Abstract
Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)n(talphay(n))(n)+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)n(talphay(n))(n) - µ,α―t2n-αy = 0, where μ n, α is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. D. Ahlbrandt, D. B. Hinton and R.T. Lewis: Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators. Canad. J. Math. 33 (1981), 229-246.
C. D. Ahlbrandt, D. B. Hinton and R.T. Lewis: The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory. J. Math. Anal. Appl. 81 (1981), 234-277.
W. A. Coppel: Disconjugacy. Lectures Notes in Mathematics, No. 220. Springer Verlag, Berlin-Heidelberg, 1971.
O. Došlý: Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators. Proc. Roy. Soc. Edinburgh 119A (1991), 219-232.
O. Došlý: Conditionally oscillatory equations and spectral properties of singular differential operators. In: Proc. Conf. Ordinary Diff. Equations, Poprad 1994. pp. 23-31.
O. Došlý: Oscillation criteria for self-adjoint linear differential equations. Math. Nachr. 166 (1994), 141-153.
O. Došlý: Nehari-type oscillation criteria for self-adjoint linear differential equations. J. Math. Anal. Appl. 182 (1994), 69-89.
O. Došlý: Oscillation and spectral properties of a class of singular self-adjoint differential operators. Math. Nach. 188 (1997), 49-68.
O. Došlý: Oscillation and spectral properties of self-adjoint differential operators. Nonlinear Anal. 30 (1997), 1375-1384.
O. Došlý and F. Fiedler: A remark on Nehari-type criteria for self-adjoint differential equations. Comment. Math. Univ. Carolin. 32 (1991), 447-462.
O. Došlý and J. Komenda: Principal solutions and conjugacy criteria for self-adjoint differential equations. Arch. Math. 31 (1995), 217-238.
O. Došlý and J. Osi£ka: Kneser-type oscillation criteria for self-adjoint, two term, differential equations. Georgian J. Math. 2 (1995), 241-258.
F. Fiedler: Oscillation criteria for a class of 2n-order ordinary differential operators. J. Differential Equations 42 (1982), 155-185.
F. Fiedler: Oscillation criteria for a special class of 2n-order ordinary differential equations. Math. Nachr. 131 (1987), 205-218.
I. M. Glazman: Direct Methods of Qualitative Analysis of Singular Differential Operators. Davey, Jerusalem, 1965.
D. B. Hinton and R.T. Lewis: Discrete spectra criteria for singular differential operators with middle terms. Math. Proc. Cambridge Philos. Soc. 77 (1975), 337-347.
D. B. Hinton and R.T. Lewis: Singular differential operators with spectra discrete and bounded below. Proc. Roy. Soc. Edinburgh 84A (1979), 117-134.
W. Kratz: Quadratic Functionals in Variational Analysis and Control Theory. Akademie Verlag, Berlin, 1995.
W. T. Reid: Sturmian Theory for Ordinary Diferential Equations. Springer Verlag, New York-Heidelberg-Berlin, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Došlý, O., Osička, J. Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations. Czechoslovak Mathematical Journal 52, 833–849 (2002). https://doi.org/10.1023/B:CMAJ.0000027237.34494.49
Issue Date:
DOI: https://doi.org/10.1023/B:CMAJ.0000027237.34494.49