In this paper, we study a new class of functions, which we call (omega,c)-asymptotically periodic functions. This collection includes asymptotically periodic, asymptotically antiperiodic, asymptotically Bloch-periodic, and unbounded functions. We prove that the set conformed by these functions is a Banach space with a suitable norm. Furthermore, we show several properties of this class of functions as the convolution invariance. We present some examples and a composition result. As an application, we prove the existence and uniqueness of (omega,c)-asymptotically periodic mild solutions to the first-order abstract Cauchy problem on the real line. Also, we establish some sufficient conditions for the existence of positive (omega,c)-asymptotically periodic solutions to the Lasota-Wazewska equation with unbounded oscillating production of red cells.