On Hilfer fractional difference operator

Abstract

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.

Keywords

• Hilfer fractional difference
• Discrete Mittag-Leffler functions
• Delta Laplace transform
• Gronwall’s inequality
• Existence and uniqueness
• Ulam stability of initial value problem