# The nonlinear limit-point/limit-circle problem for higher order equations

Miroslav Bartušek; Zuzana Došlá; John R. Graef

Archivum Mathematicum (1998)

- Volume: 034, Issue: 1, page 13-22
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topBartušek, Miroslav, Došlá, Zuzana, and Graef, John R.. "The nonlinear limit-point/limit-circle problem for higher order equations." Archivum Mathematicum 034.1 (1998): 13-22. <http://eudml.org/doc/248206>.

@article{Bartušek1998,

abstract = {We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation \[ y^\{(n)\} + r(t)f(y,y^\{\prime \}, \dots , y^\{(n-1)\}) = 0. \]
The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.},

author = {Bartušek, Miroslav, Došlá, Zuzana, Graef, John R.},

journal = {Archivum Mathematicum},

keywords = {Higher order equations; nonlinear limit-point; nonlinear limit-circle; higher-order equations; nonlinear limit-point; nonlinear limit-circle},

language = {eng},

number = {1},

pages = {13-22},

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

title = {The nonlinear limit-point/limit-circle problem for higher order equations},

url = {http://eudml.org/doc/248206},

volume = {034},

year = {1998},

}

TY - JOUR

AU - Bartušek, Miroslav

AU - Došlá, Zuzana

AU - Graef, John R.

TI - The nonlinear limit-point/limit-circle problem for higher order equations

JO - Archivum Mathematicum

PY - 1998

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

VL - 034

IS - 1

SP - 13

EP - 22

AB - We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation \[ y^{(n)} + r(t)f(y,y^{\prime }, \dots , y^{(n-1)}) = 0. \]
The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

LA - eng

KW - Higher order equations; nonlinear limit-point; nonlinear limit-circle; higher-order equations; nonlinear limit-point; nonlinear limit-circle

UR - http://eudml.org/doc/248206

ER -

## References

top- M. Bartušek, Z. Došlá, On the limit-point/limit-circle problem for nonlinear third order differential equations, Math. Nachr. 187 (1997), 5–18. (1997) MR1471135
- M. Bartušek Z. Došlá, and J. R. Graef, On ${L}^{2}$ and limit-point type solutions of fourth order differential equations, Appl. Anal. 60 (1996), 175–187. (1996) MR1623388
- M. Bartušek Z. Došlá, and J. R. Graef, Limit-point type results for nonlinear fourth order differential equations, Nonlinear Anal. 28 (1997), 779–792. (1997) MR1422183
- M. Bartušek Z. Došlá, and J. R. Graef, Nonlinear limit-point type solutions of $n$th order differential equations, J. Math. Anal. Appl. 209 (1997), 122–139. (1997) MR1444516
- N. Dunford, J. T. Schwartz, Linear Operators; Part II: Spectral Theory, Wiley, New York, (1963). (1963) Zbl0128.34803MR1009163
- M. S. P. Eastham, The limit-$2n$ case of symmetric differential operators of order $2n$, Proc. London Math. Soc. (3) 38 (1979), 272–294. (1979) Zbl0398.34021MR0531163
- M. S. P. Eastham, C. G. M. Grudniewicz, Asymptotic theory and deficiency indices for fourth and higher order self-adjoint equations: a simplified approach, in: Ordinary and Partial Differential Equations (W. N. Everitt and B. D. Sleeman, eds.), Lecture Notes in Math. Vol 846 Springer Verlag, New York, 1981, pp. 88–99. (1981) Zbl0514.34047MR0610637
- W. N. Everitt, On the limit-point classification of fourth-order differential equations, J. London Math. Soc. 44 (1969), 273–281. (1969) Zbl0162.39201MR0235187
- M. V. Fedorjuk, Asymptotic method in the theory of one-dimensional singular differential operators, Trudy Mosk. Matem. Obsch. 15 (1966), 296–345. (1966) MR0208060
- J. R. Graef, Limit circle criteria and related properties for nonlinear equations, J. Differential Equations 35 (1980), 319–338. (1980) Zbl0441.34024MR0563385
- J. R. Graef, Limit circle type results for sublinear equations, Pacific J. Math. 104 (1983), 85–94. (1983) Zbl0535.34024MR0683730
- J. R. Graef, Some asymptotic properties of solutions of ${\left(a\left(t\right){x}^{\text{'}}\right)}^{\text{'}}-q\left(t\right)f\left(x\right)=r\left(t\right)$, in: Differential Equations: Qualitative Theory (Szeged, 1984), Colloquia Mathematica Societatis János Bolyai, Vol. 47, North-Holland, Amsterdam, 1987, pp. 347–359. (1984) MR0890550
- J. R. Graef, P. W. Spikes, On the nonlinear limit-point/limit-circle problem, Nonlinear Anal. 7 (1983), 851–871. (1983) Zbl0535.34023MR0709039
- R. M. Kauffman T. T. Read, and A. Zettl, The Deficiency Index Problem for Powers of Ordinary Differential Expressions, Lecture Notes in Math. Vol. 621, Springer-Verlag, New York, 1977. (1977) MR0481243
- R. M. Kauffman, On the limit-$n$ classification of ordinary differential operators with positive coefficients, Proc. London Math. Soc. 35 (1977), 496–526. (1977) Zbl0382.47025MR0460780
- M. A. Naimark, Linear Differential Operators, Part II, George Harrap & Co., London, 1968. (1968) Zbl0227.34020
- R. B. Paris, A. D. Wood, On the ${L}^{2}$ nature of solutions of $n$-th order symmetric differential equations and McLeod’s conjecture, Proc. Roy. Soc. Edinburgh 90A (1981), 209–236. (1981) Zbl0483.34014MR0647603
- B. Schultze, On singular differential operators with positive coefficients, Proc. Roy. Soc. Edinburgh 10A (1992), 361–365. (1992) Zbl0767.34058MR1159190
- P. W. Spikes, On the integrability of solutions of perturbed nonlinear differential equations, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 309–318. (1977) MR0514125
- P. W. Spikes, Criteria of limit circle type for nonlinear differential equations, SIAM J. Math. Anal. 10 (1979), 456–462. (1979) Zbl0413.34033MR0529063
- H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220–269. (1910) MR1511560

## NotesEmbed ?

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