# A linearized stability theorem for nonlinear delay fractional differential equations

## TL;DR

This paper establishes a linearized stability criterion for nonlinear delay fractional differential equations using a novel approach involving diagonalization, Mittag-Leffler functions, and fixed point theory.

## Contribution

It introduces a new stability theorem for fractional delay differential equations based on linearization and advanced mathematical techniques.

## Key findings

- Proves asymptotic stability of equilibria via linearization.
- Develops a method converting linear parts into diagonal form.
- Utilizes Mittag-Leffler functions and Lyapunov--Perron operator.

## Abstract

In this paper, we prove a theorem of linearized asymptotic stability for fractional differential equations with a time delay. More precisely, using the method of linearization of a nonlinear equation along an orbit (Lyapunov's first method), we show that an equilibrium of a nonlinear Caputo fractional differential equation with a time delay is asymptotically stable if its linearization at the equilibrium is asymptotically stable. Our approach based on a technique which converts the linear part of the equation into a diagonal one. Then using properties of generalized Mittag-Leffler functions, the construction of an associated Lyapunov--Perron operator and the Banach contraction mapping theorem, we obtain the desired result.