Some spectral properties of Sturm-Liouville problem with a spectral parameter in the boundary conditions. (English) Zbl 1513.34121
Summary: In this paper we consider Sturm-Liouville problem with a spectral parameter in the boundary conditions. We associate this problem with a self-adjoint operator in a Pontryagin space. Using analytic methods, we investigate locations, multiplicities of eigenvalues and oscillatory properties of eigenfunctions of this problem.
MSC:
| 34B24 | Sturm-Liouville theory |
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 34B07 | Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
References:
| [1] | Aliev, Z.S.: Basis properties of the root functions of an eigenvalue problem with a spectral parameter in the boundary conditions, Dokl. Math. 82(1), 583-586 (2010). · Zbl 1213.34105 |
| [2] | Aliev, Z.S.: On basis properties of root functions of a boundary value problem con-taining a spectral parameter in the boundary conditions, Dokl. Math. 87(2), 137-139 (2013). · Zbl 1277.34116 |
| [3] | Aliev, Z.S., Aghayev, E.A.: The basis properties of the system of root functions of Sturm-Liouville problem with spectral parameter in the boundary condition, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 7(4), 63-72 (2009). · Zbl 1228.34138 |
| [4] | Aliev, Z.S., Dunyamalieva, A.A.: Basis properties of root functions of the Sturm-Liouville problem with a spectral parameter in the boundary conditions, Dokl. Math. 88(1), 441-445 (2013). · Zbl 1283.34076 |
| [5] | Aliyev, Z.S., Kerimov, N.B., Mekhrabov, V.A.: On the convergence of expansions in eigenfunctions of a boundary value problem with a spectral parameter in the boundary conditions, I, Differ. Equ. 56(2), 143-157 (2020). · Zbl 1444.34035 |
| [6] | Ben Amara, J. and Shkalikov, A.A.: The Sturm-Liouville problem with physical and spectral parameters in the boundary conditions, Math. Notes 66(1-2), 127-134 (1999). · Zbl 0958.34023 |
| [7] | Iokhvidov, I.S., Krein, M.G.: Spectral theory of operators in space with indefinite met-ric. I, Tr. Mosk. Mat. Obs. 5, 367-432 (1956). · Zbl 0072.13401 |
| [8] | Kapustin, N.Yu.: Oscillation properties of solutions to a nonselfadjoint spectral prob-lem with spectral parameter in the boundary condition, Differ. Equ. 35(8), 1031-1034 (1999). · Zbl 1009.34076 |
| [9] | Kapustin, N.Yu.: On a spectral problem arising in a mathematical model of torsional vibrations of a rod with pulleys at the ends, Differ. Equ. 41(10) 1490-1492 (2005). · Zbl 1355.74049 |
| [10] | N.Yu. Kapustin N.Yu., Moiseev, E.I.: On the spectral problem from the theory of the parabolic-hyperbolic heat equation, Dokl. Math. 55(1), 80-83 (1997). · Zbl 0965.34070 |
| [11] | Kapustin N.Yu., Moiseev, E.I.: The basis property in Lp of the system of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition, Differ. Equ. 36(10), 1498-1501 (2000). · Zbl 1005.34075 |
| [12] | Kerimov, N.B., Allakhverdiev, T.I.: On a boundary value problem. I, Differ. Equ., 29(1), 45-50 (1993). · Zbl 0802.34027 |
| [13] | Kerimov, N.B., Poladov R.Q.: Basis properties of the system of eigenfunctions in the Sturm-Liouville problem with a spectral parameter in the boundary conditions, Dokl. Math. 85(1), 8-13 (2012). · Zbl 1246.34026 |
| [14] | Levitan B.M., Sargsjan I.S.: Introduction to spectral theory: Selfadjoint ordinary differ-ential operators, Translations of Mathematical Monographs, vol. 39, Amer. Math. Soc., Providence, Rhode Island, 1975. · Zbl 0302.47036 |
| [15] | Moiseev, E.I., Kapustin N.Yu.: On the singularities of the root space of one spectral problem with a spectral parameter in the boundary condition, Dokl. Math. 66(1), 14-18 (2002). · Zbl 1259.34085 |
| [16] | Naimark M.A.: Linear Differential Operators, Ungar, New York (1967). · Zbl 0219.34001 |
| [17] | Pontryagin, L.S.: Hermitian operators in a space with indefinite metric, Izv. Akad. Nauk SSSR Ser. Mat. 8, 243-280 (1944). · Zbl 0061.26004 |
| [18] | Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics, Moscow, Nauka, (1972). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
