Explicit representation of discrete fractional resolvent families in Banach spaces

TL;DR

This paper introduces a new discrete fractional resolvent family in Banach spaces, providing a framework to analyze solutions of discrete fractional difference equations with potential applications in various mathematical and engineering fields.

Contribution

It develops the concept of discrete fractional resolvent families generated by linear operators in Banach spaces, offering new tools for studying fractional difference equations.

Findings

Defined the discrete fractional resolvent family $ ext{S}_{ ext{α,β}}^n$ in Banach spaces.

Established properties of the resolvent family.

Provided a method for existence and uniqueness of solutions to fractional difference equations.

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.