On Convexity, Monotonicity and Positivity Analysis for Discrete Fractional Operators Defined Using Exponential Kernels

Pshtiwan Othman Mohammed,Ravi P. Agarwal,Y. S. Hamed,Ohud Almutairi

doi:10.3390/fractalfract6020055

Abstract

This article deals with analysing the positivity, monotonicity and convexity of the discrete nabla fractional operators with exponential kernels from the sense of Riemann and Caputo operators. These operators are called discrete nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional operators. Further, some of our results concern sequential nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional differences, such as ∇aCFRμ∇bCFCυh(x), for various values of start points a and b, and for orders υ and μ in different ranges. Three illustrative examples of the main lemmas in the case of Riemann–Liouville are given at the end of the article.

Highlights

Fractional Operators Defined UsingThe discrete fractional calculus (DFC) is a field of mathematical analysis and it is a new branch of continuous fractional calculus that is responsible for studying the discrete operators of the sum and difference of fractional order on domains of discrete functions.The definitions of DFC models with singular and nonsingular kernels have been studied by researchers in recent years
Monotonicity, positivity and convexity analyses of discrete fractional operators have been perhaps one of the most fundamental studies in the context of discrete fractional calculus due to a wide variety of representations, forms and applications. Due to these new challenges and trends of discrete fractional calculus, many scholars are devoting their attention to establishing new monotonicity results for discrete generalized nabla fractional operators with discrete singular and non-singular kernels on both time scales Z
The remaining sections of this article are organized as follows: Section 2 is dedicated to recalling the basic concept of discrete CFC fractional operators and some related properties including the relationship between discrete nabla CFC and CFR fractional differences; Section 3 deals with the analysis of the results of the discrete nabla CF fractional operators: Section 3.1 is dedicated to the analysis of positivity and monotonicity results; Section 3.2 is dedicated to the analysis of monotonicity and υ-convexity results; Section 3.3 is dedicated to the analysis of convexity results in some specific domains; Section 4 includes the discussion of results by means of presenting some fractional difference initial value problems; we present a brief conclusion and expectations in the last section

Summary

Introduction

Fractional Operators Defined UsingThe discrete fractional calculus (DFC) is a field of mathematical analysis and it is a new branch of continuous fractional calculus that is responsible for studying the discrete operators of the sum and difference of fractional order on domains of discrete functions.The definitions of DFC models with singular and nonsingular kernels have been studied by researchers in recent years (see [1,2,3,4,5,6]). Monotonicity, positivity and convexity analyses of discrete fractional operators have been perhaps one of the most fundamental studies in the context of discrete fractional calculus due to a wide variety of representations, forms and applications. Due to these new challenges and trends of discrete fractional calculus, many scholars are devoting their attention to establishing new monotonicity results for discrete generalized nabla fractional operators with discrete singular and non-singular kernels on both time scales Z and h Z

Objectives

Results

Conclusion