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]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: a, b] → R ^n×n with bounded total variation components, and h: BV_s(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(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s(a, b],R ⁿ ) → a, b] (i = 1, 2) and B: BV_s(a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .