Existence and Globally Asymptotic Stability of Equilibrium...

Research ArticleExistence and Globally Asymptotic Stability of EquilibriumSolution for Fractional-Order Hybrid BAM Neural Networkswith Distributed Delays and Impulses

Hai Zhang,1 Renyu Ye,1,2 Jinde Cao,3,4 and Ahmed Alsaedi5

1School of Mathematics and Computation Science, Anqing Normal University, Anqing 246133, China2College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China3School of Mathematics and Research Center for Complex Systems and Network Sciences, Southeast University,Nanjing 210096, China4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia5Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics,King Abdulaziz University, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Jinde Cao; [email protected]

Received 7 May 2017; Revised 10 July 2017; Accepted 20 July 2017; Published 13 September 2017

Academic Editor: Amr Elsonbaty

Copyright © 2017 Hai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including thedistributed delays, impulsive effects, and two different fractional-order derivatives between the 𝑈-layer and 𝑉-layer are taken intoaccount synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existenceand uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed offractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained,which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method isthat one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to showthe validity and feasibility of the theoretical results.

1. Introduction

Since fractional derivatives are nonlocal and have weaklysingular kernels, the subject of fractional calculus has beenattracting attention and interest in various fields of diffu-sion [1], physics [2], market dynamics [3], engineering [4],control system [5], biological system [6], financial system[7], epidemic model [8], and so on. At the same time,fractional-order differential equations have been proved to bean excellent tool in themodelling ofmany phenomena [9–11].Recently, some important advances on dynamical behaviorssuch as chaos phenomena, Hopf bifurcation, synchronizationcontrol, and stabilization problems for fractional-order sys-tems or fractional-order practical models have been reportedin [12–16]. These proposed results show the superiority and

importance of fractional calculus and effectively motivate thedevelopment of new applied fields.

Note that various classes of neural networks such asHopfield neural networks [17, 18], recurrent neural networks[19, 20], cellular neural networks [21], Cohen-Grossbergneural networks [22], and bidirectional associative memory(BAM) neural networks [23–25] have been widely used insolving some signal processing, optimization, and imageprocessing problems. In the last few years, some researchershave introduced fractional operators to neural networks toform fractional-order neural models [26–30], which couldbetter describe the dynamical behaviors of the neurons. As animportant dynamic behavior, stability is one of the most con-cerned problems for any dynamic system. For example, Songand Cao [26] have established some sufficient conditions to

HindawiComplexityVolume 2017, Article ID 6875874, 13 pageshttps://doi.org/10.1155/2017/6875874

https://doi.org/10.1155/2017/6875874

2 Complexity

ensure the existence and uniqueness of the nontrivial solutionby using the contraction mapping principle, Krasnoselskiifixed point theorem, and the inequality technique, in whichuniform stability conditions of fractional-order neural net-works are also derived in fixed time-intervals. Note that time-delay (see [23–25, 31–37]) is a common phenomenon andis inevitable in practice, which often exists in almost everyneural network and has an important effect on the stabilityand performance of system.

There are also several recent results discussing the topicsincluding stability analysis for fractional-order dynamicalsystems in [38, 39]. For instance, the stability problems ofmain concern for control theory in finite-dimensional linearfractional-order systems have been considered [38], in whichboth internal and external stabilities for fractional-order dif-ferential systems in state-space form have been studied. Forfractional-order differential systems in polynomial represen-tation, the external stability has been thoroughly discussed.In [39], Matouk has investigated the stability conditions of aclass of fractional-order hyperchaotic systems; then the sta-bility conditions have been applied to a novel fractional-orderhyperchaotic system. Based on the Routh-Hurwitz theorem,the conditions for controlling hyperchaos via feedback con-trol approach have also been derived. At the same time, thevarious kinds of stability of delayed fractional-order neuralnetworks have been extensively investigated. For example,Mittag-Leffler stability of fractional-order delayed neural net-works has been investigated by applying fractional Lyapunovdirect method [28, 30, 32].The finite-time stability of Caputofractional-order delayed neural networks has been studiedby applying Gronwall’s inequality approach and inequalityscaling techniques [33, 34]. The delay-independent stabilitycriteria of Riemann-Liouville fractional-order neutral-typedelayed neural networks have been proposed based onclassical Lyapunov functional method [35]. The uniformstability and global stability of fractional neural networkswith delay are considered based on the fractional calculustheory and analytical techniques [36]. Global 𝑜(𝑡−𝛼) stabilityand global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time-varyingdelays are discussed by using the fractional Lyapunovmethodand a Leibniz rule for fractional differentiation [37].

Although most dynamical systems are analyzed in eitherthe continuous-time or discrete-time domain, many realsystems in physics, engineering, chemistry, biology, andinformation science may experience abrupt changes as cer-tain instants during the continuous dynamical processes.This kind of impulsive behaviors can be modelled by impul-sive systems [23, 25, 29, 32, 40–42]. On the other hand,bidirectional associative memory (BAM) neural networksattract many studies due to its extensive applications in manyfields [22–25, 43–46]. In [43], Kosko first introduced hybridBAM neural network models. The remarkable feature of theproposed BAM neural networks lies in the close relation ofthe neurons between the 𝑈-layer and 𝑉-layer. That is, theneurons in one layer are fully interconnected to the one inthe other layer, but there are not any interconnections amongneurons in the same layer. It is worth mentioning that manycontributions have been made concerning the dynamics of

fractional-order BAM delayed neural networks (see [44–46]) including finite-time stability [44] and Mittag-Lefflersynchronization [45]. In [46], globally asymptotic stabilityproblem of impulsive fractional-order neural networks withdiscrete delays has been studied, yet the existence of the equi-librium solution for fractional-order BAM neural networkshas not been taken into account. On the other hand, it shouldbe pointed out that the finite-time stability and asymptoticstability in the sense of Lyapunov are different concepts,because finite-time stability does not contain Lyapunovasymptotic stability and vice versa [34, 47]. Although thesignal transmission is sometimes instantaneous modellingwith discrete delays, it may be sometimes a distributionpropagation delay over a period of time so that distributeddelays (see [20, 23, 25]) should not be ignored in the model.Compared to the advances of integer-order neural networkswith or without time delays, the research on the stabilityof fractional-order BAM delayed neural networks is still atthe stage of exploiting and developing [44–46]. To the bestof our knowledge, there are few papers on investigating theglobal stability of the fractional-order hybrid BAM neuralnetworks with both impulse and distributed delay in thecurrent literature.

Motivated by the above discussions, this paper investi-gates the existence and globally asymptotic stability of equi-librium solution for impulsive Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays.The factors of such network systems including the distributeddelays, impulses, and two different fractional-order deriva-tives between the𝑈-layer and𝑉- layer are taken into accountsynchronously. Based on the contraction mapping principle,the sufficient conditions are presented for the existence anduniqueness of the equilibrium solution for such networksystems. By constructing a suitable Lyapunov functional asso-ciated with fractional integral terms, the globally asymptoticstability criteria of the equilibrium point are derived. Theadvantage of constructing the Lyapunov functional is thatone can directly calculate its first-order derivative to checkglobal stability. A numerical example is also given to showthe validity and feasibility of the theoretical results.

This paper is organized as follows. In Section 2, werecall some definitions concerning fractional calculus anddescribe impulsive Riemann-Liouville fractional-order BAMneural networks with distributed delays. In Section 3, theexistence and uniqueness of the equilibrium solution forsuch network systems are discussed based on the contractionmapping principle. In Section 4, the globally asymptoticstability criteria of the equilibrium solution are derived. Anillustrative example is given to show the effectiveness andapplicability of the proposed results in Section 5. Finally,some concluding remarks are drawn in Section 6.

2. Preliminaries and Model Description

In this section, we recall the definitions of fractional calculusand several basic lemmas. Moreover, we describe a classof impulsive fractional-order hybrid BAM neural networkmodels with distributed delays.

Complexity 3

Definition 1 (see [10]). The Riemann-Liouville fractionalintegral of order 𝑞 for a function 𝑓 is defined as

𝑡0𝐷−𝑞𝑡𝑓 (𝑡) = 1Γ (𝑞) ∫

𝑡

𝑡0

(𝑡 − 𝑠)𝑞−1 𝑓 (𝑠) 𝑑𝑠, (1)where 𝑞 > 0, 𝑡 ⩾ 𝑡0. The Gamma function Γ(𝑞) is defined bythe integral

Γ (𝑧) = ∫+∞0

𝑠𝑧−1𝑒−𝑠𝑑𝑠, (Re (𝑧) > 0) . (2)Currently, there exist several definitions about the frac-

tional derivative of order 𝑞 > 0 includingGrünwald-Letnikov(GL) definition, Riemann-Liouville (RL) definition, andCaputo definition [9–11]. In this paper, our consideration isthe fractional-order neural networks with Riemann-Liouvillederivative, whose definition and properties are given below.

Definition 2 (see [10]). The Riemann-Liouville fractionalderivative of order 𝑞 for a function 𝑓 is defined as

RL𝑡0𝐷𝑞𝑡𝑓 (𝑡) = 𝑑𝑚𝑑𝑡𝑚 [ 𝑡0𝐷−(𝑚−𝑞)𝑡 𝑓 (𝑡)]

= 1Γ (𝑚 − 𝑞) 𝑑𝑚

𝑑𝑡𝑚 ∫𝑡

𝑡0

(𝑡 − 𝑠)𝑚−𝑞−1 𝑓 (𝑠) 𝑑𝑠,(3)

where 0 ⩽ 𝑚 − 1 < 𝑞 < 𝑚, 𝑚 ∈ Z+.In particular, for 𝛼 ∈ (0, 1) case, the Riemann-Liouville

fractional derivative of order 𝛼 for a constant 𝑥∗ isRL0𝐷𝛼𝑡 𝑥∗ = 𝑡−𝛼Γ (1 − 𝛼)𝑥∗. (4)

Lemma 3 (see [10]). If 𝑓(𝑡), 𝑔(𝑡) ∈ C𝑚[𝑡0, 𝑏], and 𝑚 − 1 ⩽𝑝 < 𝑚 ∈ Z+, then(1) RL𝑡0𝐷𝑞𝑡 (𝐿1𝑓(𝑡) + 𝐿2𝑔(𝑡)) = 𝐿1RL𝑡0𝐷𝑞𝑡𝑓(𝑡) + 𝐿2RL𝑡0𝐷𝑞𝑡𝑔(𝑡),𝐿1, 𝐿2 ∈ R, 𝑞 > 0;(2) 𝑡0𝐷−𝑝𝑡 ( 𝑡0𝐷−𝑞𝑡 𝑓(𝑡)) = 𝑡0𝐷−(𝑝+𝑞)𝑡 𝑓(𝑡), 𝑝, 𝑞 > 0;(3) RL𝑡0𝐷𝑝𝑡 ( 𝑡0𝐷−𝑞𝑡 𝑓(𝑡)) = RL𝑡0𝐷𝑝−𝑞𝑡 𝑓(𝑡), 𝑝 > 𝑞 > 0;(4) RL𝑡0𝐷𝑝𝑡 ( 𝑡0𝐷−𝑞𝑡 𝑓(𝑡)) = 𝑡0𝐷−(𝑞−𝑝)𝑡 𝑓(𝑡), 𝑞 > 𝑝 > 0.The following lemmas will be used in the proof of our

main results.

Lemma4 (contractionmapping principle [48]). Suppose that(𝑋, 𝜌) is a completemetric space,Φ : 𝑋 → 𝑋, and there is somereal number 0 < 𝑘 < 1 such that

𝜌 (Φ (𝑥) , Φ (𝑦)) ⩽ 𝑘𝜌 (𝑥, 𝑦) , ∀𝑥, 𝑦 ∈ 𝑋; (5)then there is a unique point 𝑥0 ∈ 𝑋 such that Φ(𝑥0) = 𝑥0.Lemma5 (fractional Barbalat lemma [42]). If∫𝑡

𝑡0𝑤(𝑠)𝑑𝑠has a

finite limit as 𝑡 → +∞, and RL𝑡0𝐷𝛼𝑡 𝑤(𝑡) is bounded, then𝑤(𝑡) →0 as 𝑡 → +∞, where 0 < 𝛼 < 1.

In this paper, we consider the Riemann-Liouville frac-tional-order hybrid BAM neural network models with dis-tributed delay and impulsive effects described by the follow-ing states equations:

RL0𝐷𝛼𝑡 𝑥𝑖 (𝑡) = −𝑎𝑖𝑥𝑖 (𝑡) +

𝑚∑𝑗=1

𝑏𝑖𝑗𝑓𝑗 (𝑦𝑗 (𝑡))

+ 𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑑𝑠 + 𝐼𝑖,

𝑡 > 0, 𝑡 ̸= 𝑡𝑘,Δ𝑥𝑖 (𝑡𝑘) = 𝛾(1)𝑘 (𝑥𝑖 (𝑡𝑘)) ,

𝑖 = 1, 2, . . . , 𝑛; 𝑘 = 1, 2, . . . ,RL0𝐷𝛽𝑡 𝑦𝑗 (𝑡) = −𝑐𝑗𝑦𝑗 (𝑡) +

𝑛∑𝑖=1

𝑑𝑗𝑖𝑔𝑖 (𝑥𝑖 (𝑡))

+ 𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) 𝑑𝑠 + 𝐽𝑗,

𝑡 > 0, 𝑡 ̸= 𝑡𝑘,Δ𝑦𝑗 (𝑡𝑘) = 𝛾(2)𝑘 (𝑦𝑗 (𝑡𝑘)) ,

𝑗 = 1, 2, . . . , 𝑚; 𝑘 = 1, 2, . . . ,

(6)

where 𝑈 = {𝑥1, 𝑥2, . . . , 𝑥𝑛} and 𝑉 = {𝑦1, 𝑦2, . . . , 𝑦𝑚} are twolayers in the BAMmodel (6);𝑥𝑖(𝑡) and𝑦𝑗(𝑡) are state variablesof 𝑖th neuron in the 𝑈-layer and 𝑗th neuron in the 𝑉-layer,respectively; RL0𝐷𝛼𝑡 𝑥𝑖(⋅) and RL0𝐷𝛽𝑡 𝑦𝑗(⋅) denote an 𝛼 and a 𝛽order Riemann-Liouville fractional-order derivative of 𝑥𝑖(⋅)and 𝑦𝑗(⋅), respectively; the constants 𝛼 and 𝛽 satisfy 0 < 𝛼 0 and 𝑐𝑗 > 0 denote decay coefficients ofsignals from neurons 𝑥𝑖 to 𝑦𝑗, respectively; 𝑓𝑖 and 𝑔𝑗 are theneuron activation functions; 𝑏𝑖𝑗, 𝑑𝑗𝑖, 𝑟𝑖𝑗(𝑡) and 𝑝𝑗𝑖(𝑡) representthe weight coefficients of the neurons; 𝐼𝑖 and 𝐽𝑗 denote theexternal inputs of 𝑈-layer and 𝑉-layer, respectively; 𝜏 >0 denotes the maximum possible transmission delay fromneuron to another. Moreover, impulsive moments {𝑡𝑘 | 𝑘 =1, 2, . . .} satisfy 0 = 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑘 < ⋅ ⋅ ⋅ , 𝑡𝑘 → +∞as 𝑘 → +∞, andΔ𝑥𝑖 (𝑡𝑘) = 𝑥𝑖 (𝑡+𝑘 ) − 𝑥𝑖 (𝑡−𝑘 ) ,

𝑥𝑖 (𝑡+𝑘 ) = lim𝜀→0+

𝑥𝑖 (𝑡𝑘 + 𝜀) , 𝑥𝑖 (𝑡−𝑘 ) = 𝑥𝑖 (𝑡𝑘) ,Δ𝑦𝑗 (𝑡𝑘) = 𝑦𝑗 (𝑡+𝑘 ) − 𝑦𝑗 (𝑡−𝑘 ) ,

𝑦𝑗 (𝑡+𝑘 ) = lim𝜀→0+

𝑦𝑗 (𝑡𝑘 + 𝜀) , 𝑦𝑗 (𝑡−𝑘 ) = 𝑦𝑗 (𝑡𝑘) ,(7)

where 𝑥𝑖(𝑡+𝑘 ) and 𝑥𝑖(𝑡−𝑘 ) represent the right and left limits of𝑥𝑖(𝑡) at 𝑡 = 𝑡𝑘, respectively; 𝑥𝑖(𝑡−𝑘 ) = 𝑥𝑖(𝑡𝑘) and 𝑦𝑗(𝑡−𝑘 ) = 𝑦𝑗(𝑡𝑘)imply that 𝑥𝑖(𝑡) and 𝑦𝑗(𝑡) are both left continuous at 𝑡 =𝑡𝑘. The initial conditions associated with Riemann-Liouville

4 Complexity

fractional-order network system (6) can be expressed as (see[9–11])

0𝐷−(1−𝛼)𝑡 𝑥𝑖 (𝑡) = 𝜑𝑖 (𝑡) ,0𝐷−(1−𝛼)𝑡 𝑦𝑗 (𝑡) = 𝜓𝑗 (𝑡) ,

𝑖 = 1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , 𝑚, 𝑡 ∈ [−𝜏, 0] .(8)

Throughout this paper, we assume that the neuron acti-vation functions 𝑓𝑗, 𝑔𝑖 and impulsive operators 𝛾(1)𝑘 (𝑥𝑖(𝑡𝑘)),𝛾(2)𝑘(𝑦𝑗(𝑡𝑘)) satisfy the following conditions:(H1) For 𝑖 = 1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , 𝑚, the functions 𝑟𝑖𝑗(⋅)

and 𝑝𝑗𝑖(⋅) are continuous on [0, 𝜏]. Thus, there exist positiveconstants 𝑅𝑖𝑗, 𝑃𝑗𝑖 ∈ R+ such that𝑟𝑖𝑗 (𝑠) ⩽ 𝑅𝑖𝑗,𝑝𝑗𝑖 (𝑠) ⩽ 𝑃𝑗𝑖,

∀𝑠 ∈ [0, 𝜏] .(9)

(H2) The neuron activation functions 𝑓𝑗(⋅), 𝑔𝑖(⋅) (𝑖 =1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , 𝑚) are Lipschitz continuous. That is,there exist positive constants 𝐹𝑗, 𝐺𝑗 ∈ R+ such that𝑓𝑗 (𝑥) − 𝑓𝑗 (𝑦) ⩽ 𝐹𝑗 𝑥 − 𝑦 ,𝑔𝑖 (𝑥) − 𝑔𝑖 (𝑦) ⩽ 𝐺𝑖 𝑥 − 𝑦 ,

∀𝑥, 𝑦 ∈ R.(10)

(H3)The impulsive operators 𝛾(1)𝑘(𝑥𝑖(𝑡𝑘)) and 𝛾(2)𝑘 (𝑦𝑗(𝑡𝑘))

satisfy

𝛾(1)𝑘 (𝑥𝑖 (𝑡𝑘)) = −𝜆(1)𝑖𝑘 (𝑥𝑖 (𝑡𝑘) − 𝑥∗𝑖 ) ,𝑖 = 1, 2, . . . , 𝑛; 𝑘 = 1, 2, . . . ,

𝛾(2)𝑘 (𝑦𝑗 (𝑡𝑘)) = −𝜆(2)𝑗𝑘 (𝑦𝑗 (𝑡𝑘) − 𝑦∗𝑗 ) ,𝑗 = 1, 2, . . . , 𝑚; 𝑘 = 1, 2, . . . ,

(11)

where 𝜆(1)𝑖𝑘∈ (0, 2) (𝑖 = 1, 2, . . . , 𝑛; 𝑘 = 1, 2, . . .), and 𝜆(2)

𝑗𝑘∈(0, 2) (𝑗 = 1, 2, . . . , 𝑚; 𝑘 = 1, 2, . . .).

Remark 6. The purpose of this paper is to investigate theexistence and globally asymptotic stability conditions ofthe equilibrium solution for fractional-order BAM networkmodel (6). In discussing the stability of neural networks,the neuron activation functions are usually assumed to bebounded, monotonic [23], and differential [36, 37]. In system(6), the neuron activation functions are not necessarilybounded, monotonic, and differential.Therefore, the globallyasymptotic stability criteria are more general and less conser-vative in this paper.

3. Existence of Equilibrium Solution

In this section, the sufficient conditions for the existenceand uniqueness of the equilibrium solution of system (6) arederived based on the contraction mapping principle [48].

Similar to integer-order differential systems, we firstdefine the equilibrium solution of fractional-order networksystems. It should be pointed out that Riemann-Liouvillefractional-order derivative of a nonzero constant is notequal to zero, which leads to the remarkable difference ofthe equilibrium solution between integer-order systems andRiemann-Liouville fractional-order systems.

Definition 7. Aconstant vector (𝑥∗𝑇, 𝑦∗𝑇)𝑇 = (𝑥∗1 , 𝑥∗2 , . . . , 𝑥∗𝑛 ,𝑦∗1 , 𝑦∗2 , . . . , 𝑦∗𝑚)𝑇 ∈ R𝑛+𝑚 is an equilibrium solution of system(6) if and only if 𝑥∗ = (𝑥∗1 , 𝑥∗2 , . . . , 𝑥∗𝑛 )𝑇 and 𝑦∗ = (𝑦∗1 , 𝑦∗2 , . . . ,𝑦∗𝑚)𝑇 satisfy the following equations:

RL0𝐷𝛼𝑡 {𝑥∗𝑖 } = −𝑎𝑖𝑥∗𝑖 +

𝑚∑𝑗=1

𝑏𝑖𝑗𝑓𝑗 (𝑦∗𝑗 )

+ 𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦∗𝑗 ) 𝑑𝑠 + 𝐼𝑖,

𝑖 = 1, 2, . . . , 𝑛,RL0𝐷𝛽𝑡 {𝑦∗𝑗 } = −𝑐𝑗𝑦∗𝑗 +

𝑛∑𝑖=1

𝑑𝑗𝑖𝑔𝑖 (𝑥∗𝑖 )

+ 𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) 𝑔𝑖 (𝑥∗𝑖 ) 𝑑𝑠 + 𝐽𝑗,

𝑗 = 1, 2, . . . , 𝑚,

(12)

and the impulsive jumps 𝛾(1)𝑘(𝑥𝑖(𝑡𝑘)) and 𝛾(2)𝑘 (𝑦𝑗(𝑡𝑘)) are

assumed to satisfy

𝛾(1)𝑘 (𝑥∗𝑖 ) = 0,𝛾(2)𝑘 (𝑦∗𝑗 ) = 0,

𝑖 = 1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , 𝑚; 𝑘 = 1, 2, . . . .(13)

In what follows, we use the following vector norm ofR𝑛+𝑚:

‖𝑢‖ = 𝑛+𝑚∑𝑖=1

𝑢𝑖 , 𝑢 = (𝑢1, 𝑢2, . . . , 𝑢𝑛+𝑚)𝑇 ∈ R𝑛+𝑚. (14)

Theorem 8. Suppose that conditions (H1)–(H3) hold; thenthere exists a unique equilibrium solution for system (6), if thefollowing inequalities simultaneously hold for a small enoughconstant 𝜀 > 0

𝜔1 = max1⩽𝑖⩽𝑛

{{{𝜀Γ (1 − 𝛼) ⋅ 1𝑎𝑖 +

𝐺𝑖𝑎𝑖𝑚∑𝑗=1

[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}< 1,

Complexity 5

𝜔2 = max1⩽𝑗⩽𝑚

{ 𝜀Γ (1 − 𝛽) ⋅ 1𝑐𝑗 +𝐹𝑗𝑐𝑗𝑛∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]}< 1.

(15)

Proof. According to Definition 2, for 𝛼, 𝛽 ∈ (0, 1), theRiemann-Liouville fractional-order derivatives of the con-stants 𝑢∗𝑖 and V∗𝑗 can be written as the following forms:

RL0𝐷𝛼𝑡 𝑢∗𝑖 = 𝑡−𝛼Γ (1 − 𝛼)𝑢∗𝑖 ,

RL0𝐷𝛽𝑡 V∗𝑗 = 𝑡−𝛽Γ (1 − 𝛽)V∗𝑗 ,

𝑖 = 1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , 𝑚.(16)

Define a mapping Φ : R𝑛+𝑚 → R𝑛+𝑚, where u = (𝑢1, . . . , 𝑢𝑛,V1, . . . , V𝑚)𝑇 ∈ R𝑛+𝑚 and

Φ (u) =

[[[[[[[[[[[[[[[[[[[[[[

𝑚∑𝑗=1

𝑏1𝑗𝑓𝑗 (V𝑗𝑐𝑗 ) +𝑚∑𝑗=1

∫𝜏0𝑟1𝑗 (𝑠) 𝑓𝑗 (V𝑗𝑐𝑗 )𝑑𝑠 + 𝐼1 −

𝑡−𝛼Γ (1 − 𝛼) 𝑢1𝑎1...𝑚∑𝑗=1

𝑏𝑛𝑗𝑓𝑗 (V𝑗𝑐𝑗 ) +𝑚∑𝑗=1

∫𝜏0𝑟𝑛𝑗 (𝑠) 𝑓𝑗 (V𝑗𝑐𝑗 )𝑑𝑠 + 𝐼𝑛 −

𝑡−𝛼Γ (1 − 𝛼) 𝑢𝑛𝑎𝑛𝑛∑𝑖=1

𝑑1𝑖𝑔𝑖 (𝑢𝑖𝑎𝑖 ) +𝑛∑𝑖=1

∫𝜏0𝑝1𝑖 (𝑠) 𝑔𝑖 (𝑢𝑖𝑎𝑖 )𝑑𝑠 + 𝐽1 −

𝑡−𝛽Γ (1 − 𝛽) V1𝑐1...𝑛∑𝑖=1

𝑑𝑚𝑖𝑔𝑖 (𝑢𝑖𝑎𝑖 ) +𝑛∑𝑖=1

∫𝜏0𝑝𝑚𝑖 (𝑠) 𝑔𝑖 (𝑢𝑖𝑎𝑖 )𝑑𝑠 + 𝐽𝑚 −

𝑡−𝛽Γ (1 − 𝛽) Vm𝑐𝑚

]]]]]]]]]]]]]]]]]]]]]]

. (17)

Consider ∀u = (𝑢1, . . . , 𝑢𝑛, V1, . . . , V𝑚)𝑇 ∈ R𝑛+𝑚; then itfollows from (14) that

‖Φ (u) − Φ (u)‖ ⩽ 𝑛∑𝑖=1

𝑚∑𝑗=1

{𝑏𝑖𝑗 [𝑓𝑗 (V𝑗𝑐𝑗 ) − 𝑓𝑗 (V𝑗𝑐𝑗 )] + ∫

𝜏

0𝑟𝑖𝑗 (𝑠) [𝑓𝑗 (V𝑗𝑐𝑗 ) − 𝑓𝑗 (

V𝑗𝑐𝑗 )]𝑑𝑠}

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

{𝑑𝑗𝑖 [𝑔𝑖 (𝑢𝑖𝑎𝑖 ) − 𝑔𝑖 (𝑢𝑖𝑎𝑖 )] + ∫

𝜏

0𝑝𝑗𝑖 (𝑠) [𝑔𝑖 (𝑢𝑖𝑎𝑖 ) − 𝑔𝑖 (

𝑢𝑖𝑎𝑖 )]𝑑𝑠}

+ 𝑛∑𝑖=1

𝑡−𝛼Γ (1 − 𝛼) [𝑢𝑖𝑎𝑖 −

𝑢𝑖𝑎𝑖 ] +𝑚∑𝑗=1

𝑡−𝛽Γ (1 − 𝛽) [

V𝑗𝑐𝑗 −V𝑗𝑐𝑗 ] .

(18)

According to (H1)-(H2), one has

‖Φ (u) − Φ (u)‖ ⩽ 𝑛∑𝑖=1

𝑚∑𝑗=1

{𝑏𝑖𝑗 𝐹𝑗V𝑗 − V𝑗𝑐𝑗

+ ∫𝜏0

𝑟𝑖𝑗 (𝑠) 𝐹𝑗V𝑗 − V𝑗𝑐𝑗

𝑑𝑠}

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

{𝑑𝑗𝑖 𝐺𝑖𝑢𝑖 − 𝑢𝑖𝑎𝑖

+ ∫𝜏0

𝑝𝑗𝑖 (𝑠) 𝐺𝑖𝑢𝑖 − 𝑢𝑖𝑎𝑖

𝑑𝑠} +𝑡−𝛼Γ (1 − 𝛼)

⋅max1⩽𝑖⩽𝑛

{ 1𝑎𝑖} ⋅𝑛∑𝑖=1

𝑢𝑖 − 𝑢𝑖 + 𝑡−𝛽Γ (1 − 𝛽) ⋅ max1⩽𝑗⩽𝑚{ 1𝑐𝑗}⋅ 𝑚∑𝑗=1

V𝑗 − V𝑗 ,(19)

For 𝛼, 𝛽 ∈ (0, 1), we have lim𝑡→+∞𝑡−𝛼 = 0, lim𝑡→+∞𝑡−𝛽 = 0.Therefore, there exists a small enough constant 𝜀 > 0 suchthat 𝑡−𝛼 < 𝜀, 𝑡−𝛽 < 𝜀. Thus, it follows from (19) that

‖Φ (u) − Φ (u)‖ ⩽ 𝑛∑𝑖=1

𝑚∑𝑗=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗𝑐𝑗 𝐹𝑗

V𝑗 − V𝑗]+ 𝑚∑𝑗=1

𝑛∑𝑖=1

[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖𝑎𝑖 𝐺𝑖

𝑢𝑖 − 𝑢𝑖] + 𝜀Γ (1 − 𝛼)

6 Complexity

⋅max1⩽𝑖⩽𝑛

{ 1𝑎𝑖} ⋅𝑛∑𝑖=1

𝑢𝑖 − 𝑢𝑖 + 𝜀Γ (1 − 𝛽) ⋅ max1⩽𝑗⩽𝑚{ 1𝑐𝑗}

⋅ 𝑚∑𝑗=1

V𝑗 − V𝑗⩽ max1⩽𝑗⩽𝑚

{ 𝜀Γ (1 − 𝛽) ⋅ 1𝑐𝑗 +𝐹𝑗𝑐𝑗𝑛∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]}

⋅ 𝑚∑𝑗=1

V𝑗 − V𝑗

+max1⩽𝑖⩽𝑛

{{{𝜀Γ (1 − 𝛼) ⋅ 1𝑎𝑖 +


[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}⋅ 𝑛∑𝑖=1

𝑢𝑖 − 𝑢𝑖 .(20)

Let 𝑘 = max{𝜔1, 𝜔2}, where 𝜔1 and 𝜔2 are defined in (15).Hence, we have

‖Φ (u) − Φ (u)‖ ⩽ 𝑘[[𝑛∑𝑖=1

𝑢𝑖 − 𝑢𝑖 +𝑚∑𝑗=1

V𝑗 − V𝑗]]= 𝑘 ‖u − u‖ .

(21)

Thus, it follows from (15) that 0 < 𝑘 < 1, which implies thatΦ : R𝑛+𝑚 → R𝑛+𝑚 is a contraction mapping.Therefore, fromLemma 4, there exists a unique fixed point of the map Φ :R𝑛+𝑚 → R𝑛+𝑚, such thatΦ(u∗) = u∗. Thus, from (17), we get

𝑚∑𝑗=1

𝑏𝑖𝑗𝑓𝑗 (V∗𝑗𝑐𝑗 ) +

𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) 𝑓𝑗 (V

∗𝑗𝑐𝑗 )𝑑𝑠 + 𝐼𝑖

− 𝑡−𝛼Γ (1 − 𝛼)𝑢∗𝑖𝑎𝑖 = 𝑢∗𝑖 , 𝑖 = 1, 2, . . . , 𝑛,

𝑛∑𝑖=1

𝑑𝑗𝑖𝑔𝑖 (𝑢∗𝑖𝑎𝑖 ) +𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) 𝑔𝑖 (𝑢∗𝑖𝑎𝑖 )𝑑𝑠 + 𝐽𝑗

− 𝑡−𝛽Γ (1 − 𝛽)V∗𝑗𝑐𝑗 = V∗𝑗 , 𝑗 = 1, 2, . . . , 𝑚.

(22)

Let 𝑥∗𝑖 = 𝑢∗𝑖 /𝑎𝑖, 𝑦∗𝑗 = V∗𝑗 /𝑐𝑗; then it follows from (22) that𝑚∑𝑗=1

𝑏𝑖𝑗𝑓𝑗 (𝑦∗𝑗 ) + 𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦∗𝑗 ) 𝑑𝑠 + 𝐼𝑖

− 𝑡−𝛼Γ (1 − 𝛼)𝑥∗𝑖 = 𝑎𝑖𝑥∗𝑖 , 𝑖 = 1, 2, . . . , 𝑛,𝑛∑𝑖=1

𝑑𝑗𝑖𝑔𝑖 (𝑥∗𝑖 ) + 𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) 𝑔𝑖 (𝑥∗𝑖 ) 𝑑𝑠 + 𝐽𝑗

− 𝑡−𝛽Γ (1 − 𝛽)𝑦∗𝑗 = 𝑐𝑗𝑦∗𝑗 , 𝑗 = 1, 2, . . . , 𝑚;

(23)

that is𝑚∑𝑗=1

𝑏𝑖𝑗𝑓𝑗 (𝑦∗𝑗 ) + 𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) 𝑓𝑗 (𝑦∗𝑗 ) 𝑑𝑠 + 𝐼𝑖 − 𝑎𝑖𝑥∗𝑖

= RL0𝐷𝛼𝑡 {𝑥∗𝑖 } , 𝑖 = 1, 2, . . . , 𝑛,𝑛∑𝑖=1

𝑑𝑗𝑖𝑔𝑖 (𝑥∗𝑖 ) + 𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) 𝑔𝑖 (𝑥∗𝑖 ) 𝑑𝑠 + 𝐽𝑗 − 𝑐𝑗𝑦∗𝑗

= RL0𝐷𝛽𝑡 {𝑦∗𝑗 } , 𝑗 = 1, 2, . . . , 𝑚.

(24)

According to (H3), we know that

𝛾(1)𝑘 (𝑥∗𝑖 ) = 0,𝛾(2)𝑘 (𝑦∗𝑗 ) = 0,

𝑖 = 1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , 𝑚; 𝑘 = 1, 2, . . . .(25)

Thus, it follows from Definition 7 that (𝑥∗1 , 𝑥∗2 , . . . , 𝑥∗𝑛 , 𝑦∗1 ,𝑦∗2 , . . . , 𝑦∗𝑚)𝑇 ∈ R𝑛+𝑚 is a unique equilibrium solution forsystem (6). The proof is complete.

The following corollary is the direct result of Theorem 8.

Corollary 9. Suppose that conditions (H1)–(H3) hold; thenthere exists a unique equilibrium solution for system (6), if thefollowing inequalities simultaneously hold for a small enoughconstant 𝜀 > 0

min1⩽𝑖⩽𝑛

{{{𝑎𝑖 − 𝜀Γ (1 − 𝛼) − 𝐺𝑖

𝑚∑𝑗=1

[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}> 0,

min1⩽𝑗⩽𝑚

{𝑐𝑗 − 𝜀Γ (1 − 𝛽) − 𝐹𝑗𝑛∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]} > 0.(26)

Remark 10. Theorem 8 and Corollary 9 reveal that theconditions of existence and uniqueness of the equilibriumsolution for system (6) are based on the contraction mappingprinciple, which can be expressed in terms of the algebraicinequalities. The conditions of existence and uniqueness ofthe equilibrium point for system (6) reflect the close relationbetween the coefficients, neuron activation functions, andtime-delay of network parameters, which are also dependenton the orders 𝛼 and 𝛽 of Riemann-Liouville derivatives. Onthe other hand, if we only assume that (H1)–(H3) hold, thenthere exists at least an equilibrium solution for system (6)by applying Schauder fixed point theorem, whose proof isomitted here.

4. Globally Asymptotic Stability Criteria

In this section, by constructing a novel Lyapunov functional,we obtain the sufficient conditions to ensure the globallyasymptotic stability of the equilibrium solution for system (6)based on fractional Barbalat theorem and classical Lyapunovstability theory.

Complexity 7

Theorem 11. Suppose that conditions (H1)–(H3) hold; then aunique equilibrium solution for system (6) is globally asymptot-ically stable, if the following inequalities simultaneously hold fora small enough constant 𝜀 > 0

𝜂1 = min1⩽𝑖⩽𝑛

{{{𝑎𝑖 − 𝐺𝑖 𝑚∑

𝑗=1

[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}> 𝜀Γ (1 − 𝛼) ,

𝜂2 = min1⩽𝑗⩽𝑚

{𝑐𝑗 − 𝐹𝑗 𝑛∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]} > 𝜀Γ (1 − 𝛽) .(27)

Proof. From Corollary 9, there exists a unique equilibriumsolution (𝑥∗𝑇, 𝑦∗𝑇)𝑇 for system (6). By using the variabletransformation method, we can shift the equilibrium pointto the origin. Let 𝑢𝑖(𝑡) = 𝑥𝑖(𝑡) − 𝑥∗𝑖 , V𝑗(𝑡) = 𝑦𝑗(𝑡) − 𝑦∗𝑗 ; thensystem (6) is transformed into

RL0𝐷𝛼𝑡 𝑢𝑖 (𝑡)= −𝑎𝑖𝑢𝑖 (𝑡) + 𝑚∑

𝑗=1

𝑏𝑖𝑗 [𝑓𝑗 (𝑦𝑗 (𝑡)) − 𝑓𝑗 (𝑦∗𝑗 )]

+ 𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) [𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) − 𝑓𝑗 (𝑦∗𝑗 )] 𝑑𝑠,

𝑡 ̸= 𝑡𝑘,𝑢𝑖 (𝑡+𝑘 ) = (1 − 𝜆(1)𝑖𝑘 ) 𝑢𝑖 (𝑡−𝑘 ) ,

𝑖 = 1, 2, . . . , 𝑛; 𝑘 = 1, 2, . . . ,RL0𝐷𝛽𝑡 V𝑗 (𝑡)= −𝑐𝑗V𝑗 (𝑡) + 𝑛∑

𝑖=1

𝑑𝑗𝑖 [𝑔𝑖 (𝑥𝑖 (𝑡)) − 𝑔𝑖 (𝑥∗𝑖 )]

+ 𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) [𝑔𝑖𝑥𝑖 (𝑡 − 𝑠) − 𝑔𝑖 (𝑥∗𝑖 )] 𝑑𝑠,

𝑡 ̸= 𝑡𝑘,V𝑗 (𝑡+𝑘 ) = (1 − 𝜆(2)𝑗𝑘 ) V𝑗 (𝑡−𝑘 ) ,

𝑗 = 1, 2, . . . , 𝑚; 𝑘 = 1, 2, . . . .

(28)

Construct a novel Lyapunov functional composed of frac-tional-order integral and definite integral terms:

𝑉 (𝑡) = 0𝐷−(1−𝛼)𝑡 [𝑛∑𝑖=1

𝑢𝑖 (𝑡)]

+ 0𝐷−(1−𝛽)𝑡 [[𝑚∑𝑗=1

V𝑗 (𝑡)]]

+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗 𝑏𝑖𝑗 ∫𝑡

𝑡−𝜏

V𝑗 (𝑠) 𝑑𝑠

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖 𝑑𝑗𝑖 ∫𝑡

𝑡−𝜏

𝑢𝑖 (𝑠) 𝑑𝑠

+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗𝑅𝑖𝑗 ∫𝜏0∫𝑡𝑡−𝑠

V𝑗 (𝜂) 𝑑𝜂 𝑑𝑠

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0∫𝑡𝑡−𝑠

𝑢𝑖 (𝜂) 𝑑𝜂 𝑑𝑠.(29)

The time derivative of 𝑉(𝑡) along the trajectories of system(6) can be calculated, which are carried out for the followingcases.

Case 1. For 𝑡 ̸= 𝑡𝑘, from Lemma 3, we obtain𝑑+𝑉 (𝑡)𝑑𝑡 = RL0𝐷𝛼𝑡 [

𝑛∑𝑖=1

𝑢𝑖 (𝑡)] + RL0𝐷𝛽𝑡 [[𝑚∑𝑗=1

V𝑗 (𝑡)]]+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗 𝑏𝑖𝑗 [V𝑗 (𝑡) − V𝑗 (𝑡 − 𝜏)]

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖 𝑑𝑗𝑖 [𝑢𝑖 (𝑡) − 𝑢𝑖 (𝑡 − 𝜏)]

+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗𝑅𝑖𝑗 ∫𝜏0[V𝑗 (𝑡) − V𝑗 (𝑡 − 𝑠)] 𝑑𝑠

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0[𝑢𝑖 (𝑡) − 𝑢𝑖 (𝑡 − 𝑠)] 𝑑𝑠.

(30)

An application of Definition 2 yields

RL0𝐷𝛼𝑡 𝑢𝑖 (𝑡) = sgn (𝑢𝑖 (𝑡)) ⋅ (RL0𝐷𝛼𝑡 𝑢𝑖 (𝑡)) ,

RL0𝐷𝛽𝑡 V𝑗 (𝑡) = sgn (V𝑗 (𝑡)) ⋅ (RL0𝐷𝛽𝑡 V𝑗 (𝑡)) ,

(31)

where sgn(⋅) denotes the standard signum function. Thus,(30) can be rewritten as

𝑑+𝑉 (𝑡)𝑑𝑡 =𝑛∑𝑖=1

sgn (𝑢𝑖 (𝑡)) [RL0𝐷𝛼𝑡 (𝑢𝑖 (𝑡))]

+ 𝑚∑𝑗=1

sgn (V𝑗 (𝑡)) [RL0𝐷𝛽𝑡 (V𝑗 (𝑡))]

+ 𝑛∑𝑖=1

𝑚∑𝑗=1


8 Complexity

+ 𝑚∑𝑗=1

𝑛∑𝑖=1


+ 𝑛∑𝑖=1

𝑚∑𝑗=1


+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0[𝑢𝑖 (𝑡) − 𝑢𝑖 (𝑡 − 𝑠)] 𝑑𝑠.

(32)

Combining (28) and (32) yields

𝑑+𝑉 (𝑡)𝑑𝑡 =𝑛∑𝑖=1

sgn (𝑢𝑖 (𝑡)){{{−𝑎𝑖𝑢𝑖 (𝑡)

+ 𝑚∑𝑗=1

𝑏𝑖𝑗 [𝑓𝑗 (𝑦𝑗 (𝑡)) − 𝑓𝑗 (𝑦∗𝑗 )]

+ 𝑚∑𝑗=1

∫𝜏0𝑟𝑖𝑗 (𝑠) [𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) − 𝑓𝑗 (𝑦∗𝑗 )] 𝑑𝑠}}}

+ 𝑚∑𝑗=1

sgn (V𝑗 (𝑡)) {−𝑐𝑗V𝑗 (𝑡)

+ 𝑛∑𝑖=1

𝑑𝑗𝑖 [𝑔𝑖 (𝑥𝑖 (𝑡)) − 𝑔𝑖 (𝑥∗𝑖 )]

+ 𝑛∑𝑖=1

∫𝜏0𝑝𝑗𝑖 (𝑠) [𝑔𝑖𝑥𝑖 (𝑡 − 𝑠) − 𝑔𝑖 (𝑥∗𝑖 )] 𝑑𝑠}

+ 𝑛∑𝑖=1

𝑚∑𝑗=1


+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖 𝑑𝑗𝑖 [𝑢𝑖 (𝑡) − 𝑢𝑖 (𝑡 − 𝜏)] +𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗𝑅𝑖𝑗⋅ ∫𝜏0[V𝑗 (𝑡) − V𝑗 (𝑡 − 𝑠)] 𝑑𝑠 +

𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖⋅ ∫𝜏0[𝑢𝑖 (𝑡) − 𝑢𝑖 (𝑡 − 𝑠)] 𝑑𝑠.

(33)

By computations, we have

𝑑+𝑉 (𝑡)𝑑𝑡 ⩽ −𝑛∑𝑖=1

𝑎𝑖 𝑢𝑖 (𝑡) −𝑛∑𝑗=𝑚

𝑐𝑗 V𝑗 (𝑡)+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗 𝑏𝑖𝑗 V𝑗 (𝑡 − 𝜏)+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖 𝑑𝑗𝑖 𝑢𝑖 (𝑡 − 𝜏)

+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗𝑅𝑖𝑗 ∫𝜏0

V𝑗 (𝑡 − 𝑠) 𝑑𝑠

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0

𝑢𝑖 (𝑡 − 𝑠) 𝑑𝑠

+ 𝑛∑𝑖=1

𝑚∑𝑗=1


+ 𝑚∑𝑗=1

𝑛∑𝑖=1


+ 𝑛∑𝑖=1

𝑚∑𝑗=1


+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0[𝑢𝑖 (𝑡) − 𝑢𝑖 (𝑡 − 𝑠)] 𝑑𝑠

⩽ − 𝑛∑𝑖=1

𝑎𝑖 𝑢𝑖 (𝑡) −𝑛∑𝑗=𝑚

𝑐𝑗 V𝑗 (𝑡)+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗 𝑏𝑖𝑗 V𝑗 (𝑡)+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖 𝑑𝑗𝑖 𝑢𝑖 (𝑡)+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗𝑅𝑖𝑗 ∫𝜏0

V𝑗 (𝑡) 𝑑𝑠

+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0

𝑢𝑖 (𝑡) 𝑑𝑠

⩽ 𝑛∑𝑖=1

{{{−𝑎𝑖 + 𝐺𝑖 𝑚∑

𝑗=1

(𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖)}}}𝑢𝑖 (𝑡)

+ 𝑚∑𝑗=1

{−𝑐𝑗 + 𝐹𝑗 𝑛∑𝑖=1

(𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗)} V𝑗 (𝑡)⩽ − 𝜀Γ (1 − 𝛼)

𝑛∑𝑖=1

𝑢𝑖 (𝑡)− 𝜀Γ (1 − 𝛽)

𝑚∑𝑗=1

V𝑗 (𝑡) , 𝑡 ̸= 𝑡𝑘,(34)

which implies that 𝑑+𝑉(𝑡)/𝑑𝑡 ⩽ 0 as 𝑡 ̸= 𝑡𝑘. Hence, for any𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘), we get𝑉 (𝑡) + ∫𝑡𝑘

𝑡𝑘−1

[[

𝜀Γ (1 − 𝛼)𝑛∑𝑖=1

𝑢𝑖 (𝑠)

+ 𝜀Γ (1 − 𝛽)𝑚∑𝑗=1

V𝑗 (𝑠)]]𝑑𝑠 ⩽ 𝑉 (𝑡+𝑘−1) .

(35)

Complexity 9

Case 2. For 𝑡 = 𝑡𝑘, from (29), one has𝑉 (𝑡+𝑘 ) = 0𝐷−(1−𝛼)𝑡+

𝑘

[ 𝑛∑𝑖=1

𝑢𝑖 (𝑡+𝑘 )]

+ 0𝐷−(1−𝛽)𝑡+𝑘

[[𝑚∑𝑗=1

V𝑗 (𝑡+𝑘 )]]+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗 𝑏𝑖𝑗 ∫𝑡+𝑘

𝑡+𝑘−𝜏


+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖 𝑑𝑗𝑖 ∫𝑡+𝑘

𝑡+𝑘−𝜏


+ 𝑛∑𝑖=1

𝑚∑𝑗=1

𝐹𝑗𝑅𝑖𝑗 ∫𝜏0∫𝑡+𝑘𝑡+𝑘−𝑠


+ 𝑚∑𝑗=1

𝑛∑𝑖=1

𝐺𝑖𝑃𝑗𝑖 ∫𝜏0∫𝑡+𝑘𝑡+𝑘−𝑠

𝑢𝑖 (𝜂) 𝑑𝜂 𝑑𝑠.

(36)

From (H3), we get

𝑉 (𝑡+𝑘 ) = 0𝐷−(1−𝛼)𝑡+𝑘

[ 𝑛∑𝑖=1

1 − 𝜆(1)𝑖𝑘 𝑢𝑖 (𝑡−𝑘 )]

+ 0𝐷−(1−𝛽)𝑡+𝑘

[[𝑚∑𝑗=1

1 − 𝜆(2)𝑗𝑘 V𝑗 (𝑡−𝑘 )]]+ 𝑛∑𝑖=1

𝑚∑𝑗=1


𝑡+𝑘−𝜏


+ 𝑚∑𝑗=1

𝑛∑𝑖=1


𝑡+𝑘−𝜏

𝑢i (𝑠) 𝑑𝑠

+ 𝑛∑𝑖=1

𝑚∑𝑗=1



+ 𝑚∑𝑗=1

𝑛∑𝑖=1


𝑢𝑖 (𝜂) 𝑑𝜂 𝑑𝑠.

(37)

Note that the inequalities |1−𝜆(1)𝑖𝑘| < 1 and |1−𝜆(2)

𝑗𝑘| < 1 hold;

then

𝑉 (𝑡+𝑘 ) ⩽ 0𝐷−(1−𝛼)𝑡−𝑘

[ 𝑛∑𝑖=1

𝑢𝑖 (𝑡−𝑘 )]

+ 0𝐷−(1−𝛽)𝑡−𝑘

[[𝑚∑𝑗=1

V𝑗 (𝑡−𝑘 )]]+ 𝑛∑𝑖=1

𝑚∑𝑗=1


𝑡+𝑘−𝜏


+ 𝑚∑𝑗=1

𝑛∑𝑖=1


𝑡+𝑘−𝜏


+ 𝑛∑𝑖=1

𝑚∑𝑗=1



+ 𝑚∑𝑗=1

𝑛∑𝑖=1


𝑢𝑖 (𝜂) 𝑑𝜂 𝑑𝑠 = 𝑉 (𝑡−𝑘 )= 𝑉 (𝑡𝑘) .

(38)

Let 𝑈(𝑡) = ∑𝑛𝑖=1 |𝑢𝑖(𝑡)| + ∑𝑚𝑗=1 |V𝑗(𝑡)|, for any 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘);then we have the following estimations:

𝑉 (𝑡) ⩽ −∫𝑡𝑡𝑘−1

𝑈 (𝑠) 𝑑𝑠 + 𝑉 (𝑡+𝑘−1)⩽ −∫𝑡𝑡𝑘−1

𝑈 (𝑠) 𝑑𝑠 + 𝑉 (𝑡−𝑘−1)⩽ −∫𝑡𝑡𝑘−2

𝑈 (𝑠) 𝑑𝑠 + 𝑉 (𝑡−𝑘−2) ⩽ ⋅ ⋅ ⋅⩽ −∫𝑡0𝑈 (𝑠) 𝑑𝑠 + 𝑉 (0) ;

(39)

Thus, we can get the following inequality:

𝑉 (𝑡) + ∫𝑡0𝑈 (𝑠) 𝑑𝑠 ⩽ 𝑉 (0) , (40)

which implies that lim𝑡→+∞𝑈(𝑡) is bounded. From (28),|RL0𝐷𝛼𝑡 𝑢𝑖(𝑡)| and |RL0𝐷𝛽𝑡 V𝑗(𝑡)| are also bounded. FromLemma5,we have lim𝑡→+∞∑𝑛𝑖=1 |𝑢𝑖(𝑡)| = 0 and lim𝑡→+∞∑𝑚𝑗=1 |V𝑗(𝑡)| =0.Therefore, according to Lyapunov stability theory, a uniqueequilibrium solution (𝑥∗𝑇, 𝑦∗𝑇)𝑇 for system (6) is globallyasymptotically stable. This completes the proof.

The following corollary is the direct result of Theorem 11.

Corollary 12. Suppose that (H1)–(H3) hold; then a uniqueequilibrium solution for system (6) is globally asymptoticallystable, if the following inequalities simultaneously hold for asmall enough constant 𝜀 > 0

𝜔1 = max1⩽𝑖⩽𝑛

{{{𝜀Γ (1 − 𝛼) ⋅ 1𝑎𝑖 +


[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}< 1,

𝜔2 = max1⩽𝑗⩽𝑚

{ 𝜀Γ (1 − 𝛽) ⋅ 1𝑐𝑗 +𝐹𝑗𝑐𝑗𝑛∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]}< 1.

(41)

Remark 13. Different from fractional Lyapunov functionalmethod in [30, 32, 37], an appropriate Lyapunov functionalcomposed of fractional integral and definite integral termsin the proof of Theorem 11 is presented, and we only

10 Complexity

need to calculate its first-order derivative to derive stabilityconditions. As discussed in [35], in general speaking, it isvery difficult to calculate the fractional-order derivatives of aLyapunov functional.Themain advantage of our constructedmethod is that we can avoid computing the fractional-orderderivatives of the Lyapunov functional.

Remark 14. The globally asymptotic stability criteria of aunique equilibrium solution for system (6) are described bythe algebraic inequalities, which are dependent on the orders𝛼 and 𝛽 of fractional derivatives and reflect the close relationbetween the coefficients, neuron activation functions, andtime-delay of network parameters. Moreover, the globallyasymptotic stability criteria are more easily checked andcontribute to reducing the computational burden.

Remark 15. When𝛼 = 𝛽 = 1, system (6) is reduced to integer-order BAM neural networks with distributed delays andimpulses [23]. Note that the Riemann-Liouville derivativeis a continuous operator of the order (see [9–11]); then wecan obtain globally asymptotic stability criteria for impulsiveinteger-order hybrid BAMneural networks from the proof ofTheorem 11.

Remark 16. In [33, 34, 44], the authors have focused onstudying the finite-time stability of fractional-order delayedneural networks. However, it should be pointed out that thefinite-time stability and asymptotic stability in the sense ofLyapunov are different concepts, because finite-time stabilitydoes not contain Lyapunov asymptotic stability and vice versa[34, 47]. This is also the motivation of this paper.

5. An Illustrative Example

In this section, an example for impulsive fractional-orderhybrid BAM neural networks with distributed delays is givento illustrate the effectiveness and feasibility of the theoreticalresults.

Example 17. Consider the four-state Riemann-Liouville frac-tional-order hybrid BAM neural network model with dis-tributed delays and impulsive effects described by

RL0𝐷0.2𝑡 𝑥1 (𝑡) = −0.7𝑥1 (𝑡) − 0.2𝑓1 (𝑦1 (𝑡))

+ 0.1𝑓2 (𝑦2 (𝑡))+ 2∫0.20𝑠𝑓1 (𝑦1 (𝑡 − 𝑠)) 𝑑𝑠

+ ∫0.20𝑠𝑓2 (𝑦2 (𝑡 − 𝑠)) 𝑑𝑠,

RL0𝐷0.2𝑡 𝑥2 (𝑡) = −0.6𝑥2 (𝑡) + 0.3𝑓1 (𝑦1 (𝑡))

+ 0.2𝑓2 (𝑦2 (𝑡))+ ∫0.20𝑠𝑓1 (𝑦1 (𝑡 − 𝑠)) 𝑑𝑠

− ∫0.20𝑠3𝑓2 (𝑦2 (𝑡 − 𝑠)) 𝑑𝑠,

Δ𝑥𝑖 (𝑡𝑘) = −0.3 (𝑥𝑖 (𝑡𝑘) − 𝑥∗𝑖 ) ,𝑖 = 1, 2; 𝑘 = 1, 2, . . . ,

RL0𝐷0.6𝑡 𝑦1 (𝑡) = −0.7𝑦1 (𝑡) + 0.4𝑔1 (𝑦1 (𝑡))

+ 0.2𝑔2 (𝑦2 (𝑡))− ∫0.20𝑠𝑔1 (𝑦1 (𝑡 − 𝑠)) 𝑑𝑠

+ ∫0.20𝑠2𝑔2 (𝑦2 (𝑡 − 𝑠)) 𝑑𝑠,

RL0𝐷0.6𝑡 𝑦2 (𝑡) = −0.6𝑦2 (𝑡) + 0.1𝑔1 (𝑦1 (𝑡))

− 0.3𝑔2 (𝑦2 (𝑡))+ ∫0.20𝑠2𝑔1 (𝑦1 (𝑡 − 𝑠)) 𝑑𝑠

+ ∫0.20𝑠𝑔2 (𝑦2 (𝑡 − 𝑠)) 𝑑𝑠,

Δ𝑦𝑗 (𝑡𝑘) = −0.4 (𝑦𝑗 (𝑡𝑘) − 𝑦∗𝑖 ) ,𝑗 = 1, 2; 𝑘 = 1, 2, . . . ,

(42)

where 𝛼 = 0.2, 𝛽 = 0.6, 𝜏 = 0.2, 𝑎1 = 𝑐1 = 0.7, 𝑎2 = 𝑐2 = 0.6,𝑏11 = −0.2, 𝑏12 = 0.1, 𝑏21 = 0.3, 𝑏22 = 0.2, 𝑑11 = 0.4, 𝑑12 = 0.2,𝑑21 = 0.1, 𝑑22 = −0.3, 𝑟11(𝑠) = 2𝑠, 𝑟12(𝑠) = 𝑠, 𝑟21(𝑠) = 𝑠,𝑟22(𝑠) = −𝑠3, 𝑝11(𝑠) = −𝑠, 𝑝12(𝑠) = 𝑠2, 𝑝21(𝑠) = 𝑠2, 𝑝22(𝑠) = 𝑠,and

𝑓𝑗 (𝑦𝑗) = 12 (𝑦𝑗 + 1 − 𝑦𝑗 − 1) , 𝑗 = 1, 2,𝑔𝑖 (𝑥𝑖) = 12 (𝑥𝑖 + 1 − 𝑥𝑖 − 1) , 𝑖 = 1, 2.

(43)

From (43), we know that 𝐹1 = 𝐹2 = 𝐺1 = 𝐺2 = 1. Since𝑓1(0) = 𝑓2(0) = 0,𝑔1(0) = 𝑔2(0) = 0, then (𝑥∗1 , 𝑥∗2 , 𝑦∗1 , 𝑦∗1 )𝑇 =(0, 0, 0, 0)𝑇 is an equilibrium solution for system (42). Next,we applyTheorem 11 or Corollary 12 to check the uniquenessand global asymptotic stability of the equilibrium point forsystem (42).

In fact, by computations, one can get that𝑅11 = 0.4,𝑅12 =𝑅21 = 0.2, 𝑅22 = 0.008, 𝑃11 = 𝑃22 = 0.2, and 𝑃12 = 𝑃21 = 0.04.Choosing a positive constant 𝜀 = 0.04 > 0, thenwe can obtain

𝜂1 = min1⩽𝑖⩽2

{{{𝑎𝑖 − 𝐺𝑖 2∑

𝑗=1

[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}= 0.052

> 𝜀Γ (1 − 𝛼) = 0.044,𝜂2 = min1⩽𝑗⩽2

{𝑐𝑗 − 𝐹𝑗 2∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]} = 0.116> 𝜀Γ (1 − 𝛽) = 0.045,

Complexity 11

10 20 30 40 500

Time (s)

10 20 30 40 500

Time (s)

200 400 600 8000

Time (s)

200 400 600 8000

Time (s)

−1

−0.5

0

0.5

1

1.5x1(t),

with

di�

eren

t ini

tial c

ondi

tions

−2

−1.5

−1

−0.5

0

0.5

y2(t),

with

di�

eren

t ini

tial c

ondi

tions

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

y1(t),

with

di�

eren

t ini

tial c

ondi

tions

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x2(t),

with

di�

eren

t ini

tial c

ondi

tions

Figure 1: State trajectories of BAM neural network (42) with 𝛼 = 0.2; 𝛽 = 0.6 under different initial conditions.

𝜔1 = max1⩽𝑖⩽2

{{{𝜀Γ (1 − 𝛼) ⋅ 1𝑎𝑖 +

𝐺𝑖𝑎𝑖2∑𝑗=1

[𝑑𝑗𝑖 + 𝜏𝑃𝑗𝑖]}}}= 0.856 < 1,

𝜔2 = max1⩽𝑗⩽2

{ 𝜀Γ (1 − 𝛽) ⋅ 1𝑐𝑗 +𝐹𝑗𝑐𝑗2∑𝑖=1

[𝑏𝑖𝑗 + 𝜏𝑅𝑖𝑗]}= 0.722 < 1.

(44)

Thus, the conditions of Theorem 11 or Corollary 12 are satis-fied. For numerical simulations, Figure 1 depicts the state tra-jectories of system (42) under different initial conditions with𝛼 = 0.2, 𝛽 = 0.6. It can be directly observed that the uniqueequilibrium solution (0, 0, 0, 0)𝑇 for system (42) is globallyasymptotically stable with 𝛼 = 0.2, 𝛽 = 0.6. Therefore, thenumerical simulations further confirm the theoretical resultsof this paper.

6. Conclusions

In this paper, the sufficient conditions for the existence anduniqueness of the equilibrium solution are presented based

on the contractionmapping principle. By constructing a suit-able Lyapunov functional composed of fractional integral anddefinite integral terms, we calculate its first-order derivativeto derive global asymptotic stability of the equilibrium point.The constructed method avoids calculating the fractional-order derivative of the Lyapunov functional. Furthermore,the presented results are described as the algebraic inequal-ities, which are convenient and feasible to verify the existenceand asymptotic stability of the equilibrium solution. Forfurther research, it is interesting and challenging to discussthe chaos phenomena,Hopf bifurcation, and synchronizationcontrol problem for fractional-ordermemristor-based hybridBAM neural networks with leakage, time-varying, and dis-tributed delays.

Conflicts of Interest

The authors declare that there are no conflicts of interest withregard to the publication of this paper.

Acknowledgments

This work is jointly supported by the National NaturalScience Fund of China (11301308, 61573096, and 61272530),

12 Complexity

the 333 Engineering Fund of Jiangsu Province of China(BRA2015286), the Natural Science Fund of Anhui Provinceof China (1608085MA14), the Key Project of Natural ScienceResearch of Anhui Higher Education Institutions of China(gxyqZD2016205 and KJ2015A152), and the Natural ScienceYouth Fund of Jiangsu Province of China (BK20160660).

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