On Hilfer fractional difference operator

TL;DR

This paper introduces a new fractional Hilfer difference operator, develops its properties, and applies it to establish existence, uniqueness, and stability results for fractional difference equations, along with a modified Gronwall inequality.

Contribution

The paper presents a novel definition of the fractional Hilfer difference operator and explores its properties and applications in fractional difference equations.

Findings

Established conditions for existence and uniqueness of solutions.

Proved Ulam-Hyers and Ulam-Hyers-Rassias stability.

Presented a modified Gronwall's inequality for discrete calculus.

Abstract

In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial condition. We acquire some conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability. Modified Gronwall's inequality is presented for discrete calculus with the delta difference operator.