On the solvability of the periodic problem for systems of linear generalized ordinary differential equations. (English) Zbl 1398.34036

Ashordia, Malkhaz; Chania, Medea; Kuciai, Malkhaz

Mem. Differ. Equ. Math. Phys. 74, 7-26 (2018).

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

Keywords:

nonlocal boundary value problem; periodic problem; linear systems; generalized ordinary differential equations; unique solvability; effective conditions