An explicit formula of the parameter dependence of de partial derivatives of the Green's functions related to arbitrary two-point boundary conditions

TL;DR

This paper derives an explicit integral formula for how Green's functions' partial derivatives depend on parameters in two-point boundary value problems, simplifying proofs of their monotonicity and extending applicability.

Contribution

It provides a new explicit integral expression for parameter dependence of Green's functions' derivatives, improving previous monotonicity results under broader conditions.

Findings

Derived an explicit integral formula for Green's function derivatives.

Simplified proof of monotonicity of Green's functions' derivatives.

Extended results to various differential equations and boundary conditions.

Abstract

In this paper we obtain an explicit formula of the parameter dependence of the partial derivatives of the Green's functions related to two-point boundary conditions. Such expression follows as an integral of both kernels times the difference of the corresponding parameters of each Green's function. As a direct consequence, we deduce a simpler proof of the monotony of the constant sign of the partial derivative of a Green's function with respect to a real parameter. As a consequence, we improve the results obtained in \cite{C1}, where the monotone dependence was proved for the constant sign Green's function (not for any ot its partial derivatives) and under weaker assumptions on the Green's function. The arguments are valid for any other types of Ordinary Differential Equations coupled to Nonlocal Conditions. Moreover, analogous ideas could be developed for Partial and Fractional Differential Equations.