Spectral Properties of Discrete Sturm-Liouville Problems with two Squared Eigenparameter-Dependent Boundary Conditions
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
SPECTRAL PROPERTIES OF DISCRETE STURM-LIOUVILLE PROBLEMS WITH TWO SQUARED EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS∗
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Chenghua GAO (
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Yali WANG (
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Li LV (
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China E-mail : [email protected]; [email protected]; [email protected] Abstract In this article, we consider a discrete right-definite Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions. By constructing some new Lagrange-type identities and two fundamental functions, we obtain not only the existence, the simplicity, and the interlacing properties of the real eigenvalues, but also the oscillation properties, orthogonality of the eigenfunctions, and the expansion theorem. Finally, we also give a computation scheme for computing eigenvalues and eigenfunctions of specific eigenvalue problems. Key words
Discrete Sturm-Liouville problems; squared eigenparameter-dependent boundary conditions; interlacing; oscillation properties; orthogonality
2010 MR Subject Classification
1
39A10; 39A12
Introduction
Spectra of the eigenvalue problems with eigenparameter-dependent boundary conditions have been discussed in a long history. In 1820s, Poisson [42] derived an ODE model with eigenparameter-dependent boundary conditions from a pendulum problem. After that, lots of eigenvalue problems with eigenparameter-dependent boundary conditions are derived from different subjects, such as, biology, engineering, heat conduction, and so on; see, for instance, [1, 6, 8, 14, 20, 21, 40, 46, 47]. Therefore, many excellent and interesting eigenvalue results for both the continuous eigenvalue problems and the discrete eigenvalue problems with eigenparameterdependent boundary conditions have been obtained; see, for instance, [1–6, 8–12, 14–18, 20, 21, 23–27, 29, 30, 36]. In [15–18, 37, 38], the authors developed several general eigenvalue theories in different spaces for boundary value problems with eigenparameter-dependent boundary value conditions. Meanwhile, by using Pr¨ ufer transformation, Binding et al. [9–12] obtained SturmLiouville theories of the second-order continuous eigenvalue problems with eigenparameter∗ Received
December 27, 2018; revised July 13, 2019. The authors are supported by National Natural Sciences Foundation of China (11961060, 11671322), and the Key Project of Natural Sciences Foundation of Gansu Province (18JR3RA084). † Corresponding author
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
dependent boundary conditions, including the existence, simplicity, and interlacing properties of eigenvalues and the oscillation properties of corresponding eigenfunctions. Furthermore, using different methods, Aliyev [2–6, 36] also obtained a series of basic properties of systems of root functions in different spaces for several eigenvalue problems with eigenparameter-dependent boundary conditions. For the discrete case, there are also several i
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