Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique

Guo-Cheng Wu,Dumitru Baleanu,Kai-Teng Wu,Thabet Abdeljawad,Jinliang Liu

doi:10.15388/na.2019.6.5

Abstract

A class of semilinear fractional difference equations is introduced in this paper. The fixed point theorem is adopted to find stability conditions for fractional difference equations. The complete solution space is constructed and the contraction mapping is established by use of new equivalent sum equations in form of a discrete Mittag-Leffler function of two parameters. As one of the application, finite-time stability is discussed and compared. Attractivity of fractional difference equations is proved, and Mittag-Leffler stability conditions are provided. Finally, the stability results are applied to fractional discrete-time neural networks with and without delay, which show the fixed point technique’s efficiency and convenience.

Highlights

This paper aims to address this problem and propose a new kind of discrete-time neural network by use of discrete fractional calculus on an isolated time scale [1,2,3,4,5,6, 9, 24]
Many efforts have been made to the stability theory of fractional differential equations
Fractional difference equations can be considered a class of generalized difference equations

Summary

Preliminaries

We use the following definitions in this paper. Definition 1. (See [6, 9, 24].) Let Na := {a, a + 1, a + 2, . . . }. u : Na → R and 0 < ν be given. The νth-order Riemann–Liouville difference is given by. We revisit some basics in stability theory in the discrete fractional calculus. We assume |λ| < 1 for the convergence of the discrete Mittag-Leffler function eν(λ, t − a). (ii) the Mittag-Leffler function is asymptotically stable [7]: eν(λ, t − a) → 0, t → +∞, where −1 < λ < 0; (iii) the discrete Mittag-Leffler function of two parameter is defined by eν,ν (λ, t − a) =. We note the following results hold: t−ν eν,ν λ, t − σ(s) eν (λ, t a). If t ∈ Na+1, These lemmas and remarks are useful to analyze stability conditions by the fixed point technique in the rest.

Picard’s method

Application to finite-time stability

Mittag-Leffler stability

Numerical examples

Conclusions