The eigenvalue Characterization for the constant Sign Green's Functions of $(k,n-k)$ problems

TL;DR

This paper characterizes the parameter ranges for which the Green's function of certain linear differential operators with boundary conditions maintains a constant sign, using eigenvalue analysis without explicitly computing the Green's function.

Contribution

It provides a novel eigenvalue-based method to determine constant sign intervals of Green's functions for general linear operators with $(k,n-k)$ boundary conditions, avoiding direct Green's function calculation.

Findings

Identifies intervals of parameter M with constant sign Green's functions.

Shows the sign behavior depends on the parity of (n-k).

Provides explicit eigenvalue-based criteria for sign determination.

Abstract

This paper is devoted to the study of the sign of the Green's function related to a general linear $n^{\rm th}$-order operator, depending on a real parameter, $T_n[M]$, coupled with the $(k,n-k)$ boundary value conditions. If operator $T_n[\bar M]$ is disconjugate for a given $\bar M$, we describe the interval of values on the real parameter $M$ for which the Green's function has constant sign. One of the extremes of the interval is given by the first eigenvalue of operator $T_n[\bar M]$ satisfying $(k,n-k)$ conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of $(k-1,n-k+1)$ and $(k+1,n-k-1)$ problems. Moreover if $n-k$ is even (odd) the Green's function cannot be non-positive (non-negative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green's functions for particular operators. Our method avoids the necessity of calculating the expression of the Green's function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator $T_n[\bar M]$ for a given $\bar M$ cannot be eliminated.