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

Research Article

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

Year 2017, , 535 - 543, 30.09.2017

Yaşar Çakmak Baki Keskin

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

Cited By: 2

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.

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

Year 2017, , 535 - 543, 30.09.2017

Yaşar Çakmak Baki Keskin

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

Cited By: 2

Abstract

Bu makalede, sonlu sayıda süreksizlik koşullarına sahip
ve sınır koşulları spektral parametreye lineer şekilde bağlı Sturm–Liouville
operatörlerin ters spektral problemlerinin çözümü için bazı teklik teoremleri
ispatlayacağız.

Keywords

Ters problem, Sturm-Liouville operatör, Weyl fonksiyonu, Prüfer Açısı

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.

There are 27 citations in total.

Details

Journal Section	Articles
Authors	Yaşar Çakmak Baki Keskin
Publication Date	September 30, 2017
Submission Date	August 3, 2017
Acceptance Date	September 14, 2017
Published in Issue	Year 2017

Cite

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

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