The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points

Hayati Olğar; Oktay Mukhtarov

doi:10.54187/jnrs.1569468

EN

The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points

Abstract

In this study, we consider a spectral problem for one boundary value problem with discontinuities at two interior points. The boundary conditions involve a spectral parameter. We consider some compact, positive, self-adjoint operators to reduce the spectral problem to an operator-pencil equation. Then, it was proven that this operator-pencil is positive definite, the spectrum is discrete, and the system of weak eigenfunctions forms a Riesz basis of the appropriate Sobolev space.

Keywords

Spectral problem, Boundary conditions, Jump conditions, Eigenvalues, Weak eigenfunctions

Ethical Statement

No approval from the Board of Ethics is required.

References

B. P. Belinskiy, J. W. Hiestand, J. V. Matthews, Piecewise uniform optimal design of a bar with an attached mass, Electronic Journal of Differential Equations 2015 (2015) Article Number 206 17 pages.
N. J. Guliyev, Schr¨odinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, Journal of Mathematical Physics 60 (6) (2019) 063501.
D. B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition, Quarterly Journal of Mathematics (30) (1979) 33–42.
A. Kawano, A. Morassi and R. Zaera, Detecting a prey in a spider orb-web from in-plane vibration, SIAM Journal on Applied Mathematics 81 (6) (2021) 2297-2322.
J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Mathematische Zeitschrift 133 (1973) 301–312.
A. Yakar, Z. Akdoğan, On the fundamental solutions of a discontinuous fractional boundary value problem, Advances in Difference Equations, 2017 (2017) Article ID 378 15 pages.
Z. Akdoğan, A. Yakar, M. Demirci, Discontinuous fractional Sturm-Liouville problems with transmission conditions, Applied Mathematics and Computation 350 (2019) 1–10.
B. P. Allahverdiev, H. Tuna, Eigenfunction expansion for singular Sturm-Liouville problems with transmission conditions, Electronic Journal of Differential Equations 3 (2019) 1–10.
M. Yücel, O. Mukhtarov, K. Aydemir, Computation of eigenfunctions of nonlinear boundaryvalue-transmission problems by developing some approximate techniques, Boletim da Sociedade Paranaense de Matem´atica 3 (41) (2023) 1–12.
P. A. Binding, P. J. Browne, B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II, Journal of Computational and Applied Mathematics (148) (2002) 147–168.

M. Kandemir, O. Sh. Mukhtarov, Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions, Electronic Journal of Differential Equations 2017 (11) (2017) 1–12.
Y. A. Küçükevcilioğlu, E. Bayram , G. G. ¨Ozbey, On the spectral and scattering properties of eigenparameter dependent discrete impulsive Sturm-Liouville equations, Turkish Journal of Mathematics 45(2) (2021) 988–1000.
O. Sh. Mukhtarov, M. Yücel, K. Aydemir, A new generalization of the differential transform method for solving boundary value problems, Journal of New Results in Science 10 (2) (2021) 49–58.
O.Sh. Mukhtarov, H. Olğar, K. Aydemir, Resolvent operator and spectrum of new type boundary value problems, Filomat 29 (7) (2015) 1671–1680.
O. Sh. Mukhtarov, M. Yücel, K. Aydemir, Treatment a new approximation method and its justification for Sturm-Liouville problems, Complexity 2020 (2020) Article ID 8019460 8 pages.
H. Olğar, Self-adjointness and positiveness of the differential operators generated by new type Sturm-Liouville problems, Cumhuriyet Science Journal 40 (1) (2019) 24–34.
E. Şen, M. Açıkgöz, S. Aracı, Spectral problem for Sturm-Liouville operator with retarded argument which contains a spectral parameter in the boundary condition, Ukrainian Mathematical Journal 68(8) (2017) 1263–1277.
E. Uğurlu, K. Taş, A new method for dissipative dynamic operator with transmission conditions, Complex Analysis and Operator Theory 12(4) (2018) 1027–1055.
A. Wang, J. Sun, X. Hao, S. Yao, Completeness of eigenfunctions of Sturm-Liouville problems with transmission conditions, Methods and Application of Analysis 16 (3) (2009) 299–312.
N. B. Kerimov, R. G. Poladov, Basis properties of the system of eigenfunctions in the Sturm- Liouville problem with a spectral parameter in the boundary conditions, Doklady Mathematics (2012) 85 (1) 8–13.
O. Sh. Mukhtarov, K. Aydemir, Basis properties of the eigenfunctions of two-interval SturmLiouville problems, Analysis and Mathematical Physics 9 (2019) 1363–1382.
H. Olğar, O. Sh. Mukhtarov, Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions, Journal of Mathematical Physics 58 (2017) Article ID 042201.
H. Olğar, O. Sh. Mukhtarov, K. Aydemir, Some properties of eigenvalues and generalized eigenvectors of one boundary value problem, Filomat 32 (3) (2018) 911–920.
A. M. Sarsenbi, A. A. Tengaeva, On the basis properties of root functions of two generalized eigenvalue problems, Differential Equations 48 (2) (2012) 306–308.
M. V. Keldysh, On the characteristics values and characteristics functions of a certain class of nonselfadjoint equations (in Russian), Doklady Akademii Nauk (77) (1951) 11–14.
I. Gohberg, S. Goldberg, Basic operator theory, Birkhauser, Boston-Basel-Stuttgart, 1981.
O. A. Ladyzhenskaia, The boundary value problems of mathematical physics, Springer-Verlag, New York 1985.
A. S. Markus, Introduction to the spectral theory of polynomial pencils, Translation of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1988.
L. Rodman, An introduction to operator polynomials, Birkha¨user Verlag, Boston, Massachusetts, 1989.
B. P. Belinskiy, J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Mathematics (56) (1998) 521–541.
E. Kreyszig, Introductory functional analysis with application, New-York, 1978.

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis, Applied Mathematics (Other)

Journal Section

Research Article

Authors

Hayati Olğar ^*
0000-0003-4732-1605
Türkiye

Oktay Mukhtarov
0000-0001-7480-6857
Türkiye

Publication Date

December 31, 2024

Submission Date

October 17, 2024

Acceptance Date

November 24, 2024

Published in Issue

Year 2024 Volume: 13 Number: 3

DOI

https://doi.org/10.54187/jnrs.1569468

IZ

https://izlik.org/JA43EL55LF

APA

Olğar, H., & Mukhtarov, O. (2024). The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points. Journal of New Results in Science, 13(3), 221-231. https://doi.org/10.54187/jnrs.1569468

AMA

1.Olğar H, Mukhtarov O. The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points. JNRS. 2024;13(3):221-231. doi:10.54187/jnrs.1569468

Chicago

Olğar, Hayati, and Oktay Mukhtarov. 2024. “The Basis Property of Generalized Eigenfunctions for One Boundary Value Problem With Discontinuities at Two Interior Points”. Journal of New Results in Science 13 (3): 221-31. https://doi.org/10.54187/jnrs.1569468.

EndNote

Olğar H, Mukhtarov O (December 1, 2024) The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points. Journal of New Results in Science 13 3 221–231.

IEEE

[1]H. Olğar and O. Mukhtarov, “The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points”, JNRS, vol. 13, no. 3, pp. 221–231, Dec. 2024, doi: 10.54187/jnrs.1569468.

ISNAD

Olğar, Hayati - Mukhtarov, Oktay. “The Basis Property of Generalized Eigenfunctions for One Boundary Value Problem With Discontinuities at Two Interior Points”. Journal of New Results in Science 13/3 (December 1, 2024): 221-231. https://doi.org/10.54187/jnrs.1569468.

JAMA

1.Olğar H, Mukhtarov O. The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points. JNRS. 2024;13:221–231.

MLA

Olğar, Hayati, and Oktay Mukhtarov. “The Basis Property of Generalized Eigenfunctions for One Boundary Value Problem With Discontinuities at Two Interior Points”. Journal of New Results in Science, vol. 13, no. 3, Dec. 2024, pp. 221-3, doi:10.54187/jnrs.1569468.

Vancouver

1.Hayati Olğar, Oktay Mukhtarov. The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points. JNRS. 2024 Dec. 1;13(3):221-3. doi:10.54187/jnrs.1569468