Pseudo (\(\omega ,c\))-periodic solutions to Volterra difference equations in Banach spaces

In this paper, we consider a new recurrent sequence in Banach spaces, which is called pseudo ( \(\omega ,c\) $\omega ,c$ )-periodic sequence. This notion is the discretized pseudo ( \(\omega ,c\) $\omega ,c$ )-periodic function and covers the pseudo \(\omega \) $\omega$ -(anti-)peridoic sequence and pseudo ( \(\omega ,c\) $\omega ,c$ )-Bloch periodic sequence in Banach spaces. We prove the completeness, convolution and superposition theorems for the pseudo ( \(\omega ,c\) $\omega ,c$ )-periodic sequence in abstract spaces. We also consider some existence results for pseudo ( \(\omega ,c\) $\omega ,c$ )-periodic sequential solutions to a Volterra difference equation of convolution type in Banach spaces under some different conditions. Some numerical simulations are given to illustrate the existence results.