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On boundary value problems for systems of nonlinear generalized ordinary differential equations. (English) Zbl 06770119
Summary: A general theorem (principle of a priori boundedness) on solvability of the boundary value problem \[ \text{d}x=\text{d}A(t)\cdot f(t,x),\quad h(x)=0 \] 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 \(c_0\in\mathbb{R}^n\).

34K10 Boundary value problems for functional-differential equations
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