Consider the class of the differential equations in the form t (x t h t B t x t k t t ( ) + = + ( )) ( ) ( ) ( ), + . (1) By using Massera-type theorem, we demonstrate the existence and uniqueness of periodic solutions for the equation (1), with the operator B t X D B X ( ) : ( ) → is - periodic and possibly unbounded on a Banach space X
the function h gets the value in the Banach space D(B), the function k attains the value in the Banach space X and is -periodic.