ON THE GENERAL AND MULTIPOINT BOUNDARY VALUE PROBLEMS FOR LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS, LINEAR IMPULSE AND LINEAR DIFFERENCE SYSTEMS

The system of the generalized linear ordinary differential equations dx(t) = dA(t) + x(t) + df(t) is considered with general l(x) = c(0), multipoint Sigma(n0)(j=1) L(j)x(t(j)) = C-0, and CauchyNicoletti type x(i)(l(i)) = l(i)(x(1),..., x(n)) (i = 1,..., n) boundary value conditions, where Lambda : [a, b] -> R-nxn and f : [a, b] -> R-n are, respectively, matrix and vector -functions with bounded total variation components on the closed interval [a, b], c(0) = (co(i))(i=1)(n) is an element of R-n, t(i) is an element of [a, b] (i = 1,..., n(n(o))), n(o) is a fixed natural number, L-j is an element of R-nxn (j = 1,...,n(o)), xi is the i-th component of x, and l and l(i) (i = 1,, n) are linear operators. Effective sufficient, among them spectral, conditions are obtained for the unique solvability of these problems. The obtained results are realized for the linear impulsive system dx/dt = P(t)x + q(t), x(Tk+) - x(Tk-) = Gkx(Tk) gk (k = 1, 2,...), where P is an element of E L([a, b], R-nxn), q is an element of L([a, b]R-n), Gk is an element of R-n and T-k is an element of [a,b] (k = 1, 2,...), and linear difference system Delta y(k - 1)=G(1)(k - 1)y(k - 1) + G(2)(k)y(k) + G3(k)y(k + 1) + g0(k) (k = 1,..., mo), where G(j)(k) is an element of R-nxn, go(k) is an element of R-n (j = 1, 2, 3; k = 1,..., m(0)).