Skip to main content
Advances in Continuous and Discrete Models

Theory and Modern Applications

Download PDF

Eigenvalue comparisons for boundary value problems of the discrete beam equation

Advances in Difference Equations volume 2006, Article number: 081025 (2006) Cite this article

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]

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.

    MathSciNet  MATH  Google 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.

    Article  MathSciNet  MATH  Google 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/S0002-9904-1976-14062-1

    Article  MathSciNet  MATH  Google 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.

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  8. Varga RS: Matrix Iterative Analysis. Prentice-Hall, New Jersey; 1962:xiii+322.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kennesaw State University, Kennesaw, GA, 30144, USA

    Jun Ji & Bo Yang

Authors
  1. Jun Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Ji.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Ji, J., Yang, B. Eigenvalue comparisons for boundary value problems of the discrete beam equation. Adv Differ Equ 2006, 081025 (2006). https://doi.org/10.1155/ADE/2006/81025

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

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

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation