Eigenvalue comparisons for boundary value problems of the discrete beam equation
© Jun Ji and Yang 2006
Received: 29 September 2005
Accepted: 24 February 2006
Published: 24 July 2006
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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Varga RS: Matrix Iterative Analysis. Prentice-Hall, New Jersey; 1962:xiii+322.Google Scholar
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