Finite time stability of fractional delay differential equations

In this article, we discuss a new coupled system of fractional differential equations with integral boundary conditions $$\textstyle\begin{cases} D^{\alpha}u(t)+f(t,v(t))=a,\quad 0< t< 1,\\ D^{\beta}v(t)+g(t,u(t))=b,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\int_{0}^{1} \phi(t)u(t)\,dt,\\ v(0)=0,\qquad v(1)=\int_{0}^{1} \psi(t)v(t)\,dt, \end{cases} $$
 where $1< \alpha,\beta\le2, f,g \in C([0,1]\times(-\infty,+\infty ),(-\infty,+\infty)), \phi,\psi\in L^{1}[0,1]$
 , $a,b$
 are constants and D denotes the usual Riemann-Liouville fractional derivative. Based upon a fixed point theorem of increasing φ-
 $(h,e)$
 -concave operators, we establish the existence and uniqueness of solutions for the new coupled system dependent on two constants. And then the obtained result is well demonstrated with the aid of an interesting example.

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Unique solutions for a new coupled system of fractional differential equations

In this paper, coexistence and stability of multiple equilibrium points of fractional-order recurrent neural networks are addressed. Several sufficient conditions are derived for ascertaining the existence of $ {\prod _{i=1}^{n}(2K_{i}+1)}$ equilibrium points ( $ {K_{i}\geq 0}$ ) and the local Mittag–Leffler stability of $ {\prod _{i=1}^{n}(K_{i}+1)}$ equilibrium points of them by using the geometrical properties of activation functions and algebraic properties of nonsingular ${M}$ -matrix. In contrast with many existing results, the derived results cover both mono-stability and multistability, and the activation functions herein could be nonmonotonic and nonlinear in any open interval. In addition, three numerical examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.

Multiple Mittag–Leffler Stability of Fractional-Order Recurrent Neural Networks

We study a nonlocal boundary value problem of Hadamard type coupled sequential fractional differential equations supplemented with coupled strip conditions (nonlocal Riemann-Liouville integral boundary conditions). The nonlinearities in the coupled system of equations depend on the unknown functions as well as their lower order fractional derivatives. We apply Leray-Schauder alternative and Banach’s contraction mapping principle to obtain the existence and uniqueness results for the given problem. An illustrative example is also discussed.

A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional “differintegral” equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind. Received February 1, 2019; revised October 7, 2019; accepted October 29, 2019. 2020 Mathematics Subject Classification: primary 26A33, 33B15, 33C05, 33C20, 33E12, 34A25, 44A10, secondary 33C65, 34A05, 34A08.

https://www.koreascience.or.kr:443/article/JAKO202011464411170.pdf

Yuruo Zhang

Papers

On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives