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On the solvability of the periodic problem for systems of linear generalized ordinary differential equations. (English) Zbl 1398.34036
From the text: In the present paper, we investigate the solvability for the system of linear generalized ordinary differential equations \[ dx(t)= dA(t)\cdot x(t)+ df(t)\tag{1} \] with the \(\omega\)-periodic \((\omega>0)\) condition \[ x(t+\omega)= x(t)\quad\text{for }t\in\mathbb{R},\tag{2} \] where \(A= (a_{ik})^n_{i,k=1}: \mathbb{R}\to \mathbb{R}^{n\times n}\) and \(f= (f_i)^n_{i=1}:\mathbb{R}\to\mathbb{R}^n\) are, respectively, the matrix- and the vector-functions with hounded variation components on every closed interval from \(\mathbb{R}\), and \(\omega\) is a fixed positive number.
We establish the Green type theorem on the solvability of problem (1), (2) and the representation of a solution of the problem. In addition, we give effective necessary and sufficient conditions (of spectral type) for the unique solvability of the problem.

34B15 Nonlinear boundary value problems for ordinary differential equations
34A30 Linear ordinary differential equations and systems, general
34C25 Periodic solutions to ordinary differential equations
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