Uniqueness Theorems for Sturm-Liouville Operator with Parameter Dependent Boundary Conditions and Finite Number Of Transmission Conditions

Yaşar Çakmak; Baki Keskin

doi:10.17776/csj.340505

EN TR

Uniqueness Theorems for Sturm-Liouville Operator with Parameter Dependent Boundary Conditions and Finite Number Of Transmission Conditions

Abstract

In this paper, we prove some uniqueness theorems for the solution of inverse spectral problems of Sturm–Liouville operators with boundary conditions depending linearly on the spectral parameter and with a finite number of transmission conditions.

Keywords

Inverse problem,Sturm-Liouville operator,Weyl function,Prüfer Angle

References

[1]. Ambartsumyan, V. A., Uber eine frage der eigenwerttheorie, Zeitschrift für Physik, 1929, 53, 690–695.
[2]. Borg, G., Eine umkehrung der Sturm–Liouvilleschen eigenwertaufgabe. Bestimmung der differentialgleichung durch die eigenwerte, Acta Math., 1946, 78, 1–96.
[3]. Benedek, A. and Panzone, R., On inverse eigenvalue problems for a second-order differential equations with parameter contained in the boundary conditions, Notas de Algebra y Analisis, 1980, 9, 1–13.
[4]. Binding, P. A., Browne, P. J. and Watson, B. A., Inverse spectral problems for Sturm–Liouville equations with eigenparameter dependent boundary conditions, J. London Math. Soc., 2000, 62, 161–182.
[5]. Browne, P. J. and Sleeman, B. D., A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems, Inverse Problems, 1997, 13, 1453-1462.
[6]. Chugunova, M. V., Inverse spectral problem for the Sturm–Liouville operator with eigenvalue parameter dependent boundary conditions, Oper. Theory: Adv. Appl., 2001, 123 (Basel: Birkhauser), 187–94.
[7]. Fulton, C. T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, 1977, A77, 293-308.
[8]. Fulton, C. T., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, 1980, A87, 1-34.

[9]. Keskin, B., Ozkan, A. S. and Yalçın, N., Inverse spectral problems for discontinuous Sturm-Liouville operator with eigenparameter dependent boundary conditions, Commun. Fac. Sci. Univ. Ank. Series A1, 2011, 60(1), 1, 15–25.
[10]. Ozkan, A. S. and Keskin, B., Spectral problems for Sturm–Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter, Inverse Problems in Science and Engineering, 2012, 20(6), 799–808.
[11]. Ozkan, A. S., Keskin, B. and Cakmak, Y., Double Discontinuous Inverse Problems for Sturm-Liouville Operator with Parameter-Dependent Conditions, Abstract and Applied Analysis, 2013, Article ID 794262, p.7.
[12]. Walter, J., Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z., 1973, 133, 301-312.
[13]. Wang, Y. P., Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter, Mathematical Methods in the Applied Sciences, 2013, 36(7), 857-868.
[14]. Yang, C. F. and Huang, Z. Y., A half-inverse problem with eigenparameter dependent boundary conditions, Numerical Functional Analysis and Optimization, 2010, 31(6), 754–762.
[15]. Binding, P. A., Browne, P. J. and Seddighi, K., Sturm–Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc.,1993, 2(37), 57–72.
[16]. Binding, P. A. and Browne, P. J., Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter-dependent boundary conditions, Proc. R. Soc. Edinburgh A, 1997, 127, 1123-1136.
[17]. Binding, P. A., Browne, P. J. and Watson, B. A., Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl., 2004, 291, 246–261.
[18]. Mennicken, R., Schmid, H. and Shkalikov, A. A., On the eigenvalue accumulation of Sturm-Liouville problems depending nonlinearly on the spectral parameter, Math. Nachr., 1998, 189, 157-170.
[19]. Schmid, H. and Tretter, C., Singular Dirac systems and Sturm–Liouville problems nonlinear in the spectral parameter, Journal of Differential Equations, 2002, 181(2), 511-542.
[20]. Altinisik, N., Kadakal, M. and Mukhtarov, O. Sh., Eigenvalues and eigenfunctions of discontinuous Sturm–Liouville problems with eigenparameter dependent boundary conditions, Acta Math. Hungar., 2004, 102(1–2), 159–175.
[21]. Amirov, R. Kh., Ozkan, A. S. and Keskin, B., Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions, Integral Transforms and Special Functions, 2009, 20(8), 607-618.
[22]. Freiling, G. and Yurko, V., Inverse Sturm–Liouville problems and their applications, Nova Science, New York, 2001.
[23]. Hald, O. H., Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 1984, 37, 539-577.
[24]. Ozkan, A. S., Inverse Sturm–Liouville problems with eigenvalue dependent boundary and discontinuity conditions, Inverse Probl. Sci. Eng., 2012, 20(6), 857–868.
[25]. Yurko, V. A., Integral transforms connected with discontinuous boundary value problems, Integral Transforms Spec. Funct., 2000, 10, 141–164.
[26]. Yurko ,V. A., Method of spectral mappings in the inverse problem theory, Inverse Ill-posed Problems Ser., VSP, Utrecht, 2002.
[27]. Guldu, Y., Inverse Eigenvalue Problems for a Discontinuous Sturm-Liouville Operator with Two Discontinuities, Boundary Value Problems, 2013, DOI:10.1186/1687-2770-2013-209.

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Yaşar Çakmak

Baki Keskin

Publication Date

September 30, 2017

Submission Date

August 3, 2017

Acceptance Date

September 14, 2017

Published in Issue

Year 1970 Volume: 38 Number: 3

DOI

https://doi.org/10.17776/csj.340505

IZ

https://izlik.org/JA78HU82FX

Cite

RIS / Bibtex

APA

Çakmak, Y., & Keskin, B. (2017). Uniqueness Theorems for Sturm-Liouville Operator with Parameter Dependent Boundary Conditions and Finite Number Of Transmission Conditions. Cumhuriyet Science Journal, 38(3), 535-543. https://doi.org/10.17776/csj.340505

Cited By

Computation of Trace and Nodal Points of Eigenfunctions for a Sturm-Liouville Problem with Retarded Argument

Cumhuriyet Science Journal

https://doi.org/10.17776/csj.416602

An Inverse Three Spectra Problem for Parameter-Dependent and Jumps Conformable Sturm–Liouville Operators

Bulletin of the Malaysian Mathematical Sciences Society

https://doi.org/10.1007/s40840-023-01610-2

Uniqueness Theorems for Sturm-Liouville Operator with Parameter Dependent Boundary Conditions and Finite Number Of Transmission Conditions

Uniqueness Theorems for Sturm-Liouville Operator with Parameter Dependent Boundary Conditions and Finite Number Of Transmission Conditions

Abstract

Keywords

Sonlu Sayıda Süreksizlik Koşullarına Sahip ve Sınır Koşulları Parametreye Bağlı Sturm-Liouville Problemi için Teklik Teoremleri

Öz

Anahtar Kelimeler

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite

Cited By

Computation of Trace and Nodal Points of Eigenfunctions for a Sturm-Liouville Problem with Retarded Argument

An Inverse Three Spectra Problem for Parameter-Dependent and Jumps Conformable Sturm–Liouville Operators