Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions

TL;DR

This paper derives explicit Green's functions for general n-th order linear differential equations with non-local boundary conditions, providing comparison principles and conditions for solution uniqueness, with an illustrative example.

Contribution

It provides a novel explicit expression for Green's functions in complex boundary value problems with non-local conditions, extending previous methods.

Findings

Explicit Green's function formula depends on two-point problem Green's function

Comparison principles for solutions are established

Conditions for solution uniqueness are identified

Abstract

In this paper we obtain the explicit expression of the Green's function related to a general $n$ order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, a $n$ dimensional parameter dependence is also assumed. Moreover, some comparison principles are obtained. The explicit expression depends on the value of the Green's function related to the two-point homogeneous problem, that is, we are assuming that when all the parameters involved on the boundary conditions take the value zero then the problem has a unique solution which is characterized by the corresponding Green's function $g$. The expression of the Green's function $G$ of the general problem is given as a function of $g$ and the real parameters considered at the boundary conditions. It is important to show that, in order to ensure the uniqueness of solutions of the linear considered problem, we must assume a non resonant additional condition on the considered problem, which depends on the non local conditions and the corresponding parameters. We point out that the assumption of the uniqueness of solutions of the two-point homogeneous problem is not a necessary condition to ensure the solution of the general case. Of course, in this situation the expression we are looking for must be obtained in a different manner. To show the applicability of the obtained results, a particular example is given.