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Research Article Existence and Global Exponential Stability of Almost Automorphic Solution for Clifford-Valued High-Order Hopfield Neural Networks with Leakage Delays Bing Li 1 and Yongkun Li 2 School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan , China Department of Mathematics, Yunnan University, Kunming, Yunnan , China Correspondence should be addressed to Yongkun Li; [email protected] Received 13 May 2019; Accepted 1 July 2019; Published 14 July 2019 Academic Editor: Dimitri Volchenkov Copyright © 2019 Bing Li and Yongkun Li. is 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. In this paper, we study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by direct method. at is to say, we do not decompose the systems under consideration into real-valued systems, but we directly study Clifford-valued systems. Our methods and results are new. Finally, an example is given to illustrate our main results. 1. Introduction In recent years, high-order Hopfield neural networks have become the object of intensive analysis by many scholars because of their stronger approximation characteristics, faster convergence speed, larger storage capacity, and higher fault tolerance than low-order Hopfield neural networks. A lot of excellent research results about their dynamic characteristics have been obtained [1–14]. On the one hand, the Clifford algebra was proposed by the British mathematician William K. Clifford [15] in 1878 and is a generalization of the plural, quaternion, and Glassman algebra. Currently, Clifford algebra has been widely used in various fields such as neural computing, computer and robot vision, image and signal processing, and control problems. Studies have shown that Clifford-valued neural networks are superior to commonly used real-valued neural networks [16, 17], so they have become an active research field in recent years. However, because the multiplication of Clifford numbers does not satisfy the commutative law, it has brought great difficulties to the research of Clifford- valued neural networks. erefore, the current results on the dynamics of Clifford-valued neural networks are still very rare. At present, only a few papers have been published on the dynamics of Clifford-valued neural networks [18–22]. It is worth mentioning that the results of these mentioned papers are obtained by decomposing Clifford-valued neural networks into real-valued networks. erefore, it is meaning- ful to study the dynamics of Clifford-valued neural networks by direct method. On the other hand, it is well known that periodic and almost periodic oscillations are important dynamic behaviors of neural networks. Almost automorphy is an extension of almost periodicity and plays an important role in better understanding of almost periodicity. erefore, almost auto- morphic oscillation is more complex than almost periodic oscillation. Considering the interaction between neurons in a neural network is very complex, it is meaningful to study the almost automorphic oscillation of neural networks. In addition, time delays are inevitable and may affect and change the dynamic behavior of dynamic systems [23, 24]. erefore, neural networks with various delays have been extensively studied. In particular, since K. Gopalsamy [25] first studied the stability of neural networks with leakage delays, a lot of research has been done on neural networks with leakage delays [6, 11, 12, 26]. However, there is no research on the Clifford-valued neural network with leakage delays. Hindawi Complexity Volume 2019, Article ID 6751806, 13 pages https://doi.org/10.1155/2019/6751806

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Research ArticleExistence and Global Exponential Stability of AlmostAutomorphic Solution for Clifford-Valued High-Order HopfieldNeural Networks with Leakage Delays

Bing Li1 and Yongkun Li 2

1School of Mathematics and Computer Science Yunnan Minzu University Kunming Yunnan 650500 China2Department of Mathematics Yunnan University Kunming Yunnan 650091 China

Correspondence should be addressed to Yongkun Li yklieynueducn

Received 13 May 2019 Accepted 1 July 2019 Published 14 July 2019

Academic Editor Dimitri Volchenkov

Copyright copy 2019 Bing Li and Yongkun Li This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper we study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-orderHopfield neural networks by direct method That is to say we do not decompose the systems under consideration into real-valuedsystems but we directly study Clifford-valued systems Our methods and results are new Finally an example is given to illustrateour main results

1 Introduction

In recent years high-order Hopfield neural networks havebecome the object of intensive analysis by many scholarsbecause of their stronger approximation characteristics fasterconvergence speed larger storage capacity and higher faulttolerance than low-order Hopfield neural networks A lot ofexcellent research results about their dynamic characteristicshave been obtained [1ndash14]

On the one hand the Clifford algebra was proposedby the British mathematician William K Clifford [15] in1878 and is a generalization of the plural quaternion andGlassman algebra Currently Clifford algebra has beenwidelyused in various fields such as neural computing computerand robot vision image and signal processing and controlproblems Studies have shown that Clifford-valued neuralnetworks are superior to commonly used real-valued neuralnetworks [16 17] so they have become an active researchfield in recent years However because the multiplicationof Clifford numbers does not satisfy the commutative lawit has brought great difficulties to the research of Clifford-valued neural networks Therefore the current results on thedynamics of Clifford-valued neural networks are still veryrare At present only a few papers have been published on

the dynamics of Clifford-valued neural networks [18ndash22]It is worth mentioning that the results of these mentionedpapers are obtained by decomposing Clifford-valued neuralnetworks into real-valued networksTherefore it is meaning-ful to study the dynamics of Clifford-valued neural networksby direct method

On the other hand it is well known that periodic andalmost periodic oscillations are important dynamic behaviorsof neural networks Almost automorphy is an extension ofalmost periodicity and plays an important role in betterunderstanding of almost periodicity Therefore almost auto-morphic oscillation is more complex than almost periodicoscillation Considering the interaction between neurons ina neural network is very complex it is meaningful to studythe almost automorphic oscillation of neural networks

In addition time delays are inevitable and may affect andchange the dynamic behavior of dynamic systems [23 24]Therefore neural networks with various delays have beenextensively studied In particular since K Gopalsamy [25]first studied the stability of neural networks with leakagedelays a lot of research has been done on neural networkswith leakage delays [6 11 12 26] However there is noresearch on the Clifford-valued neural network with leakagedelays

HindawiComplexityVolume 2019 Article ID 6751806 13 pageshttpsdoiorg10115520196751806

2 Complexity

Inspired by the above analysis and discussion the mainpurpose of this paper is to study the existence and globalexponential stability of almost automorphic solutions forClifford-valued high-order Hopfield neural networks withleakage delays by direct method that is we will study theconsidered Clifford-valued neural networks directly insteadof converting them to real-valued ones As far aswe know thisis the first paper to study the almost automorphic solutionsof Clifford-valued high-order Hopfield neural networks withleakage delays In addition this is the first paper to studyalmost automorphic solutions of Clifford neural networks bydirect method So our methods and results of this paper arenew Besides ourmethods proposed in this paper can be usedto study the problem of almost automorphic solutions forother types of Clifford-valued neural networks

This paper is organized as follows In Section 2 weintroduce some basic definitions and lemmas and give amodel description In Section 3 we study the existence ofalmost automorphic solutions for Clifford-valued high-orderHopfield neural networks with leakage delays In Section 4we investigate the global exponential stability of almostautomorphic solutions of the neural networks In Section 5an example is given to demonstrate the proposed resultsFinally we draw a brief conclusion in Section 6

2 Preliminaries and Model Description

The real Clifford algebra over R119898 is defined as

A = sum119860sube12119898

119886119860119890119860 119886119860 isin R (1)

where 119890119860 = 119890ℎ1119890ℎ2 sdot sdot sdot 119890ℎ] with 119860 = ℎ1ℎ2 sdot sdot sdot ℎ] 1 le ℎ1 ltℎ2 lt sdot sdot sdot lt ℎ] le 119898 Moreover 1198900 = 1198900 = 1 and119890ℎ ℎ = 1 2 119898 are said to be Clifford generators andsatisfy 1198902119894 = minus1 119894 = 1 2 119898 and 119890119894119890119895 + 119890119895119890119894 = 0 119894 =119895 119894 119895 = 1 2 119898 For convenience we will denote theproduct of Clifford generators 119890ℎ1 119890ℎ2 119890ℎ] as 119890ℎ1ℎ2 sdotsdotsdotℎ] LetΠ = 0 1 2 119860 12 sdot sdot sdot 119898 then it is easy to see thatA = sum119860 119886119860119890119860 119886119860 isin R where sum119860 is short for sum119860isinΠ anddim A = 2119898

For 119909 = sum119860 119909119860119890119860 isin A we define 119909A = max119860isinΠ|119909119860|and for 119909 = (1199091 1199092 119909119899)119879 isin A119899 we define 119909A119899 =max1le119901le119899119909119901A

The derivative of 119909(119905) = sum119860 119909119860(119905)119890119860 is given by (119905) =sum119860 119860(119905)119890119860 For more knowledge about Clifford algebra werefer the reader to [27]

In this paper we are concerned with the followingClifford-valued high-order Hopfield neural network withdelays in the leakage term

119901 (119905) = minus119886119901 (119905) 119909119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119909119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119897 (119909119897 (119905 minus ]119901119902119897 (119905))) + 119868119901 (119905) 119901 = 1 2 119899(2)

where 119909119901(119905) isin A corresponds to the state of the 119901th unit attime 119905 119886119901(119905) isin R+ represents the rate with which the 119901th unitwill reset its potential to the resting state when disconnectedfrom the network and external inputs at time 119905 119887119901119902(119905) isinA denotes the strength of 119902th unit on 119901th unit at time 119905119888119901119902119897(119905) isin A is the second-order synaptic weight of the neuralnetworks 119891119902(119905) 119892119902(119905) isin A denote the activation functions119868119901(119905) isin A is the external input on the 119901th at time 119905 and120578119901(119905) 120590119901119902119897(119905) ]119901119902119897(119905) isin R+ denote the transmission delays

Remark 1 When 119898 the number of the generators of Aequals 0 1 and 2 system (2) degenerates into real-valuedcomplex-valued and quaternion-valued systems respec-tively

We will adopt the following notation119886minus119901 = inf119905isinR

119886119901 (119905) 119887+119901119902 = sup

119905isinR

10038171003817100381710038171003817119887119901119902 (119905)10038171003817100381710038171003817A 119888+119901119902119897 = sup

119905isinR

10038171003817100381710038171003817119888119901119902119897 (119905)10038171003817100381710038171003817A 120578+119901 = sup

119905isinR120578119901 (119905)

120590+119901119902119897 = sup119905isinR

120590119901119902119897 (119905) ]+119901119902119897 = sup

119905isinR120590119901119902119897 (119905)

120585 = max1le119901119902119897le119899

120578+119901 120590+119901119902119897 ]+119901119902119897

(3)

The initial value of system (2) is given by119909119901 (119904) = 120593119901 (119904) isin A119901 (119904) = 119901 (119904) isin A119904 isin [minus120585 0] (4)

where 120593119901 isin 1198621([minus120585 0]A)Denote by 119880119862(RR) the set of all uniformly continuous

functions from R to R Let 119861119862(RA119899) denote the set ofall bounded continuous functions from R to A119899 Then(119861119862(RA119899) sdot 0) is a Banach space with the norm 1198910 =max1le119901le119899sup119905isinR119891119901(119905)A where 119891 = (1198911 1198912 119891119901)119879 isin119861119862(RA119899)Definition 2 A function 119891 isin 119861119862(RA119899) is said to be almostautomorphic if for every sequence of real numbers (1199041015840119899)119899isinNthere exists a subsequence (119904119899)119899isinN such that119892 (119905) fl lim

119899997888rarrinfin119891 (119905 + 119904119899) (5)

is well defined for each 119905 isin R and

lim119899997888rarrinfin

119892 (119905 minus 119904119899) = 119891 (119905) (6)

for each 119905 isin R

Complexity 3

From the above definition similar to the proofs of thecorresponding results in [28] it is not difficult to prove thefollowing two lemmas

Lemma 3 If 120572 isin R 119891 119892 isin 119860119860(RA119899) then 120572119891 119891 + 119892 119891119892 isin119860119860(RA119899)Lemma 4 Let 119891 isin 119862(AA119899) satisfy the Lipschitz conditionand 120593 isin 119860119860(RA) then 119891(120593(sdot)) isin 119860119860(RA119899)Lemma 5 If 119909 isin 1198621(RA) 119909 1199091015840 isin 119860119860(RA) 120578 isin119860119860(RR) cap 119880119862(RR) then 119909(sdot minus 120578(sdot)) isin 119860119860(RA)Proof Since 1199091015840 isin 119860119860(RA) and 120578 isin 119880119862(RR) 119909(119905 minus 120578(119905)) isuniformly continuous for 119905 isin R Hence for each 120576 gt 0 thereexists a positive number 120575 = 1205762 such that |1199051minus1199052| lt 120575 implies1003817100381710038171003817119909 (1199051) minus 119909 (1199052)1003817100381710038171003817A lt 120576 (7)

Since 119909 120578 isin 119860119860(RA) for every sequence of real numbers(1199041015840119899)119899isinN there exists a subsequence (119904119899)119899isinN such that

lim119899997888rarrinfin

119909 (119905 + 119904119899) fl 119909 (119905) lim119899997888rarrinfin

119909 (119905 minus 119904119899) = 119909 (119905) lim119899997888rarrinfin

120578 (119905 + 119904119899) fl 120578 (119905) lim119899997888rarrinfin

120578 (119905 minus 119904119899) = 120578 (119905)(8)

for every 119905 isin R Therefore there exists a positive integer 119873such that

lim119899997888rarrinfin

1003817100381710038171003817119909 (119905 + 119904119899) minus 119909 (119905)1003817100381710038171003817A lt 1205762 lim119899997888rarrinfin

1003816100381610038161003816120578 (119905 + 119904119899) minus 120578 (119905)1003816100381610038161003816 lt 1205762 (9)

for 119899 gt 119873 and 119905 isin R Hence we have1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905 + 119904119899)) minus 119909 (119905 minus 120578 (119905))1003817100381710038171003817Ale 1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905 + 119904119899)) minus 119909 (119905 + 119904119899 minus 120578 (119905))1003817100381710038171003817A+ 1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905)) minus 119909 (119905 minus 120578 (119905))1003817100381710038171003817A lt 1205762 + 1205762= 120576(10)

Consequently 119909(119905+119904119899minus120578(119905+119904119899)) converges to 119909(119905minus120578(119905)) foreach 119905 isin R Similarly we can obtain that 1199091(119905minus119904119899minus120578(119905minus119904119899))converges to 119909(119905minus120578(119905)) for each 119905 isin R Therefore 119909(sdotminus120578(sdot)) isin119860119860(RA) The proof is complete

Throughout this paper we make the following assump-tions

(H1) For 119901 119902 119897 = 1 2 119899 119886119901 120578119901 120590119901119902119897 ]119901119902119897 isin119860119860(RR+) cap 119880119862(RR) and min1le119901le119899inf 119905isinR119886119901(119905) gt0 119887119901119902 119888119901119902119897 119868119901 isin A119860(RA)

(H2) For 119902 = 1 2 119899 119891119902 119892119902 ℎ119902 isin 119862(AA) and thereexist positive constants 119871119891119902 119871119892119902 119871ℎ119902 119866119901 such that

10038171003817100381710038171003817119891119902 (119909) minus 119891119902 (119910)10038171003817100381710038171003817A le 119871119891119902 1003817100381710038171003817119909 minus 1199101003817100381710038171003817A 10038171003817100381710038171003817119892119902 (119909) minus 119892119902 (119910)10038171003817100381710038171003817A le 119871119892119902 1003817100381710038171003817119909 minus 1199101003817100381710038171003817A 10038171003817100381710038171003817119892119901 (119909)10038171003817100381710038171003817A le 119866119901(11)

Moreover for 119902 = 1 2 119899 119891119901(0) = 119892119901(0) = 0(H3) max1le119901le119899(1119886minus119901 )119872119901 (1 + 119886+119901119886minus119901 )119872119901 fl 119903 lt 1 where119872119901 = 119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902 + 119866119902119871119892119897 )

119901 = 1 2 119899 (12)

3 The Existence of AlmostAutomorphic Solutions

In this section we study the existence of almost automorphicsolutions by the contracting mapping principle

Let

Y = 119909 isin 1198621 (RA119899) 119909 1199091015840 isin 119860119860 (RA119899) X = 120593 = (1205931 1205932 120593119899)119879 | 120593 isin Y 119901 = 1 2 119899 (13)

For any 120593 = (1205931 1205932 120593119899)119879 isin X we define thenorm of 120593 as 120593X = max1205930 12059310158400 where 1205930 =max1le119901le119899sup119905isinR120593119901(119905)A thenX is a Banach space

Let 1205930(119905) = (int119905minusinfin 119890minusint119905119904 1198861(119906)1198891199061198681(119904)119889119904int119905minusinfin 119890minusint119905119904 1198862(119906)1198891199061198682(119904)119889119904 int119905minusinfin 119890minusint119905119904 119886119899(119906)119889119906119868119899(119904)119889119904)119879 andtake a positive number 119877 gt 1205930X

Set

X0

= 120593 = (1205931 1205932 120593119899)119879 isin X 1003817100381710038171003817120593 minus 12059301003817100381710038171003817X le 1199031198771 minus 119903 (14)

and then for every 120593 isin X0 we have 120593X le 120593 minus 1205930X +1205930X le 119903119877(1 minus 119903) + 119877 = 119877(1 minus 119903)Theorem 6 Assume that (1198671)-(1198673) holden system (2) hasat least one almost automorphic solution inX0

Proof Firstly it is easy to check that if 119909 = (1199091 1199092 119909119899)119879 isinX is a solution of the integral equation

4 Complexity

119909119901 (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119909119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119909119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (119909119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(15)

where 119901 = 1 2 119899 then 119909 is also a solution of system (2) Secondly we define an operator 119879 X 997888rarr 119861119862(RA119899) by119879120593 = (1198791120593 1198792120593 119879119899120593)119879 (16)

where 120593 isin X 119901 = 1 2 119899(119879119901120593) (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(17)

We will prove that 119879mapsX into itself To this end let

Γ119901 (119904) = 119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))

sdot 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904) 119901 = 1 2 119899(18)

Then by Lemmas 3ndash5 Γ119901 isin 119860119860(RA) We will prove that119879Γ119901 isin 119860119860(RA)Let (1199041015840119899)119899isinN be a sequence of real numbers since 119886119901 isin119860119860(RR) and Γ119901 isin 119860119860(RA) we can extract a subsequence(119904119899)119899isinN of (1199041015840119899)119899isinN such that for each 119905 isin R

lim119899997888rarr+infin

119886119901 (119905 + 119904119899) fl 119886119901 (119905) lim

119899997888rarr+infin119886119901 (119905 minus 119904119899) = 119886119901 (119905) 119901 = 1 2 119899

(19)

and

lim119899997888rarr+infin

Γ119901 (119905 + 119904119899) fl Γ119901 (119905) lim

119899997888rarr+infinΓ119901 (119905 minus 119904119899) = Γ119901 (119905) 119901 = 1 2 119899

(20)

Set

(119879119901Γ119901) (119905) = int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (21)

Complexity 5

and then we have10038171003817100381710038171003817(119879119901Γ119901) (119905 + 119904119899) minus (119879119901Γ119901) (119905)10038171003817100381710038171003817A= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905+119904119899119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905119904minus119904119899 119886119901(120575+119904119899)119889120575Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119904 119886119901(120575)119889120575Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

le 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

le int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575 10038171003817100381710038171003817(Γ1119901 (119904 + 119904119899) minus Γ119901 (119904))10038171003817100381710038171003817A 119889119904

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin(119890minusint119905119906 119886119901(120575+119904119899)119889120575 minus 119890minusint119905119906 119886119901(120575)119889120575) Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A 119901 = 1 2 119899

(22)

By the Lebesgue dominated convergence theorem we obtainthat lim119899997888rarr+infin(119879119901Γ119901)(119905 + 119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 =1 2 119899 Similarly one can prove that lim119899997888rarr+infin(119879119901Γ119901)(119905 minus119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 = 1 2 119899 Hence 119879119901Γ119901 isin119860119860(RA) 119901 = 1 2 119899

Noticing that

(119879119901120593)1015840 (119905) = Γ119901 (119905) minus 119886119901 (119905) int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (23)

similar to the above we can prove that (119879119901120593)1015840 isin119860119860(RA) 119901 = 1 2 119899 Therefore 119879 maps X intoitself

Thirdly we will prove that 119879 is a self-mapping fromX0 toX0 In fact for each 120593 isin X0 we have

1003817100381710038171003817(119879120593) minus 120593010038171003817100381710038170 = max1le119901le119899

sup119905isinR

10038171003817100381710038171003817(119879119901120593) (119905) minus 1205930 (119905)10038171003817100381710038171003817Ale max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)

10038171003817100381710038171003817119901 (119906) 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum119902=1

10038171003817100381710038171003817119887119901119902 (119904)sdot 119891119902 (120593119902 (119904))10038171003817100381710038171003817A + 119899sum

119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))times 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))10038171003817100381710038171003817A]119889119904le max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906(119886+119901 100381710038171003817100381710038171205931015840100381710038171003817100381710038170

sdot 120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 100381710038171003817100381712059310038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902 100381710038171003817100381712059310038171003817100381710038170)

le max1le119901le119899

1119886minus119901 (119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899 1119886minus11990111987211990110038171003817100381710038171205931003817100381710038171003817X

(24)

6 Complexity

and

10038171003817100381710038171003817(Ψ120593 minus 1205930)1015840100381710038171003817100381710038170 le max1le119901le119899

sup119905isinR

10038171003817100381710038171003817100381710038171003817100381710038171003817119886119901 (119905) int

119905

119905minus120578119901(119905120593119901(119905))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119905) 119891119902 (120593119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (120593119902 (119905 minus 120590119901119902119897 (119905 120593119902 (119905)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119905 120593119897 (119905))))10038171003817100381710038171003817100381710038171003817100381710038171003817A + 10038171003817100381710038171003817100381710038171003817100381710038171003817int

119905

minusinfin119886119901 (119905)

sdot 119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904120593119901(119904))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904 120593119902 (119904)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904 120593119897 (119904))))] 11988911990410038171003817100381710038171003817100381710038171003817100381710038171003817A

le max1le119901le119899

(119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X + 119886+119901119886minus119901 (119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899(1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X

(25)

Thus we obtain

1003817100381710038171003817(119879120593 minus 1205930)1003817100381710038171003817119883= max

1le119901le119899sup119905isinR

1003817100381710038171003817(119879120593 minus 1205930)10038171003817100381710038170 sup119905isinR

10038171003817100381710038171003817(119879120593 minus 1205930)1015840100381710038171003817100381710038170le max

1le119901le119899 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X = 119903 10038171003817100381710038171205931003817100381710038171003817X

le 1199031198771 minus 119903 (26)

Fourthly we will prove that 119879 is a contracting mapping Infact for any 120593 120595 isin X0 we have that

1003817100381710038171003817119879120593 minus 11987912059510038171003817100381710038170 le max1le119901le119899

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)[119901 (119906) minus 119901 (119906)] 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum

119902=1

10038171003817100381710038171003817119887119901119902 (119904) [119891119902 (120593119902 (119904)) minus 119891119902 (120595119902 (119904))]10038171003817100381710038171003817A+ 119899sum119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) [119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) minus 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))]10038171003817100381710038171003817A]119889119904le max

1le119901le119899int119905

minusinfin119890minus119886minus119901 (119905minus119904)119889119904 [119886+119901120578+119901 10038171003817100381710038171003817119902 minus 119902100381710038171003817100381710038170 + 119899sum

119902=1119887+119901119902119871119891119902 10038171003817100381710038171003817120593119902 minus 120595119902100381710038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 [119866119902119871119892119897 1003817100381710038171003817120593 minus 12059510038171003817100381710038170 + 119866119897119871119892119902 1003817100381710038171003817120593 minus 12059510038171003817100381710038170]]

le max1le119901le119899

1119886minus1199011198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X

(27)

and 10038171003817100381710038171003817(119879120593 minus 119879120595)1015840100381710038171003817100381710038170 le max1le119901le119899

(1 + 119886+119901119886minus119901 )1198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X

le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X (28)

Hence by (1198673) 119879 is a contracting mapping principle There-fore there exists a unique fixed-point 120593lowast isin X0 such that

119879120593lowast = 120593lowast which implies that system (2) has an almostautomorphic solution inX0 The proof is complete

4 Global Exponential Stability

In this section we investigate the global exponential stabilityof almost automorphic solutions by reduction to absurdity

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 2: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

2 Complexity

Inspired by the above analysis and discussion the mainpurpose of this paper is to study the existence and globalexponential stability of almost automorphic solutions forClifford-valued high-order Hopfield neural networks withleakage delays by direct method that is we will study theconsidered Clifford-valued neural networks directly insteadof converting them to real-valued ones As far aswe know thisis the first paper to study the almost automorphic solutionsof Clifford-valued high-order Hopfield neural networks withleakage delays In addition this is the first paper to studyalmost automorphic solutions of Clifford neural networks bydirect method So our methods and results of this paper arenew Besides ourmethods proposed in this paper can be usedto study the problem of almost automorphic solutions forother types of Clifford-valued neural networks

This paper is organized as follows In Section 2 weintroduce some basic definitions and lemmas and give amodel description In Section 3 we study the existence ofalmost automorphic solutions for Clifford-valued high-orderHopfield neural networks with leakage delays In Section 4we investigate the global exponential stability of almostautomorphic solutions of the neural networks In Section 5an example is given to demonstrate the proposed resultsFinally we draw a brief conclusion in Section 6

2 Preliminaries and Model Description

The real Clifford algebra over R119898 is defined as

A = sum119860sube12119898

119886119860119890119860 119886119860 isin R (1)

where 119890119860 = 119890ℎ1119890ℎ2 sdot sdot sdot 119890ℎ] with 119860 = ℎ1ℎ2 sdot sdot sdot ℎ] 1 le ℎ1 ltℎ2 lt sdot sdot sdot lt ℎ] le 119898 Moreover 1198900 = 1198900 = 1 and119890ℎ ℎ = 1 2 119898 are said to be Clifford generators andsatisfy 1198902119894 = minus1 119894 = 1 2 119898 and 119890119894119890119895 + 119890119895119890119894 = 0 119894 =119895 119894 119895 = 1 2 119898 For convenience we will denote theproduct of Clifford generators 119890ℎ1 119890ℎ2 119890ℎ] as 119890ℎ1ℎ2 sdotsdotsdotℎ] LetΠ = 0 1 2 119860 12 sdot sdot sdot 119898 then it is easy to see thatA = sum119860 119886119860119890119860 119886119860 isin R where sum119860 is short for sum119860isinΠ anddim A = 2119898

For 119909 = sum119860 119909119860119890119860 isin A we define 119909A = max119860isinΠ|119909119860|and for 119909 = (1199091 1199092 119909119899)119879 isin A119899 we define 119909A119899 =max1le119901le119899119909119901A

The derivative of 119909(119905) = sum119860 119909119860(119905)119890119860 is given by (119905) =sum119860 119860(119905)119890119860 For more knowledge about Clifford algebra werefer the reader to [27]

In this paper we are concerned with the followingClifford-valued high-order Hopfield neural network withdelays in the leakage term

119901 (119905) = minus119886119901 (119905) 119909119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119909119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119897 (119909119897 (119905 minus ]119901119902119897 (119905))) + 119868119901 (119905) 119901 = 1 2 119899(2)

where 119909119901(119905) isin A corresponds to the state of the 119901th unit attime 119905 119886119901(119905) isin R+ represents the rate with which the 119901th unitwill reset its potential to the resting state when disconnectedfrom the network and external inputs at time 119905 119887119901119902(119905) isinA denotes the strength of 119902th unit on 119901th unit at time 119905119888119901119902119897(119905) isin A is the second-order synaptic weight of the neuralnetworks 119891119902(119905) 119892119902(119905) isin A denote the activation functions119868119901(119905) isin A is the external input on the 119901th at time 119905 and120578119901(119905) 120590119901119902119897(119905) ]119901119902119897(119905) isin R+ denote the transmission delays

Remark 1 When 119898 the number of the generators of Aequals 0 1 and 2 system (2) degenerates into real-valuedcomplex-valued and quaternion-valued systems respec-tively

We will adopt the following notation119886minus119901 = inf119905isinR

119886119901 (119905) 119887+119901119902 = sup

119905isinR

10038171003817100381710038171003817119887119901119902 (119905)10038171003817100381710038171003817A 119888+119901119902119897 = sup

119905isinR

10038171003817100381710038171003817119888119901119902119897 (119905)10038171003817100381710038171003817A 120578+119901 = sup

119905isinR120578119901 (119905)

120590+119901119902119897 = sup119905isinR

120590119901119902119897 (119905) ]+119901119902119897 = sup

119905isinR120590119901119902119897 (119905)

120585 = max1le119901119902119897le119899

120578+119901 120590+119901119902119897 ]+119901119902119897

(3)

The initial value of system (2) is given by119909119901 (119904) = 120593119901 (119904) isin A119901 (119904) = 119901 (119904) isin A119904 isin [minus120585 0] (4)

where 120593119901 isin 1198621([minus120585 0]A)Denote by 119880119862(RR) the set of all uniformly continuous

functions from R to R Let 119861119862(RA119899) denote the set ofall bounded continuous functions from R to A119899 Then(119861119862(RA119899) sdot 0) is a Banach space with the norm 1198910 =max1le119901le119899sup119905isinR119891119901(119905)A where 119891 = (1198911 1198912 119891119901)119879 isin119861119862(RA119899)Definition 2 A function 119891 isin 119861119862(RA119899) is said to be almostautomorphic if for every sequence of real numbers (1199041015840119899)119899isinNthere exists a subsequence (119904119899)119899isinN such that119892 (119905) fl lim

119899997888rarrinfin119891 (119905 + 119904119899) (5)

is well defined for each 119905 isin R and

lim119899997888rarrinfin

119892 (119905 minus 119904119899) = 119891 (119905) (6)

for each 119905 isin R

Complexity 3

From the above definition similar to the proofs of thecorresponding results in [28] it is not difficult to prove thefollowing two lemmas

Lemma 3 If 120572 isin R 119891 119892 isin 119860119860(RA119899) then 120572119891 119891 + 119892 119891119892 isin119860119860(RA119899)Lemma 4 Let 119891 isin 119862(AA119899) satisfy the Lipschitz conditionand 120593 isin 119860119860(RA) then 119891(120593(sdot)) isin 119860119860(RA119899)Lemma 5 If 119909 isin 1198621(RA) 119909 1199091015840 isin 119860119860(RA) 120578 isin119860119860(RR) cap 119880119862(RR) then 119909(sdot minus 120578(sdot)) isin 119860119860(RA)Proof Since 1199091015840 isin 119860119860(RA) and 120578 isin 119880119862(RR) 119909(119905 minus 120578(119905)) isuniformly continuous for 119905 isin R Hence for each 120576 gt 0 thereexists a positive number 120575 = 1205762 such that |1199051minus1199052| lt 120575 implies1003817100381710038171003817119909 (1199051) minus 119909 (1199052)1003817100381710038171003817A lt 120576 (7)

Since 119909 120578 isin 119860119860(RA) for every sequence of real numbers(1199041015840119899)119899isinN there exists a subsequence (119904119899)119899isinN such that

lim119899997888rarrinfin

119909 (119905 + 119904119899) fl 119909 (119905) lim119899997888rarrinfin

119909 (119905 minus 119904119899) = 119909 (119905) lim119899997888rarrinfin

120578 (119905 + 119904119899) fl 120578 (119905) lim119899997888rarrinfin

120578 (119905 minus 119904119899) = 120578 (119905)(8)

for every 119905 isin R Therefore there exists a positive integer 119873such that

lim119899997888rarrinfin

1003817100381710038171003817119909 (119905 + 119904119899) minus 119909 (119905)1003817100381710038171003817A lt 1205762 lim119899997888rarrinfin

1003816100381610038161003816120578 (119905 + 119904119899) minus 120578 (119905)1003816100381610038161003816 lt 1205762 (9)

for 119899 gt 119873 and 119905 isin R Hence we have1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905 + 119904119899)) minus 119909 (119905 minus 120578 (119905))1003817100381710038171003817Ale 1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905 + 119904119899)) minus 119909 (119905 + 119904119899 minus 120578 (119905))1003817100381710038171003817A+ 1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905)) minus 119909 (119905 minus 120578 (119905))1003817100381710038171003817A lt 1205762 + 1205762= 120576(10)

Consequently 119909(119905+119904119899minus120578(119905+119904119899)) converges to 119909(119905minus120578(119905)) foreach 119905 isin R Similarly we can obtain that 1199091(119905minus119904119899minus120578(119905minus119904119899))converges to 119909(119905minus120578(119905)) for each 119905 isin R Therefore 119909(sdotminus120578(sdot)) isin119860119860(RA) The proof is complete

Throughout this paper we make the following assump-tions

(H1) For 119901 119902 119897 = 1 2 119899 119886119901 120578119901 120590119901119902119897 ]119901119902119897 isin119860119860(RR+) cap 119880119862(RR) and min1le119901le119899inf 119905isinR119886119901(119905) gt0 119887119901119902 119888119901119902119897 119868119901 isin A119860(RA)

(H2) For 119902 = 1 2 119899 119891119902 119892119902 ℎ119902 isin 119862(AA) and thereexist positive constants 119871119891119902 119871119892119902 119871ℎ119902 119866119901 such that

10038171003817100381710038171003817119891119902 (119909) minus 119891119902 (119910)10038171003817100381710038171003817A le 119871119891119902 1003817100381710038171003817119909 minus 1199101003817100381710038171003817A 10038171003817100381710038171003817119892119902 (119909) minus 119892119902 (119910)10038171003817100381710038171003817A le 119871119892119902 1003817100381710038171003817119909 minus 1199101003817100381710038171003817A 10038171003817100381710038171003817119892119901 (119909)10038171003817100381710038171003817A le 119866119901(11)

Moreover for 119902 = 1 2 119899 119891119901(0) = 119892119901(0) = 0(H3) max1le119901le119899(1119886minus119901 )119872119901 (1 + 119886+119901119886minus119901 )119872119901 fl 119903 lt 1 where119872119901 = 119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902 + 119866119902119871119892119897 )

119901 = 1 2 119899 (12)

3 The Existence of AlmostAutomorphic Solutions

In this section we study the existence of almost automorphicsolutions by the contracting mapping principle

Let

Y = 119909 isin 1198621 (RA119899) 119909 1199091015840 isin 119860119860 (RA119899) X = 120593 = (1205931 1205932 120593119899)119879 | 120593 isin Y 119901 = 1 2 119899 (13)

For any 120593 = (1205931 1205932 120593119899)119879 isin X we define thenorm of 120593 as 120593X = max1205930 12059310158400 where 1205930 =max1le119901le119899sup119905isinR120593119901(119905)A thenX is a Banach space

Let 1205930(119905) = (int119905minusinfin 119890minusint119905119904 1198861(119906)1198891199061198681(119904)119889119904int119905minusinfin 119890minusint119905119904 1198862(119906)1198891199061198682(119904)119889119904 int119905minusinfin 119890minusint119905119904 119886119899(119906)119889119906119868119899(119904)119889119904)119879 andtake a positive number 119877 gt 1205930X

Set

X0

= 120593 = (1205931 1205932 120593119899)119879 isin X 1003817100381710038171003817120593 minus 12059301003817100381710038171003817X le 1199031198771 minus 119903 (14)

and then for every 120593 isin X0 we have 120593X le 120593 minus 1205930X +1205930X le 119903119877(1 minus 119903) + 119877 = 119877(1 minus 119903)Theorem 6 Assume that (1198671)-(1198673) holden system (2) hasat least one almost automorphic solution inX0

Proof Firstly it is easy to check that if 119909 = (1199091 1199092 119909119899)119879 isinX is a solution of the integral equation

4 Complexity

119909119901 (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119909119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119909119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (119909119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(15)

where 119901 = 1 2 119899 then 119909 is also a solution of system (2) Secondly we define an operator 119879 X 997888rarr 119861119862(RA119899) by119879120593 = (1198791120593 1198792120593 119879119899120593)119879 (16)

where 120593 isin X 119901 = 1 2 119899(119879119901120593) (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(17)

We will prove that 119879mapsX into itself To this end let

Γ119901 (119904) = 119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))

sdot 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904) 119901 = 1 2 119899(18)

Then by Lemmas 3ndash5 Γ119901 isin 119860119860(RA) We will prove that119879Γ119901 isin 119860119860(RA)Let (1199041015840119899)119899isinN be a sequence of real numbers since 119886119901 isin119860119860(RR) and Γ119901 isin 119860119860(RA) we can extract a subsequence(119904119899)119899isinN of (1199041015840119899)119899isinN such that for each 119905 isin R

lim119899997888rarr+infin

119886119901 (119905 + 119904119899) fl 119886119901 (119905) lim

119899997888rarr+infin119886119901 (119905 minus 119904119899) = 119886119901 (119905) 119901 = 1 2 119899

(19)

and

lim119899997888rarr+infin

Γ119901 (119905 + 119904119899) fl Γ119901 (119905) lim

119899997888rarr+infinΓ119901 (119905 minus 119904119899) = Γ119901 (119905) 119901 = 1 2 119899

(20)

Set

(119879119901Γ119901) (119905) = int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (21)

Complexity 5

and then we have10038171003817100381710038171003817(119879119901Γ119901) (119905 + 119904119899) minus (119879119901Γ119901) (119905)10038171003817100381710038171003817A= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905+119904119899119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905119904minus119904119899 119886119901(120575+119904119899)119889120575Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119904 119886119901(120575)119889120575Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

le 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

le int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575 10038171003817100381710038171003817(Γ1119901 (119904 + 119904119899) minus Γ119901 (119904))10038171003817100381710038171003817A 119889119904

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin(119890minusint119905119906 119886119901(120575+119904119899)119889120575 minus 119890minusint119905119906 119886119901(120575)119889120575) Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A 119901 = 1 2 119899

(22)

By the Lebesgue dominated convergence theorem we obtainthat lim119899997888rarr+infin(119879119901Γ119901)(119905 + 119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 =1 2 119899 Similarly one can prove that lim119899997888rarr+infin(119879119901Γ119901)(119905 minus119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 = 1 2 119899 Hence 119879119901Γ119901 isin119860119860(RA) 119901 = 1 2 119899

Noticing that

(119879119901120593)1015840 (119905) = Γ119901 (119905) minus 119886119901 (119905) int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (23)

similar to the above we can prove that (119879119901120593)1015840 isin119860119860(RA) 119901 = 1 2 119899 Therefore 119879 maps X intoitself

Thirdly we will prove that 119879 is a self-mapping fromX0 toX0 In fact for each 120593 isin X0 we have

1003817100381710038171003817(119879120593) minus 120593010038171003817100381710038170 = max1le119901le119899

sup119905isinR

10038171003817100381710038171003817(119879119901120593) (119905) minus 1205930 (119905)10038171003817100381710038171003817Ale max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)

10038171003817100381710038171003817119901 (119906) 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum119902=1

10038171003817100381710038171003817119887119901119902 (119904)sdot 119891119902 (120593119902 (119904))10038171003817100381710038171003817A + 119899sum

119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))times 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))10038171003817100381710038171003817A]119889119904le max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906(119886+119901 100381710038171003817100381710038171205931015840100381710038171003817100381710038170

sdot 120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 100381710038171003817100381712059310038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902 100381710038171003817100381712059310038171003817100381710038170)

le max1le119901le119899

1119886minus119901 (119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899 1119886minus11990111987211990110038171003817100381710038171205931003817100381710038171003817X

(24)

6 Complexity

and

10038171003817100381710038171003817(Ψ120593 minus 1205930)1015840100381710038171003817100381710038170 le max1le119901le119899

sup119905isinR

10038171003817100381710038171003817100381710038171003817100381710038171003817119886119901 (119905) int

119905

119905minus120578119901(119905120593119901(119905))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119905) 119891119902 (120593119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (120593119902 (119905 minus 120590119901119902119897 (119905 120593119902 (119905)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119905 120593119897 (119905))))10038171003817100381710038171003817100381710038171003817100381710038171003817A + 10038171003817100381710038171003817100381710038171003817100381710038171003817int

119905

minusinfin119886119901 (119905)

sdot 119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904120593119901(119904))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904 120593119902 (119904)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904 120593119897 (119904))))] 11988911990410038171003817100381710038171003817100381710038171003817100381710038171003817A

le max1le119901le119899

(119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X + 119886+119901119886minus119901 (119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899(1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X

(25)

Thus we obtain

1003817100381710038171003817(119879120593 minus 1205930)1003817100381710038171003817119883= max

1le119901le119899sup119905isinR

1003817100381710038171003817(119879120593 minus 1205930)10038171003817100381710038170 sup119905isinR

10038171003817100381710038171003817(119879120593 minus 1205930)1015840100381710038171003817100381710038170le max

1le119901le119899 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X = 119903 10038171003817100381710038171205931003817100381710038171003817X

le 1199031198771 minus 119903 (26)

Fourthly we will prove that 119879 is a contracting mapping Infact for any 120593 120595 isin X0 we have that

1003817100381710038171003817119879120593 minus 11987912059510038171003817100381710038170 le max1le119901le119899

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)[119901 (119906) minus 119901 (119906)] 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum

119902=1

10038171003817100381710038171003817119887119901119902 (119904) [119891119902 (120593119902 (119904)) minus 119891119902 (120595119902 (119904))]10038171003817100381710038171003817A+ 119899sum119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) [119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) minus 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))]10038171003817100381710038171003817A]119889119904le max

1le119901le119899int119905

minusinfin119890minus119886minus119901 (119905minus119904)119889119904 [119886+119901120578+119901 10038171003817100381710038171003817119902 minus 119902100381710038171003817100381710038170 + 119899sum

119902=1119887+119901119902119871119891119902 10038171003817100381710038171003817120593119902 minus 120595119902100381710038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 [119866119902119871119892119897 1003817100381710038171003817120593 minus 12059510038171003817100381710038170 + 119866119897119871119892119902 1003817100381710038171003817120593 minus 12059510038171003817100381710038170]]

le max1le119901le119899

1119886minus1199011198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X

(27)

and 10038171003817100381710038171003817(119879120593 minus 119879120595)1015840100381710038171003817100381710038170 le max1le119901le119899

(1 + 119886+119901119886minus119901 )1198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X

le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X (28)

Hence by (1198673) 119879 is a contracting mapping principle There-fore there exists a unique fixed-point 120593lowast isin X0 such that

119879120593lowast = 120593lowast which implies that system (2) has an almostautomorphic solution inX0 The proof is complete

4 Global Exponential Stability

In this section we investigate the global exponential stabilityof almost automorphic solutions by reduction to absurdity

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 3: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Complexity 3

From the above definition similar to the proofs of thecorresponding results in [28] it is not difficult to prove thefollowing two lemmas

Lemma 3 If 120572 isin R 119891 119892 isin 119860119860(RA119899) then 120572119891 119891 + 119892 119891119892 isin119860119860(RA119899)Lemma 4 Let 119891 isin 119862(AA119899) satisfy the Lipschitz conditionand 120593 isin 119860119860(RA) then 119891(120593(sdot)) isin 119860119860(RA119899)Lemma 5 If 119909 isin 1198621(RA) 119909 1199091015840 isin 119860119860(RA) 120578 isin119860119860(RR) cap 119880119862(RR) then 119909(sdot minus 120578(sdot)) isin 119860119860(RA)Proof Since 1199091015840 isin 119860119860(RA) and 120578 isin 119880119862(RR) 119909(119905 minus 120578(119905)) isuniformly continuous for 119905 isin R Hence for each 120576 gt 0 thereexists a positive number 120575 = 1205762 such that |1199051minus1199052| lt 120575 implies1003817100381710038171003817119909 (1199051) minus 119909 (1199052)1003817100381710038171003817A lt 120576 (7)

Since 119909 120578 isin 119860119860(RA) for every sequence of real numbers(1199041015840119899)119899isinN there exists a subsequence (119904119899)119899isinN such that

lim119899997888rarrinfin

119909 (119905 + 119904119899) fl 119909 (119905) lim119899997888rarrinfin

119909 (119905 minus 119904119899) = 119909 (119905) lim119899997888rarrinfin

120578 (119905 + 119904119899) fl 120578 (119905) lim119899997888rarrinfin

120578 (119905 minus 119904119899) = 120578 (119905)(8)

for every 119905 isin R Therefore there exists a positive integer 119873such that

lim119899997888rarrinfin

1003817100381710038171003817119909 (119905 + 119904119899) minus 119909 (119905)1003817100381710038171003817A lt 1205762 lim119899997888rarrinfin

1003816100381610038161003816120578 (119905 + 119904119899) minus 120578 (119905)1003816100381610038161003816 lt 1205762 (9)

for 119899 gt 119873 and 119905 isin R Hence we have1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905 + 119904119899)) minus 119909 (119905 minus 120578 (119905))1003817100381710038171003817Ale 1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905 + 119904119899)) minus 119909 (119905 + 119904119899 minus 120578 (119905))1003817100381710038171003817A+ 1003817100381710038171003817119909 (119905 + 119904119899 minus 120578 (119905)) minus 119909 (119905 minus 120578 (119905))1003817100381710038171003817A lt 1205762 + 1205762= 120576(10)

Consequently 119909(119905+119904119899minus120578(119905+119904119899)) converges to 119909(119905minus120578(119905)) foreach 119905 isin R Similarly we can obtain that 1199091(119905minus119904119899minus120578(119905minus119904119899))converges to 119909(119905minus120578(119905)) for each 119905 isin R Therefore 119909(sdotminus120578(sdot)) isin119860119860(RA) The proof is complete

Throughout this paper we make the following assump-tions

(H1) For 119901 119902 119897 = 1 2 119899 119886119901 120578119901 120590119901119902119897 ]119901119902119897 isin119860119860(RR+) cap 119880119862(RR) and min1le119901le119899inf 119905isinR119886119901(119905) gt0 119887119901119902 119888119901119902119897 119868119901 isin A119860(RA)

(H2) For 119902 = 1 2 119899 119891119902 119892119902 ℎ119902 isin 119862(AA) and thereexist positive constants 119871119891119902 119871119892119902 119871ℎ119902 119866119901 such that

10038171003817100381710038171003817119891119902 (119909) minus 119891119902 (119910)10038171003817100381710038171003817A le 119871119891119902 1003817100381710038171003817119909 minus 1199101003817100381710038171003817A 10038171003817100381710038171003817119892119902 (119909) minus 119892119902 (119910)10038171003817100381710038171003817A le 119871119892119902 1003817100381710038171003817119909 minus 1199101003817100381710038171003817A 10038171003817100381710038171003817119892119901 (119909)10038171003817100381710038171003817A le 119866119901(11)

Moreover for 119902 = 1 2 119899 119891119901(0) = 119892119901(0) = 0(H3) max1le119901le119899(1119886minus119901 )119872119901 (1 + 119886+119901119886minus119901 )119872119901 fl 119903 lt 1 where119872119901 = 119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902 + 119866119902119871119892119897 )

119901 = 1 2 119899 (12)

3 The Existence of AlmostAutomorphic Solutions

In this section we study the existence of almost automorphicsolutions by the contracting mapping principle

Let

Y = 119909 isin 1198621 (RA119899) 119909 1199091015840 isin 119860119860 (RA119899) X = 120593 = (1205931 1205932 120593119899)119879 | 120593 isin Y 119901 = 1 2 119899 (13)

For any 120593 = (1205931 1205932 120593119899)119879 isin X we define thenorm of 120593 as 120593X = max1205930 12059310158400 where 1205930 =max1le119901le119899sup119905isinR120593119901(119905)A thenX is a Banach space

Let 1205930(119905) = (int119905minusinfin 119890minusint119905119904 1198861(119906)1198891199061198681(119904)119889119904int119905minusinfin 119890minusint119905119904 1198862(119906)1198891199061198682(119904)119889119904 int119905minusinfin 119890minusint119905119904 119886119899(119906)119889119906119868119899(119904)119889119904)119879 andtake a positive number 119877 gt 1205930X

Set

X0

= 120593 = (1205931 1205932 120593119899)119879 isin X 1003817100381710038171003817120593 minus 12059301003817100381710038171003817X le 1199031198771 minus 119903 (14)

and then for every 120593 isin X0 we have 120593X le 120593 minus 1205930X +1205930X le 119903119877(1 minus 119903) + 119877 = 119877(1 minus 119903)Theorem 6 Assume that (1198671)-(1198673) holden system (2) hasat least one almost automorphic solution inX0

Proof Firstly it is easy to check that if 119909 = (1199091 1199092 119909119899)119879 isinX is a solution of the integral equation

4 Complexity

119909119901 (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119909119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119909119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (119909119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(15)

where 119901 = 1 2 119899 then 119909 is also a solution of system (2) Secondly we define an operator 119879 X 997888rarr 119861119862(RA119899) by119879120593 = (1198791120593 1198792120593 119879119899120593)119879 (16)

where 120593 isin X 119901 = 1 2 119899(119879119901120593) (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(17)

We will prove that 119879mapsX into itself To this end let

Γ119901 (119904) = 119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))

sdot 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904) 119901 = 1 2 119899(18)

Then by Lemmas 3ndash5 Γ119901 isin 119860119860(RA) We will prove that119879Γ119901 isin 119860119860(RA)Let (1199041015840119899)119899isinN be a sequence of real numbers since 119886119901 isin119860119860(RR) and Γ119901 isin 119860119860(RA) we can extract a subsequence(119904119899)119899isinN of (1199041015840119899)119899isinN such that for each 119905 isin R

lim119899997888rarr+infin

119886119901 (119905 + 119904119899) fl 119886119901 (119905) lim

119899997888rarr+infin119886119901 (119905 minus 119904119899) = 119886119901 (119905) 119901 = 1 2 119899

(19)

and

lim119899997888rarr+infin

Γ119901 (119905 + 119904119899) fl Γ119901 (119905) lim

119899997888rarr+infinΓ119901 (119905 minus 119904119899) = Γ119901 (119905) 119901 = 1 2 119899

(20)

Set

(119879119901Γ119901) (119905) = int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (21)

Complexity 5

and then we have10038171003817100381710038171003817(119879119901Γ119901) (119905 + 119904119899) minus (119879119901Γ119901) (119905)10038171003817100381710038171003817A= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905+119904119899119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905119904minus119904119899 119886119901(120575+119904119899)119889120575Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119904 119886119901(120575)119889120575Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

le 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

le int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575 10038171003817100381710038171003817(Γ1119901 (119904 + 119904119899) minus Γ119901 (119904))10038171003817100381710038171003817A 119889119904

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin(119890minusint119905119906 119886119901(120575+119904119899)119889120575 minus 119890minusint119905119906 119886119901(120575)119889120575) Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A 119901 = 1 2 119899

(22)

By the Lebesgue dominated convergence theorem we obtainthat lim119899997888rarr+infin(119879119901Γ119901)(119905 + 119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 =1 2 119899 Similarly one can prove that lim119899997888rarr+infin(119879119901Γ119901)(119905 minus119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 = 1 2 119899 Hence 119879119901Γ119901 isin119860119860(RA) 119901 = 1 2 119899

Noticing that

(119879119901120593)1015840 (119905) = Γ119901 (119905) minus 119886119901 (119905) int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (23)

similar to the above we can prove that (119879119901120593)1015840 isin119860119860(RA) 119901 = 1 2 119899 Therefore 119879 maps X intoitself

Thirdly we will prove that 119879 is a self-mapping fromX0 toX0 In fact for each 120593 isin X0 we have

1003817100381710038171003817(119879120593) minus 120593010038171003817100381710038170 = max1le119901le119899

sup119905isinR

10038171003817100381710038171003817(119879119901120593) (119905) minus 1205930 (119905)10038171003817100381710038171003817Ale max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)

10038171003817100381710038171003817119901 (119906) 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum119902=1

10038171003817100381710038171003817119887119901119902 (119904)sdot 119891119902 (120593119902 (119904))10038171003817100381710038171003817A + 119899sum

119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))times 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))10038171003817100381710038171003817A]119889119904le max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906(119886+119901 100381710038171003817100381710038171205931015840100381710038171003817100381710038170

sdot 120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 100381710038171003817100381712059310038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902 100381710038171003817100381712059310038171003817100381710038170)

le max1le119901le119899

1119886minus119901 (119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899 1119886minus11990111987211990110038171003817100381710038171205931003817100381710038171003817X

(24)

6 Complexity

and

10038171003817100381710038171003817(Ψ120593 minus 1205930)1015840100381710038171003817100381710038170 le max1le119901le119899

sup119905isinR

10038171003817100381710038171003817100381710038171003817100381710038171003817119886119901 (119905) int

119905

119905minus120578119901(119905120593119901(119905))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119905) 119891119902 (120593119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (120593119902 (119905 minus 120590119901119902119897 (119905 120593119902 (119905)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119905 120593119897 (119905))))10038171003817100381710038171003817100381710038171003817100381710038171003817A + 10038171003817100381710038171003817100381710038171003817100381710038171003817int

119905

minusinfin119886119901 (119905)

sdot 119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904120593119901(119904))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904 120593119902 (119904)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904 120593119897 (119904))))] 11988911990410038171003817100381710038171003817100381710038171003817100381710038171003817A

le max1le119901le119899

(119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X + 119886+119901119886minus119901 (119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899(1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X

(25)

Thus we obtain

1003817100381710038171003817(119879120593 minus 1205930)1003817100381710038171003817119883= max

1le119901le119899sup119905isinR

1003817100381710038171003817(119879120593 minus 1205930)10038171003817100381710038170 sup119905isinR

10038171003817100381710038171003817(119879120593 minus 1205930)1015840100381710038171003817100381710038170le max

1le119901le119899 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X = 119903 10038171003817100381710038171205931003817100381710038171003817X

le 1199031198771 minus 119903 (26)

Fourthly we will prove that 119879 is a contracting mapping Infact for any 120593 120595 isin X0 we have that

1003817100381710038171003817119879120593 minus 11987912059510038171003817100381710038170 le max1le119901le119899

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)[119901 (119906) minus 119901 (119906)] 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum

119902=1

10038171003817100381710038171003817119887119901119902 (119904) [119891119902 (120593119902 (119904)) minus 119891119902 (120595119902 (119904))]10038171003817100381710038171003817A+ 119899sum119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) [119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) minus 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))]10038171003817100381710038171003817A]119889119904le max

1le119901le119899int119905

minusinfin119890minus119886minus119901 (119905minus119904)119889119904 [119886+119901120578+119901 10038171003817100381710038171003817119902 minus 119902100381710038171003817100381710038170 + 119899sum

119902=1119887+119901119902119871119891119902 10038171003817100381710038171003817120593119902 minus 120595119902100381710038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 [119866119902119871119892119897 1003817100381710038171003817120593 minus 12059510038171003817100381710038170 + 119866119897119871119892119902 1003817100381710038171003817120593 minus 12059510038171003817100381710038170]]

le max1le119901le119899

1119886minus1199011198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X

(27)

and 10038171003817100381710038171003817(119879120593 minus 119879120595)1015840100381710038171003817100381710038170 le max1le119901le119899

(1 + 119886+119901119886minus119901 )1198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X

le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X (28)

Hence by (1198673) 119879 is a contracting mapping principle There-fore there exists a unique fixed-point 120593lowast isin X0 such that

119879120593lowast = 120593lowast which implies that system (2) has an almostautomorphic solution inX0 The proof is complete

4 Global Exponential Stability

In this section we investigate the global exponential stabilityof almost automorphic solutions by reduction to absurdity

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 4: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

4 Complexity

119909119901 (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119909119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119909119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (119909119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(15)

where 119901 = 1 2 119899 then 119909 is also a solution of system (2) Secondly we define an operator 119879 X 997888rarr 119861119862(RA119899) by119879120593 = (1198791120593 1198792120593 119879119899120593)119879 (16)

where 120593 isin X 119901 = 1 2 119899(119879119901120593) (119905)= int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904)] 119889119904(17)

We will prove that 119879mapsX into itself To this end let

Γ119901 (119904) = 119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))

sdot 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) + 119868119901 (119904) 119901 = 1 2 119899(18)

Then by Lemmas 3ndash5 Γ119901 isin 119860119860(RA) We will prove that119879Γ119901 isin 119860119860(RA)Let (1199041015840119899)119899isinN be a sequence of real numbers since 119886119901 isin119860119860(RR) and Γ119901 isin 119860119860(RA) we can extract a subsequence(119904119899)119899isinN of (1199041015840119899)119899isinN such that for each 119905 isin R

lim119899997888rarr+infin

119886119901 (119905 + 119904119899) fl 119886119901 (119905) lim

119899997888rarr+infin119886119901 (119905 minus 119904119899) = 119886119901 (119905) 119901 = 1 2 119899

(19)

and

lim119899997888rarr+infin

Γ119901 (119905 + 119904119899) fl Γ119901 (119905) lim

119899997888rarr+infinΓ119901 (119905 minus 119904119899) = Γ119901 (119905) 119901 = 1 2 119899

(20)

Set

(119879119901Γ119901) (119905) = int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (21)

Complexity 5

and then we have10038171003817100381710038171003817(119879119901Γ119901) (119905 + 119904119899) minus (119879119901Γ119901) (119905)10038171003817100381710038171003817A= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905+119904119899119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905119904minus119904119899 119886119901(120575+119904119899)119889120575Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119904 119886119901(120575)119889120575Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

le 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

le int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575 10038171003817100381710038171003817(Γ1119901 (119904 + 119904119899) minus Γ119901 (119904))10038171003817100381710038171003817A 119889119904

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin(119890minusint119905119906 119886119901(120575+119904119899)119889120575 minus 119890minusint119905119906 119886119901(120575)119889120575) Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A 119901 = 1 2 119899

(22)

By the Lebesgue dominated convergence theorem we obtainthat lim119899997888rarr+infin(119879119901Γ119901)(119905 + 119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 =1 2 119899 Similarly one can prove that lim119899997888rarr+infin(119879119901Γ119901)(119905 minus119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 = 1 2 119899 Hence 119879119901Γ119901 isin119860119860(RA) 119901 = 1 2 119899

Noticing that

(119879119901120593)1015840 (119905) = Γ119901 (119905) minus 119886119901 (119905) int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (23)

similar to the above we can prove that (119879119901120593)1015840 isin119860119860(RA) 119901 = 1 2 119899 Therefore 119879 maps X intoitself

Thirdly we will prove that 119879 is a self-mapping fromX0 toX0 In fact for each 120593 isin X0 we have

1003817100381710038171003817(119879120593) minus 120593010038171003817100381710038170 = max1le119901le119899

sup119905isinR

10038171003817100381710038171003817(119879119901120593) (119905) minus 1205930 (119905)10038171003817100381710038171003817Ale max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)

10038171003817100381710038171003817119901 (119906) 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum119902=1

10038171003817100381710038171003817119887119901119902 (119904)sdot 119891119902 (120593119902 (119904))10038171003817100381710038171003817A + 119899sum

119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))times 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))10038171003817100381710038171003817A]119889119904le max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906(119886+119901 100381710038171003817100381710038171205931015840100381710038171003817100381710038170

sdot 120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 100381710038171003817100381712059310038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902 100381710038171003817100381712059310038171003817100381710038170)

le max1le119901le119899

1119886minus119901 (119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899 1119886minus11990111987211990110038171003817100381710038171205931003817100381710038171003817X

(24)

6 Complexity

and

10038171003817100381710038171003817(Ψ120593 minus 1205930)1015840100381710038171003817100381710038170 le max1le119901le119899

sup119905isinR

10038171003817100381710038171003817100381710038171003817100381710038171003817119886119901 (119905) int

119905

119905minus120578119901(119905120593119901(119905))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119905) 119891119902 (120593119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (120593119902 (119905 minus 120590119901119902119897 (119905 120593119902 (119905)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119905 120593119897 (119905))))10038171003817100381710038171003817100381710038171003817100381710038171003817A + 10038171003817100381710038171003817100381710038171003817100381710038171003817int

119905

minusinfin119886119901 (119905)

sdot 119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904120593119901(119904))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904 120593119902 (119904)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904 120593119897 (119904))))] 11988911990410038171003817100381710038171003817100381710038171003817100381710038171003817A

le max1le119901le119899

(119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X + 119886+119901119886minus119901 (119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899(1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X

(25)

Thus we obtain

1003817100381710038171003817(119879120593 minus 1205930)1003817100381710038171003817119883= max

1le119901le119899sup119905isinR

1003817100381710038171003817(119879120593 minus 1205930)10038171003817100381710038170 sup119905isinR

10038171003817100381710038171003817(119879120593 minus 1205930)1015840100381710038171003817100381710038170le max

1le119901le119899 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X = 119903 10038171003817100381710038171205931003817100381710038171003817X

le 1199031198771 minus 119903 (26)

Fourthly we will prove that 119879 is a contracting mapping Infact for any 120593 120595 isin X0 we have that

1003817100381710038171003817119879120593 minus 11987912059510038171003817100381710038170 le max1le119901le119899

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)[119901 (119906) minus 119901 (119906)] 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum

119902=1

10038171003817100381710038171003817119887119901119902 (119904) [119891119902 (120593119902 (119904)) minus 119891119902 (120595119902 (119904))]10038171003817100381710038171003817A+ 119899sum119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) [119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) minus 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))]10038171003817100381710038171003817A]119889119904le max

1le119901le119899int119905

minusinfin119890minus119886minus119901 (119905minus119904)119889119904 [119886+119901120578+119901 10038171003817100381710038171003817119902 minus 119902100381710038171003817100381710038170 + 119899sum

119902=1119887+119901119902119871119891119902 10038171003817100381710038171003817120593119902 minus 120595119902100381710038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 [119866119902119871119892119897 1003817100381710038171003817120593 minus 12059510038171003817100381710038170 + 119866119897119871119892119902 1003817100381710038171003817120593 minus 12059510038171003817100381710038170]]

le max1le119901le119899

1119886minus1199011198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X

(27)

and 10038171003817100381710038171003817(119879120593 minus 119879120595)1015840100381710038171003817100381710038170 le max1le119901le119899

(1 + 119886+119901119886minus119901 )1198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X

le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X (28)

Hence by (1198673) 119879 is a contracting mapping principle There-fore there exists a unique fixed-point 120593lowast isin X0 such that

119879120593lowast = 120593lowast which implies that system (2) has an almostautomorphic solution inX0 The proof is complete

4 Global Exponential Stability

In this section we investigate the global exponential stabilityof almost automorphic solutions by reduction to absurdity

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 5: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Complexity 5

and then we have10038171003817100381710038171003817(119879119901Γ119901) (119905 + 119904119899) minus (119879119901Γ119901) (119905)10038171003817100381710038171003817A= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905+119904119899119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119890minusint119905119904minus119904119899 119886119901(120575+119904119899)119889120575Γ119901 (119904) 119889119904

minus int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

= 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119904 119886119901(120575)119889120575Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A

le 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906 + 119904119899) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575Γ119901 (119906) 119889119906

minus int119905

minusinfin119890minusint119905119906 119886119901(120575)119889120575Γ119901 (119906) 11988911990610038171003817100381710038171003817100381710038171003817A

le int119905

minusinfin119890minusint119905119906 119886119901(120575+119904119899)119889120575 10038171003817100381710038171003817(Γ1119901 (119904 + 119904119899) minus Γ119901 (119904))10038171003817100381710038171003817A 119889119904

+ 10038171003817100381710038171003817100381710038171003817int119905

minusinfin(119890minusint119905119906 119886119901(120575+119904119899)119889120575 minus 119890minusint119905119906 119886119901(120575)119889120575) Γ119901 (119904) 11988911990410038171003817100381710038171003817100381710038171003817A 119901 = 1 2 119899

(22)

By the Lebesgue dominated convergence theorem we obtainthat lim119899997888rarr+infin(119879119901Γ119901)(119905 + 119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 =1 2 119899 Similarly one can prove that lim119899997888rarr+infin(119879119901Γ119901)(119905 minus119904119899) = (119879119901Γ119901)(119905) for each 119905 isin R 119901 = 1 2 119899 Hence 119879119901Γ119901 isin119860119860(RA) 119901 = 1 2 119899

Noticing that

(119879119901120593)1015840 (119905) = Γ119901 (119905) minus 119886119901 (119905) int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906Γ119901 (119904) 119889119904

119901 = 1 2 119899 (23)

similar to the above we can prove that (119879119901120593)1015840 isin119860119860(RA) 119901 = 1 2 119899 Therefore 119879 maps X intoitself

Thirdly we will prove that 119879 is a self-mapping fromX0 toX0 In fact for each 120593 isin X0 we have

1003817100381710038171003817(119879120593) minus 120593010038171003817100381710038170 = max1le119901le119899

sup119905isinR

10038171003817100381710038171003817(119879119901120593) (119905) minus 1205930 (119905)10038171003817100381710038171003817Ale max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)

10038171003817100381710038171003817119901 (119906) 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum119902=1

10038171003817100381710038171003817119887119901119902 (119904)sdot 119891119902 (120593119902 (119904))10038171003817100381710038171003817A + 119899sum

119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904)))times 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))10038171003817100381710038171003817A]119889119904le max

1le119901le119899sup119905isinR

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906(119886+119901 100381710038171003817100381710038171205931015840100381710038171003817100381710038170

sdot 120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 100381710038171003817100381712059310038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902 100381710038171003817100381712059310038171003817100381710038170)

le max1le119901le119899

1119886minus119901 (119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899 1119886minus11990111987211990110038171003817100381710038171205931003817100381710038171003817X

(24)

6 Complexity

and

10038171003817100381710038171003817(Ψ120593 minus 1205930)1015840100381710038171003817100381710038170 le max1le119901le119899

sup119905isinR

10038171003817100381710038171003817100381710038171003817100381710038171003817119886119901 (119905) int

119905

119905minus120578119901(119905120593119901(119905))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119905) 119891119902 (120593119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (120593119902 (119905 minus 120590119901119902119897 (119905 120593119902 (119905)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119905 120593119897 (119905))))10038171003817100381710038171003817100381710038171003817100381710038171003817A + 10038171003817100381710038171003817100381710038171003817100381710038171003817int

119905

minusinfin119886119901 (119905)

sdot 119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904120593119901(119904))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904 120593119902 (119904)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904 120593119897 (119904))))] 11988911990410038171003817100381710038171003817100381710038171003817100381710038171003817A

le max1le119901le119899

(119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X + 119886+119901119886minus119901 (119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899(1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X

(25)

Thus we obtain

1003817100381710038171003817(119879120593 minus 1205930)1003817100381710038171003817119883= max

1le119901le119899sup119905isinR

1003817100381710038171003817(119879120593 minus 1205930)10038171003817100381710038170 sup119905isinR

10038171003817100381710038171003817(119879120593 minus 1205930)1015840100381710038171003817100381710038170le max

1le119901le119899 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X = 119903 10038171003817100381710038171205931003817100381710038171003817X

le 1199031198771 minus 119903 (26)

Fourthly we will prove that 119879 is a contracting mapping Infact for any 120593 120595 isin X0 we have that

1003817100381710038171003817119879120593 minus 11987912059510038171003817100381710038170 le max1le119901le119899

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)[119901 (119906) minus 119901 (119906)] 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum

119902=1

10038171003817100381710038171003817119887119901119902 (119904) [119891119902 (120593119902 (119904)) minus 119891119902 (120595119902 (119904))]10038171003817100381710038171003817A+ 119899sum119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) [119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) minus 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))]10038171003817100381710038171003817A]119889119904le max

1le119901le119899int119905

minusinfin119890minus119886minus119901 (119905minus119904)119889119904 [119886+119901120578+119901 10038171003817100381710038171003817119902 minus 119902100381710038171003817100381710038170 + 119899sum

119902=1119887+119901119902119871119891119902 10038171003817100381710038171003817120593119902 minus 120595119902100381710038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 [119866119902119871119892119897 1003817100381710038171003817120593 minus 12059510038171003817100381710038170 + 119866119897119871119892119902 1003817100381710038171003817120593 minus 12059510038171003817100381710038170]]

le max1le119901le119899

1119886minus1199011198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X

(27)

and 10038171003817100381710038171003817(119879120593 minus 119879120595)1015840100381710038171003817100381710038170 le max1le119901le119899

(1 + 119886+119901119886minus119901 )1198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X

le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X (28)

Hence by (1198673) 119879 is a contracting mapping principle There-fore there exists a unique fixed-point 120593lowast isin X0 such that

119879120593lowast = 120593lowast which implies that system (2) has an almostautomorphic solution inX0 The proof is complete

4 Global Exponential Stability

In this section we investigate the global exponential stabilityof almost automorphic solutions by reduction to absurdity

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 6: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

6 Complexity

and

10038171003817100381710038171003817(Ψ120593 minus 1205930)1015840100381710038171003817100381710038170 le max1le119901le119899

sup119905isinR

10038171003817100381710038171003817100381710038171003817100381710038171003817119886119901 (119905) int

119905

119905minus120578119901(119905120593119901(119905))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119905) 119891119902 (120593119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (120593119902 (119905 minus 120590119901119902119897 (119905 120593119902 (119905)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119905 120593119897 (119905))))10038171003817100381710038171003817100381710038171003817100381710038171003817A + 10038171003817100381710038171003817100381710038171003817100381710038171003817int

119905

minusinfin119886119901 (119905)

sdot 119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904120593119901(119904))119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (120593119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904) 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904 120593119902 (119904)))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904 120593119897 (119904))))] 11988911990410038171003817100381710038171003817100381710038171003817100381710038171003817A

le max1le119901le119899

(119886+119901120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X + 119886+119901119886minus119901 (119886+119901120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897119866119897119871119892119902)10038171003817100381710038171205931003817100381710038171003817X le max

1le119901le119899(1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X

(25)

Thus we obtain

1003817100381710038171003817(119879120593 minus 1205930)1003817100381710038171003817119883= max

1le119901le119899sup119905isinR

1003817100381710038171003817(119879120593 minus 1205930)10038171003817100381710038170 sup119905isinR

10038171003817100381710038171003817(119879120593 minus 1205930)1015840100381710038171003817100381710038170le max

1le119901le119899 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )11987211990110038171003817100381710038171205931003817100381710038171003817X = 119903 10038171003817100381710038171205931003817100381710038171003817X

le 1199031198771 minus 119903 (26)

Fourthly we will prove that 119879 is a contracting mapping Infact for any 120593 120595 isin X0 we have that

1003817100381710038171003817119879120593 minus 11987912059510038171003817100381710038170 le max1le119901le119899

int119905

minusinfin119890minusint119905119904 119886119901(119906)119889119906 [100381710038171003817100381710038171003817100381710038171003817119886119901 (119904) int119904

119904minus120578119901(119904)[119901 (119906) minus 119901 (119906)] 119889119906100381710038171003817100381710038171003817100381710038171003817A + 119899sum

119902=1

10038171003817100381710038171003817119887119901119902 (119904) [119891119902 (120593119902 (119904)) minus 119891119902 (120595119902 (119904))]10038171003817100381710038171003817A+ 119899sum119902=1

119899sum119897=1

10038171003817100381710038171003817119888119901119902119897 (119904) [119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904))) minus 119892119902 (120593119902 (119904 minus 120590119901119902119897 (119904))) 119892119897 (120593119897 (119904 minus ]119901119902119897 (119904)))]10038171003817100381710038171003817A]119889119904le max

1le119901le119899int119905

minusinfin119890minus119886minus119901 (119905minus119904)119889119904 [119886+119901120578+119901 10038171003817100381710038171003817119902 minus 119902100381710038171003817100381710038170 + 119899sum

119902=1119887+119901119902119871119891119902 10038171003817100381710038171003817120593119902 minus 120595119902100381710038171003817100381710038170 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 [119866119902119871119892119897 1003817100381710038171003817120593 minus 12059510038171003817100381710038170 + 119866119897119871119892119902 1003817100381710038171003817120593 minus 12059510038171003817100381710038170]]

le max1le119901le119899

1119886minus1199011198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X

(27)

and 10038171003817100381710038171003817(119879120593 minus 119879120595)1015840100381710038171003817100381710038170 le max1le119901le119899

(1 + 119886+119901119886minus119901 )1198721199011003817100381710038171003817120593 minus 1205951003817100381710038171003817X

le 119903 1003817100381710038171003817120593 minus 1205951003817100381710038171003817X (28)

Hence by (1198673) 119879 is a contracting mapping principle There-fore there exists a unique fixed-point 120593lowast isin X0 such that

119879120593lowast = 120593lowast which implies that system (2) has an almostautomorphic solution inX0 The proof is complete

4 Global Exponential Stability

In this section we investigate the global exponential stabilityof almost automorphic solutions by reduction to absurdity

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 7: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Complexity 7

Definition 7 Let 119909 = (1199091 1199092 119909119899)119879 be an almostautomorphic solution of system (2) with the initial value120593 = (1205931 1205932 120593119899)119879 isin 119862([minus120585 0]A119899) and let 119910 =(1199101 1199102 119910119899)119879 be an arbitrary solution of system (2) withthe initial value 120595 = (1205951 1205952 120595119899)119879 isin 119862([minus120585 0]A119899)respectively If there exist positive constants 120582 and 119872 suchthat 1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 le 1198721003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 119890minus120582119905 forall119905 gt 0 (29)

where1003817100381710038171003817119909 (119905) minus 119910 (119905)10038171003817100381710038171 = max 1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817A119899 10038171003817100381710038171003817(119909 (119905) minus 119910 (119905))101584010038171003817100381710038171003817A119899 1003817100381710038171003817120593 minus 1205951003817100381710038171003817120585 = max sup119905isin[minus1205850]

max1le119901le119899

10038171003817100381710038171003817120593119901 (119905) minus 120595119901 (119905)10038171003817100381710038171003817A sup

119905isin[minus1205850]max1le119901le119899

100381710038171003817100381710038171003817(120593119901 (119905) minus 120595119901 (119905))1015840100381710038171003817100381710038171003817A (30)

Then the almost automorphic solution of system (2) is said tobe globally exponentially stable

Theorem 8 Assume that (1198671)-(1198673) holden system (2) hasa unique pseudo almost automorphic solution that is globallyexponentially stable

Proof ByTheorem6 system (2) has a pseudo almost periodicsolution Let 119909(119905) be an almost automorphic solution of (2)with the initial value 120593(119905) and let 119910(119905) be an arbitrary solutionwith the initial value 120595(119905) Set 119911119901(119905) = 119909119901(119905) minus 119910119901(119905) 120601119901(119905) =120593119901(119905) minus 120595119901(119905) and we have

119901 (119905) = minus119886119901 (119905) 119911119901 (119905 minus 120578119901 (119905)) + 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905))

+ 119899sum119902=1

119899sum119897=1119888119901119902119897 (119905) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905)))

sdot 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) 119901 = 1 2 119899(31)

where119891119902 (119911119902 (119905)) = 119891119902 (119909119902 (119905)) minus 119891119902 (119910119902 (119905)) 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905)))= 119892119902 (119909119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119909119902 (119905 minus ]119901119902119897 (119905)))minus 119892119902 (119910119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119910119902 (119905 minus ]119901119902119897 (119905))) (32)

Let Θ119901 and Δ119901 be defined by

Θ119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901120578+119901119890120596120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120596

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (33)

and

Δ 119901 (120596) = 119886minus119901 minus 120596 minus (119886+119901 + 119886minus119901 minus 120596)(119886+119901120578+119901119890120596120578+119901+ 119899sum119902=1119887+119901119902119871119891119902119890120596 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119871119892119902119890120596120590+119901119902119897 + 119871119892119897 119890120596]+119901119902119897)) (34)

where 119901 = 1 2 119899When 120596 = 0 we getΘ119901 (0) gt 0and Δ119901 (0) gt 0119901 = 1 2 119899 (35)

Since Θ119901(120596) Δ119901(120596) are continuous on [0 +infin) andΘ119901(120596) Δ119901(120596) 997888rarr minusinfin as 120596 997888rarr +infin there exist 120576119901 120576lowast119901 gt 0such that Θ119901(120576119901) = Δ(120576lowast119901) = 0 and Θ119901(120576) gt 0 for 120576 isin (0 120576119901)and Δ(120576lowast119901) gt 0 for 120576 isin (0 120576lowast119901) 119901 = 1 2 119899 Let120572 = min1le119901le119899120596119901 120576lowast119901 then we haveΘ119901 (120572) ge 0Δ 119901 (120572) ge 0119901 = 1 2 119899 (36)

So we can take a positive constant 120582 isin(0min120572 119886minus1 119886minus2 119886minus119899 ) Obviously we haveΘ119901 (120582) gt 0Θ119901 (120582) gt 0119901 = 1 2 119899 (37)

which implies that1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (38)

(1 + 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) lt 1 (39)

where 119901 = 1 2 119899 Let 119872 = max1le119901le119899119886minus119901119872119901 From(1198673) we have119872 gt 1 Hence for119901 = 1 2 119899 we can obtain1119872 le 1119886minus119901 minus 120582 times (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902119890120582

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120596120590+119901119902119897 + 119866119902119871119892119897 119890120596]+119901119902119897)) (40)

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 8: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

8 Complexity

By (31) we have

119901 (119905) + 119886119901 (119905) 119911119901 (119905) = 119886119901 (119905) int119905

119905minus120578119901(119905)119901 (119904) 119889119904

+ 119899sum119902=1119887119901119902 (119905) 119891119902 (119911119902 (119905)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119905)

sdot 119892119902 (119911119902 (119905 minus 120590119901119902119897 (119905))) 119892119902 (119911119902 (119905 minus ]119901119902119897 (119905))) (41)

Multiplying (41) by 119890int1199040 119886119901(119906)119889119906 and integrating on [0 119905] wehave

119911119901 (119905) = 120601119901 (0) 119890minusint1199050 119886119901(119906)119889119906+ int119905

0119890minusint119905119904 119886119901(119906)119889119906 [119886119901 (119904) int119904

119904minus120578119901(119904)119901 (119906) 119889119906 + 119899sum

119902=1119887119901119902 (119904) 119891119902 (119911119902 (119904)) + 119899sum

119902=1

119899sum119897=1119888119901119902119897 (119904)

sdot 119892119902 (119911119902 (119904 minus 120590119901119902119897 (119904))) 119892119902 (119911119902 (119904 minus ]119901119902119897 (s)))] 119889119904119901 = 1 2 119899

(42)

It is easy to see that

119911 (119905)1 = 1003817100381710038171003817120601 (119905)10038171003817100381710038171 le 10038171003817100381710038171206011003817100381710038171003817120585 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin (minus120585 0] (43)

We claim that

119911 (119905)1 le 11987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [0 +infin) (44)

To prove (44) we show that for any 120573 gt 1 the followinginequality holds

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 gt 0 (45)

If (45) is false then there must be some 1199051 gt 0 such that

1003817100381710038171003817119911 (1199051)10038171003817100381710038171 = max1le119901le119899

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A 10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A= 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (46)

and

119911 (119905)1 le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582119905 119905 isin [minus120585 1199051] (47)

By (39) (42) (46) and (47) we have

10038171003817100381710038171003817119911119901 (1199051)10038171003817100381710038171003817A le 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1199051119886minus119901 + 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus120582119904 int1199051

0119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)] 119889119904

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051120573119872 + 1119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901+ 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))(1

minus 119890(120582minus119886minus119901 )1199051)] le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [119890(120582minus119886minus119901 )1199051( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + 1119886minus119901 minus 120582

sdot 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

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Page 9: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Complexity 9

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 [ 1119886minus119901 minus 120582 119899sum119902=1

(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902

+ 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897))]

lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(48)

Finding the derivative of (42) then by (39) (46) and (47) wehave

10038171003817100381710038171003817119901 (1199051)10038171003817100381710038171003817A le 119886+119901 10038171003817100381710038171206011003817100381710038171003817120585 119890minus1198861199011199051 + 119886+119901120578+119901119890minus120582(1199051minus120578+119901 )12057311987210038171003817100381710038171206011003817100381710038171003817120585+ 119890minus120582119905112057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum

119902=1119887+119901119902119871119891119902

+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119899sum119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 119890minus120582(1199051minus]+119901119902119897))+ 12057311987210038171003817100381710038171206011003817100381710038171003817120585 int1199051

0119886+119901119890minus(1199051minus119904)119886minus119901 [119886+119901120578+119901119890minus120582(119904minus120578+119901 ) + 119890minus120582119904 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890minus120582(1199051minus120590+119901119902119897)

+ 119866119902119871119892119897 eminus120582(1199051minus]+119901119902119897))] 119889119904le 12057311987210038171003817100381710038171206011003817100381710038171003817120585sdot 119890minus1205821199051 [[

119886+119901119890(120582minus119886minus119901 )1199051120573119872 + 119886+119901119886minus119901 minus 120582 (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))(1 minus 119890(120582minus119886minus119901 )1199051)]]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus120582t1 [119886+119901119890(120582minus119886minus119901 )1199051 ( 1119872minus 1119886minus119901 minus 120582 119899sum

119902=1(119886+119901120578+119901119890120582120578+119901 + 119899sum

119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897 + 119866119902119871119892119897 119890120582]+119901119902119897)) + (1

+ 119886+119901119886minus119901 minus 120582)(119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897))]le 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 (1 + 119886+119901119886minus119901 minus 120582)

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 10: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

10 Complexity

sdot (119886+119901120578+119901119890120582120578+119901 + 119899sum119902=1119887+119901119902119871119891119902 + 119899sum

119902=1

119899sum119897=1119888+119901119902119897 (119866119897119871119892119902119890120582120590+119901119902119897

+ 119866119902119871119892119897 119890120582]+119901119902119897)) lt 12057311987210038171003817100381710038171206011003817100381710038171003817120585 119890minus1205821199051 119901 = 1 2 119899(49)

From (48) and (49) we have 119911(1199051)1 lt 120573119872120601120585119890minus1205821199051 whichcontradicts equality (46) Hence (45) holds Letting 120573 997888rarr 1then (44) holds Therefore the almost automorphic solutionof (2) is globally exponentially stable The proof is complete

5 Example

In this section we give an example to show the feasibility ofour results obtained in this paper

Example 1 In system (2) let119898 = 119899 = 2 and take

119891119902 (119909119902) = 1221198900 sin1199090119902 + 1201198901 sin1199091119902 + 1211198902 sin1199092119902 + 12711989012 sin 11990912119902 119902 = 1 2119892119902 (119909119902) = 1161198900 sin1199090119902 + 1201198901 sin1199091119902 + 1151198902 sin1199092119902 + 11811989012 sin 11990912119902 119902 = 1 2(1198861 (119905)1198862 (119905)) = ( 1 + 01 sinradic211990512 + 02 cosradic3119905) (1205781 (119905)1205782 (119905)) = ( 015 + 002 sinradic2119905016 + 0012 cosradic3119905)

(11988711 (119905) 11988712 (119905)11988721 (119905) 11988722 (119905)) = ( 011198900 sinradic6119905 + 021198901 sinradic6119905 0131198900 + 0111989012 sinradic71199050151198900 + 011198901 cosradic5119905 + 0211989012 cosradic2119905 0111198900 + 021198902 sinradic3119905) (119888111 (119905) 119888112 (119905)119888121 (119905) 119888122 (119905)) = ( 0211198900 sinradic6119905 + 0121198901 sinradic3119905 0111198900 + 01211989012 sinradic21199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic5119905 0121198900 + 0121198902 sinradic3119905) (119888211 (119905) 119888212 (119905)119888221 (119905) 119888222 (119905)) = ( 0111198900 sinradic6119905 + 0121198901 sinradic5119905 0111198900 + 01211989012 sinradic71199050141198900 + 0111198901 cosradic5119905 + 01211989012 cos 2radic3119905 0121198900 + 0121198902 sinradic3119905) (120590111 (119905) 120590112 (119905)120590121 (119905) 120590122 (119905)) = (0002 sinradic6119905 + 001 2 minus sin 1199051 minus 01 cosradic7119905 0001 sinradic2119905 + 001) (120590211 (119905) 120590212 (119905)120590221 (119905) 120590222 (119905)) = (0012 sinradic3119905 + 002 01 minus 003 cos 1199052 minus sin 1199052 001 sinradic6119905 + 0011) (]111 (119905) ]112 (119905)]121 (119905) ]122 (119905)) = (002 sinradic5119905 + 003 1 minus 07 sin 1199053 minus 2 cosradic2119905 0001 sinradic7119905 + 001)

(]211 (119905) ]212 (119905)]221 (119905) ]222 (119905)) = (0002 sinradic3119905 + 002 5 minus 3 cos 11990501 minus 007 sin 119905 001 sinradic8119905 + 0015)

(1198761 (119905)1198762 (119905)) = ( 1151198900 sin 2radic5119905 + 1201198901 sinradic3119905 + 1121198902 cosradic3119905 + 11511989012 sinradic71199051151198900 sin 2radic3119905 + 1121198901 sinradic6119905 + 1101198902 cosradic7119905 + 12011989012sin2radic3119905)

(50)

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 11: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Complexity 11

0 10 20 30 40 50 60 70 80 90 100t

0 10 20 30 40 50 60 70 80 90 100t

0

005

01

0

005

01

Ｒ01(t)

Ｒ02(t)

Ｒ11(t)

Ｒ12(t)

Ｒ0 Ｊ

(t) p

=12

Ｒ1 Ｊ

(t) p

=12

minus005

minus01

minus005

minus01

Figure 1 Curves of 1199090119901(119905) and 1199091119901(119905) 119901 = 1 2

0

005

01

0

005

01

0 10 20 30 40 50 60 70 80 90 100t

Ｒ21(t)

Ｒ22(t)

0 10 20 30 40 50 60 70 80 90 100t

Ｒ121 (t)

Ｒ122 (t)

Ｒ2 Ｊ

(t) p

=12

Ｒ12

Ｊ(t)

p=1

2

minus005

minus005

minus01

Figure 2 Curves of 1199092119901(119905) and 11990912119901 (119905) 119901 = 1 2

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 12: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

12 Complexity

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

0

0

005

01

0 0101

minus005 minus005

minus005minus005

minus01 minus01

minus01

minus01 minus01minus01 minus01

minus01minus01minus01minus01

minus01

Ｒ2(t)

Ｒ2(t)Ｒ

2(t)

Ｒ1(t) Ｒ

1(t)

Ｒ1(t)

Ｒ0(t) Ｒ

0(t)

Ｒ0(t)

Ｒ1(t)Ｒ2(t)

Ｒ12(t)

Ｒ12(t)

Ｒ12(t)

Figure 3 Curves of 1199090(119905) 1199091(119905) 1199092(119905) and 11990912(119905) in 3-dimensional space for stable case

By computing we get 119871119891119902 = 120 119871119892119902 = 115 119866119902 = 0067119902 = 1 2 119886minus1 = 09 119886minus2 = 1 119886+1 = 11 119886+2 = 14 120578+1 = 017 120578+2 =0172 119887+11 = 02 119887+12 = 013 119887+21 = 02 119887+22 = 02 119888+111 = 021119888+112 = 012 119888+121 = 014 119888+122 = 012 119888+211 = 012 119888+212 = 012119888+221 = 014 119888+222 = 0121198721 = 020881198722 = 02653 and119903 = max

1le119901le2 1119886minus119901119872119901 (1 + 119886+119901119886minus119901 )119872119901 asymp 06366 lt 1 (51)

Therefore all of the conditions of Theorem 8 are satisfiedHence system (2) has one almost automorphic solution thatis globally exponentially stable (see Figures 1ndash3)

6 Conclusion

In this paper we obtained the existence and global exponen-tial stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by directmethod Our methods and results are new The methodsproposed in this paper can be used to study the problemof almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays suchas Clifford-valued BAM networks Clifford-valued cellularneural networks andClifford-valued shunting inhibitory cel-lular neural networks Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our futurework

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant No 11861072

References

[1] B Xu X Liu and X Liao ldquoGlobal asymptotic stability ofhigh-order Hopfield type neural networks with time delaysrdquoComputers amp Mathematics with Applications vol 45 no 10-11pp 1729ndash1737 2003

[2] X Liu K Teo and B Xu ldquoExponential stability of impulsivehigh-order hopfield-type neural networks with time-varyingdelaysrdquo IEEE Transactions on Neural Networks and LearningSystems vol 16 no 6 pp 1329ndash1339 2005

[3] X-Y Lou and B-T Cui ldquoNovel global stability criteria for high-order Hopfield-type neural networks with time-varying delaysrdquoJournal of Mathematical Analysis and Applications vol 330 no1 pp 144ndash158 2007

[4] H Xiang K M Yan and B Y Wang ldquoExistence and globalexponential stability of periodic solution for delayed high-orderHopfield-type neural networksrdquo Physics Letters vol 352 no 4-5 pp 341ndash349 2006

[5] L Duan L Huang and Z Guo ldquoStability and almost peri-odicity for delayed high-order Hopfield neural networks withdiscontinuous activationsrdquo Nonlinear Dynamics vol 77 no 4pp 1469ndash1484 2014

[6] C Aouiti M S Mrsquohamdi J Cao and A Alsaedi ldquoPiecewisepseudo almost periodic solution for impulsive generalised high-order hopfield neural networks with leakage delaysrdquo NeuralProcessing Letters vol 45 no 2 pp 615ndash648 2017

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 13: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Complexity 13

[7] C Aouiti ldquoOscillation of impulsive neutral delay generalizedhigh-order Hopfield neural networksrdquo Neural Computing andApplications vol 29 no 9 pp 477ndash495 2018

[8] L Duan L Huang Z Guo and X Fang ldquoPeriodic attractorfor reactionndashdiffusion high-order Hopfield neural networkswith time-varying delaysrdquo Computers amp Mathematics withApplications vol 73 no 2 pp 233ndash245 2017

[9] A M Alimi C Aouiti F Cherif F Dridi and M S MrsquohamdildquoDynamics and oscillations of generalized high-order Hopfieldneural networks with mixed delaysrdquo Neurocomputing vol 321pp 274ndash295 2018

[10] Y Li J Qin and B Li ldquoAnti-periodic solutions for quaternion-valued high-order hopfield neural networks with time-varyingdelaysrdquo Neural Processing Letters vol 49 no 3 pp 1217ndash12372019

[11] Y K Li X F Meng and L L Xiong ldquoPseudo almost periodicsolutions for neutral type high-order Hopfield neural networkswith mixed time-varying delays and leakage delays on timescalesrdquo International Journal ofMachine Learning and Cybernet-ics vol 8 no 6 pp 1915ndash1927 2017

[12] C Aouiti and E A Assali ldquoStability analysis for a class ofimpulsive high-order Hopfield neural networks with leakagetime-varying delaysrdquo Neural Computing and Applications pp1ndash23 2018

[13] L Zhao Y Li and B Li ldquoWeighted pseudo-almost automorphicsolutions of high-order Hopfield neural networks with neutraldistributed delaysrdquo Neural Computing and Applications vol 29no 7 pp 513ndash527 2018

[14] N Huo B Li and Y Li ldquoExistence and exponential stabilityof anti-periodic solutions for inertial quaternion-valued high-order hopfield neural networks with state-dependent delaysrdquoIEEE Access vol 7 pp 60010ndash60019 2019

[15] W K Clifford ldquoApplications of grassmannrsquos extensive algebrardquoAmerican Journal ofMathematics vol 1 no 4 pp 350ndash358 1878

[16] J Pearson and D Bisset ldquoNeural networks in the Clifforddomainrdquo in Proceedings of IEEE World Congress on Computa-tional Intelligence (WCCI) International Conference on NeuralNetworks (ICNN) vol 3 pp 1465ndash1469Orlando FLUSA 1994

[17] S Buchholz and G Sommer ldquoOn Clifford neurons and Cliffordmulti-layer perceptronsrdquo Neural Networks vol 21 no 7 pp925ndash935 2008

[18] Y Liu P Xu J Lu and J Liang ldquoGlobal stability of Clifford-valued recurrent neural networks with time delaysrdquo NonlinearDynamics vol 84 no 2 pp 767ndash777 2016

[19] J Zhu and J Sun ldquoGlobal exponential stability of Clifford-valued recurrent neural networksrdquo Neurocomputing vol 173pp 685ndash689 2016

[20] Y Li and J Xiang ldquoExistence and global exponential sta-bility of anti-periodic solution for Clifford-valued inertialCohenndashGrossberg neural networks with delaysrdquo Neurocomput-ing vol 332 pp 259ndash269 2019

[21] Y Li and J Xiang ldquoGlobal asymptotic almost periodic synchro-nization of clifford-valued cnns with discrete delaysrdquo Complex-ity vol 2019 Article ID 6982109 13 pages 2019

[22] Y Li J Xiang and B Li ldquoGlobally asymptotic almost auto-morphic synchronization of clifford-valued RNNs with delaysrdquoIEEE Access vol 7 pp 54946ndash54957 2019

[23] C Huang Z Yang T Yi and X Zou ldquoOn the basins ofattraction for a class of delay differential equations with non-monotone bistable nonlinearitiesrdquo Journal of Differential Equa-tions vol 256 no 7 pp 2101ndash2114 2014

[24] L Duan H Wei and L Huang ldquoFinite-time synchronizationof delayed fuzzy cellular neural networks with discontinuousactivationsrdquo Fuzzy Sets and Systems vol 361 pp 56ndash70 2019

[25] K Gopalsamy ldquoLeakage delays in BAMrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 2 pp 1117ndash1132 2007

[26] Y Li and XMeng ldquoExistence and global exponential stability ofpseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on timescalesrdquo Complexity vol 2017 Article ID 9878369 15 pages 2017

[27] F Brackx R Delanghe and F SommenClifford Analysis vol 76ofResearchNotes inMathematics Pitman Advanced PublishingProgram Boston MA USA 1982

[28] T Diagana Almost Automorphic Type and Almost Periodic TypeFunctions in Abstract Spaces Springer New York NY USA2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 14: Existence and Global Exponential Stability of Almost ...downloads.hindawi.com/journals/complexity/2019/6751806.pdf · ResearchArticle Existence and Global Exponential Stability of

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


EXISTENCE, UNIQUENESS, AND STABILITY OF OSCILLATIONS IN ...€¦ · Abstract. We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a

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Research Article Exponential Stability of Neutral Stochastic …downloads.hindawi.com/journals/mpe/2014/907982.pdf · 2019. 7. 31. · 3. Exponential Stability of NSFDE with Two-Time-Scale

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ASYMPTOTIC STABILITY OF FRACTIONAL IMPULSIVE NEUTRAL ... · Using xed point approach, Luo [20] studied ... stability and exponential stability of second-order stochastic evolution

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Exponential stability of time-varying linear systems · Exponential stability of time-varying linear systems Adrian T. Hill∗ Achim Ilchmann† May 9, 2009 Abstract This paper considers

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