Discrete Sturm-Liouville problems with parameter in the boundary conditions

B. J. Harmsen, A. Li

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10 Scopus citations

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 languageEnglish
Pages (from-to)969-981
Number of pages13
JournalJournal of Difference Equations and Applications
Volume8
Issue number11
DOIs
StatePublished - 1 Nov 2002

Keywords

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

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