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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.

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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.

    MathSciNet  Article  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

    MathSciNet  Article  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.

    MathSciNet  Article  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 

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Authors and Affiliations

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

    Jun Ji & Bo Yang

Authors
  1. Jun Ji
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  2. Bo Yang
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Correspondence to Jun Ji.

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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.

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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

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

Keywords

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