On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) center dot f(t, x), h(x) = 0 is established, where f: [a, b]xR (n) -> R (n) is a vector-function belonging to the Carath,odory class corresponding to the matrix-function A: [a, b] -> R (nxn) with bounded total variation components, and h: BVs([a, b],R (n) ) -> R (n) is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t(1)(x)) = B(x) center dot x(t (2)(x))+c (0), where t (i): BVs([a, b],R (n) ) -> [a, b] (i = 1, 2) and B: BVs([a, b], R (n) ) -> R (n) are continuous operators, and c (0) a R (n) .