Abstract
This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each eigenpair to be determined, the recursive application of the operator associated to the equivalent integral problem to functions belonging to such a cone, and the calculation of the Collatz–Wielandt numbers of the resulting functions.
Highlights
The results presented in this paper allow finding the n smallest eigenvalues of boundary value problems with sign-regular Green functions, as well as the following ones provided that certain conditions on the functions a j ( x ) of L
The procedure is sequential in the sense that it requires running it for the first p − 1 eigenvalues in order to use it to calculate the p-th one
It can be summarized in the following algorithm, which assumes the knowledge of the p − 1 previous eigenfunctions φi and the p − 1 previous adjoint eigenfunctions ψi : 1
Summary
Stepanov provided necessary and sufficient conditions for the Green function of (6) to be sign-regular in [3] The research on this topic was continued by Pokornyi and his collaborators due to its relationship with the theory of differential equations in networks [6,15]. For self-completeness, let us recall that, given a Banach space B, a cone P ⊂ B is a non-empty closed set defined by the conditions: If u, v ∈ P, cu + dv ∈ P for any real numbers c, d ≥ 0. 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.
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