Explicit representation of discrete fractional resolvent families in Banach spaces

Abstract

In this paper we introduce a discrete fractional resolvent family $\{S_{\alpha,\beta}^n\}_{n\in\mathbb{N}_0}$ generated by a closed linear operator in a Banach space $X$ for a given $\alpha,\beta>0.$ Moreover, we study its main properties and, as a consequence, we obtain a method to study the existence and uniqueness of the solutions to discrete fractional difference equations in a Banach space.