The boundedness and the existence of periodic solutions to the functional equation (1) \(x'(t)=F(t,x(t+s))\) is considered, where \(F\in {\mathcal C}(R\times C_ g,R^ n)\) and \(-h\leq s\leq 0\). Using the Lyapunov functional method conditions are established under which solutions to (1) are uniformly bounded and uniformly ultimately bounded with respect to unbounded \((C_ g)\) initial function spaces. The Lyapunov functional is also used for proving the existence of periodic solutions to (1).