Articles

SPECTRAL PROPERTIES OF DISCRETE STURM-LIOUVILLE PROBLEMS WITH TWO SQUARED EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

  • Chenghua GAO ,
  • Yali WANG ,
  • Li LV
Expand
  • Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received date: 2018-12-27

  Revised date: 2019-07-13

  Online published: 2020-07-17

Supported by

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).

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.

Cite this article

Chenghua GAO , Yali WANG , Li LV . SPECTRAL PROPERTIES OF DISCRETE STURM-LIOUVILLE PROBLEMS WITH TWO SQUARED EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS[J]. Acta mathematica scientia, Series B, 2020 , 40(3) : 755 -781 . DOI: 10.1007/s10473-020-0312-5

References

[1] Ahn H J. On random transverse vibrations of rotating beam with tip mass. Quart J Mech Appl Math, 1983, 39:97-109
[2] Aliyev Z S. Basis properties in Lp of systems of root functions of a spectral problem with spectral parameter in a boundary condition. Differential Equations, 2011, 47:766-777
[3] Aliyev Z S. On basis properties of root functions of a boundary value problem containing a spectral parameter in the boundary conditions. Dokl Math, 2013, 87:137-139
[4] Aliyev Z S, Dunyamalieva A A. Defect basis property of a system of root functions of a Sturm-Liouville problem with spectral parameter in the boundary conditions. Differ Equ, 2015, 51:1249-1266
[5] Aliyev Z S, Dunyamalieva A A. Basis properties of root functions of the Sturm-Liouville problem with a spectral parameter in the boundary conditions. (Russian) Dokl Akad Nauk, 2013, 451:487-491; translation in Dokl Math, 2013, 88:441-445
[6] Aliyev Z S, Guliyeva S B. Properties of natural frequencies and harmonic bending vibrations of a rod at one end of which is concentrated inertial load. J Differential Equations, 2017, 263:5830-5845
[7] Atkinson F. Discrete and Continuous Boundary Problems. New York:Academic Press, 1964
[8] Belinskiy B, Dauer J P, Xu Y. Inverse scattering of accustic waves in an oceas with ice cover. Appl Anal, 1996, 61:255-283
[9] Binding P. A hierarchy of Sturm-Liouville problems. Math Meth Appl Sci, 2003, 26:349-357
[10] Bhattacharyya T, Binding P, Seddighi K. Two-parameter right definite Sturm-Liouville problems with eigenparameter-dependent boundary conditions. Proc Edinburgh Math Soc, 2001, 131:45-58
[11] Binding P, Browne Patrick J. Application of two parameter eigencurves to Sturm-Liouville problems with eigenparameter-dependent boundary condition. Proc Edinburgh Math Soc, 1995, 125:1205-1218
[12] Binding P, Browne Patrick J, Seddighi K. Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc Edinburgh Math Soc, 1993, 37:57-72
[13] Bohner M, DoŠlý O, Kratz W. An oscillation theorem for discrete eigenvalue problems. Rocky Mountain J Math, 2003, 33:1233-1260
[14] Currie S, Love D. Hierarchies of difference boundary value problems II-Application. Quaest Math, 2014, 37:371-392
[15] Curgus B, Dijksma A, Read T. The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces. Linear Algebra Appl, 2001, 329:97-136
[16] Dijksma A, Langer H, H S V de Snoo. Symmetric Sturm-Liouville operators with eigenvalue dependending boundary conditions. CMS Conf Proc, 1987, 8:87-116
[17] Dijksma A, Langer H, H S V de Snoo. Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary condition. Math Nachr, 1993, 161:107-154
[18] Dijksma A, Langer H. Operator theory and ordinary differential operators, Lectures on operator theory and its applications. Fields Inst Monogr, 1996, 3:73-139
[19] Došlý O, Kratz W. Oscilation theorems for symplectic difference systems. J Difference Equ Appl, 2007, 13:585-605
[20] Fulton C. Two-point boundary value problems with eigenvalue parameter contained in the boundary condition. Proc Edniburgh Math Soc, 1977, 77:293-308
[21] Fulton C, Pruess S. Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions. J Math Anal Appl, 1979, 71:431-462
[22] Gao C H. On the linear and nonlinear discrete second-order Neumann boundary value problems. Appl Math Comput, 2014, 233:62-71
[23] Gao C H, Li X L, Ma R Y. Eigenvalues of a linear fourth-order differential operator with squared spectral parameter in a boundary condition. Mediterr J Math, 2018, 15:107
[24] Gao C H, Ma R Y. Eigenvalues of discrete Sturm-Liouville problems with eigenparameter dependent boundary conditions. Linear Algebra Appl, 2016, 503:100-119
[25] Gao C H, Li X L, Zhang F. Eigenvalues of discrete Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions. Quaest Math, 2018, 41:773-797
[26] Gao C H, Ma R Y, Zhang F. Spectrum of discrete left definite Sturm-Liouville problems with eigenparameter-dependent boundary conditions. Linear Multilinear Algebra, 2017, 65:1905-1923
[27] Gao C H, Ma R Y. Eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight. Linear Algebra Appl, 2015, 467:40-56
[28] Hartman P. Difference equations:Disconjugacy, principal solutions, Green's functions, complety monotonicity. Trans Amer Math Soc, 1978, 246:1-30
[29] Harmsen B J, Li A. Discrete Sturm-Liouville problems with nonlinear parameter in the boundary conditions. J Difference Equ Appl, 2007, 13:639-653
[30] Harmsen B J, Li A. Discrete Sturm-Liouville problems with parameter in the boundary conditions. J Difference Equ. Appl, 2002, 8:969-981
[31] Jirari A. Second-order Sturm-Liouville difference equations and orthogonal polynomials. Mem Amer Math Soc, 2004, 294:104-112
[32] Kapustin N Yu. On the uniform convergence of the Fourier series for a spectral problem with squared spectral parameter in the boundary condition. Differ Equ, 2010, 46:1507-1510
[33] KapustinN Yu. On the uniform convergence in the class C1 of the Fourier series for a spectral problem with squared spectral parameter in the boundary condition. Differ Equ, 2011, 47:1408-1413
[34] Kapustin N Yu. On the basis property of the system of eigenfunctions of a problem with squared spectral parameter in a boundary condition. Differ Equ, 2015, 51:1274-1279
[35] Kelley W G, Peterson A C. Difference Equations:An Introduction with Applications (2nd ed). CA:Academic Press, 2001
[36] Kerimov N B, Poladov R G. Basis properties of the system of eigenfunctions in the Sturm-Liouville problem with a spectral parameter in the boundary conditions. Dokl Math, 2012, 85:8-13
[37] Koprubasi T, Yokus N. Quadratic eigenparameter dependent discrete Sturm-Liouville equations with spectral singularities. Appl Math Comput, 2014, 244:57-62
[38] Koprubasi Turhan, Mohapatra R N. Spectral properties of generalized eigenparameter dependent discrete Sturm-Liouville type equation. Quaest Math, 2017, 40:491-505
[39] Kratz W. Discrete Oscillation. J Difference Equ Appl, 2003, 9:135-147
[40] Langer R E. A problem in diffusion or in the flow of heat for a solid in contact with fluid. Tohoku Math J, 1932, 35:360-375
[41] Ma R Y, Gao C H, Lu Y Q. Spectrum theory of second-order difference equations with indefinite weight. J Spectr Theory, 2018, 8:971-985
[42] Poisson M. Sur la manière d'éxperimer les fonctions par des series de quantités, et sur l'usage de cette transformation dans la résolution de différents problèms. Paris:Ecole Polytechnique de Paris, 1820, 18emé cahier, Vol. XI
[43] Shi Y M, Chen S Z. Spectral theory of second-order vector difference equations. J Math Anal Appl, 1999, 239:195-212
[44] Sun H Q, Shi Y. Eigenvalues of second-order difference equations with coupled boundary conditions. Linear Algebra Appl, 2006, 414:361-372
[45] Wang Y, Shi Y M. Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. J Math Anal Appl, 2005, 309:56-69
[46] Gao C H. Solutions to discrete multiparameter periodic boundary value problems involving the p-Laplacian via critical point theory. Acta Mathematica Scientia, 2014, 34B(4):1225-1236
[47] Luo H. Spectral theory of linear weighted Sturm-Liouville eigenvalue problems. Acta Mathematica Scientia, 2017, 37B(3):427-449
[48] Gao C H, Lv L, Wang Y L. Spectra of a discrete Sturm-Liouville problem with eigenparameterdependent boundary conditions in Pontryagin space. Quaestiones Mathematicae, 2019. DOI:10.2989/16073606.2019.1680456
Options
Outlines

/