On periodic solutions of two-dimensional nonautonomous differential systems

Nonimprovable conditions are established for the existence and uniqueness of an ω-periodic solution of the nonautonomous differential systemsui′=pi1(t)u1+pi2(t)u2+qi(t)(i=1,2)andui′=fi(t,u1,u2)(i=1,2),where pik:R→R, qi:R→R(i,k=1,2) are ω-periodic functions, Lebesgue integrable on [0,ω], and fi:R×R2→R(i=1,2) are functions from the Carathéodory class such thatfi(t+ω,x1,x2)≡fi(t,x1,x2)(i=1,2).