Eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary and interface conditions
Abstract
In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.
Keyword : Sturm-Liouville problems, interface conditions, continuity, differential expression
How to Cite
Zheng, J., Li, K., & Zheng, Z. (2023). Eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary and interface conditions. Mathematical Modelling and Analysis, 28(3), 374–392. https://doi.org/10.3846/mma.2023.17094
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Y. Bai, W. Wang, K. Li and Z. Zheng. Eigenvalues of a class of eigenparameter dependent third-order differential operators. Journal of Nonlinear Mathematical Physics, 29(3):477–492, 2022. https://doi.org/10.1007/s44198-022-00032-1
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N.J. Guliyev. Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter. Journal of Mathematical Physics, 60(6):063501, 2019. https://doi.org/10.1063/1.5048692
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Mathematische Nachrichten, 188(1):173–201, 1997. https://doi.org/10.1002/mana.19971880111
Q. Kong and A. Zettl. Dependence of eigenvalues of Sturm-Liouville problems on the boundary. Journal of Differential Equations, 126(2):389–407, 1996. https://doi.org/10.1006/jdeq.1996.0056
Q. Kong and A. Zettl. Eigenvalues of regular Sturm-Liouville problems. Journal of Differential Equations, 131(1):1–19, 1996. https://doi.org/10.1006/jdeq.1996.0154
K. Li, J. Sun and X. Hao. Eigenvalues of regular fourth-order Sturm-Liouville problems with transmission conditions. Mathematical Methods in the Applied Sciences, 40(10):3538–3551, 2017. https://doi.org/10.1002/mma.4243
A.V. Likov and Y. Mikhailov. The Theory of Heat and Mass Transfer. Qosenergaizdat, 1963. (in Russian)
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O.Sh. Mukhtarov and K. Aydemir. Oscillation properties for nonclassical Sturm-Liouville problems with additional transmission conditions. Mathematical Modelling and Analysis., 26(3):432–443, 2021. https://doi.org/10.3846/mma.2021.13216
O.Sh. Mukhtarov and K. Aydemir. Two-linked periodic Sturm-Liouville problems with transmission conditions. Mathematical Methods in the Applied Sciences, 44(18):14664–14676, 2021. https://doi.org/10.1002/mma.7734
H. Olğar and O.Sh. Mukhtarov. Weak eigenfunctions of two-interval SturmLiouville problems together with interaction conditions. Journal of Mathematical Physics, 58(4):042201, 2017. https://doi.org/10.1063/1.4979615
E. Şen. A class of second-order differential operators with eigenparameterdependent boundary and transmission conditions. Mathematical Methods in the Applied Sciences, 37(18):2952–2961, 2014. https://doi.org/10.1002/mma.3033
E. Tunç and O.Sh. Mukhtarov. Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions. Applied Mathematics and Computation, 157(2):347–355, 2004. https://doi.org/10.1016/j.amc.2003.08.039
E. Uğurlu. Regular third-order boundary value problems. Applied Mathematics and Computation, 343:247–257, 2019. https://doi.org/10.1016/j.amc.2018.09.046
E. Uğurlu. Third-order boundary value transmission problems. Turkish Journal of Mathematics, 43(3):1518–1532, 2019. https://doi.org/10.3906/mat-1812-36
E. Uğurlu and E. Bairamov. Spectral analysis of eigenparameter dependent boundary value transmission problems. Journal of Mathematical Analysis and Applications, 413(1):482–494, 2014. https://doi.org/10.1016/j.jmaa.2013.11.022
N.N. Voitovich, B.Z. Katsenelbaum and A.N. Sivov. Generalized method of eigen-vibration in the theory of diffraction. Nakua, Mockow., 373(22):1901– 2000, 1997. https://doi.org/10.1016/j.physleta.2009.03.060
X.-C. Xu and C.-F Yang. Inverse spectral problems for the Sturm-Liouville operator with discontinuity. Journal of Differential Equations, 262(3):3093– 3106, 2017. https://doi.org/10.1016/j.jde.2016.11.024
A. Zettl. Sturm-Liouville Theory. American Mathematical Society, 2005.
H-Y. Zhang, J-J. Ao and D. Mu. Eigenvalues of discontinuous third-order boundary value problems with eigenparameter-dependent boundary conditions. Journal of Mathematical Analysis and Applications., 506(2):125680, 2022. https://doi.org/10.1016/j.jmaa.2021.125680
M. Zhang and K. Li. Dependence of eigenvalues of Sturm–Liouville problems with eigenparameter dependent boundary conditions. Applied Mathematics and Computation, 378:125214, 2020. https://doi.org/10.1016/j.amc.2020.125214
M. Zhang, J. Sun and A. Zettl. Eigenvalues of limit-point Sturm–Liouville problems. Journal of Mathematical Analysis and Applications, 419(1):627–642, 2014. https://doi.org/10.1016/j.jmaa.2014.05.021
M. Zhang and Y. Wang. Dependence of eigenvalues of Sturm-Liouville problems with interface conditions. Applied Mathematics and Computation, 265:31–39, 2015. https://doi.org/10.1016/j.amc.2015.05.002
Z. Akdoğan, M. Demirci and O.Sh. Mukhtarov. Green function of discontinuous boundary-value problem with transmission conditions. Mathematical Methods in the Applied Sciences, 30(14):1719–1738, 2007. https://doi.org/10.1002/mma.867
B.P. Allahverdiev. Spectral analysis of singular Hamiltonian systems with an eigenparameter in the boundary condition. Electronic Journal of Differential Equations, 2019(2):1–14, 2019. Available from Internet: https://digital.library.txstate.edu/handle/10877/14645
K. Aydemir and O.Sh. Mukhtarov. Class of Sturm-Liouville problems with eigenparameter dependent transmission conditions. Numerical Functional Analysis and Optimization, 38(10):1260–1275, 2017. https://doi.org/10.1080/01630563.2017.1316995
Y. Bai, W. Wang, K. Li and Z. Zheng. Eigenvalues of a class of eigenparameter dependent third-order differential operators. Journal of Nonlinear Mathematical Physics, 29(3):477–492, 2022. https://doi.org/10.1007/s44198-022-00032-1
P. Bailey, W. Everitt and A. Zettl. The SLEIGN2 Sturm-Liouville code. ACM Transactions on Mathematical Software, 27(2):143–192, 2001. https://doi.org/10.1145/383738.383739
P.A. Binding, P.J. Browne and B.A. Watson. Equivalence of inverse SturmLiouville problems with boundary conditions rationally dependent on the eigenparameter. Journal of Mathematical Analysis and Applications, 291(1):246–261, 2004. https://doi.org/10.1016/j.jmaa.2003.11.025
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 77(3-4):293–308, 1977. https://doi.org/10.1017/S030821050002521X
N.J. Guliyev. Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter. Journal of Mathematical Physics, 60(6):063501, 2019. https://doi.org/10.1063/1.5048692
Q. Kong, H. Wu and A. Zettl. Dependence of eigenvalues on the problem.
Mathematische Nachrichten, 188(1):173–201, 1997. https://doi.org/10.1002/mana.19971880111
Q. Kong and A. Zettl. Dependence of eigenvalues of Sturm-Liouville problems on the boundary. Journal of Differential Equations, 126(2):389–407, 1996. https://doi.org/10.1006/jdeq.1996.0056
Q. Kong and A. Zettl. Eigenvalues of regular Sturm-Liouville problems. Journal of Differential Equations, 131(1):1–19, 1996. https://doi.org/10.1006/jdeq.1996.0154
K. Li, J. Sun and X. Hao. Eigenvalues of regular fourth-order Sturm-Liouville problems with transmission conditions. Mathematical Methods in the Applied Sciences, 40(10):3538–3551, 2017. https://doi.org/10.1002/mma.4243
A.V. Likov and Y. Mikhailov. The Theory of Heat and Mass Transfer. Qosenergaizdat, 1963. (in Russian)
O.Sh. Mukhtarov and K. Aydemir. Minimization principle and generalized Fourier series for discontinuous Sturm-Liouville systems in direct sum spaces. Journal of Applied Analysis & Computation, 8(5):1511–1523, 2018. https://doi.org/10.11948/2018.1511
O.Sh. Mukhtarov and K. Aydemir. Oscillation properties for nonclassical Sturm-Liouville problems with additional transmission conditions. Mathematical Modelling and Analysis., 26(3):432–443, 2021. https://doi.org/10.3846/mma.2021.13216
O.Sh. Mukhtarov and K. Aydemir. Two-linked periodic Sturm-Liouville problems with transmission conditions. Mathematical Methods in the Applied Sciences, 44(18):14664–14676, 2021. https://doi.org/10.1002/mma.7734
H. Olğar and O.Sh. Mukhtarov. Weak eigenfunctions of two-interval SturmLiouville problems together with interaction conditions. Journal of Mathematical Physics, 58(4):042201, 2017. https://doi.org/10.1063/1.4979615
E. Şen. A class of second-order differential operators with eigenparameterdependent boundary and transmission conditions. Mathematical Methods in the Applied Sciences, 37(18):2952–2961, 2014. https://doi.org/10.1002/mma.3033
E. Tunç and O.Sh. Mukhtarov. Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions. Applied Mathematics and Computation, 157(2):347–355, 2004. https://doi.org/10.1016/j.amc.2003.08.039
E. Uğurlu. Regular third-order boundary value problems. Applied Mathematics and Computation, 343:247–257, 2019. https://doi.org/10.1016/j.amc.2018.09.046
E. Uğurlu. Third-order boundary value transmission problems. Turkish Journal of Mathematics, 43(3):1518–1532, 2019. https://doi.org/10.3906/mat-1812-36
E. Uğurlu and E. Bairamov. Spectral analysis of eigenparameter dependent boundary value transmission problems. Journal of Mathematical Analysis and Applications, 413(1):482–494, 2014. https://doi.org/10.1016/j.jmaa.2013.11.022
N.N. Voitovich, B.Z. Katsenelbaum and A.N. Sivov. Generalized method of eigen-vibration in the theory of diffraction. Nakua, Mockow., 373(22):1901– 2000, 1997. https://doi.org/10.1016/j.physleta.2009.03.060
X.-C. Xu and C.-F Yang. Inverse spectral problems for the Sturm-Liouville operator with discontinuity. Journal of Differential Equations, 262(3):3093– 3106, 2017. https://doi.org/10.1016/j.jde.2016.11.024
A. Zettl. Sturm-Liouville Theory. American Mathematical Society, 2005.
H-Y. Zhang, J-J. Ao and D. Mu. Eigenvalues of discontinuous third-order boundary value problems with eigenparameter-dependent boundary conditions. Journal of Mathematical Analysis and Applications., 506(2):125680, 2022. https://doi.org/10.1016/j.jmaa.2021.125680
M. Zhang and K. Li. Dependence of eigenvalues of Sturm–Liouville problems with eigenparameter dependent boundary conditions. Applied Mathematics and Computation, 378:125214, 2020. https://doi.org/10.1016/j.amc.2020.125214
M. Zhang, J. Sun and A. Zettl. Eigenvalues of limit-point Sturm–Liouville problems. Journal of Mathematical Analysis and Applications, 419(1):627–642, 2014. https://doi.org/10.1016/j.jmaa.2014.05.021
M. Zhang and Y. Wang. Dependence of eigenvalues of Sturm-Liouville problems with interface conditions. Applied Mathematics and Computation, 265:31–39, 2015. https://doi.org/10.1016/j.amc.2015.05.002