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SETSCI - Volume 3 (2018)
ISAS2018-Winter - 2nd International Symposium on Innovative Approaches in Scientific Studies, Samsun, Turkey, Nov 30, 2018

Hilbert Space Method for Sturm-Liouville Problems with Discontinuities
O. Sh.  Mukhtarov 1*, Hayati Olgar2, M. Kandemir3, Kadriye Aydemir4
1AzerbaijanNational Academy of Sciences , Bakü, Azerbijan
2Gaziosmanpaşa University, Tokat, Turkey
3Amasya University, Amasya, Turkey
4Amasya University, Amasya, Turkey
* Corresponding author:
Published Date: 2019-01-14   |   Page (s): 1163-1167   |    202     9

ABSTRACT Boundary-value problems with eigenparameter dependent boundary conditions is one of the most extensively developing
fields in pure and applied mathematics. First, we cite the works of Walter and Fulton, both of which have extensively bibliographies,
a discussion of physical applications. Walter had given an Hilbert space formulation of such type eigenvalue problems and obtained
the expansion result using the selfadjointness of the operator associated with the Sturm-Liouville problem. In recent years there
has been increasing interest of Sturm-Liouville type problems with supplementary transmission conditions. For example, the first
author of this study and some others have developed the spectral theory of discontinuous Sturm-Liouville problems with additional
transmission conditions at the interior points. The concept of generalized solutions in a Hilbert space allows the eigenvalue problem
to be reduced to an operator-pencil equation. In this study, we prove that the weak eigenfunctions of the Sturm-Liouville problem
with additional transmission conditions form a Riesz basis of the corresponding Hilbert space. First, the generalized solution of
the Sturm-Liouville problem is defined as an element of a direct sum spaces with satisfies some integral equalities. Second, using
the Riesz representation theorem these equalities are reduced to an operator-pencil equation. Finally, it is established that the
eigenfunctions of the original boundary-value-transmission problem form a Riesz basis of suitable Hilbert space.  
KEYWORDS Sturm-Liouville problems, transmission conditions, eigenvalue, discontinuity, Hilbert space
REFERENCES [1] B. P. Allahverdiev, E. Bairamov and E. Ugurlu, Eigenparameter dependent Sturm-Liouville problems in boudary conditions with transmission conditions, J. Math. Anal. Appl. 401, 388-396 (2013)
[2] K. Aydemir and O. Mukhtarov, Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions, Advances in Difference Equations, 2016:76 (2016)
[3] K. Aydemir, H. Olgar, O. Sh. Mukhtarov and F. S. Muhtarov, Di ˘ fferential operator equations with interface conditions in modified direct sum spaces, Filomat, 32:3 (2018), 921-931.
[4] B. P. Belinskiy and J. P. Dauer, On a regular Sturm - Liouville problem on a finite interval with the eigenvalue parameter appearing linearly in the boundary conditions, Spectral theory and computational methods of Sturm Liouville problem. Eds. D. Hinton and P. W. Schaefer, 1997.
[5] B. P. Belinskiy and J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Math. 56 (1998), 521-541.
[6] C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edin. 77A(1977), P. 293-308.
[7] I. C. Gohberg, M. G. Krein, Introduction to The Theory of Linear Non-Selfadjoint Operators, Translation of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, Rhode Island, 1969.
[9] M. Kandemir and O. Sh. Mukhtarov, Nonlocal Sturm-Liouville Problems with Integral Terms in the Boundary Conditions, Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.
[10] M. V. Keldysh, On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Russian Mathematical Surveys, 26(4), 15-44 (1971).
[11] E.Kreyszig, Introductory Functional Analysis Whit Application, New-York, 1978.
[12] O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York 1985.
[13] O. Sh. Mukhtarov, H. Olgar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value ˇ Problems, Filomat, 29:7 (2015), 1671-1680.
[14] O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Sib. Math. J., Vol. 46, 681-694 (2005)
[15] O. Sh. Mukhtarov and M. Kandemir, Asymptotic behaviour of eigenvalues for the discontinuous boundaryvalue problem with Functional-Transmissin conditions, Acta Mathematica Scientia vol 22 B(3), pp.335-345 (2002)
[16] O.Sh. Mukhtarov and S. Yakubov, Problems for ordinary differential equations with transmission conditions, Appl. Anal., Vol 81, 1033-1064 (2002)
[17] H. Olgar and O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together ˇ With Interaction Conditions, Journal of Mathematical Physics, 58, 042201 (2017) DOI: 10.1063/1.4979615.
[18] H. Olgar, O. Sh. Mukhtarov and K. Aydemir, Some properties of eigenvalues and generalized eigenvectors ˇ of one boundary value problem, Filomat, 32:3 (2018), 911-920.
[19] L. Rodman, An Introduction to Operator Polynomials, Birkhauser Verlag, Boston, Massachusetts, 1989.
[20] E. C. Titchmarsh, Eigenfunctions Expansion Associated with Second Order Differential Equations I, second edn. Oxford Univ. Press, London, 1962.
[21] J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133(1973), 301-312.
[22] A. Wang, J. Sun and A. Zettl, Two interval Sturm - Liouville operators in modified Hilbert spaces, J. Math. Anal. Appl., 328, 390-399 (2007)

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