The author develops an apparatus for proving the existence of periodic solutions to differential equations \(y'(t)=Ay(t)+f(t,y(t))\) and to differential inclusions \(y'(t)\in Ay(t)+F(t,y(t))\). Here, \(A\) is a constant \(n\times n\) matrix, \(f\) is a Carathéodory function, and \(F\) is a Carathéodory multifunction. If the equation \(y'(t)=Ay(t)\) has no nonzero periodic solutions with period \(\omega>0\) and the function \(f\) is time-periodic with period \(\omega\), then the differential equation has a periodic solution with period \(\omega\). The same result holds in case of the differential inclusion.