Asymptotic Expressions of Fourth Order Sturm-Liouville Operator with Conjugate Conditions

Abstract

In this paper, it is studied the asymptotic expression of fourth order differential operator with periodic boundary conditions. For this operator, it is also considered conjugate boundary conditions at x=0 which shows discontinuity. For this purpose, firstly asymptotic expression of solutions areobtained. Then by using the the asymptotic formulas of fundamental solutions, asymptotic expression of eigenvalues and eigenfunctions are presented. It is also dealt with the asymptotic expression of same operator with antiperiodic boundary conditions and conjugate conditions

Keywords

• Periodic boundary conditions
• fourth order problem
• eigenvalues
• eigenfunctions

References

1. Agarwal R.P, 1989. On fourth order boundary value problems arising in beam analysis, Differential and Integral Equations, 2(1):91-110.
2. Bonanno G, Bella BD, 2008. A boundary value problem for fourth-order elastic beam equations, Journal of Mathematical Analysis and Applications, 343:1166-1176.
3. Baranets’kyi YO, Kalenyuk PI, Kolyasa LI., 2018. Spectral Properties of Nonself-Adjoint Nonlocal -Boundary-Value Problems for the Operator of Differentiation of Even Order, Ukrainian Mathematical Journal, 70:851-865.
4. Baskakov AG, Polyakov DM, 2017. The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential, Matematicheskii Sbornik, 208(1):3-47.
5. Cabri O. 2019.On the Riesz basis property of the root functions of a discontinuous boundary problem, Mathematical Methods in Applied Sciences,6733-6740.
6. Cabri O, Mamedov KhR, 2020.Riesz basisness of root functions of a Sturm Liouville operator with conjugate conditions, Lobachevskii Journal of Mathematics, 41(1):1-6.
7. Cabri O, Mamedov, KhR, 2020. On the Riesz Basisness of Root Functions of a Sturm–Liouville Operator with Conjugate Conditions, Lobachevskii Journal of Mathematics, 41(9):1784–1790.
8. Coskun H, 2003. On the spectrum of a second-order periodic differential equation, Rocky Mountain Journal of Mathematics, 33:1261-1277

9. Djakov P, Mityagin B, 2006. Instability zones of periodic one dimensional Schr¨odinger and dirac operators. Uspekhi Math.Nauk, 61(4):663-776.
10. Dunford N, Schwartz JT, 1970. Linear Operators, Prt.3 Spectral Operators, Wiley, Newyork.
11. Gasymov MG, Guseinov IM, Nabiev IM, 1990. An inverse problem for the Sturm-Liouville operator with nonseparable self-adjoint boundary conditions, Siberian Mathematical Journal, 31:910–918.
12. Gesztesy F, Tkachenko VA, 2012. Schauder and Riesz basis criterion for non-self-adjoint Schrodinger operators with periodic and antiperiodic boundary conditions Journal of Differential Equations, 253(2):400-437. Gupta C, 1988.Solvability of a fourth order boundary value problem with periodic boundary conditions, International Journal of Mathematics and Mathematical Sciences, 11(2): 275-284.
13. Jwamer KH, Hawsar AH, 2015. Accurate asymptotic formulas for eigenvalues of a boundaryvalue problem of fourth order, Mathematics and Statistics, 3(3):71-74.
14. Kerimov NB, Mamedov KhR, 1998.On the Riesz basis property of the root functions in certain regular boundary value problems, Mathematical Notes, 64(4):483-487.
15. Kurbanov VM, 2006. A theorem on equivalent bases for a differential operator, Doklady Akademics,406(1):17-20.
16. Levitan BM, Sargsyan, IS, 1991. Sturm Liouville and Dirac Operators, Kluver Academic Publisher, Netherlands.
17. Makin AS, 2006. Convergence of expansions in the root functions of periodic boundary value problems, Doklady Mathematics, 73(1):71-76.
18. Mamedov KhR, 1996. On spectrally of differential operator of second order, Proceeding of Institute of Mathematics and Mechanics Acad, 5:526-558.
19. Mamedov KhR, 1996.Completeness and minimality of a half of the set of eigenfunctions for the biharmonic equation in a half-strip, Mathematical Notes, 60:344-396.
20. Mamedov KhR, Menken H, 2008.On the basisness in L2(0; 1) of the root functions in not strongly regular boundary value problems, European Journal of Pure and Applied Math., 1(2):51-60.
21. Mamedov KhR, 2010.On the Basis Property in Lp(0; 1) of the Root Functions of a Class NonSelf Adjoint Sturm-Lioville Operators, European Journal of Pure and Applied Math., 3(5): 881-838.
22. Marchenko VA, 1977. Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev
23. Menken H, 2010. Accurate asymptotic formulas for eigenvalues and eigenfunctions of a boundarybalue problem of fourth order, Boundary Value Problems.
24. Muravei LA, 1967. Riesz bases in L2(−1; 1), Proceedings of the Steklov Institute of Mathematics,91:113-131.
25. Mitrokhin SI, 2010. Spectral properties of a fourth-order differential operator with integrable coefficients, Proceedings of the Steklov Institute of Mathematics, 270:184-193.
26. Nabiyev IM, 2007. The Inverse quasiperiodic problem for a diffusion operator, Doklady Mathematics, 76:527-529.
27. Naimark MA, 1967. Linear differential operators, Part I. Frederick Ungar, Newyork.
28. Tikhonov AN, Samarskii AA, 1963. Equations of Mathematical Physics, Dover Publications, New York.
29. Yao Q, 2004.Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Applied Mathematics Letters, 17: 237-243.