### Abstract

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

Original language | English |
---|---|

Pages (from-to) | 969-981 |

Number of pages | 13 |

Journal | Journal of Difference Equations and Applications |

Volume | 8 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2002 |

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### Keywords

- Boundary value problem
- Difference equation
- Discrete Sturm-Liouville problem
- Eigenvalue
- Parameter in the boundary conditions

### Cite this

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*Journal of Difference Equations and Applications*, vol. 8, no. 11, pp. 969-981. https://doi.org/10.1080/1023619021000048869

**Discrete Sturm-Liouville problems with parameter in the boundary conditions.** / Harmsen, B. J.; Li, A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Discrete Sturm-Liouville problems with parameter in the boundary conditions

AU - Harmsen, B. J.

AU - Li, A.

PY - 2002/11/1

Y1 - 2002/11/1

N2 - This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

AB - This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

KW - Boundary value problem

KW - Difference equation

KW - Discrete Sturm-Liouville problem

KW - Eigenvalue

KW - Parameter in the boundary conditions

UR - http://www.scopus.com/inward/record.url?scp=0036851484&partnerID=8YFLogxK

U2 - 10.1080/1023619021000048869

DO - 10.1080/1023619021000048869

M3 - Article

AN - SCOPUS:0036851484

VL - 8

SP - 969

EP - 981

JO - Journal of Difference Equations and Applications

JF - Journal of Difference Equations and Applications

SN - 1023-6198

IS - 11

ER -