ON SPECTRAL PROPERTIES FOR A REGULAR STURM-LIOUVILLE PROBLEM WITH RETARDED ARGUMENT

Year 2014, Volume: 2 Issue: 1, 63 - 74, 01.06.2014

Erdoğan Şen AZADBAYRAMOV SERKANARACI Mehmetaçikgöz

https://izlik.org/JA86SE76DW

Abstract

In this work we study a discontinuous boundary-value problemwith retarded argument which contains a spectral parameter in the transmission conditions. We firstly prove the existence theorem and then obtainasymptotic representation of eigenvalues and eigenfunctions

Keywords

Differential equation with retarded argument , transmission conditions;asymptotics of eigenvalues and eigenfunctions

References

• S.B. Norkin, On boundary problem of Sturm-Liouville type for second-order differential equa- tion with retarded argument, Izv. Vys´s.U´cebn. Zaved. Matematika, no 6(7) (1958) 203-214 (Russian).
• S.B. Norkin, Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, Vol. 31, AMS, Providence, RI, (1972).
• R. Bellman, K.L. Cook, Differential-difference equations, New York Academic Press London (1963).
• Q. Yang, W. Wang, Asymptotic behavior of a differential operator with discontinuities at two points, Mathematical Methods in the Applied Sciences 34 (2011) 373-383.
• O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694.
• O. Sh. Mukhtarov, M. Kadakal, F.S¸. Muhtarov, Eigenvalues and normalized eigenfunctions of discontinuous Sturm–Liouville problem with transmission conditions, Rep. Math. Phys. 54 (2004) 41–56.
• N. Altını¸sık, O. Mukhtarov, M. Kadakal, Asymptotic formulas for eigenfunctions of the Sturm-Liouville problems with eigenvalue parameter in the boundary conditions, Kuwait journal of Science and Engineering, 39 (1A) (2012) 1-17.
• M. Kadakal, O. Sh. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl. 54 (2007) 1367-1379.
• M. Kadakal, O. Sh. Mukhtarov, F.S¸. Muhtarov, Some spectral problems of Sturm-Liouville problem with transmission conditions, Iranian Journal of Science and Technology, 49(A2) (2005) 229-245.
• Z. Akdo˘gan, M. Demirci, O. Sh. Mukhtarov, Green function of discontinuous boundary-value problem with transmission conditions, Mathematical Methods in the Applied Sciences 30 (2007) 1719-1738.
• C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh, A 77 (1977), 293-308.
• J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133 (1973) 301–312.
• Kh. R. Mamedov, On a basic problem for a second order differential equation with a discontin- uous coefficient and spectral parameter in the boundary conditions, Geometry, Integrability and Quantisation, 7 (2006) 218-226.
• A. Bayramov, S. C
• alıs.kan and S. Uslu, Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument, Applied Mathematics and Computation 191 (2007) 592-600.
• M. Bayramoglu, K. K¨okl¨u, O. Baykal, On the spectral properties of the regular Sturm- Liouville problem with the lag argument for which its boundary condition depends on the spectral parameter, Turk. J. Math., 26 (4) (2002) 421-431.
• E. S¸en 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, Mathematical and Computer Modelling, 54 (2011) 3090-3097.
• E. S¸en and A. Bayramov, On calculation of eigenvalues and eigenfunctions of a Sturm- Liouville type problem with retarded argument which contains a spectral parameter in the boundary condition, Journal of Inequalities and Applications 2011:113 (2011) 9 pages.
• A. Bayramov and E. S¸en, On a Sturm–Liouville type problem with retarded argument, Math- ematical Methods in the Applied Sciences, 36 (2013).39-48.
• E. S¸en and A. Bayramov, Asymptotic formulations of the eigenvalues and eigenfunctions for a boundary value problem, Mathematical Methods in the Applied Sciences, 36 (2013) 1512-1519.
• Department of Mathematics, Faculty of Arts and Science, Namik Kemal University, 59030, Tekirda˘g, Turkey and Department of Mathematics Engineering, Istanbul Tech- nical University, Maslak, 34469 Istanbul, Turkey
• E-mail address: [e-mail]
• Department of Mathematics Education, Faculty of Education, Recep Tayyip Erdogan University, Rize, Turkey
• E-mail address: [e-mail]
• Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu Univer- sity, 27410 Gaziantep, Turkey
• E-mail address: [e-mail]
• Department of Mathematics, University of Gaziantep, Gaziantep 27310, Turkey
• E-mail address: [e-mail]