On periodic-type solutions of systems of linear ordinary differential equations
The author considers existence and uniqueness of an \(\omega\)-periodic solution of the linear differential system \[ \frac {dx_i}{dt}={\mathcal P}_{i_1}(t)x_1+{\mathcal P}_{i_2}(t)x_2+q_i(t), \quad i=1,2, \] satisfying the conditions \[ x_i(t+\omega)=\Lambda_ix_i(t)\text{ for }t\in\mathbb{R},\quad i=1,2, \] where \(\omega>0\); \(\Lambda_i\in \mathbb{R}^{n_i\times n_i},\) \(i=1,2\), are nonsingular matrices; \(n_1,n_2\) natural numbers; \[ {\mathcal P}_{i_k}:\mathbb{R}\to \mathbb{R}^{n_i\times n_k},\quad i,k=1,2, \] and \(q_i:\mathbb{R}\to \mathbb{R}^{n_i}\), \(i=1,2\), matrix and vector functions, whose components are Lebesgue integrable on each compact interval.
scientific article; zbMATH DE number 2163661
scientific article