SPECTRAL ANALYSIS OF BOUNDARY VALUE PROBLEMS WITH RETARDED ARGUMENT

In this paper, by modifying some techniques of [S.B. Norkin, Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, Vol. 31, AMS, Providence, RI, 1972]and suggesting own approaches we find asymptotic formulas for the eigenvalues and eigen functions of boundary value problems of Sturm-Liouville type for the second order differential equation with retarded argument

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  • Norkin, S.B., Diğerential equations of the second order with retarded argument, Translations of Mathematical Monographs, Vol. 31, AMS, Providence, RI, 1972.
  • Norkin, S.B., On boundary problem of Sturm-Liouville type for second-order diğ erential equa- tion with retarded argument, Izv. Vys´s. U´cebn. Zaved. Matematika, no 6(7) (1958) 203-214 (Russian).
  • Kamenskii, G. A., On the asymptotic behaviour of solutions of linear diğ erential equations of the second order with retarded argument, Uch. Zap. Mosk. Gos. Univ., 165 (1954) 195-204 (Russian).
  • Bayramo¼glu, M., K. Köklü, O. Baykal, On the spectral properties of the regular Sturm- Liouville Problem with the lag argument for which its boundary conditions depends on the spectral parameter, Turk. J. Math., 26 (2002) 421-431.
  • ¸Sen, E. and A. Bayramov, Asymptotic formulations of the eigenvalues and eigenfunctions for a boundary value problem, Math. Method. Appl. Sci., 36 (2013) 1512-1519.
  • ¸Sen, E. and A. Bayramov, Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the
  • boundary condition, Math. Comput. Model., 54 (11-12) (2011) 3090-3097.
  • Bayramov, A., On asymptotic of the eigenvalues and eigenfunctions for the problem of Sturm- Liouville with the lag argument and the spectral parameter in the boundary condition, Trans. Acad. Sci. Azerb., Ser. Phys.-Tech. Math. Sci., 18 (3-4) (1998) 6-11.
  • Bayramov, A., S. C. alıs.kan and S. Uslu, Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument, Appl. Math. Comput., 191 (2007) 592-600.
  • Yang, C., Trace and inverse problem of a discontinuous Sturm–Liouville operator with re- tarded argument, J. Math. Anal. Appl., 395 (1) (2012) 30-41.
  • Levitan,B. M., Expansion in characteristic functions of diğerential equations of the second order, GITTL, Moscow, 1950 (Russian).
  • Bairamov, E., E. Ugurlu, On the characteristic values of the real component of a dissipative boundary value transmission problem, Appl. Math. Comput., 218 (2012) 9657-9663.
  • Bairamov, E., E, Ugurlu, E, The determinants of dissipative Sturm-Liouville operators with transmission conditions, Math. Comput. Model., 53 (5/6) (2011) 805-813.
  • Mamedov, Kh. R., On Boundary Value Problem with Parameter in Boundary Conditions, Spectral Theory of Operator and Its Applications, 11 (1997) 117-121 (Russian).
  • Fulton, C. and S. Pruess, Eigenvalue and eigenfunction asymptotics for regular Sturm- Liouville problems, J. Math. Anal. Appl., 188 (1) (1994) 297–340.
  • Aydemir, K., O. Sh. Mukhtarov, Variational principles for spectral analysis of one Sturm- Liouville problem with transmission conditions, Advances in Diğerence Equations, 2016 (2016) 1-14.
  • Mukhtarov, O. Sh., K. Aydemir, Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scientia, 35 (3) (2015) 639-649.
  • Mukhtarov, O. Sh., M. Kadakal, F. S. Muhtarov, On discontinuous Sturm-Liouville problems with transmission conditions, J. Math. Kyoto Univ., 44 (4) (2004) 779-798.
  • Kadakal, M., O. Sh. Mukhtarov, Discontinuous Sturm-Liouville problems containing eigen- parameter in the boundary conditions, 22 (5) (2006) 1519-1528.
  • Current address : Erdo¼gan ¸Sen: Department of Mathematics, Namık Kemal University, 59030, Tekirda¼g, Turkey
  • E-mail address : erdogan.math@gmail.com
  • Current address : Azad Bayramov: Azerbaijan State Pedagogical University, 1000, Baku, Azer- baijan
  • E-mail address : azadbay@gmail.com