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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.

MSC:
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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