On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

Pshtiwan Othman Mohammed,Faraidun Kadir Hamasalh,Thabet Abdeljawad

doi:10.3390/fractalfract5030116

Pshtiwan Othman Mohammed, Faraidun Kadir Hamasalh + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract5030116

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Abstract

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

Highlights

IntroductionSeveral authors have begun to study monotonicity analysis in the context of discrete fractional calculus, especially in fractional difference equations
We focus on implementing monotonicity analysis for the discrete delta CF
To obtain the mean value theorem with discrete fractional difference terms, monotonicity analysis is considered for the discrete delta Caputo–Fabrizio fractional operators

Summary

Introduction

Several authors have begun to study monotonicity analysis in the context of discrete fractional calculus, especially in fractional difference equations. They have often obtained some υ-increasing and υ-decreasing results for the discrete nabla and delta operators. Atici and Uyanik [11] obtained several monotonicity analysis results for the discrete nabla Riemann–Liouville fractional operators on the time scale Z. Abdeljawad and Abdallaa [15] used the dual identities to obtain some monotonicity results for the discrete nabla and delta Riemann–Liouville and Caputo fractional operators on the time scale Z. Established new monotonicity results for discrete generalized nabla Attangana–Baleanu fractional operators with discrete generalized Mittag–Leffler kernels on the time scale Z.

Preliminaries and Basic Concepts

Discrete Delta Caputo–Fabrizio Fractional Sums

Results on Discrete Monotonicity Analysis

Fractional Difference Mean Value Theorem

Conclusions