THE GENERAL BOUNDARY VALUE PROBLEMS FOR LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS, LINEAR IMPULSIVE DIFFERENTIAL AND ORDINARY DIFFERENTIAL SYSTEMS. NUMERICAL SOLVABILITY

For the system of generalized linear ordinary differential equations, the boundary value problem dx = dA(t).x + df(t) (t is an element of I), l(x) = c(0) is considered, where I = [a, b] is a closed interval, A : I -> R-nxn and f : I -> R-n are, respectively, the matrix- and vector-functions with components of bounded variation, l is a linear bounded vector-functional, c(0) is an element of R-n. Under a solution of the system is understood a vector-function x : I -> R-n with components of bounded variation satisfying the corresponding integral equality, where the integral is understood in the Kurzweil sense. Along with a number of questions, such as solvability, construction of solutions, etc., we investigate the problem of the well-posedness. Effective sufficient conditions, as well as effective necessary and sufficient conditions, are established for each of these problems. The obtained results are realized for the above boundary value problem for linear impulsive system dx/dt = P(t)x + q(t), x(tau(l)+) - x(tau(l)-) = G(tau(l))x(tau(l)) + u(tau(l)) (l = 1, 2, ...), where P and q are, respectively, the matrix- and vector-functions with Lebesgue integrable components, tau(l) (l = 1, 2, ...) are the points of impulse actions, and G(tau(l)) and u(tau(l)) (l = 1, 2, ...) are the matrix- and vector-functions of discrete variables. Using the well-posedness results, the effective sufficient conditions, as well as the effective necessary and sufficient conditions, are established for the convergence of difference schemes to the solution of linear boundary value problem for impulsive systems of differential equations, as well for ordinary differential equations. The analogous results are obtained for the stability of difference schemes.