Existence of \(2^n\) periodic solutions of a system of \(n\) differential equations of first order (Q1781391)

The author considers the problem of the existence of a solution for the system of ordinary differential equations \[ u_i'= f_i(t,u_1,\dots, u_n),\quad 0< t< 1,\quad i= 1,\dots, n, \] with two-point boundary conditions \[ u_i(0)= \Delta_i u_i(1), \] where \(\Delta_i\in \mathbb{R}\), \(f_i(t, u_1,\dots, u_n)\), \(i= 1,\dots, n,\) satisfies the Carathéodory conditions for \(0\leq t\leq 1\) and \(-\infty< u_k<+\infty\), \(k=1,\dots, n\).