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On existence of solutions to the periodic boundary value problem for a nonlinear system of generalized ordinary differential equations. (English) Zbl 0934.34010

The author considers an \(\omega\)-periodic problem for the nonlinear system \[ dx(t)= dA(t)\cdot f(t,x(t)),\tag{1} \] where \(\omega\) is a positive number, and \(A= (a_{ik})^n_{i,k= 1}\), \(A(t)= A^{(1)}(t)- A^{(2)}(t)\), \(A^{(m)}\) \((m= 1,2)\) are \(\omega\)-periodic matrix functions such that the restrictions of their components on \([0,\omega]\) are nondecreasing and have bounded total variation on \([0,\omega]\). \(f= (f_k)^n_{k=1}\), \(f_k: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\), \(k= 1,\dots, n\), are \(\omega\)-periodic in \(t\) and their restrictions on \([0,\omega]\) fulfil the Carathéodory conditions on \([0,\omega]\times \mathbb{R}^n\) with respect to the measures \(\mu(a_{ik})\) for every \(i\in \{1,\dots, n\}\).
Under growth conditions on \(f\) and inequalities for \(A^{(m)}\) the existence of at least one \(\omega\)-periodic solution to (1) is presented.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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