Boundary value problem for system of finite-difference with averaging equations

The boundary value problem for the system of linear nonhomogeneous differential equations with generalized coefficients is considered
$$ \begin{cases} \dot{X}(t)=\dot{L}(t)X(t)+\dot{F}(t), \\ M_{1}X(0)+M_{2}X(b)=Q, \end{cases} $$
where $t\in T=[0,b], L:T\rightarrow \mathbb{R}^{p\times p}$ и $F:T\rightarrow \mathbb{R}^{p}$ are right-continuous matrix and vector valued functions of bounded variation; $M_{1}, M_{2}\in \mathbb{R}^{p\times p}, Q\in \mathbb{R}^{p}$ are defined matrices and vector. The problem is investigated with the help of the corresponding finite-difference with averaging equation behavior studying. The definition of the fundamental matrix, corresponding to the finite-difference with averaging equation is introduced. The theorem of the existence and uniqueness of the finite-difference with averaging boundary value problem, corresponding to the described system is proved.