Existence of $T/k$-Periodic Solutions of a Nonlinear Nonautonomous System Whose Matrix Has a Multiple Eigenvalue

A system of $n$th-order ordinary differential equations with relay nonlinearity and periodic perturbation function on the right-hand side is studied. The matrix of the system has real nonzero eigenvalues, among which there is at least one positive and one multiple eigenvalue. A nonsingular transformation that reduces the matrix of the system to Jordan form is used. Continuous periodic solutions with two switching points in the phase space of the system are considered. It is assumed that the period of the perturbation function is a multiple of the periods of these solutions. Necessary conditions for the existence of such solutions are established. An existence theorem for a solution of period equal to the period of the perturbation function is proved. A numerical example confirming the obtained results is presented.