77
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A new type Sturm-Liouville problem with an abstract linear operator contained in the equation

&
Pages 1931-1948 | Received 02 Jul 2021, Published online: 27 Oct 2021

References

  • B.P. Allahverdiev, E. Bairamov, and E. Uğurlu, Eigenparameter dependent SturmLiouville problems in boundary conditions with transmission conditions, Journal of Mathematical Analysis and Applications 401 (2013), 388–396. doi: 10.1016/j.jmaa.2012.12.020
  • B.P. Allahverdiev and H. Tuna, TitchmarshWeyl Theory for Dirac Systems with Transmission Conditions, Mediterr. J. Math. 15 (2018), 151. doi: 10.1007/s00009-018-1197-6
  • F.V. Atkinson, On bounds for the Titchmarsh-Weyl m-coecients and for spectral functions for second-order differential equations, Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 97 (1984), 1–7.
  • K. Aydemir and O.Sh. Mukhtarov, Completeness Of One Two-Interval Boundary Value Problem Wth Transmsson Condtons, Miskolc Mathematical Notes 15(2) (2014), 293-303. doi: 10.18514/MMN.2014.1229
  • P.A. Binding, P.J. Browne, and B.A. Watson, Sturm-Liouville problems with boundary conditions rationallay dependent on the eigenparameter II, J. Comput. Appl. Math. 148 (2002), 147–169. doi: 10.1016/S0377-0427(02)00579-4
  • J.R. Cannon and G.H. Meyer, On a Diffusion in a Fractured Medium, SIAM J. Appl. Math. 3 (1971), 434–448. doi: 10.1137/0120047
  • E.A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955.
  • C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh, Section A: Mathematics77A (1977), 293–308. doi: 10.1017/S030821050002521X
  • F. Gesztesy, C. Macedo, and L. Streit, An exactly solvable periodic Schröedinger operator, J. Phys. A: Math. Gen. 18 (1985), 503–507. doi: 10.1088/0305-4470/18/9/003
  • I.C. Gohberg and M.G. Krein, Introduction to the theory of linear non-self-adjoint operator, Transl. Math. Monographs, Vol. 18, Amer. Math. Soc., Providence, Rhode Island, 1969.
  • D.B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter contained in the boundary condition, Quart. J. Math. Oxford 30 (1979), 33–42. doi: 10.1093/qmath/30.1.33
  • Z. İsmailov and P. Ipek, Selfadjoint Singular Differential Operators of First Order and Their Spectrum, Electronic Journal of Differential Equations 2016 (2016), 1–9.
  • M. Kandemir and O.Sh. Mukhtarov, Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions, Electronic Journal of Differential Equations 2017(11) (2017), 1–12.
  • M. Kandemir, O.Sh. Mukhtarov, and Y.Y. Yakubov, Irregular boundary value problems with discontinuous coefficients and the eigenvalue parameter, Mediterr, J. Math. 6 (2009), 317–338.
  • T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin/New York, 1984.
  • M.V. Keldysh, On the completeness of the eigenfunctions of some classes of non-self adjoint linear operators, Soviet Math. Dokl. 77 (1951), 11-14; Russian Math. Survey 26 (1971), 15-44.
  • M.R. Lancia and M.A. Vivaldi, On the Regularity of the Solutions for Transmission Problems, Advances in Mathematical Sciences and Applications 13 (2002), 455-466.
  • B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac operators, Mathematics and its Applications (Soviet Series), Vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1991.
  • A.S. Markus and V.I. Matsayev, Comparison theorems for spectra of linear Operators and spectral asymptotics, Tr. Mosk. Mat. Obshch. 45 (1982), 133–181; English transl., Trans. Moscow Math. Soc. 1 (1984), 139–188.
  • O.Sh. Mukhtarov and K. Aydemir, The Eigenvalue Problem with Interaction Conditions at One Interior Singular Point, Filomat 31(17) (2017), 5411–5420. doi: 10.2298/FIL1717411M
  • O.Sh. Mukhtarov and S. Yakubov, Problems for ordinary differential equations with transmission conditions, Applicable Analysis 81 (2002), 1033–1064. doi: 10.1080/0003681021000029853
  • M.A. Naimark, Linear differential operators, Part II: Linear differential operators in Hilbert space, Frederick Ungar Publishing Co., New York, 1968.
  • H. Olğar and O.Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together With Interaction Conditions, Journal of Mathematical Physics 58 (2017), 042201, DOI: 10.1063/1.4979615.
  • H. Olğar, O.Sh. Mukhtarov, and K. Aydemir, Some Properties of Eigenvalues and Generalized Eigenvectors of One Boundary Value Problem, Filomat 32(13) (2018), 911–920. doi: 10.2298/FIL1803911O
  • H. Pham Huy and E. Sanchez-Palencia, Phénoménes des transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl. 47 (1974), 284-309. doi: 10.1016/0022-247X(74)90023-7
  • E. Şen, Spectral analysis of discontinuous boundary-value problems with retarded argument, Journal of Mathematical Physics Analysis Geometry 14(1) (2018), 78–99.
  • C. Sturm and J. Liouville, Extrait d’un Mémoire sur le développement des fonctions en séries dont les différents termes sont assujettis à satisfaire à une même équation différentielle linéaire, contenant un paramétre variable, J. Math. Pures Appl. 2 (1837), 220-223.
  • A.N. Tikhonov and A.A. Samarskii, Equations of Mathematical Physics, Pergamon, Oxford and New York, 1963.
  • E.C. Titchmarsh, Eigenfunctions Expansion Associated with Second Order Differential Equations I, Second edn., Oxford Univ. Press, London, 1962.
  • E. Uğurlu and E. Bairamov, Krein’s theorem for the dissipative operators with finite impulsive effects, Numer. Funct. Anal. Optim. 36(2) (2015), 256–270. doi: 10.1080/01630563.2014.970642
  • B. Van Brunt, G.C. Wake, and H.K. Kim, On a singular Sturm-Liouville problem involving an advanced functional differential equation, European Journal of Applied Mathematics 23(2) (2012), 201–244. doi: 10.1017/S0956792511000301
  • J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133 (1973), 301–312. doi: 10.1007/BF01177870

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.