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

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## Abstract

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem d*x* = d*A*(*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*(t_{1}(*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}.

### Keywords

system of nonlinear generalized ordinary differential equations Kurzweil-Stieltjes integral general boundary value problem solvability principle of a priori bound-edness### MSC 2010

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