The principal eigenvalue of some nth order linear boundary value problems

Pedro Almenar,Lucas Jódar

doi:10.1186/s13661-021-01561-2

Abstract

The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz–Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.

Highlights

Let L be a disconjugate linear differential operator of nth order on an interval [a, b] which, according to a well-known theorem of Pólya [1], can be factored as a product of operators of first order asL0y = ρ0y, Liy = ρi(Li–1y), i = 1, . . . , n, (1)Ly = Lny, where ρi > 0, ρ0ρ1 · · · ρn = 1, and ρi ∈ Cn–i[a, b]
The purpose of this paper is to provide an iterative procedure to: 1. Calculate the smallest or principal eigenvalue of problem (2) when the boundary conditions are poised
2 The sign of the quasi-derivatives of the Green function we study the signs of the quasi-derivatives of the Green function of the problem

Summary

Introduction

Let L be a disconjugate linear differential operator of nth order on an interval [a, b] which, according to a well-known theorem of Pólya [1], can be factored as a product of operators of first order as. If (–1)n–kz(x) ≥ 0, x ∈ [a, b] with (–1)n–kz(x) > 0 in a subset of [a, b] of positive measure, and Z0{α, β} = 2, given that (–1)n–kp(x) > 0 a.e. on [a, b] and imax > 1 (the total number of boundary conditions is n and two of them are set on L0z), the previous theorem shows that the number of zeroes of L1Mz on (a, b) is exactly one, so that (–1)n–kL0Mz has only one maximum on that interval This allows extending [24, Theorem 14] to the case Z0{α, β} = 2 by means of the following theorems. The latter assertions are a result of [24, Theorems 7 and 8] and Theorem 5

Example Let us consider the problem

Conclusions

Methods