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 $c_0\in\mathbb{R}^n$., Malkhaz Ashordia., and Obsahuje bibliografické odkazy
The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense dx(t) = dA0(t) · x(t) + df0(t), x(t0) = c0 (t ∈ I) with a unique solution x0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems dx(t) = dAk(t) · x(t) + dfk(t), x(tk) = ck (k = 1, 2, . . .) to have a unique solution xk for any sufficiently large k such that xk(t) → x0(t) uniformly on I. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.