Some Properties of Green's Matrix of Nonlinear Boundary Value Problem of First Order Differential

This paper discusses Green's matrix of nonlinear boundary value problem of first-order differential system with rectangular coeffisients, especially about its properties. In this case, the differential equation of the form  with boundary conditions of the form   and  which  is a real  matrix with  whose entries are continuous on  and . ,  are nonsingular matrices such that  and  are constant vectors. To get the Green's matrix and the assosiated generalized Green's matrix, we change the boundary condition problem into an equivalent  differential equation by using the properties of the  Moore-Penrose generalized inverse, then  its solution is found by using method of variation of parameters. The last we prove  that the defined matrices  satisfy the properties of green's function. The result is the corresponding the Green's matrix and the assosiated generalized Green's matrix have the property of Green's functions with the jump-discontinuity.


Introduction
Green's matrix is a function. It is used to represent certain terms in the solution of differential system. In the special case, for one differential equation, it is called Green's function. Talking about Green's function, beside it is a solution of the homogeneous differential equation also there are some the related properties. Among others are the continuity, the derivative and the uniqueness of it. As a solution of the homogeneous differential equation, Green's function has discontinuity at certain points. The jumdiscontinuity of Green's function can have different magnitude. The difference follow the form of differential equations or regarded variable [2,5,6,7]. In the other side in [9] , it has been proved uniqueness the solution of nonlinear boundary value problem of firstorder differential system with rectangular coefficients by using method of iterative monotone.
Inspired by [2,5,6,7,9], in this research will be discussed the property of the Green's function that hold on the Green's matrix of nonlinear boundary value problem of first-order differential system with rectangular coefficients. The objective of this research is to prove that the Green's matrix and the assosiated generalized Green's matrix of the relevant boundary value problem satisfy the property of Green's function.

Research Method
To prove the properties of Green's function that hold to the Green's matrix and generalized Green's matrix done in four steps. Firstly is to change a nonlinear boundary value problem of first-order differential system with rectangular coefficients into a differential equation that equivalent by using the property of inverse of Moore-Penrose matrix. Then to find the solution by using method of variation of parameters. After that defining Green's matrix and Green's matrix generalized associate. The last proving the properties of Green's function possessed by them.

Result and Discussion 1. Prelimineries
We consider a nonlinear boundary problem of first-order differential system with rectangular coefficients of the form ( ) , + ( ) = ( , ) (1.1) with boundary condition of the form where ( ) is a real × matrix with > that entries are continuous on = [ 0 , ] and ∈ [ × ℝ , ℝ ]. 1 , 2 are nonsingular matrices, and are constant vectors.
We will give some definitions and theories of Moore-Penrose matrix inverse .
Consider an equation

Main Result
In this part will be proved the property of Green'n function that satisfied by Green's matrix. Before it we define the Green's matrix and the assosiated generalized Green's matrix as follow. The continuity and existence of its first derivative as well as an upward jumpdiscontinuity of ( , ) at the points = can be proved similarly. We leave it.

Proof:
For arbitrary, let ( , ) is others Green's matrix that has properties of Theorem 2.2 and Theorem 2.3. Write ( , ) = ( , ) − ( , ). It will be shown that ( , ) = . It is obvious that ( , ) has the first derivative continue with respect to in interval It means that is unique. The uniqueness of can be proved similarly. We leave it.

Conclusion
In this paper we have the conclusion that the Green's matrix and the assosiated generalized Green's matrix satisfy the property of Green's function and they have an upward jump-discontinuity of unit matrix magnitude at the points = .