Opuscula Math. 43, no. 5 (2023), 651-661
One boundary value problem including a spectral parameter in all boundary conditions
Ayşe Kabataş
Abstract.
In this paper, asymptotic formulae for solutions and Green's function of a boundary value problem are investigated when the equation and the boundary conditions contain a spectral parameter.
Keywords:
boundary value problems, asymptotics, Green's functions.
Mathematics Subject Classification:
34B08, 34B27, 34L10.
References ▼
- [1] K. Aydemir, Boundary value problems with eigenvalue-dependent boundary and transmission condition, Bound. Value Probl. 2014 (2014), Article no. 131. https://doi.org/10.1186/1687-2770-2014-131
- [2] E. Başkaya, Asymptotics of eigenvalues for Sturm-Liouville problem including eigenparameter-dependent boundary conditions with integrable potential, NTMSCI 6 (2018), 39-47. https://doi.org/10.20852/ntmsci.2018.292
- [3] E. Başkaya, Asymptotics of eigenvalues for Sturm-Liouville problem including quadratic eigenvalue in the boundary condition, NTMSCI 6 (2018), 76-82. https://doi.org/10.20852/ntmsci.2018.296
- [4] E. Başkaya, Asymptotics of eigenvalues for Sturm-Liouville problem with eigenparameter dependent-boundary conditions, NTMSCI 6 (2018), 247-257. https://doi.org/10.20852/ntmsci.2018.288
- [5] E. Başkaya, Periodic and semi-periodic eigenvalues of Hill's equation with symmetric double well potential, TWMS J. App. Eng. Math. 10 (2020), 346-352.
- [6] E. Başkaya, Asymptotic eigenvalues of regular Sturm-Liouville problems with spectral parameter-dependent boundary conditions and symmetric single well potential, Turk. J. Math. Comput. Sci. 13 (2021), 44-50. https://doi.org/10.47000/tjmcs.851839
- [7] E. Başkaya, On the gaps of Neumann eigenvalues for Hill's equation with symmetric double well potential, Tbillisi Math. J. 8 (2021), 139-145.
- [8] P.A. Binding, P.J. Browne, B.A. Watson, Sturm-Liouville problems with reducible boundary conditions, Proc. Edinburgh Math. Soc. 49 (2006), 593-608. https://doi.org/10.1017/S0013091505000131
- [9] H. Coşkun, E. Başkaya, Asymptotics of eigenvalues of regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential, Math. Scand. 107 (2010), 209-223. https://doi.org/10.7146/math.scand.a-15152
- [10] H. Coşkun, E. Başkaya, Asymptotics of eigenvalues for Sturm-Liouville problem with eigenvalue in the boundary condition for differentiable potential, APAM 16 (2018), 7-19. https://doi.org/10.22457/apam.v16n1a2
- [11] H. Coşkun, E. Başkaya, A. Kabataş, Instability intervals for Hill's equation with symmetric single well potential, Ukr. Math. J. 71 (2019), 977-983. https://doi.org/10.1007/s11253-019-01692-x
- [12] H. Coşkun, A. Kabataş, Asymptotic approximations of eigenfunctions for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential, Math. Scand. 113 (2013), 143-160.
- [13] H. Coşkun, A. Kabataş, Green's function of regular Sturm-Liouville problem having eigenparameter in one boundary condition, Turkish J. Math. and Comput. Sci. 4 (2016), 1-9.
- [14] H. Coşkun, A. Kabataş, E. Başkaya, On Green's function for boundary value problem with eigenvalue dependent quadratic boundary condition, Bound. Value Probl. 2017 (2017), Article no. 71. https://doi.org/10.1186/s13661-017-0802-0
- [15] D.G. Duffy, Green’s Functions with Applications, Chapman Hall/CRC, New York, 2015. https://doi.org/10.1201/9781315371412
- [16] C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293-308. https://doi.org/10.1017/S030821050002521X
- [17] B.J. Harris, The form of the spectral functions associated with Sturm-Liouville problems with continuous spectrum, Mathematika 44 (1997), 149-161. https://doi.org/10.1112/S0025579300012043
- [18] H. Hochstadt, Estimates on the stability intervals for Hill's equation, Proc. Am. Math. Soc. 14 (1963), 930-932.
- [19] Ch.G. Ibadzadeh, I.M. Nabiev, Reconstruction of the Sturm-Liouville operator with nonseparated boundary conditions and a spectral parameter in the boundary condition, Ukr. Math. J. 69 (2018), 1416-1423. https://doi.org/10.1007/s11253-018-1440-0
- [20] A. Kabataş, Eigenfunction and Green's function asymptotics for Hill's equation with symmetric single well potential, Ukr. Math. J. 74 (2022), 218-231. https://doi.org/10.1007/s11253-022-02059-5
- [21] A. Kabataş, On eigenfunctions of Hill's equation with symmetric double well potential, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 71 (2022), 634-649. https://doi.org/10.31801/cfsuasmas.974409
- [22] N.B. Kerimov, Kh.R. Mamedov, On one boundary value problem with a spectral parameter in the boundary conditions, Sib. Math. J. 40 (1999), 281-290. https://doi.org/10.1007/s11202-999-0008-5
- [23] H.Y. Zhang, J.J. Ao, M.L. Li, Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and interface conditions, Mediterr. J. Math. 19 (2022), Article no. 90. https://doi.org/10.1007/s00009-021-01943-x
About the authors ▼
Ayşe Kabataş
Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey
About the article ▼
Communicated by Alexander Gomilko.
Received: 2023-01-04.
Revised: 2023-05-02.
Accepted: 2023-05-09.
Published online: 2023-06-24.
Cite this article as ▼
Ayşe Kabataş, One boundary value problem including a spectral parameter in all boundary conditions, Opuscula Math. 43, no. 5 (2023), 651-661,
https://doi.org/10.7494/OpMath.2023.43.5.651.
Download this article's citation as: