Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

Jianfang Qin; Kun Li; Zhaowen Zheng; Jinming Cai; Jianfang Qin; Kun Li; Zhaowen Zheng; Jinming Cai

doi:10.3934/math.2022512

AIMS Mathematics

2022, Volume 7, Issue 5: 9247-9260. doi: 10.3934/math.2022512

Previous Article Next Article

Research article

Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

Received: 25 November 2021 Revised: 13 February 2022 Accepted: 22 February 2022 Published: 09 March 2022
MSC : 34B05, 34L30, 47E05

This paper is concerned with a fourth-order differential operator with eigenparameter dependent boundary conditions. We prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch. Furthermore, the differential expressions of the eigenvalues with respect to each of parameters are given.
- fourth-order differential operator,
- eigenparameter-dependent boundary condition,
- dependence of eigenvalue,
- differential expression
Citation: Jianfang Qin, Kun Li, Zhaowen Zheng, Jinming Cai. Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions[J]. AIMS Mathematics, 2022, 7(5): 9247-9260. doi: 10.3934/math.2022512

Related Papers:

Abstract

This paper is concerned with a fourth-order differential operator with eigenparameter dependent boundary conditions. We prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch. Furthermore, the differential expressions of the eigenvalues with respect to each of parameters are given.

References

[1]	A. A. Abramov, A. L. Duischko, N. B. Konyukhova, T. V. Pak, B. S. Pariiskii, Evaluation of prolate spheroidal function by solving the corresponding differential equations, U.S.S.R. Comput. Math. Math. Phys., 24 (1984), 1–11. https://doi.org/10.1016/0041-5553(84)90110-1 doi: 10.1016/0041-5553(84)90110-1
[2]	A. Zettl, Sturm-Liouville theory, Providence: Mathematical Surveys and Monographs, American Mathematics Society, 2005. http://dx.doi.org/10.1090/surv/121
[3]	P. A. Binding, P. J. Browne, B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Comput. Appl. Math., 148 (2002), 147–168. https://doi.org/10.1016/S0377-0427(02)00579-4 doi: 10.1016/S0377-0427(02)00579-4
[4]	E. Uǧurlu, E. Bairamov, Spectral analysis of eigenparameter dependent boundary value transmission problems, J. Math. Anal. Appl., 413 (2014), 482–494. https://doi.org/10.1016/j.jmaa.2013.11.022 doi: 10.1016/j.jmaa.2013.11.022
[5]	J. Cai, Z. Zheng, Matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions and transmission conditions, Math. Method. Appl. Sci., 41 (2018), 3495–3508. https://doi.org/10.1002/mma.4842 doi: 10.1002/mma.4842
[6]	L. Zhang, J. Ao, On a class of inverse Sturm-Liouville problems with eigenparameter-dependent boundary conditions, Appl. Math. Comput., 362 (2019), 124553. https://doi.org/10.1016/j.amc.2019.06.067 doi: 10.1016/j.amc.2019.06.067
[7]	C. Gao, Y. Wang, L. Lv, Spectral properties of discrete Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions, Acta Math. Sci., 40 (2020), 755–781. https://doi.org/10.1007/s10473-020-0312-5 doi: 10.1007/s10473-020-0312-5
[8]	J. B. Amara, A. A. Vladimirov, On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions, J. Math. Sci., 150 (2008), 2317–2325. https://doi.org/10.1007/s10958-008-0131-z doi: 10.1007/s10958-008-0131-z
[9]	M. Möller, B. Zinsou, Spectral asymptotics of self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions, Complex Anal. Oper. Th., 6 (2012), 799–818. https://doi.org/10.1007/s11785-011-0162-1 doi: 10.1007/s11785-011-0162-1
[10]	S. Currie, A. D. Love, Hierarchies of difference boundary value problems II-Applications, Quaest. Math., 37 (2014), 371–392. https://doi.org/10.1080/10236198.2013.778841 doi: 10.1080/10236198.2013.778841
[11]	C. Gao, X. Li, R. Ma, Eigenvalues of a linear fourth-order differential operator with squared spectral parameter in a boundary condition, Mediterr. J. Math., 15 (2018), 1–14. https://doi.org/10.1007/s00009-018-1148-2 doi: 10.1007/s00009-018-1148-2
[12]	Q. Kong, A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differ. Equ., 131 (1996), 1–19. https://doi.org/10.1006/jdeq.1996.0154 doi: 10.1006/jdeq.1996.0154
[13]	Q. Kong, A. Zettl, Dependence of eigenvalues of Sturm-Liouville problems on the boundary, J. Differ. Equ., 126 (1996), 389–407. https://doi.org/10.1006/jdeq.1996.0056 doi: 10.1006/jdeq.1996.0056
[14]	Q. Kong, H. Wu, A. Zettl, Dependence of the nth Sturm-Lioouville eigenvalue on the problem, J. Differ. Equ., 156 (1999), 328–354. https://doi.org/10.1006/jdeq.1998.3613 doi: 10.1006/jdeq.1998.3613
[15]	O. Sh. Mukhtarov, E. Tunc, Eigenvalue problems for Sturm-Liouville equations with transmission conditions, Israel J. Math., 144 (2004), 367–380. https://doi.org/10.1007/BF02916718 doi: 10.1007/BF02916718
[16]	M. Zhang, Y. Wang, Dependence of eigenvalues of Sturm-Liouville problems with interface conditions, Appl. Math. Comput., 265 (2015), 31–39. https://doi.org/10.1016/j.amc.2015.05.002 doi: 10.1016/j.amc.2015.05.002
[17]	K. Li, J. Sun, X. Hao, Eigenvalues of regular fourth-order Sturm-Liouville problems with transmission conditions, Math. Method. Appl. Sci., 40 (2017), 3538–3551. https://doi.org/10.1002/mma.4243 doi: 10.1002/mma.4243
[18]	E. Uǧurlu, Regular third-order boundary value problems, Appl. Math. Comput., 343 (2019), 247–257. https://doi.org/10.1016/j.amc.2018.09.046 doi: 10.1016/j.amc.2018.09.046
[19]	M. Zhang, K. Li, Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions, Appl. Math. Comput., 378 (2020), 125214. https://doi.org/10.1016/j.amc.2020.125214 doi: 10.1016/j.amc.2020.125214
[20]	B. Zinsou, Dependence of eigenvalues of fourth-order boundary value problems with transmission conditions, Rocky Mt. J. Math., 50 (2020), 369–381. https://doi.org/10.1216/rmj.2020.50.369 doi: 10.1216/rmj.2020.50.369
[21]	L. Greenberg, M. Marletta, The code SLEUTH for solving fourth order Sturm-Liouville problems, ACM T. Math. Software, 23 (1997), 453–493. https://doi.org/10.1145/279232.279231 doi: 10.1145/279232.279231
[22]	P. B. Bailey, W. N. Everitt, A. Zettl, The SLEIGN2 Sturm-Liouville code, ACM T. Math. Software, 27 (2001), 143–192. https://doi.org/10.1145/383738.383739 doi: 10.1145/383738.383739
[23]	Z. S. Aliyev, S. B. Guliyeva, Propreties of natural frequencies and harmonic bending vibrations of a rod at one end of which is concentrated inertial load, J. Differ. Equ., 263 (2017), 5830–5845. https://doi.org/10.1016/j.jde.2017.07.002 doi: 10.1016/j.jde.2017.07.002
[24]	Z. S. Aliyev, G. T. Mamedova, Some properties of eigenfunctions for the equation of vibrating beam with a spectral parameter in the boundary conditions, J. Differ. Equ., 269 (2020), 1383–1400. https://doi.org/10.1016/j.jde.2020.01.010 doi: 10.1016/j.jde.2020.01.010
[25]	X. Zhang, The self-adjointness, dissipation and spectrum analysis of some classes high order differential operators with discontinuity (in Chinese), Ph. D. thesis, Inner Mongolia University, 2013, 38–52.
[26]	J. Liu, Spectral theory of ordinary differential operators (in Chinese), Beijing: Science Press, 2009.
[27]	Q. Kong, A. Zettl, Linear ordinary differential equations, in inequalities and applications, Singapore: World Scientific Series in Applicable Analysis, 1994. https://doi.org/10.1142/9789812798879_0031
[28]	J. Dieudonné, Foundations of modern analysis, New York: Academis Press, 1969. https://doi.org/10.2307/3613136

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)