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 <i>X</i>(<i>t</i>) = <i>L</i>(<i>t</i>)<i>X</i>(<i>t</i>) + <i>F</i>(<i>t</i>), <i>M</i><sub>1</sub><i>X</i>(0) + <i>M</i><sub>2</sub><i>X</i>(<i>b</i>) = <i>Q</i>, where <i>t</i> ∈ <i>T</i> = [0, <i>b</i>], <i>L</i> : <i>T</i> → R<sup>pxp</sup> and <i>F</i> : <i>T</i> → R<sup>p</sup> are right-continuous matrix and vector valued functions of bounded variation; <i>M</i><sub>1</sub>, <i>M</i><sub>2</sub> ∈ R<sup>pxp</sup>, Q ∈ R<sup>p</sup> 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.