## 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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