Spectral Characterization of the Constant Sign Derivatives of Green's Function Related to Two Point Boundary Value Conditions

In this paper we will consider a n-th order linear operator Tn[M]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{n}[M]$$\end{document}, depending on a real parameter M, coupled to different two-point boundary conditions, and we will study the set of parameters for which certain partial derivatives of the related Green's function are of constant sign. We will do it without using the explicit expression of the Green's function. In particular, the set of parameters for which the derivatives of the Green's function have constant sign will be an interval whose extremes are characterized as the first eigenvalues of the studied operator related to suitable boundary conditions. As a consequence of the main result, we will be able to give sufficient conditions to ensure that the derivatives of the Green's function cannot be nonpositive (nonnegative). These characterizations and the obtained results can be used to deduce the existence of solutions of nonlinear problems under additional conditions on the nonlinear part. In order to illustrate the obtained results, some examples are given.