlý O., Elbert Á., Integral characterization of the principal solution of half-linear second order differential equations, Studia Sci. Math. Hungar. 36 (2000), 455–469. MR1798750
- Došlý O., Řehák P., Half-linear Differential Equations, North-Holland, Mathematics Studies 202, Elsevier, Amsterdam, 2005. Zbl1090.34001MR2158903
- Došlý O., Řezníčková J., Regular half-linear second order differential equations, Arch. Math. (Brno) 39 (2003), 233–245. Zbl1119.34029MR2010724
- Elbert Á., On the half-linear second order differential equations, Acta Math. Hungar. 49 (1987), 487–508. (1987) Zbl0656.34008MR0891061
- Elbert Á., Kusano T., Principal solutions of non-oscillatory half-linear differential equations, Adv. Math. Sci. Appl. 8 2 (1998), 745–759. (1998) Zbl0914.34031MR1657164
- Fan X., Li W. T., Zhong C., A classification scheme for positive solutions of second order iterative differential equations, Electron. J. Differential Equations 25 (2000), 1–14.
- Hartman P., Ordinary Differential Equations, 2nd ed., Birkhäuser, Boston–Basel–Stuttgart, 1982. (1982) Zbl0476.34002MR0658490
- Hoshino H., Imabayashi R., Kusano T., Tanigawa T., On second-order half-linear oscillations, Adv. Math. Sci. Appl. 8 1 (1998), 199–216. (1998) Zbl0898.34036MR1623342
- Jaroš J., Kusano T., Tanigawa T., Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129–149. Zbl1047.34034MR1962855
- Jingfa W., On second order quasilinear oscillations, Funkcial. Ekvac. 41 (1998), 25–54. (1998) Zbl1140.34356MR1627369
- Mirzov J. D., Asymptotic Properties of Solutions of the Systems of Nonlinear Nonautonomous Ordinary Differential Equations, (Russian), Maikop, Adygeja Publ., 1993; the english version: Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 14 2004. (1993) MR2144761

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.