Relationship of the Green's functions related to the Hill's equation coupled to different boundary value conditions

TL;DR

This paper analyzes the properties of Green's functions associated with Hill's equation under various boundary conditions, exploring their relationships and implications for solving nonlinear problems using fixed point theory.

Contribution

It introduces a method to express Green's functions of different boundary conditions as linear combinations, facilitating comparison and analysis for nonlinear problem solutions.

Findings

Green's functions can be expressed as linear combinations of each other.

Properties of Green's functions with constant sign are characterized.

Existence of solutions to nonlinear problems is established using fixed point theory.

Abstract

In this paper we will deduce several properties of the Green's functions related to the Hill's equation coupled to various boundary value conditions. In particular, the idea is to study the Green's functions of the second order differential operator coupled to Neumann, Dirichlet, Periodic and Mixed boundary conditions, by expressing the Green's function of a given problem as a linear combination of the Green's function of the other ones. This will allow us to compare different Green's functions when they have constant sign. Finally, such properties of the Green's function of the linear problem will be fundamental to deduce the existence of solutions to the nonlinear problem. The results are derived from the fixed point theory applied to related operators defined on suitable cones in Banach spaces.