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## On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition

#### Emir Ali MARİS [1] , Sertaç GÖKTAŞ [2]

The spectral problem
$-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1$
$y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),$
is considered, where $\lambda$ is a spectral parameter, $q(x)\in{{L}_{1}}(0,1)$ is a complex-valued function, $a$ and $b$ are arbitrary complex numbers which satisfy the condition $|a|+|b|\ne 0$. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in ${{L}_{p}}(0,1)$) of the above-mentioned Sturm-Liouville problem.
eigenvalues, eigenfunctions, minimal system, basis
• [1] Y.N. Aliyev, On the basis properties of Sturm-Liouville problems with decreasing affine boundary conditions, Proc. IMM of NAS, 24, 35–52, 2006.
• [2] Y.N. Aliyev and N.B. Kerimov, The basis property of Sturm-Liouville problems with boundary conditions depending quadratically on the eigenparameter, Arab. J. Sci. Eng. 33 (1A), 123–136, 2008.
• [3] N.K. Bary, Treatise on Trigonometric Series, Vol II., Macmillian, New York, 1964.
• [4] M.A. Evgrafov, Analytic Function (in Russian), Nauka, Moskow, 1965; trans. W.B. Saunders Comp., Philadephia and London, 1966.
• [5] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Moscow, 1965; Trans. Math. Monogr., Amer. Math. Soc., Rhode Island, 18, 1969.
• [6] S. Goktas, N.B. Kerimov, and E.A. Maris, On the uniform convergence of spectral expansions for a spectral problem with a boundary condition rationally depending on the eigenparameter, J. Korean Math. Soc. 54 (4), 1175–1187, 2017.
• [7] T. Gulsen, E. Yilmaz, and H. Koyunbakan, An inverse nodal problem for differential pencils with complex spectral parameter dependent boundary conditions, New Trends Math. Sci. 5 (1), 137–144, 2017.
• [8] N.Yu. Kapustin and E.I. Moiseev, The basis property in of the systems of eigenfunctions corresponding to two problems with a spectral parameter in the boundary conditions, Diff. Eq. 36 (10), 1498–1501, 2000.
• [9] B.S. Kashin and A.A. Saakyan, Orthogonal Series, Trans. Math. Monogr., Amer. Math. Soc. Providence, 75, 1989.
• [10] N.B. Kerimov and Y.N. Aliyev, The basis property in $L_p(0, 1)$ of the boundary value problem rationally dependent on the eigenparameter, Studia Math. 174 (2), 201–212, 2006.
• [11] N.B. Kerimov and Kh.R. Mamedov, On one boundary value problem with a spectral parameter in the boundary conditions, Siberian Math. J. 40 (2), 325–335, 1999.
• [12] N.B. Kerimov and E.A. Maris, On the basis properties and convergence of expansions in terms of eigenfunctions for a spectral problem with a spectral parameter in the boundary condition, Proc. IMM of NAS (Special Issue) 40, 245–258, 2014.
• [13] N.B. Kerimov and E.A. Maris, On the uniform convergence of the Fourier Series for one spectral problem with a spectral parameter in a boundary condition, Math. Methods Appl. Sci. 39 (9), 2298–2309, 2016.
• [14] N.B. Kerimov and E.A. Maris, On the Uniform Convergence of Fourier Series Expansions for Sturm-Liouville Problems with a Spectral Parameter in the Boundary Conditions, Results Math. 73 (3), 102, 2018.
• [15] N.B. Kerimov and V.S. Mirzoev, On the basis properties of one spectral problem with a spectral parameter in a boundary condition, Siberian Math. J. 44 (5), 813–816, 2003.
• [16] N.B. Kerimov and R.G. Poladov, Basis properties of the system of eigenfunctions in the Sturm- Liouville problem with a spectral parameter in the boundary conditions, Dokl. Math. 85 (1), 8–13, 2015.
• [17] N.B. Kerimov, S. Goktas, and E.A. Maris, Uniform convergence of the spectral expansions in terms of root functions for a spectral problem, Electron. J. Differ. Equ. 80, 1–14, 2016.
• [18] B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Kluwer Academic Publishers: Netherlands, 1991.
• [19] Kh.R. Mamedov, On one boundary value problem with parameter in the boundary conditions, Spectr Theory Oper. Appl. 11, 117–121, 1997 (in Russian).
• [20] Kh.R. Mamedov, On a basic problem for a second order differential equation with a discontinuous coefficient and a spectral parameter in the boundary conditions, Proc. Seventh Internat. Conf. Geometry, Integrability and Quantization, Institute of Biophysics and Biomedical Engineering Bulgarian Academy of Sciences, 218–225, 2006.
• [21] D.B. Marchenkov, On the convergence of spectral expansions of functions for problems with a spectral parameter in a boundary condition, Diff. Eq. 41, 1496–1500, 2005.
• [22] D.B. Marchenkov, Basis property in $L_p(0, 1)$ of the system of eigenfunctions corresponding to a problem with a spectral parameter in the boundary condition, Diff. Eq. 42 (6), 905–908, 2006.
• [23] A. Neamaty and Sh. Akbarpoor, Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition, Inverse Probl. Sci. Eng. 25 (7), 978–994, 2017.
• [24] I. Singer, Bases in Banach Spaces I, Springer-Verlag Berlin Heidelberg, New York, 1970.
• [25] E. Yilmaz and H. Koyunbakan, Reconstruction of potential function and its derivatives for Sturm-Liouville problem with eigenvalues in boundary condition, Inverse Prob. Sci. Eng. 18 (7), 935–944, 2010.
• [26] A. Zygmund, Trigonometric Series, Vol. II, 2nd Ed., Cambridge University Press, New York, 1959.
Primary Language en Mathematics Mathematics Orcid: 0000-0001-7620-8754Author: Emir Ali MARİS Institution: MERSIN UNIVERSITYCountry: Turkey Orcid: 0000-0001-7842-6309Author: Sertaç GÖKTAŞ (Primary Author)Institution: MERSIN UNIVERSITYCountry: Turkey Publication Date : August 6, 2020
 Bibtex @research article { hujms479445, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1373 - 1382}, doi = {10.15672/hujms.479445}, title = {On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition}, key = {cite}, author = {Mari̇s, Emir Ali and Göktaş, Sertaç} } APA Mari̇s, E , Göktaş, S . (2020). On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition . Hacettepe Journal of Mathematics and Statistics , 49 (4) , 1373-1382 . DOI: 10.15672/hujms.479445 MLA Mari̇s, E , Göktaş, S . "On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1373-1382 Chicago Mari̇s, E , Göktaş, S . "On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1373-1382 RIS TY - JOUR T1 - On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition AU - Emir Ali Mari̇s , Sertaç Göktaş Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.479445 DO - 10.15672/hujms.479445 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1373 EP - 1382 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.479445 UR - https://doi.org/10.15672/hujms.479445 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition %A Emir Ali Mari̇s , Sertaç Göktaş %T On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.479445 %U 10.15672/hujms.479445 ISNAD Mari̇s, Emir Ali , Göktaş, Sertaç . "On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition". Hacettepe Journal of Mathematics and Statistics 49 / 4 (August 2020): 1373-1382 . https://doi.org/10.15672/hujms.479445 AMA Mari̇s E , Göktaş S . On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1373-1382. Vancouver Mari̇s E , Göktaş S . On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1373-1382.

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