New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel.

Abstract

Discrete fractional calculus (DFC) use to analyse nonlocal behaviour of models has acquired great importance in recent years. The aim of this paper is to address the discrete fractional operator underlying discrete Atangana-Baleanu (AB)-fractional operator having $\hbar$-discrete generalized Mittag-Leffler kernels in the sense of Riemann type (ABR). In this strategy, we use the $\hbar$-discrete AB-fractional sums in order to obtain the Gr\"{u}ss type and certain other related variants having discrete generalized $\hbar$-Mittag-Leffler function in the kernel. Meanwhile, several other variants found by means of Young, weighted-arithmetic-geometric mean techniques with a discretization are formulated in the time domain $\hbar\mathbb{Z}$. At first, the proposed technique is compared to discrete AB-fractional sums that uses classical approach to derive the numerous inequalities, showing how the parameters used in the proposed discrete $\hbar$-fractional sums can be estimated. Moreover, the numerical meaning of the suggested study is assessed by two examples. The obtained results show that the proposed technique can be used efficiently to estimate the response of the neural networks and dynamic loads.