Advances in Difference Equations

Impact Factor 0.335

Advances in Difference Equations Cover Image
Research Article
Open Access

Eigenvalue comparisons for boundary value problems of the discrete beam equation

Advances in Difference Equations20062006:081025

https://doi.org/10.1155/ADE/2006/81025

©  Jun Ji and Yang 2006

Received: 29 September 2005

Accepted: 24 February 2006

Published: 24 July 2006

Abstract

We study the behavior of all eigenvalues for boundary value problems of fourth-order difference equations Δ4y i = λai+2yi+2, -1≤i≤n-2, y0 = Δ2y-1 = Δy n = Δ3yn-1 = 0, as the sequence varies. A comparison theorem of all eigenvalues is established for two sequences and with a j ≥ b j , 1 ≤ j ≤ n, and the existence of positive eigenvector corresponding to the smallest eigenvalue of the problem is also obtained in this paper.

[1234567812345678]

Authors’ Affiliations

(1)
Department of Mathematics, Kennesaw State University

References

  1. Davis JM, Eloe PW, Henderson J: Comparison of eigenvalues for discrete Lidstone boundary value problems. Dynamic Systems and Applications 1999,8(3–4):381–388.MathSciNetMATHGoogle Scholar
  2. Gentry RD, Travis CC: Comparison of eigenvalues associated with linear differential equations of arbitrary order. Transactions of the American Mathematical Society 1976, 223: 167–179.MathSciNetView ArticleMATHGoogle Scholar
  3. Gentry RD, Travis CC: Existence and comparison of eigenvalues of n th order linear differential equations. Bulletin of the American Mathematical Society 1976,82(2):350–352. 10.1090/S0002-9904-1976-14062-1MathSciNetView ArticleMATHGoogle Scholar
  4. Graef JR, Yang B: Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems. Applicable Analysis 2000,74(1–2):201–214.MathSciNetView ArticleMATHGoogle Scholar
  5. Hankerson D, Peterson A: Comparison theorems for eigenvalue problems for n th order differential equations. Proceedings of the American Mathematical Society 1988,104(4):1204–1211.MathSciNetMATHGoogle Scholar
  6. Hankerson D, Peterson A: Comparison of eigenvalues for focal point problems for n th order difference equations. Differential and Integral Equations 1990,3(2):363–380.MathSciNetMATHGoogle Scholar
  7. Travis CC: Comparison of eigenvalues for linear differential equations of order 2 n . Transactions of the American Mathematical Society 1973, 177: 363–374.MathSciNetMATHGoogle Scholar
  8. Varga RS: Matrix Iterative Analysis. Prentice-Hall, New Jersey; 1962:xiii+322.Google Scholar

Copyright

© Jun Ji and Yang 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.