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.

18 pages, 2 figures