Date of Award

5-2010

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Aihua Li

Committee Member

Jonathan Cutler

Committee Member

Baojun Song

Abstract

This thesis deals with discrete second order Sturm-Liouville Boundary Value Problems (DSLBVP) where the parameter as part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. We focus on analyzing the case with cubic nonlinearity in the boundary condition. First, we describe the problem by a matrix equation with nonlinear variables such that solving the DSLBVP is equivalent to solving the matrix equation. Second, we formulate the problem as a nonlinear eigenvalue problem. We further reduce the problem to finding eigenvalues of a matrix pencil in the form A - X B . Under certain conditions, such a matrix pencil eigenvalue problem can be reduced to a regular eigenvalue problem (REP), thus allowing us to use existing computational tools to solve the problem. The main results of this thesis provide the reduction procedure and the rules to identify the cubic DSLBVPs which can be reduced to REP. We also investigate the structure of the matrix form of a DSLBVP and its effect on the reality of the eigenvalues of the problem. The last section of the thesis focuses on a class of DSLBVPs with only real eigenvalues.

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