Solution of the boundary value problem for differential equation with piecewise-constant argument of generalized type

We examine a boundary value problem for a differential equation with piecewise-constant argument of generalized type. An interval [0,T] is divided into N parts, the values of a solution at the interior points of the subintervals are treated as additional parameters, and boundary value problem for differential equation with piecewise-constant argument of generalized type is transformed to an equivalent initial value problems with parameters for differential-algebraic equations on subintervals. Differential part of this problem contains of the Cauchy problems for ordinary differential equations on the subintervals. Algebraic part of this problem contains of the algebraic equations with respect to parameters composed by boundary and continuity conditions at interior points of the partition. The coefficients and right-hand sides of this system are determined by solving the Cauchy problems for ordinary differential equations on the subintervals. We demonstrate that the solvability of boundary value problems is equivalent to the solvability of the composed systems. We propose methods for solving boundary value problems based on the construction and solution of these systems.