Existence and Multi-Stability of a Generalized ABC Fractional-Order Neural Network Model

Abstract

This study presents a fractional-order neural network model formulated using the Atangana-Baleanu-Caputo Fractional Derivative (ABC-FD) defined with respect to a generalized kernel function ϑ(t). The primary objective is to establish rigorous results on the existence, uniqueness, and stability of solutions under minimal regularity assumptions. By employing Banach's and Krasnoselskii's fixed point theorems, we prove existence and uniqueness. The stability analysis compares three regimes: Mittag-Leffler, asymptotic, and finite-time, showing that they form a hierarchy of convergence strength: asymptotic stability ensures gradual decay, Mittag-Leffler stability provides algebraic convergence, and finite-time stability guarantees exact quenching within a bounded interval. Numerical simulations of two- and three-neuron systems confirm these theoretical distinctions, illustrating the role of both the fractional order and ϑ(t) in shaping the rate and type of convergence.