The Sign of the Green Function of an n-th Order Linear Boundary Value Problem

Pedro Almenar Belenguer,Lucas Jódar

doi:10.3390/math8050673

Pedro Almenar Belenguer, Lucas Jódar

Open Access

https://doi.org/10.3390/math8050673

Copy DOI

Abstract

This paper provides results on the sign of the Green function (and its partial derivatives) of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed.

Highlights

Let J be a compact interval in R and let us consider the real disfocal differential operator L:→ C ( J ) defined by Cn ( J )Ly = y(n) ( x ) + an−1 ( x )y(n−1) ( x ) + · · · + a0 ( x )y( x ), x ∈ J, (1)where a j ( x ) ∈ C ( J ), 0 ≤ j ≤ n − 1
This paper provides results on the sign of the Green function of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions
Following Eloe and Ridenhour [1], let Ωl be the set whose members are collections of l different ordered integer indices i such that 0 ≤ i ≤ n − 1, let k ∈ N be such that 1 ≤ k ≤ n − 1, let α ∈ Ωk be the set {α1, . . . , αk } and β ∈ Ωn−k be the set { β 1, . . . , β n−k }, both associated to the homogeneous boundary conditions y(αi ) ( a) = 0, i = 1, 2, . . . , k, αi ∈ α, (2)

Summary

Introduction

Let J be a compact interval in R and let us consider the real disfocal differential operator L:. The purpose of this paper will be to provide results on the sign of G ( x, t), the Green function associated to the problem. G ( x, t) f (y(t), t)dt, x ∈ ( a, b) In most of these problems, the information about the sign of the Green function is relevant to apply other tools (fixed-point theorems, upper and lower solutions method, fixed-point index theory, etc.). When we want to highlight the dependence of the Green function of (4) on the boundary conditions (α, β) and the extremes a, b, respectively. That will be useful when we manipulate Green functions subject to different boundary conditions or different extremes. If we assume that y is a function with (n − 1)th derivative in [ a, b], we will make use of the following nomenclature associated to (α, β):.

The Sign of the Green Function and Its Derivatives on the Admissible Case

The Case of p-Alternate Boundary Conditions

The Strongly Admissible Case

Discussion