On Hilfer fractional difference operator

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.

On substantial fractional difference operator

“…We next present stability criteria for the Cauchy problem under consideration. Authors also discussed Ulam type stabilities of fractional difference equation with multipoint boundary value problem and for nonlinear Hilfer F Cauchy problem in [30,41].…”

Existence and stability results for nonlinear fractional integrodifferential coupled systems

“…Very recently, numerous monographs have appeared concerning the results of the existence and stability of Ulam-Hyers-Rassias and Ulam-Hyers of nonlinear fractional differential equations focused on Riemann-Liouville, Caputo, Hilfer, etc. The readers can refer to the papers of Wang and Xu [13], Rajan et al [14] and Haider et al [15] so on [16][17][18][19][20][21][22][23]. However, there are few research results on the existence and stability of solutions for the ψ-Hilfer fractional derivative system except for [24,25].…”

Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives

“…There are several definitions, each with its advantages and disadvantages [1,2]. One way to overcome this problem is to introduce more general concepts of fractional operators, such as the Hilfer operator [3,4], derivatives depending on another function [5][6][7], or involving arbitrary kernels [8,9].…”