Characterization of constant sign Green's function of a two point boundary value problem by means of spectral theory

TL;DR

This paper characterizes the parameter ranges where the Green's function of a general linear boundary value problem maintains a constant sign, using spectral theory to relate it to eigenvalues without explicitly computing the Green's function.

Contribution

It provides a spectral-theoretic characterization of constant sign Green's functions for general linear boundary value problems, including nonhomogeneous conditions, avoiding explicit Green's function calculation.

Findings

Identifies parameter intervals where Green's function is of constant sign.

Relates constant sign property to eigenvalues of the differential operator.

Includes examples demonstrating applicability of the spectral characterization.

Abstract

This paper is devoted to the study of the parameter's set where the Green's function related to a general linear $n^{\rm th}$-order operator, depending on a real parameter, $T_n[M]$, coupled with many different two point boundary value conditions, is of constant sign. This constant sign is equivalent to the strongly inverse positive (negative) character of the related operator on suitable spaces related to the boundary conditions. This characterization is based on spectral theory, in fact the extremes of the obtained interval are given by suitable eigenvalues of the differential operator with different boundary conditions. Moreover, we also obtain a characterization of the strongly inverse positive (negative) character on some sets, where non homogeneous boundary conditions are considered. In order to see the applicability of the obtained results, some examples are given along the paper. This method avoids the explicit calculation of the related Green's function.