Abstract Nonimprovable conditions are established for the existence and uniqueness of an ω -periodic solution of the nonautonomous differential systems u i ′ = p i 1 ( t ) u 1 + p i 2 ( t ) u 2 + q i ( t ) ( i = 1 , 2 ) and u i ′ = f i ( t , u 1 , u 2 ) ( i = 1 , 2 ) , where p ik : R → R , q i : R → R ( i , k = 1 , 2 ) are ω -periodic functions, Lebesgue integrable on [ 0 , ω ] , and f i : R × R 2 → R ( i = 1 , 2 ) are functions from the Caratheodory class such that f i ( t + ω , x 1 , x 2 ) ≡ f i ( t , x 1 , x 2 ) ( i = 1 , 2 ) .