In this paper we study a new class of functions, which we call ( ω , c ) $(\omega ,c)$ -pseudo periodic functions. This collection includes pseudo periodic, pseudo anti-periodic, pseudo 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 ( ω , c ) $(\omega ,c)$ -pseudo 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 ( ω , c ) $(\omega ,c)$ -pseudo periodic solutions to the Lasota–Wazewska equation with unbounded oscillating production of red cells.

In this paper we introduce the class of \((N,\lambda )\)-periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of \((N, \lambda )\)-periodic solutions to the following class of Volterra difference equations with infinite delay: $$ u(n+1)=\alpha \sum_{j=-\infty }^{n}a(n-j)u(j)+f \bigl(n,u(n) \bigr), \quad n \in \mathbb{Z}, \alpha \in \mathbb{C}, $$ where the kernel a and the nonlinear term f satisfy suitable conditions.