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) · f(t, x), h(x) = 0 is established, where f: [a, b]×Rn → Rn is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → Rn×n with bounded total variation components, and h: BVs([a, b],Rn) → Rn 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(t1(x)) = B(x) · x(t2(x))+c0, where ti: BVs([a, b],Rn) → [a, b] (i = 1, 2) and B: BVs([a, b], Rn) → Rn are continuous operators, and c0 ∈ Rn.