In this paper, we obtain an explicit expression for the Green’s function of a certain type of systems of differential equations subject to non-local linear boundary conditions. In such boundary conditions, the dependence on certain parameters is considered. The idea of the study is to transform the given system into another first-order differential linear system together with the two-point boundary value conditions. To obtain the explicit expression of the Green’s function of the considered linear system with non-local boundary conditions, it is assumed that the Green’s function of the homogeneous problem, that is, when all the parameters involved in the non-local boundary conditions take the value zero, exists and is unique. In such a case, the homogeneous problem has a unique solution that is characterized by the corresponding Green’s function g. The expression of the Green’s function of the given system is obtained as the sum of the function g and a part that depends on the parameters involved in the boundary conditions and the expression of function g. The novelty of our work is that in the system to be studied, the unknown functions do not appear separated neither in the equations nor in the boundary conditions. The existence of solutions of nonlinear systems with linear non-local boundary conditions is also studied. We illustrate the obtained results in this paper with examples.

在本文中，我们获得了受非局部线性边界条件影响的某类微分方程组的格林函数的显式表达式。在这种边界条件下，需要考虑对某些参数的依赖性。研究的思想是将给定系统连同两点边值条件一起变换为另一个一阶微分线性系统。为了获得所考虑的具有非局部边界条件的线性系统的格林函数的显式表达式，假设齐次问题的格林函数，即当涉及非局部边界条件的所有参数都取值为零，存在且唯一。在这种情况下，齐次问题有一个唯一的解决方案，其特征在于相应的格林函数克。给定系统的格林函数的表达式是函数g与取决于边界条件中涉及的参数和函数g的表达式的部分之和。我们工作的新颖之处在于，在要研究的系统中，未知函数在方程和边界条件中都不会出现分离。还研究了具有线性非局部边界条件的非线性系统解的存在性。我们通过示例说明本文中获得的结果。