Estimation of the smallest eigenvalue of an nth‐order linear boundary value problem

Abstract

This paper describes a recursive procedure to estimate the smallest eigenvalue of an nth‐order boundary value problem under a wide set of boundary conditions. The procedure yields lower and upper bounds for that eigenvalue as well as an estimation of the associated eigenfunction, both of which are shown to converge to their exact values as the recursion index grows. A simpler version of the procedure is also displayed for the self‐adjoint case.

Accurate Estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems

“…From here and [25, Lemma 1.16], one gets (33). Likewise, following the same reasoning as in [32,Theorems 6 and 7], one can get to (35) and, noting that for some j 0 > 0, M j p u ∈ int(K p ) for all j ≥ j 0 , also to (34).…”

“…. From the sign-regularity of G(x, t), (7) and (32), one has that the signs of ∆ p (M p u) and ∆ p (u) coincide in Ω * . This completes the proof.…”

“…Chang [25] proved it for other 1-homogeneous non-linear differential problems like p-Laplace systems, calling it the power method, so it is possible that it was known and used before. Later, the authors used it in [29][30][31] to determine the solvability of boundary value problems and in [32] to bound and estimate the principal eigenvalue of boundary value problems including higher derivatives, for which some results on the sign of the derivatives of the Green function were needed. However, to the knowledge of the authors, it has never been applied to determine the value of other eigenpairs apart from the principal one.…”

New results on the sign of the Green function of a two-point n-th order linear boundary value problem

“…If E j [a, b] > 1 for some j > K(α , β ), from Lemma 1 and ( 7) one would get z n-1 [a, b] > 1, and by Rolle's theorem there would be a change of signs of y (n) in (a, b), contradicting ( 28)-( 29). Therefore, E j [a, b] = 1 for all j > K(α , β ), and from (7), it follows z j [a, b] = Z j {α , β }j + 1. This completes the proof.…”

“…Therefore, the knowledge of the sign of G(x, t) and its derivatives can provide information about the sign of the solution y(x) and these same derivatives, at least when f does not change sign on (a, b). Likewise, there is a large amount of literature ( [4][5][6][7][8], and [9]) on the use of the sign of G(x, t) to define cones that, by means of Krein and Rutman's [10] works, allow finding information about the eigenvalues and eigenfunctions of the general problem Ly = λ μ l=0 c l (x)y (l) (x), x ∈ (a, b), y (α i ) (a) = 0, α i ∈ α; y (β i ) (b) = 0, β i ∈ β; (4) with μ ≤ n -1, c l (x) ∈ C(J) for 0 ≤ l ≤ μ, and even calculate them.…”