In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete $ℏ$ -proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete $ℏ$ -proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla $ℏ$ fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

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Vol. 30, No. 01

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Received 18 June 2020

Accepted 31 May 2021

Published: September 11, 2021

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This is an Open Access article in the “Special Issue Section on Fractal AI-Based Analyses and Applications to Complex Systems: Part II”, edited by Yeliz Karaca (University of Massachusetts Medical School, USA), Dumitru Baleanu (Cankaya University, Turkey), Majaz Moonis (University of Massachusetts Medical School, USA), Khan Muhammad (Sejong University, South Korea), Yu-Dong Zhang (University of Leicester, UK) & Osvaldo Gervasi (Perugia University, Italy) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 (CC-BY-NC-ND) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

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