Basis properties of root functions of a regular fourth order boundary value problem

Abstract

In this paper, we consider the following boundary value problem

\[ y^{(4)}+q(x) y=\lambda y,~\ \ \ 0<x<1, \]

\[ y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \]

\[ y^{(s)}(1) -( -1) ^{\sigma}y^{(s) }( 0) =0,\ \ \ s=\overline{0,2}, \]

where $\lambda $ is a spectral parameter, $q( x)\in L_{1}(0,1)$ is complex-valued function and $\sigma =0,1$. The boundary conditions of this problem are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. When $\alpha\ne 0$, we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space $L_{2}( 0,1)$. Furthermore, we show that the system of root functions forms a basis in the space $L_{p}( 0,1)$, $1<p<\infty$ $(p\neq 2)$, under the conditions $\alpha\ne 0$ and $q( x) \in W_{1}^{1}( 0,1)$.

Keywords

• fourth order eigenvalue problem,not strongly regular boundary conditions,asymptotic behavior of eigenvalues and eigenfunctions,basis properties of the system of root functions

References

1. [1] N.K. Bari, Trigonometric Series (Russian), Fizmatgiz, Moscow, 1961. English transl.: N.K. Bary, A Treatise on Trigonometric Series. Vols. I, II, MacMillan, New York, 1964.
2. [2] G.D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc. 9 (4), 373-395, 1908.
3. [3] N. Dernek and O. Veliev, On the Riesz basisness of the root functions of the nonselfadjoint Sturm-Liouville operator, Israel J. Math. 145 (1), 113-123, 2005.
4. [4] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomials as potentials, Dokl. Akad. Nauk. 436, 11-13, 2011.
5. [5] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. 351 (3), 509-540, 2011.
6. [6] P. Djakov and B. Mityagin, Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators, J. Funct. Anal. 263 (8), 2300-2332, 2012.
7. [7] P. Djakov and B. Mityagin, Instability zones of periodic 1-dimensional Schrodinger and Dirac operators, Uspekhi Mat. Nauk 61 (4), 77-182, 2006.
8. [8] N. Dunford, J.T. Schwartz, W.G. Bade and R.G. Bartle, Linear Operators, Wiley- Interscience, New York, 1971.

9. [9] F. Gesztesy and V. Tkachenko, A Schauder and Riesz basis criterion for non-selfadjoint Schrodinger operators with periodic and antiperiodic boundary conditions, J. Differential Equations 253 (2), 400-437, 2012.
10. [10] I. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, 1969.
11. [11] H. Gunes, N.B. Kerimov and U. Kaya, Spectral properties of fourth order differential operators with periodic and antiperiodic boundary conditions, Results Math. 68 (3-4), 501-518, 2015.
12. [12] N. Ionkin and E. Moiceev, Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions, Differ. Equ. 13 (2), 294-304, 1977.
13. [13] B.S. Kashin and A. Saakyan, Orthogonal Series, volume 75, American Mathematical Society, Providence, 2005.
14. [14] N.B. Kerimov and U. Kaya, Some problems of spectral theory of fourth-order differential operators with regular boundary conditions, Arab. J. Math. 3 (1), 49-61, 2014.
15. [15] N.B. Kerimov and U. Kaya, Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions, Math. Methods Appl. Sci. 37 (5), 698-710, 2014.
16. [16] N.B. Kerimov and U. Kaya, Spectral properties of some regular boundary value problems for fourth order differential operators, Cent. Eur. J. Math. 11 (1), 94-111, 2013.
17. [17] N.B. Kerimov and K.R. Mamedov, On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes 64 (4), 483-487, 1998.
18. [18] N.B. Kerimov, Unconditional basis property of a system of eigen and associated functions of a fourth-order differential operator, Dokl. Akad. Nauk 286, 803-808, 1986.
19. [19] G. Keselman, On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 2, 82-93, 1964.
20. [20] A.A. Kirac, Riesz basis property of the root functions of non-selfadjoint operators with regular boundary conditions, Int. J. Math. Anal. (Ruse) 3 (21-24), 1101-1109, 2009.
21. [21] A.S. Makin, On a class of boundary value problems for the SturmLiouville operator, Differ. Uravn. 35 (8), 1067-1076, 1999.
22. [22] A.S. Makin, Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions, Differ. Equ. 44 (5), 645-658, 2008.
23. [23] A.S. Makin, Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ. 44 (3), 341-348, 2008.
24. [24] A.S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math. 73, 71-76, 2006.
25. [25] A.S. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm- Liouville operators, Dokl. Math. 73, 15-18, 2006.
26. [26] A.S. Makin, On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ. 42 (12), 1717-1728, 2006.
27. [27] V. Mikhailov, On Riesz bases in $L_2(0, 1)$, Dokl. Akad. Nauk 144 (5), 981-984, 1962.
28. [28] M.A. Naimark, Linear Differential Operators, 2nd ed. Nauka, Moskow, 1969.
29. [29] A.A. Shkalikov, Bases formed by eigenfunctions of ordinary differential operators with integral boundary conditions, Moscow Univ. Math. Bull. 6, 12-21, 1982.
30. [30] A.A. Shkalikov and O.A. Veliev, On the Riesz basis property of the eigen and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes 85 (5-6), 647-660, 2009.
31. [31] O.A. Veliev, Asymptotic analysis of non-self-adjoint Hill operators, Open Math. J. 11 (12), 2234-2256, 2013.
32. [32] O.A. Veliev and M.T. Duman, The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. Math. Anal. Appl. 265 (1), 76-90, 2002.
33. [33] O.A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (1), 195-207, 2010.
34. [34] A. Zygmund, Trigonometric Series, volume 2, Cambridge University Press, Cambridge, 2002.