A multi-point problem for a system of differential equations with piecewise-constant argument of generalized type as a neural network model

We consider a system of ordinary differential equations with piecewise-constant argument of generalized type. An interval is divided into $N$ parts, the values of a solution at the interior points of the subintervals are considered as additional parameters, and a system of ordinary differential equations with piecewise-constant argument of generalized type is reduced to the Cauchy problems on the subintervals for linear system of ordinary differential equations with parameters. Using the solutions to these problems, new general solutions to system of differential equations with piecewiseconstant argument of generalized type are introduced and their properties are established. Based on the general solution, boundary condition, and continuity conditions of a solution at the interior points of the partition, the system of linear algebraic equations with respect to parameters is composed. Its coefficients and right-hand sides are found by solving the Cauchy problems for a linear system of ordinary differential equations on the subintervals. It is shown that the solvability of boundary value problems is equivalent to the solvability of composed systems. Methods for solving boundary value problems are proposed, which are based on the construction and solving of these systems.