Constant sign Green's function of a second order perturbed periodic problem

TL;DR

This paper derives the exact Green's function for a second order perturbed periodic problem with integral boundary conditions and characterizes the parameter set where the Green's function maintains a constant sign.

Contribution

It provides a novel explicit expression for the Green's function and analyzes its sign for a specific class of perturbed periodic boundary value problems.

Findings

Explicit Green's function expression obtained.

Parameter set for constant sign identified.

Sign properties linked to the problem's parameters.

Abstract

In this paper we are interested in obtaining the exact expression and the study of the constant sign of the Green's function related to a second order perturbed periodic problem coupled with integral boundary conditions at the extremes of the interval of definition. To obtain the expression of the Green's function related to this problem we use the theory presented in \cite{CLY} for general non-local perturbed boundary value problems. Moreover, we will characterize the parameter's set where such Green's function has constant sign. To this end, we need to consider first a related second order problem without integral boundary conditions, obtaining the properties of its Green's function and then using them to compute the sign of the one related to the main problem.