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

Abstract

top
We describe the nonlinear limit-point/limit-circle problem for the n -th order differential equation y ( n ) + r ( t ) f ( y , y ' , , 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.

How to cite

top

Bartušek, 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
  1. 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
  2. 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
  3. 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
  4. 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
  5. N. Dunford, J. T. Schwartz, Linear Operators; Part II: Spectral Theory, Wiley, New York, (1963). (1963) Zbl0128.34803MR1009163
  6. M. S. P. Eastham, The limit- 2 n case of symmetric differential operators of order 2 n , Proc. London Math. Soc. (3) 38 (1979), 272–294. (1979) Zbl0398.34021MR0531163
  7. 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
  8. W. N. Everitt, On the limit-point classification of fourth-order differential equations, J. London Math. Soc. 44 (1969), 273–281. (1969) Zbl0162.39201MR0235187
  9. M. V. Fedorjuk, Asymptotic method in the theory of one-dimensional singular differential operators, Trudy Mosk. Matem. Obsch. 15 (1966), 296–345. (1966) MR0208060
  10. J. R. Graef, Limit circle criteria and related properties for nonlinear equations, J. Differential Equations 35 (1980), 319–338. (1980) Zbl0441.34024MR0563385
  11. J. R. Graef, Limit circle type results for sublinear equations, Pacific J. Math. 104 (1983), 85–94. (1983) Zbl0535.34024MR0683730
  12. J. R. Graef, Some asymptotic properties of solutions of ( a ( t ) x ' ) ' - q ( t ) f ( x ) = r ( t ) , in: Differential Equations: Qualitative Theory (Szeged, 1984), Colloquia Mathematica Societatis János Bolyai, Vol. 47, North-Holland, Amsterdam, 1987, pp. 347–359. (1984) MR0890550
  13. J. R. Graef, P. W. Spikes, On the nonlinear limit-point/limit-circle problem, Nonlinear Anal. 7 (1983), 851–871. (1983) Zbl0535.34023MR0709039
  14. 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
  15. 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
  16. M. A. Naimark, Linear Differential Operators, Part II, George Harrap & Co., London, 1968. (1968) Zbl0227.34020
  17. 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
  18. B. Schultze, On singular differential operators with positive coefficients, Proc. Roy. Soc. Edinburgh 10A (1992), 361–365. (1992) Zbl0767.34058MR1159190
  19. 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
  20. P. W. Spikes, Criteria of limit circle type for nonlinear differential equations, SIAM J. Math. Anal. 10 (1979), 456–462. (1979) Zbl0413.34033MR0529063
  21. H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220–269. (1910) MR1511560

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.