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On the question of solvability of the periodic boundary value problem for a system of generalized ordinary differential equations. (English) Zbl 0907.34014

Sufficient conditions are given for the solvability of the \(\omega\)-periodic boundary value problem \[ dx(t)= dA(t)\cdot f(t, x(t)),\quad x(0)= x(\omega), \] where \(\omega\) is a positive number, \(A(t)= (a_{ik}(t))^n_{i, k=1}\), \(a_{ik}\equiv a^{(1)}_{ik}(t)- a^{(2)}_{ik}(t)\), \(a^{(\sigma)}_{ik}: \mathbb{R}\to \mathbb{R}\) \((\sigma= 1,2)\) are functions nondecreasing on \([0,\omega]\), \(a^{(\sigma)}_{ik}(t+ \omega)= a^{(\sigma)}_{ik}(t)+ a^{(\sigma)}_{ik}(\omega)\) for \(t\in \mathbb{R}\); \(f= (f_k)^n_{k= 1}\): \(\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is \(\omega\)-periodic with respect to the first variable such that its restriction on \([0,\omega]\times \mathbb{R}^n\) belongs to the Carathéodory classes corresponding to the matrix functions \(A^{(1)}\) and \(A^{(2)}\), \(A^{(\sigma)}(t)\equiv (a^{(\sigma)}_{ik}(t))^n_{i, k=1}\) \((\sigma= 1,2)\).

MSC:

34B05 Linear boundary value problems for ordinary differential equations
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