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

summary:A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $$ {\rm d} x={\rm d} A(t)\cdot f(t,x),\quad h(x)=0 $$ is established, where $f\colon [a,b]\times \mathbb {R}^n\to \mathbb {R}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A\colon [a,b]\to \mathbb {R}^{n\times n}$ with bounded total variation components, and $h\colon \operatorname {BV}_s([a,b],\mathbb {R}^n)\to \mathbb {R}^n$ 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 $x(t_1(x))=\mathcal {B}(x)\cdot x(t_2(x))+c_0,$ where $t_i\colon \operatorname {BV}_s([a,b],\mathbb {R}^{n})\to [a,b]$ $(i=1,2)$ and $\mathcal {B}\colon \operatorname {BV}_s([a,b],\mathbb {R}^{n})\to \mathbb {R}^n$ are continuous operators, and $c_0\in \mathbb {R}^n$