Eigenvalue comparisons for boundary value problems of the discrete beam equation

Abstract

We study the behavior of all eigenvalues for boundary value problems of fourth-order difference equations Δ4 yi = λai+2yi+2, -1 ≤ i ≤ n - 2, y0 = Δ2y-1 = Δyn = Δ3yn-1 = 0, as the sequence {ai}i=1n varies. A comparison theorem of alleigenvalues is established for two sequences {ai}i=1n and {bi}i=1n with aj ≥ bj, 1 ≤ j ≤ n, and the existence of positive eigenvector corresponding to the smallest eigenvalue of the problem is also obtained in this paper.