n阶线性微分方程耦合任意线性非局部边界条件的格林函数
Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions
摘要：
本文给出了一般$n$阶微分方程耦合非局部线性边界条件的格林函数的显式表达式。在这样的边界条件下，还假设了$n$维度参数依赖性。此外，还得到了一些比较原则。明确的表达取决于格林函数的值有关的两点齐次问题，也就是说，我们假设，当所有的参数涉及的边界条件采取的值为零，那么这个问题有一个独特的解决方案，其特征在于相应的格林函数$g$。一般问题的格林函数$G$的表达式被给出为$g$和在边界条件下考虑的实参数的函数。重要的是要表明，为了确保所考虑的线性问题的解的唯一性，我们必须假设所考虑的问题的非共振附加条件，这取决于非局部条件和相应的参数。指出两点齐次问题解的唯一性假设并不是保证一般情况解的必要条件。当然，在这种情况下，我们所寻找的表达式必须以不同的方式获得。为了说明所得结果的适用性，给出了一个特殊的例子。
英文摘要：
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.
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