Abstract
It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third-order differential equation y′′′(x) = λ y(x) are described. The general form of degenerate boundary conditions for the fourth-order differentiation operator D4 is found. Twelve classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): Are there similar problems for odd-order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): Can a spectral boundary value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
V. A. Marchenko Sturm-Liouville operators and their applications, Kiev, Naukova Dumka, (1977), 332 p. (in Russian)
A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions, Journal of Soviet Mathematics, 33 (6) (1986), 1311–1342.
V. A. Sadovnichii, Ya.T. Sultanaev, A. M. Akhtyamov, General Inverse Sturm-Liouville Problem with Symmetric Potential, Azerbaijan Journal of Mathematics, 5 (2) (2015), 96–108.
M. H. Stone Irregular differential systems of order two and the related expansion problems, Trans. Amer. Math. Soc. Vol. 29, no 1 (1927), 23–53.
V. A. Sadovnichy, B. E. Kanguzhin, On the connection between the spectrum of a differential operator with symmetric coefficients and boundary conditions, Dokl. Akad. Nauk SSSR. 267 (2) (1982), 310–313 (in Russian).
Locker J. Eigenvalues and completeness for regular and simply irregular two-point differential operators, Providence, American Mathematical Society (2008), vii, 177 p. (Memoirs of the American Mathematical Society; Vol.195, N 911).
A. Makin Two-point boundary-value problems with nonclassical asymptotics on the spectrum, Electronic Journal of Differential Equations, No. 95 (2018), 1–7.
A. M. Akhtyamov, On spectrum for differential operator of odd order, Mathematical Notes, 101 (5) (2017), 755–758.
N. Dunford, and J. T. Schwartz, Linear Operators, Part III: Spectral Operators, Wiley-Interscience, New York (1971).
A. A. Dezin, Spectral characteristics of general boundary-value problems for operator D2, Mathematical notes of the Academy of Sciences of the USSR, Vol. 37, 142–146 (1985)
B. N. Biyarov, S. A. Dzhumabaev, A criterion for the Volterra property of boundary value problems for Sturm-Liouville equations, Mathematical Notes, 56 (1) (1994), 751–753.
A. M. Akhtyamov On Degenerate Boundary Conditions in the Sturm-Liouville Problem , Differential Equations, Vol.52, No. 8, pp. 1085–1087 (2016)
A. M. Akhtyamov, Degenerate Boundary Conditions -for the Diffusion Operator, Differential Equations, Vol. 53, No.11 (2017),1515–1518.
P. Lang, and J.‘Locker, Spectral theory of two-point differential operators determined by D2. I. Spectral properties, Journal of Mathematical Analysis and Applications, Vol. 141 J.:, 538–558.
T. Sh. Kal’menov and D. Suragan, Determination of the Structure of the Spectrum of Regular Boundary Value Problems for Differential Equations by V.A. Il’in’s Method of Priori Estimates, Doklady Mathematics, Vol. 78, No. 3 92008), 913–915.
Naimark M.A.: Linear Differential Operators, Nauka, Moscow (1969) (in Russian)
V. A. Yurko, The inverse problem for differential operators of second order with regular boundary conditions, Math. Notes, 18:4 (1975), 928–932
A. Akhtyamov, M. Amram, A. Mouftakhov, On reconstruction of a matrix by its minors, International Journal of Mathematical Education in Science and Technology, Vol. 49, No. 2 (2018), 268–321.
G.‘Sh. Guseinov, Inverse spectral problems for a quadratic pencil of Sturm-Liouville operators on a finite interval, Spektralaya teoriya operatorov i ee prilozheniya (Spectral Theory of Operators and Its Applications), Baku, 1986, no. 7, pp. 51–101.
G.‘Sh. Guseinov, and I. M. Nabiev, The inverse spectral problem for pencils of differential operators, Sb. Math., 2007, vol. 198, no. 11, pp. 1579–1598.
I. M. Nabiev, A. Sh. Shukurov, Solution of inverse problems for the diffusion operator in the symmetric case, Izv. Sarat. un-that. New ser. Ser. Maths. Mechanics. Informatics, vol. 9:4 (1) (2009), 36–40 (in Russian)
E. Kamke, Reference Book in Ordinary Differential Equations, New York, Van Nostrand (19600
V. B. Lidskii, V.‘A. Sadovnichii, Regularized sums of roots of a class of entire functions, Funct. Anal. Appl., vol. 1, no. 2 (1967), 133–139.
V. B. Lidskii, V.‘A. Sadovnichii, Asymptotic formulas for the roots of a class of entire functions, Math. USSR Sb., vol. 4, no. 4 (1968), 519–527.
G. Sansone; Sopra una famiglia di cubiche con infiniti punti razionali, (Italian) Rendiconti Istituto Lombardo, 62 (1929), 354–360.
A. G. Kurosh, A Course in Higher Algebra Moscow, Nauka (1963), (in Russian).
P. Lancaster, Theory of Matrices, Academic Press, NY (1969)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Sadovnichii, V.A., Sultanaev, Y.T., Akhtyamov, A.M. (2021). On Degenerate Boundary Conditions and Finiteness of the Spectrum of Boundary Value Problems. In: Parasidis, I.N., Providas, E., Rassias, T.M. (eds) Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-030-84721-0_31
Download citation
DOI: https://doi.org/10.1007/978-3-030-84721-0_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-84720-3
Online ISBN: 978-3-030-84721-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)