We present an analog of Massera Theorem for asymptotic periodic solutions of linear equations x′(t)=A(t)x(t)+f(t),t≥0(⁎), where the family of linear operators A(t) generates a 1-periodic process (U(t,s))t≥s≥0 in a Banach space X and f is asymptotic 1-periodic in the sense that limt→∞(f(t+1)−f(t))=0. The main result says that if 1 is isolated in σ(U(1,0)) on the unit circle Γ, then (⁎) has an asymptotic 1-periodic mild solution if and only if it has an asymptotic mild solution that is bounded and uniformly continuous with precompact range. If 1∉σ(U(1,0))∩Γ, such an asymptotic 1-periodic mild solution always exists and unique within a function g(t) with limt→∞g(t)=0. Our study relies on a spectral theory of functions on the half line and the evolution semigroups associated with linear equations. The obtained results seem to be new, even in the finite dimensional case.