Two-point boundary-value problems for differential equations with generalized piecewise-constant argument

Abstract

UDC 517.9 We consider a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem.