De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation

Libor Báňa; Ondřej Došlý

Archivum Mathematicum (2014)

  • Volume: 050, Issue: 4, page 193-203
  • ISSN: 0044-8753

Abstract

top
We study the generalized half-linear second order differential equation via the associated Riccati type differential equation and Prüfer transformation. We establish a de la Vallée Poussin type inequality for the distance of consecutive zeros of a nontrivial solution and this result we apply to the “classical” half-linear differential equation regarded as a perturbation of the half-linear Euler differential equation with the so-called critical oscillation constant. In the second part of the paper we study a Dirichlet eigenvalue problem associated with the investigated half-linear equation.

How to cite

top

Báňa, Libor, and Došlý, Ondřej. "De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation." Archivum Mathematicum 050.4 (2014): 193-203. <http://eudml.org/doc/262117>.

@article{Báňa2014,
abstract = {We study the generalized half-linear second order differential equation via the associated Riccati type differential equation and Prüfer transformation. We establish a de la Vallée Poussin type inequality for the distance of consecutive zeros of a nontrivial solution and this result we apply to the “classical” half-linear differential equation regarded as a perturbation of the half-linear Euler differential equation with the so-called critical oscillation constant. In the second part of the paper we study a Dirichlet eigenvalue problem associated with the investigated half-linear equation.},
author = {Báňa, Libor, Došlý, Ondřej},
journal = {Archivum Mathematicum},
keywords = {generalized half-linear differential equation; de la Vallée Poussin inequality; half-linear Euler differential equation; Dirichlet eigenvalue problem; generalized half-linear differential equation; de la Vallée Poussin inequality; half-linear Euler differential equation; Dirichlet eigenvalue problem},
language = {eng},
number = {4},
pages = {193-203},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation},
url = {http://eudml.org/doc/262117},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Báňa, Libor
AU - Došlý, Ondřej
TI - De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 4
SP - 193
EP - 203
AB - We study the generalized half-linear second order differential equation via the associated Riccati type differential equation and Prüfer transformation. We establish a de la Vallée Poussin type inequality for the distance of consecutive zeros of a nontrivial solution and this result we apply to the “classical” half-linear differential equation regarded as a perturbation of the half-linear Euler differential equation with the so-called critical oscillation constant. In the second part of the paper we study a Dirichlet eigenvalue problem associated with the investigated half-linear equation.
LA - eng
KW - generalized half-linear differential equation; de la Vallée Poussin inequality; half-linear Euler differential equation; Dirichlet eigenvalue problem; generalized half-linear differential equation; de la Vallée Poussin inequality; half-linear Euler differential equation; Dirichlet eigenvalue problem
UR - http://eudml.org/doc/262117
ER -

References

top
  1. Agarwal, R.P., Grace, S.R., O’Regan, D., Oscillation Theory of Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers,Dordrecht-Boston-London, 2002. (2002) MR2091751
  2. Andres, J., On the criterion of de la Vallée Poussin, Publ. Math. Debrecen 45 (1994), 145–152. (1994) MR1291810
  3. Bihari, I., On the second order half-linear differential equation, Studia Sci. Math. Hungar. 3 (1968), 411–437. (1968) Zbl0167.37403MR0267190
  4. Bihari, I., 10.1007/BF02082686, Period. Math. Hungar. (1976), 117–125. (1976) MR0437847DOI10.1007/BF02082686
  5. Bognár, G., Došlý, O., 10.5486/PMD.2013.5374, Publ. Math. Debrecen 82 (2013), 451–459. (2013) Zbl1299.34122MR3034358DOI10.5486/PMD.2013.5374
  6. Cohn, J.H.E., 10.1093/qmath/39.2.173, Quart. J. Math. Oxford Ser. (2) 39 (159) (1988), 173–174. (1988) Zbl0668.34036MR0947498DOI10.1093/qmath/39.2.173
  7. de Vallée Poussin, Ch., Sur l’équation différentielle linéqire du second order. Détermination d’une intégrale par deux valuers assignés. Extension aux équasions d’ordre n , J. Math. Pures Appl. (8) (1929), 125–144. (1929) 
  8. Došlý, O., Fišnarová, S., 10.1016/j.na.2010.07.049, Nonlinear Anal. 73 (2010), 3756–3766. (2010) Zbl1207.34041MR2728552DOI10.1016/j.na.2010.07.049
  9. Došlý, O., Funková, H., Perturbations of half-linear Euler differential equation and transformations of modified Riccati equation, Abstr. Appl. Anal. 2012 (2012), 19pp. (2012) Zbl1296.34081MR2991019
  10. Došlý, O., Lomtatidze, A., Disconjugacy and disfocality criteria for singular half-linearsecond order differential equations, Ann. Polon. Math. 72 (1999), 273–284. (1999) MR1738580
  11. Došlý, O., Řehák, P., Half-Linear Differential Equations, North Holland Mathematics Studies 202, Elsevier, Amsterdam, 2005. (2005) MR2158903
  12. Došlý, O., Řezníčková, J., Conjugacy and principal solution of generalized half-linear second order differential equations, Electron. J. Qual. Theory Differ. Equ., Proc.9th Coll. QTDE 2012 (5) (2012), 1–13. (2012) MR3338524
  13. Došlý, O., Ünal, M., 10.1007/s10474-007-7120-4, Acta Math. Hungar. 120 (2008), 147–163. (2008) MR2431365DOI10.1007/s10474-007-7120-4
  14. Došlý, O., Yamaoka, N., Oscillation constants for second-order ordinary differential equations related to elliptic equations with p -Laplacian, Nonlinear Anal. 113 (2015), 115–136. (2015) MR3281849
  15. Elber, Á., Kusano, T., On differential equation y ' ' + p ( t ) | y ' | sgn y + q ( t ) y = 0 , Hiroshima Math. J. 22 (1992), 203–218. (1992) MR1160048
  16. Elbert, Á., A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. (1979) MR0680591
  17. Elbert, Á., Generalized Riccati equation for half-linear second order differentia lequations, Colloq. Math. Soc. János Bolyai 47 (1984), 227–249. (1984) MR0890544
  18. Elbert, Á., Schneider, A., 10.1007/BF03322512, Result. Math. 37 (2000), 56–83. (2000) Zbl0958.34029MR1742294DOI10.1007/BF03322512
  19. Fišnarová, S., Mařík, R., 10.1016/j.na.2011.06.025, Nonlinear Anal. 74 (2011), 6427–6433. (2011) Zbl1229.34048MR2833426DOI10.1016/j.na.2011.06.025
  20. Fišnarová, S., Mařík, R., Local estimates for modified Riccati equation in theory of half-linear differential equation, Electron. J. Qual. Theory Differ. Equ. 2012 (63) (2012), 15 pp. (2012) MR2966805
  21. Harris, B.J., On the oscillation criterion of Cohn, Quart. J. Math. Oxford Ser. (2) 42 (167) (1988), 309–9313. (1988) MR1120991
  22. Hartman, P., Wintner, A., On an oscillation criterion of De la Vallée Poussin, Quart. Appl. Math. 13 (1955), 330–332. (1955) Zbl0066.06404MR0073773
  23. Jaroš, J., Kusano, T., Tanigawa, T., 10.1016/j.na.2005.05.045, Nonlinear Anal. 64 (2006), 762–787. (2006) MR2197094DOI10.1016/j.na.2005.05.045
  24. Kiguradze, I., Půža, B., On the Vallée-Poussin problem for singular differential equations with deviating arguments, Arch. Math. (Brno) 33 (1–2) (1997), 127–138. (1997) Zbl0914.34064MR1464307
  25. Kusano, T., Manojlović, J., Tanigawa, T., 10.1016/j.camwa.2009.06.039, Comput. Math. Appl. 59 (2010), 411–425. (2010) Zbl1189.34121MR2575528DOI10.1016/j.camwa.2009.06.039
  26. Willet, D., 10.4153/CMB-1971-073-3, Canad. Math. Bull 14 (1971), 419–430. (1971) DOI10.4153/CMB-1971-073-3

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.