Abstract
In this paper a singular dissipative impulsive boundary value problem with \(n\)-impulsive points is investigated. In particular, using the Lidskiĭ’s theorem it is proved that all eigen and associated functions of this problem is complete in the Hilbert space.
Similar content being viewed by others
References
V. Lakshmikantham, D.D. Bainov, P.S. Simenov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, vol. 6 (World Scientific Publishing Co., Inc., Teaneck, NJ, 1989)
J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math. Z 133, 301–312 (1973)
D.B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition. Quart. J. Math. Oxf. Ser. 30, 33–42 (1979)
C.T. Fulton, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinburgh Sect. A 87, 1–34 (1980)
E. Tunç, O.Sh. Muhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions. Appl. Math. Comp. 157, 347–355 (2004)
Z. Akdoğan, M. Demirci, O.Sh. Mukhtarov, Green function of discontinuous boundary-value problem with transmission conditions. Math. Met. Appl. Sci. 30, 1719–1738 (2007)
O.Sh. Mukhtarov, M. Kadakal, Some spectral properties of one Sturm–Liouville type problem with discontinous weight. Siberian Math. J. 46(4), 681–694 (2005)
O.Sh. Mukhtarov, M. Kadakal, Discontinuous Sturm–Liouville problems containing eigenparameter in the boundary conditions. Acta Math. Sinica. Eng. Ser. 22(5), 1519–1528 (2006)
J.-J. Ao, J. Sun, M.-Z. Zhang, Matrix representations of Sturm–Liouville problems with transmission conditions. CAMWA 63(8), 1335–1348 (2012)
I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Israel Program for Scientific Translations, Jerusalem, 1965)
A.R. Sims, Secondary conditions for linear differential operators of the second order. J. Math. Mech. 6, 247–285 (1957)
M.V. Keldysh, On the completeness of the eigenfunctions of some classes of non self-adjoint linear operators. Soviet Math. Dokl. 77, 11–14 (1951)
I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (American Mathematical Society, Providence, 1969)
M.A. Naimark, Linear Differential Operators, 2nd edn. (Nauka, Moscow, 1969); English transl. of 1st edn., Parts 1, 2 (Ungar, New York, 1967, 1968)
B.Sz. Nagy, C. Foiaş, Harmonic Analysis of Operators on Hilbert Space (Academia Kioda, Budapest, 1970)
G.Sh. Guseinov, H. Tuncay, The determinants of perturbation connected with a dissipative Sturm–Liouville operators. J. Math. Anal. Appl. 194, 39–49 (1995)
E. Bairamov, A.M. Krall, Dissipative operators generated by the Sturm–Liouville expression in the Weyl limit circle case. J. Math. Anal. Appl. 254, 178–190 (2001)
G. Guseinov, Completeness theorem for the dissipative Sturm–Liouville operator. Doga-Tr. J. Math. 17, 48–54 (1993)
Z. Wang, H. Wu, Dissipative non-self-adjoint Sturm–Liouville operators and completeness of their eigenfunctions. J. Math. Anal. Appl. 394, 1–12 (2012)
B. P. Allahverdiev, On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl’s limit-circle case. Math. USSR Izvestiya 36, 247–262 (1991)
B.P. Allahverdiev, A dissipative singular Sturm–Liouville problem with a spectral parameter in the boundary condition. J. Math. Anal. Appl. 316, 510–524 (2006)
E. Bairamov, E. Ugurlu, The determinants of dissipative Sturm–Liouville operators with transmission conditions. Math. Comput. Model. 53, 805–813 (2011)
E. Bairamov, E. Ugurlu, On the characteristic values of the real component of a dissipative boundary value transmission problem. Appl. Math. Comput. 218, 9657–9663 (2012)
E. Bairamov, E. Ugurlu, Krein’s theorems for a dissipative boundary value transmission problem. Complex Anal. Oper. Theory. doi:10.1007/s11785-011-0180-z
F. Smithies, Integral Equations (Cambridge University Press, Cambridge, 1958)
E. Prugovečki, Quantum Mechanics in Hilbert Space, 2nd edn. (Academic Press, New York, 1981)
E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, Part 1, 2nd edn. (Oxford University Press, Oxford, 1962)
F.V. Atkinson, Discrete and Continuous Boundary Problems (Academic Press, New York, 1964)
B.J. Harris, Limit-circle criteria for second order differential expression. J. Math. Oxf. Ser. 35(2), 415–427 (1984)
W.N. Everitt, I.W. Knowles, T.T. Read, Limit-point and limit-circle criteria for Sturm–Liouville equations with intermittently negative principal coefficient. Proc. R. Soc. Edinb. Sect. A. 103, 215–228 (1986)
C.T. Fulton, Parametrization of Titchmarsh’s \( m(\lambda )\)- functions in the limit circle case. Trans. Am. Math. Soc. 229, 51–63 (1977)
A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121 (American Mathematical Society, Providence, RI, 2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Uğurlu, E., Bairamov, E. Dissipative operators with impulsive conditions. J Math Chem 51, 1670–1680 (2013). https://doi.org/10.1007/s10910-013-0172-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-013-0172-5