- Research Article
- Open Access
- Published:
Eigenvalues of the
-Laplacian and disconjugacy criteria
Journal of Inequalities and Applications volume 2006, Article number: 37191 (2006)
Abstract
We derive oscillation and nonoscillation criteria for the one-dimensional-Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.
References
- 1.
Allegretto W, Huang YX: A Picone's identity for the-Laplacian and applications. Nonlinear Analysis. Theory, Methods & Applications 1998,32(7):819–830. 10.1016/S0362-546X(97)00530-0
- 2.
Anane A, Chakrone O, Moussa M: Spectrum of one dimensional-Laplacian operator with indefinite weight. Electronic Journal of Qualitative Theory of Differential Equations 2002,2002(17):1–11.
- 3.
Barrett JH: Oscillation theory of ordinary linear differential equations. Advances in Mathematics 1969,3(4):415–509. 10.1016/0001-8708(69)90008-5
- 4.
Browder FE: Infinite dimensional manifolds and non-linear elliptic eigenvalue problems. Annals of Mathematics. Second Series 1965,82(3):459–477. 10.2307/1970708
- 5.
Coppel WA: Disconjugacy, Lecture Notes in Mathematics. Volume 220. Springer, Berlin; 1971:iv+148.
- 6.
del Pino M, Elgueta M, Manásevich R: Generalizing Hartman's oscillation result for. Houston Journal of Mathematics 1991,17(1):63–70.
- 7.
del Pino M, Manásevich R: Oscillation and nonoscillation for. Houston Journal of Mathematics 1988,14(2):173–177.
- 8.
Došlý O: Qualitative theory of half-linear second order differential equations. Mathematica Bohemica 2002,127(2):181–195.
- 9.
Došlý O, Peña S: A linearization method in oscillation theory of half-linear second-order differential equations. Journal of Inequalities and Applications 2005,2005(5):535–545. 10.1155/JIA.2005.535
- 10.
Došlý O, Řehák P: Half-Linear Differential Equations. Elsevier, Amsterdam; 2005.
- 11.
Elbert Á: Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations. In Ordinary and Partial Differential Equations (Dundee, 1982), Lecture Notes in Math.. Volume 964. Springer, Berlin; 1982:187–212.
- 12.
Fučík S, Nečas J: Ljusternik-Schnirelmann theorem and nonlinear eigenvalue problems. Mathematische Nachrichten 1972, 53: 277–289. 10.1002/mana.19720530123
- 13.
Jaroš J, Takaŝi K, Yoshida N: Picone-type inequalities for nonlinear elliptic equations and their applications. Journal of Inequalities and Applications 2001,6(4):387–404. 10.1155/S1025583401000236
- 14.
Kim W: Disconjugacy and comparison theorems for second order linear systems. SIAM Journal on Mathematical Analysis 2002, 354: 4751–4767.
- 15.
Liapounov A: Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies. Volume 17. Princeton University Press, New Jersey; 1949. reprinted from Annales de la Faculté des Sciences de Toulouse 9 (1907), 203–474, translation of original paper published in Communications of the Society of Mathematics, Kharkow (1892) reprinted from Annales de la Faculté des Sciences de Toulouse 9 (1907), 203–474, translation of original paper published in Communications of the Society of Mathematics, Kharkow (1892)
- 16.
Mirzov JD: On some analogs of Sturm's and Kneser's theorems for nonlinear systems. Journal of Mathematical Analysis and Applications 1976,53(2):418–425. 10.1016/0022-247X(76)90120-7
- 17.
Nehari Z: Oscillation criteria for second-order linear differential equations. Transactions of the American Mathematical Society 1957,85(2):428–445. 10.1090/S0002-9947-1957-0087816-8
- 18.
Pinasco JP: Lower bounds for eigenvalues of the one-dimensional-Laplacian. Abstract and Applied Analysis 2004,2004(2):147–153. 10.1155/S108533750431002X
- 19.
Riddell RC: Nonlinear eigenvalue problems and spherical fibrations of Banach spaces. Journal of Functional Analysis 1975,18(3):213–270. 10.1016/0022-1236(75)90015-4
- 20.
Shere KD: Nonoscillation of second-order, linear differential equations with retarded argument. Journal of Mathematical Analysis and Applications 1973,41(2):293–299. 10.1016/0022-247X(73)90203-5
- 21.
Takaŝi K, Naito M: On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter. Transactions of the American Mathematical Society 2002,354(12):4751–4767. 10.1090/S0002-9947-02-03079-9
- 22.
Walter W: Sturm-Liouville theory for the radial-operator. Mathematische Zeitschrift 1998,227(1):175–185. 10.1007/PL00004362
- 23.
Willett D: Classification of second order linear differential equations with respect to oscillation. Advances in Mathematics 1969,3(4):594–623. 10.1016/0001-8708(69)90011-5
- 24.
Yoshida N: Nonoscillation of elliptic differential equations of second order. Hiroshima Mathematical Journal 1974, 4: 279–284.
Author information
Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Napoli, P.L., Pinasco, J.P. Eigenvalues of the-Laplacian and disconjugacy criteria.
J Inequal Appl 2006, 37191 (2006). https://doi.org/10.1155/JIA/2006/37191
Received:
Revised:
Accepted:
Published:
Keywords
- Linear Case
- Mixed Problem
- Eigenvalue Inequality
- Nonoscillation Criterion