research article synchronization of general ... - hindawi

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 625372 14 pageshttpdxdoiorg1011552013625372

Research ArticleSynchronization of General Complex Networks withHybrid Couplings and Unknown Perturbations

Xinsong Yang1 Shuang Ai1 Tingting Su1 and Ancheng Chang2

1 Department of Mathematics Chongqing Normal University Chongqing 400047 China2Department of Mathematics Hunan Information Science Vocational College Changsha Hunan 410151 China

Correspondence should be addressed to Xinsong Yang xinsongyang163com

Received 4 January 2013 Accepted 2 February 2013

Academic Editor Chuangxia Huang

Copyright copy 2013 Xinsong Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The issue of synchronization for a class of hybrid coupled complex networks with mixed delays (discrete delays and distributeddelays) and unknown nonstochastic external perturbations is studied The perturbations do not disappear even after all thedynamical nodes have reached synchronization To overcome the bad effects of such perturbations a simple but all-powerful robustadaptive controller is designed to synchronize the complex networks evenwithout knowing a priori the functions and bounds of theperturbations Based on Lyapunov stability theory integral inequality Barbalat lemma and Schur Complement lemma rigorousproofs are given for synchronization of the complex networks Numerical simulations verify the effectiveness of the new robustadaptive controller

1 Introduction

Over the past decade complex networks have attractedmuchattention from authors of many disciplines since the pioneerworks of Watts and Strogatz [1 2] In fact many phenomenain nature and our daily life can be explained by using complexnetworks such as the Internet World Wide Web socialnetworks and neural networks A complex network can beconsidered as a graph which consists of a set of nodes andedges connecting these nodes [3]

In recent years chaos synchronization [3ndash7] has beenintensively studied due to its important applications in manydifferent areas such as secure communication biologicalsystems and information science [8ndash11] Particularly thesynchronization of all the dynamical nodes in complexnetworks has become a hot research topic [3] and severalresults have been appeared in the literature The authors of[12] studied the synchronization in complex networks withswitching topology In [13] Wu and Jiao investigated thesynchronization in complex dynamical networks with non-symmetric coupling They showed that the synchronizabilityof a dynamical network with nonsymmetric coupling is notalways characterized by its second-largest eigenvalue even

though all the eigenvalues of the nonsymmetric couplingmatrix are real Liu and Chen [14] gave some criteria forthe global synchronization of complex networks in virtualof the left eigenvector corresponding to the zero eigenvalueof the coupling matrix For a given network with identicalnode dynamics the authors of [15] showed that two keyfactors influencing the network synchronizability are thenetwork inner linking matrix and the eigenvalues of thenetwork topological matrix Some synchronization criteriawere given in [16ndash19] for coupled neural networks with orwithout delayed couplings In [20] the robust impulsivesynchronization of coupled delayed neural networks withuncertainties is considered several new criteria are obtainedto guarantee the robust synchronization via impulses

Complex networks have the properties of robustness andfragility A complex network can synchronize itself whenparameter mismatch is within some limit If parameter mis-match exceeds this limit networks cannot realize synchro-nization themselves Thus the controlled synchronization ofcoupled networks is believed to be a rather significant topicin both theoretical research and practical applications [21ndash29] Some effective control scheme has been proposed forinstance state feedback control with constant control gains

2 Abstract and Applied Analysis

impulsive control intermittent control and adaptive controlAdaptive control method receives particular attention ofresearchers in recently rears In [3] the authors studiedsynchronization in complex networks by using distributedadaptive control scheme By designing a simple adaptivecontroller authors of [23] investigated the locally and globallyadaptive synchronization of an uncertain complex dynamicalnetwork Authors in [24] investigated synchronization ofneural networks with time-varying delays and distributeddelays via adaptive control method By using the adaptivefeedback control scheme Chen and Zhou [25] studiedsynchronization of complex nondelayed networks and Caoet al [26] investigated the complete synchronization in anarray of linearly stochastically coupled identical networkswith delays By using adaptive pinning control method Zhouet al [27] studied local and global synchronization of complexnetworks without delays authors of [28 29] consideredthe global synchronization of the complex networks withnondelayed and delayed couplings and the authors of [30]investigated lag synchronization of complex networks viastate feedback pinning strategy Outer synchronization ofcomplex delayed networks with uncertain parameters wasconsidered by using adaptive coupling in [31] Howevermodels in the previous references are special that is eachof them does not consider general complex networks inwhich every dynamical node has mixed delays (discretedelay and distributed delay) and the complex networkshave nondelayed discrete-delayed and distributed-delayedcouplings

Complex networks are always affected by some unknownexternal perturbations due to environmental causes andhuman causes White noises brought by some random fluc-tuations in the course of transmission and other probabilitiescauses have received extensive attention in the literatures[21 24 32ndash35] However not all the external perturbationsare white noise and some of them may be nonlinear andnonstochastic perturbations When complex networks aredisturbed by nonlinear and nonstochastic perturbations thestates of the nodes will be changed dramatically whichwill affect the stability and synchronization of the complexnetworks Due to the fragility of complex networks if someimportant nodes are perturbed by such external perturba-tions whole states of the network will be affected or eventhe network cannot operate normally Hence how to realizesynchronization of all nodes for complex networks withuncertain nonlinear nonstochastic external perturbations isan urgent practical problem to be solved Obviously thecontrollers for stability and synchronization of stochasticperturbations are not applicable to the case of nonlinearnonstochastic perturbations especially when the functionsand bounds of the perturbations are unknown Thereforeto enhance antiperturbations capability and to realize syn-chronization of complex networks more effective controllershould be designed

Motivated by the previous analysis in this paper a classof more general complex networks is proposed The newmodel has nondelayed discrete-delayed and distributed-delayed couplings and every dynamical node has mixeddelays Unknown nonstochastic external perturbations to the

complex networks are also considered Then we study theglobal complete synchronization of the proposed model Anew simple but robust adaptive controller is designed toovercome the effects of such perturbations and synchronizethe complex networks even without knowing the exactfunctions and bounds of the perturbations Moreover theadaptive controller can also synchronize coupled systemswith stochastic perturbations since it includes existing adap-tive controller as special case Two cases are consideredall nodes or partial nodes are perturbed All nodes shouldbe controlled for the former case Pinning control schemecan also be used for the latter case Based on Lyapunovstability theory integral inequality Barbalat lemma andSchur Complement lemma rigorous proofs are given forsynchronization of the complex networks with unknownperturbations of the previous two cases It should be notedthat our new adaptive controllers can also prevent externalperturbations Therefore the new adaptive controllers arebetter than those in [23ndash29]Numerical simulations verify theeffectiveness of our theoretical results

Notations In the sequel if not explicitly stated matricesare assumed to have compatible dimensions 119868

119873denotes the

identity matrix of 119873 dimension The Euclidean norm in R119899

is denoted as sdot accordingly for vector 119909 isin R119899 119909 =

radic119909119879119909 where 119879 denotes transposition 119860 = (119886119894119895)119898times119898

denotes

a matrix of 119898 dimension 119860 = radic120582max(119860119879119860) and 119860

119904=

(12)(119860 + 119860119879) 119860 gt 0 or 119860 lt 0 denotes that the matrix 119860 is

symmetric and positive or negative definite matrix 120582min(119860119904)

is the minimum eigenvalues of the symmetric matrices 119860119904and 119860

119897denotes the matrix of the first 119897 row-column pairs of

119860 119860119888119897denotes the minor matrix of matrix 119860 by removing all

the first 119897 row-column elements of 119860The rest of this paper is organized as follows In Section 2

a class of general complex networks with mixed delaysand external perturbations is proposed Some necessaryassumptions and lemmas are also given in this section InSection 3 synchronization of the complex networks withall nodes perturbed is studied Synchronization with onlypartial nodes perturbed is considered in Section 4 Thenin Section 5 numerical simulations are given to show theeffectiveness of our results Finally in Section 6 conclusionsare given

2 Preliminaries

The general complex networks consisting of 119873 identicalnodeswith external perturbations andmixed-delay couplingsare described as

119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

Abstract and Applied Analysis 3

+ 120590119894(119905 119909119894 (119905) 119909119894 (119905 minus 120591 (119905)) int

119905

119905minus120579(119905)

119909119894 (119904) 119889119904)

+ 119877119894 119894 = 0 1 119873

(1)

where 119909119894(119905) = [119909

1198941(119905) 119909

119894119899(119905)]119879isin R119899 represents the state

vector of the 119894th node of the network at time 119905 and 119862119860 119861119863are matrices with proper dimension 119891(sdot) is a continuousvector function 119868(119905) is the external input vector 119877

119894isin R119899 is

the control input 120591(119905) gt 0 120579(119905) gt 0 are time-varying discretedelay and distributed delay respectively Constants 120572 gt 0120573 gt 0 120574 gt 0 are coupling strengths of the whole networkcorresponding to nondelay discrete delay and distributeddelay respectivelyΦΥ Λ isin R119899times119899 are inner couplingmatricesof the networks which describe the individual couplingbetween two subsystems Matrices 119880 = (119906

119894119895)119873times119873

119881 =

(V119894119895)119873times119873

119882 = (119908119894119895)119873times119873

are outer couplings of the wholenetworks satisfying the following diffusive conditions

119906119894119895ge 0 (119894 = 119895) 119906

119894119894= minus

119873

sum

119895=1119895 = 119894

119906119894119895

V119894119895ge 0 (119894 = 119895) V

119894119894= minus

119873

sum

119895=1119895 = 119894

119906119894119895

119908119894119895ge 0 (119894 = 119895) 119908

119894119894= minus

119873

sum

119895=1119895 = 119894

119908119894119895

(2)

where 119894 119895 = 1 2 119873 Vector 120590119894(119905 119909119894(119905) 119909119894(119905 minus 120591(119905))

int119905

119905minus120579(119905)119909119894(119904)119889119904) isin R119899 describes the unknown perturbation to

119894th node of the complex networks In this paper we alwaysassume that le ℎ

120591lt 1 and le ℎ

120579lt 1 120579(119905) is bounded

and we denote 120579min gt 0 the minimum of 120579(119905) and 120579max themaximum of 120579(119905)

We assume that (1) has a unique continuous solution forany initial condition in the following form

119909119894 (119904) = 120593119894 (119904) minus984858 le 119904 le 0 119894 = 0 1 2 119873 (3)

where 984858 = max 120591max 120590max and 120591max is the maximum of 120591(119905)For convenience of writing in the sequel we denote

120590119894(119905 119909119894(119905) 119909119894(119905 minus 120591(119905)) int

119905

119905minus120579(119905)119909119895(119904)119889119904) with 120590

119894(119905)

The system of an isolate node without external perturba-tion is described as

(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(4)

and 119911(119905) can be any desired state equilibrium point anontrivial periodic orbit or even a chaotic orbit

Remark 1 The nonstochastic perturbations 120590119894(119905 119909119894(119905) 119909119894(119905 minus

120591(119905)) int119905

119905minus120579(119905)119909119894(119904)119889119904) are different from stochastic ones in

the literature [21 24 32ndash35] The distinct feature of the

such stochastic perturbations is that the stochastic pertur-bations disappear when the synchronization goal is realizedHowever perturbations of this paper still exist even whencomplete synchronization has been achieved Therefore thecontrollers in most of existing papers including those in[21 24 32ndash35] are invalid for perturbations of this paper

When system (4) is perturbed then (4) turns to thefollowing system

(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

+ 120590119894(119905 119911 (119905) 119911 (119905 minus 120591 (119905)) int

119905

119905minus120579(119905)

119911 (119904) 119889119904)

(5)

Generally the state of a system will be changed when thesystem is perturbed We assume that the state of system (5)remains to be any one of the previous three states but notnecessarily the original one

The following assumptions are needed in this paper

(H1) 119891(0) equiv 0 and there exists positive constant ℎ suchthat

1003817100381710038171003817119891 (119906) minus 119891 (V)1003817100381710038171003817 le ℎ 119906 minus V for any 119906 V isin R

119899 (6)

(H2) 120590119894119896(119905 0 0 0) equiv 0 and there exist positive constants

119872119894119896such that |120590

119894119896(119905 119906 V 119908)| le 119872

119894119896for any bounded

119906 V 119908 isin R119899 119894 = 1 2 119873 119896 = 1 2 119899

Remark 2 Note that (4) unifies many well-known chaoticsystems with or without delays such as Chua system Lorenzsystem Rossler system Chen system and chaotic neuralnetworks with mixed delays [12ndash29] Hence results of thispaper are general

Remark 3 Condition (H2) is very mild We do not impose

the usual conditions such as Lipschitz condition differen-tiability on the external perturbation functions It can bediscontinuous or even impulsive functions If the state of(5) is a equilibrium point or a nontrivial periodic orbit thecondition (H

2) can be easily satisfied If the state of (5) is a

chaotic orbit the condition (H2) can also be satisfied Since

chaotic system has strange attractors there exists a boundedregion containing all attractors of it such that every orbit ofthe system never leaves them Anyway condition (H

2) can

be satisfied for equilibrium point a nontrivial periodic orbitand a chaotic orbitMoreover wewill subsequently prove thatthe complex networks (1) can be synchronized even withoutknowing the exact values of ℎ and 119872

119894119896 119894 = 1 2 119873 119896 =

1 2 119899

The aim of this paper is to synchronize all the states ofcomplex networks (1) to the following manifold

1199091 (119905) = 1199092 (119905) = sdot sdot sdot = 119909119873 (119905) = 119911 (119905) (7)

where 119911(119905) is immune to external perturbations


Lemma 4 ((Schur Complement) see [36]) The linear matrixinequality (LMI)

119878 = [

[

11987811

11987812

119878119879

1211987822

]

]

lt 0 (8)

is equivalent to any one of the following two conditions

(L1) 11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0

(L2) 11987822lt 0 119878

11minus 11987812119878minus1

22S11987912lt 0

where 11987811= 119878119879

11 11987822= 119878119879

22

Lemma 5 (see [37]) For any constant matrix119863 isin R119899times119899119863119879 =119863 gt 0 scalar 120590 gt 0 and vector function 120596 [0 120590] rarr R119899 onehas

120590int

120590

0

120596119879(119904)119863120596 (119904) d119904 ge (int

120590

0

120596 (119904) d119904)119879

119863int

120590

0

120596 (119904) d119904 (9)

provided that the integrals are all well defined

Lemma 6 ((Barbalat lemma) see [38]) If 119891(119905) R rarr R+ isa uniformly continuous function for 119905 ge 0 and if the limit of theintegral

lim119905rarrinfin

int

119905

0

119891 (119904) d119904 (10)

exists and is finite then lim119905rarrinfin

119891(119905) = 0

3 Synchronization with Allthe Nodes Perturbed

In this section we consider the case when all the nodes areperturbed To realize synchronization goal (7) we have tointroduce an isolate node (4)

Let 119890119894(119905) = 119909

119894(119905)minus119911(119905) Subtracting (4) from (1) we get the

following error dynamical system

119890119894 (119905) = 119862119890

119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120590119894 (119905) + 119877119894

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

(11)

where 119892(119890119894) = 119891(119909

119894(119905)) minus 119891(119911(119905)) 119894 = 1 2 119873

From (H1) and (H

2) we know that (11) admits a trivial

solution 119890119894(0) equiv 0 119894 = 1 2 119873 Obviously to reach

the goal (7) we have only to prove that system (11) isasymptotically stable at the origin

Theorem 7 Under the assumption conditions (H1) and (H

2)

the networks (1) are synchronized with the following adaptivecontrollers

119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(12)

where 120596 gt 1 119901119894gt 0 and 120585

119894gt 0 are arbitrary constants

respectively 119894 = 1 2 119873

Proof Define the Lyapunov function as

119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (13)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119873

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

119873

sum

119894=1

(119872119894minus 120573119894)2

120585119894

1198812 (119905) = int

119905

119905minus120591(119905)

120578119879(119904) 119876120578 (119904) 119889119904

1198813 (119905) = int

119905

119905minus120579(119905)

int

119905

120583

120578119879(119904) 119866120578 (119904) 119889119904 119889120583

(14)

120578(119905) = (1198901(119905) 119890

2(119905) 119890

119873(119905))119879 119872119894= max

1le119896le119899119872119894119896

119896119894 119894 = 1 2 119873 are constants 119876 and 119866 are symmetric

positive definite matrices and 119896119894 119876 and 119866 are to be

determinedDifferentiating 119881

1(119905) along the solution of (11) and from

(H1) and (H

2) we obtain

1 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119873

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

119873

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[

[

(119862 + 119860 ℎ minus 120572119896119894)1003817100381710038171003817119890119894(119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904

+ 120572

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120572120582min (Φ119904) 119906119894119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2


+ 120573

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817

+120574

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904]

]

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

le 120578119879(119905) ( (119862 + 119860 ℎ + 1) 119868119873

+120572 (Φ 119904

minus 119870)) 120578 (119905)

+ 120578119879(119905 minus 120591 (119905)) 119861

119879

119861120578 (119905 minus 120591 (119905))

+ (int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119863119879

119863int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(15)

where 119870 = diag (1198961 1198962 119896

119873) = (

119894119895)119873times119873

119894119895= 119906119894119895

119894 = 119895 119894119894= (120582min(Φ

119904)Φ)119906

119894119894 119861 = 119861ℎ119868

119873+ 120573Υ|119881| 119863 =

119863ℎ119868119873+ 120574Λ|119882| |119881| = (|V

119894119895|)119873times119873

|119882| = (|119908119894119895|)119873times119873

andwe have used the following deduction

119873

sum

119894=1

[119890119879

119894(119905) 120590119894 (119905) minus 120596

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 minus

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816]

le

119873

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

le minus

119873

sum

119894=1

119899

sum

119896=1

(120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 le 0

(16)

Differentiating 1198812(119905) we get

2 (119905) = 120578

119879(119905) 119876120578 (119905) minus (1 minus (119905)) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

le 120578119879(119905) 119876120578 (119905) minus (1 minus ℎ120591) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

(17)

Differentiating 1198813(119905) from Lemma 5 we have

3 (119905) = 120579 (119905) 120578

119879(119905) 119866120578 (119905) minus (1 minus (119905)) int

119905

119905minus120579(119905)

120578119879(119904) 119866120578 (119904) 119889119904

le 120579max120578119879(119905) 119866120578 (119905)

minus1 minus ℎ120579

120579min(int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119866int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(18)

Take119876 = (1(1minusℎ120591))119861119879

119861119866 = (120579min(1minusℎ120579))119863119879

119863 Fromthe definition of 119881(119905) we reach the following inequality

(119905) le 120572120578119879(119905) [

1

120572(119862 + 119860 ℎ + 1) 119868119873

+ Φ 119904

+1

120572 (1 minus ℎ120591)119861119879

119861

+120579min120579max120572 (1 minus ℎ

120579)119863119879

119863 minus 119870]120578 (119905)

(19)

Let 119896119894= 120582max((1120572)(119862 + 119860ℎ + 1)119868

119873+ Φ

119904

+

(1120572(1 minus ℎ120591))119861119879

119861 + (120579min120579max(120572(1 minus ℎ120579)))119863119879

119863) + 1 where120582max((1120572)(119862 + 119860ℎ+ 1)119868119873 + Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1 minus ℎ120579))119863119879

119863) denotes the maximum eigenvalueof (1120572)(119862 + 119860ℎ + 1)119868

119873+ Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1minusℎ120579))119863119879

119863Then from the previous inequalitywe get

(119905) le minus120572120578119879(119905) 120578 (119905) (20)

Integrating both sides of the previous equation from 0 to119905 yields

119881 (0) ge 119881 (119905) + 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 ge 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 (21)

Therefore

120572 lim119905rarrinfin

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (22)

In view of Lemma 6 and the previous inequality one caneasily get

lim119905rarrinfin

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (23)

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (24)

This completes the proof

4 Synchronization with PartialNodes Perturbed

Usually only partial nodes of complex networks are per-turbed If some important nodes are perturbed then theentire network will not work correctly Theoretically speak-ing nodes with larger degree (undirected networks) orlarger outdegree (directed networks) are more vulnerableto perturbation [39] since the states of these nodes havemore effect on networks than those with smaller degree(undirected networks) or outdegree (directed networks) Onthe other hand the real-world complex networks normally


have a large number of nodes it is usually impractical andimpossible to control a complex networks by adding thecontrollers to all nodes Therefore from both practical pointof view and the view of reducing control cost we can use thescheme of pinning control [27ndash29 40ndash42] to prevent externalperturbations and synchronize complex networks

In this section we assume that matrix 119880 is irreducible inthe sense that there is no isolate cluster in the network andthere are 119897

1nodes affected by external perturbations

Without loss of generality rearrange the order of thenodes in the network and take the first 119897 (119897 ge 119897

1) nodes to

be controlled Thus the pinning controlled network can bedescribed as

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905)

+ 119877119894 119894 = 1 2 119897

1

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 119877119894

119894 = 1198971+ 1 1198971+ 2 119897

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(25)

where 119877119894 119894 = 1 2 119897 are control inputs

minus20minus10

010

2030

minus20minus10

010

2030

minus30

60

50

40

30

20

10

0

1199111(119905)

1199112 (119905)

119911 3(119905)

Figure 1 Chaotic trajectory of (44)

Let 119890119894(119905) = 119909

119894(119905) minus 119911(119905) Subtracting (4) from (25) we

obtain the following error dynamical system

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 1 2 119897

1

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 119877119894 119894 = 119897

1+ 1 1198971+ 2 119897

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(26)

Similar to Theorem 7 to reach the goal (7) we have only toprove that system (26) is asymptotically stable at the origin


minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)

Figure 2 Chaotic trajectories of (46) with 1205901(119905) (a) 120590

2(119905) (b) 120590

3(119905) (c)

Theorem 8 Suppose that matrix 119880 is irreducible and theassumptions (H

1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)

then the complex networks (25) are synchronized with theadaptive pinning controllers

119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +

1 minus ℎ120579)(1 minus ℎ

120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868

119873+ 120573Υ|119881| and the other parameters are the

same as those of Theorem 7

Proof We define another Lyapunov function as

119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)

119896119894 119894 = 1 2 119897 are constants to be determined and 119881

2(119905)

and 1198813(119905) are defined as those in the proof of Theorem 7

In view of (H1) and (H

2) differentiating 119881

1(119905) along the

solution of (26) yields

1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)

Figure 3 Synchronization errors of (47) (a) 1198901198941(119905) (b) 119890

1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)

Figure 4 The adaptive control gains of (47) (a) the adaptive gains 120576119894 1 le 119894 le 3 (b) the adaptive gains 120573

119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897

) and the following deduc-

tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)

Combining (31) with (17) and (25) we have

119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)

where 120577(119905) = (120578119879(119905) 120578119879(119905 minus 120591(119905)) (int119905119905minus120579(119905)

120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879

12minus2 (1 minus ℎ

120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|

According to Lemma 4Π lt 0 is equivalent to

Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)

minus12 minus1 minus08 minus06 minus04 minus02 0 02 04 06 081199111(119905)

15

10

5

0

minus5

minus10

119911 2(119905)


minus12 minus1 minus08 minus06 minus04 minus02 0 02 04 061199111(119905)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix

with appropriate dimension


(a) (b)

Figure 7 WS Small-Worlds with 10 nodes In (a) each node connects 4 nodes and the rewire probability is 02 in (b) each node connects 2nodes and the rewire probability is 04

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)

Figure 8 Synchronization errors of (52) (a) 1198901198941 (b) 119890

1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897

lt 0 and there exist positiveconstants 119896

1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)

again from Lemma 4 we obtainΔ lt 0 HenceΠ lt 0 Denote120582min to be the minimum eigenvalue of minusΠ then

119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)

Integrating both sides of the previous equation from 0 to 119905yields

119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)

By Lemma 6 we obtain

lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)

Figure 9The adaptive pinning control gains of (52) (a) the adaptive pinning control gains 120576119894 119894 = 1 2 (b)The adaptive pinning control gains

120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)


When there is no external perturbation that is 120590119894(119905) =

0 119894 = 1 2 119873 one can easily get the following corollariesfrom Theorems 7 and 8 respectively We omit their proofshere

Corollary 9 Suppose that 120590119894(119905) = 0 119894 = 1 2 119873 and the

assumption condition (H1) holds Then complex networks (1)

are synchronized with the adaptive controllers (12) Moreoverthe scalar 120596 can be relaxed to any positive constant

Corollary 10 Suppose that matrix 119880 is irreducible andthe assumption (H

1) holds The complex networks (25) are

synchronized with the adaptive pinning controllers (28) if (27)holds Moreover the scalar 120596 can be relaxed to any positiveconstant

Remark 11 From the inequalities (16) and (32) one can seethat the designed adaptive controllers (12) and (28) are veryuseful They can overcome the bad effects of the uncertainnonlinear perturbations without knowing the exact functionsand bounds of the perturbations as long as the perturbedsystems are chaotic Especially when there are only partialnodes perturbed (the first 119897

1nodes in the system (25)) the

designed controllers still are effective to stabilize the errorsystem by adding them to nodes with and without suchperturbations (see the inequality (32)) Obviously in the caseof no perturbation the parameter 120596 can also be taken as 0When 120596 = 0 the controllers (12) and (28) turn out to bethe usual adaptive controller which is extensively utilizedto synchronize coupled systems with or without stochasticperturbations [8 23ndash34 40ndash42] However the controllersin [8 23ndash34 40ndash42] cannot synchronize coupled systems

with nonstochastic perturbations Therefore the designedcontrollers can deal with both stochastic and nonstochasticperturbations to the systems and hence they have betterrobustness than usual adaptive controllers

Remark 12 Model (1) can be extended to the following moregeneral complex networks

119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)

Moreover we can also consider stochastic perturbations [21]and Markovian jump [43 44] in (43) to get more generalresults For simplicity we omit the corresponding results andonly consider model (1)

5 Numerical Examples

In this section we provide two examples to illustrate the gen-eral model and the advantage of the new adaptive controller

Example 13 The Lorenz system is described as

(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)

119891(119911(119905)) = (0 minus1199111(119905)1199113(119905) 1199111(119905)1199112(119905))119879 When initial values are

taken as 1199111(0) = 08 119911

2(0) = 2 119911

3(0) = 25 chaotic trajectory

of (44) can be seen in Figure 1The following three perturbedLorenz systems are chaotic

(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)

sin 1199113(119905))119879 Chaotic trajectories of the three perturbed Lorenz

systems are showed in Figure 2 with the same initial values1199111(0) = 08 119911

2(0) = 2 119911

3(0) = 25

Now consider the following complex networks with eachnode as the previous perturbed Lorenz system

119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)

Obviously conditions (H1) and (H

2) are satisfied According

to Theorem 7 the complex networks (47) can be synchro-nized with adaptive controllers (12)

The initial conditions of the numerical simulations areas follows 120596 = 4 step = 00005 119909

1(0) = (minus2 minus1 0)

1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=

120585119894= 05 119894 = 1 2 3 Figure 3 describes the synchronization

errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows

the adaptive feedback gains Numerical simulations verify theeffectiveness of Theorem 7

Example 14 Consider the following chaotic neural networkswith mixed delays

(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)

In the case that the initial condition is chosen as 1199111(119905) = 04

1199112(119905) = 06 for all 119905 isin [minus1 0] the chaotic attractor can be

seen in Figure 5The perturbed system of (49) is

(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)

119879 The chaoticattractor of (51) can be seen in Figure 6 with 119911

1(119905) = 04

1199112(119905) = 06 for all 119905 isin [minus1 0]Now consider the following complex networks with each

node as the previous neural networks withmixed delays (49)while the second node is disturbed with the previous 120590(119905)

119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0

Figure 7 depicts the WS Small-World networks [2] cor-responding to nondelay (a) discrete delay and distributeddelay (b) The corresponding Laplacian matrices are shownas following

119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)

Take the first two nodes (corresponding to matrix 119880) tobe controlled According Theorem 8 the complex networks(52) can be synchronized with adaptive controllers (27)


The initial conditions of the numerical simulations are asfollows 120596 = 2 step = 00005 119909

119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =

1 2 Figure 8 describes the synchronization errors 119890119894119895(119905) =

119909119894119895(119905) minus 119911

119895(119905) 119894 = 1 2 10 119895 = 1 2 Figure 9 depicts the

adaptive feedback gains Numerical simulations verify theeffectiveness of Theorem 8

6 Conclusions

External perturbations to networks are unavoidable in prac-tice On the other hand many chaotic models have discretedelay and distributed delayTherefore in this paper we intro-duced a class of hybrid coupled complex networkswithmixeddelays and unknown nonstochastic external perturbations Asimple robust adaptive controller is designed to synchronizethe complex networks even without knowing a priori thebounds and the exact functions of the perturbations It shouldbe emphasized that we do not assume that the couplingmatrix is symmetric or diagonal The controller can enhancerobustness and reduce fragility of complex networks henceit has great practical significance Moreover we also verifythe effectiveness of the theoretical results by numericalsimulations

Acknowledgments

This work was jointly supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61263020 and 11101053 the Scientific Research Fund of Yun-nan Province under Grant 2010ZC150 the Scientific ResearchFund of Chongqing Normal University under Grants 940115and 12XLB031 the Key Project of ChineseMinistry of Educa-tion under Grant 211118 and the Excellent Youth Foundationof Educational Committee of Hunan Provincial under Grant10B002

References

[1] S H Strogatz ldquoExploring complex networksrdquo Nature vol 410no 6825 pp 268ndash276 2001

[2] D J Watts and S H Strogatz ldquoCollective dynamics of rdquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[3] W Yu P DeLellis G Chen M di Bernardo and J KurthsldquoDistributed adaptive control of synchronization in complexnetworksrdquo Institute of Electrical and Electronics Engineers vol57 no 8 pp 2153ndash2158 2012

[4] X Yang J Cao and J Lu ldquoSynchronization of coupled neuralnetworks with random coupling strengths and mixed proba-bilistic time-varying delaysrdquo International Journal of Robust andNonlinear Control 2012

[5] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[6] P Zhou and R Ding ldquoModified function projective synchro-nization between different dimension fractional-order chaoticsystemsrdquo Abstract and Applied Analysis vol 2012 Article ID862989 12 pages 2012

[7] Q Zhu and J Cao ldquoAdaptive synchronization of chaotic Cohen-Crossberg neural networks with mixed time delaysrdquo NonlinearDynamics vol 61 no 3 pp 517ndash534 2010

[8] M Ayati and H Khaloozadeh ldquoA stable adaptive synchroniza-tion scheme for uncertain chaotic systems via observerrdquo ChaosSolitons and Fractals vol 42 no 4 pp 2473ndash2483 2009

[9] C Huang K Cheng and J Yan ldquoRobust chaos synchro-nization of four-dimensional energy resource systems subjectto unmatched uncertaintiesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 pp 2784ndash2792 2009

[10] N N Verichev S N Verichev and M Wiercigroch ldquoAsymp-totic theory of chaotic synchronization for dissipative-coupleddynamical systemsrdquo Chaos Solitons and Fractals vol 41 no 2pp 752ndash763 2009

[11] T M Hoang and M Nakagawa ldquoA secure communicationsystem using projective-lag andor projective-anticipating syn-chronizations of coupled multidelay feedback systemsrdquo ChaosSolitons and Fractals vol 38 no 5 pp 1423ndash1438 2008

[12] L Wang and Q Wang ldquoSynchronization in complex networkswith switching topologyrdquo Physics Letters A vol 375 no 34 pp3070ndash3074

[13] J Wu and L Jiao ldquoSynchronization in complex dynamicalnetworks with nonsymmetric couplingrdquo Physica D vol 237 no19 pp 2487ndash2498 2008

[14] X Liu and T Chen ldquoSynchronization analysis for nonlinearly-coupled complex networks with an asymmetrical couplingmatrixrdquo Physica A vol 387 pp 4429ndash4439 2008

[15] Z Duan G Chen and L Huang ldquoComplex network synchro-nizability analysis and controlrdquo Physical Review E vol 76 no5 part 2 Article ID 056103

[16] W Lu and T Chen ldquoSynchronization of coupled connectedneural networks with delaysrdquo IEEE Transactions on Circuits andSystems vol 51 no 12 pp 2491ndash2503 2004

[17] G Chen J Zhou and Z Liu ldquoGlobal synchronization ofcoupled delayed neural networks and applications to chaoticCNNmodelsrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 14 no 7 pp 2229ndash22402004

[18] W Yu J Cao and G Chen ldquoLocal synchronization of acomplex network modelrdquo IEEE Transactions on Systems Manand Cybernetics B vol 39 no 1 pp 230ndash241 2009

[19] Q Song ldquoSynchronization analysis of coupled connected neuralnetworks with mixed time delaysrdquoNeurocomputing vol 72 no16ndash18 pp 3907ndash3914 2009

[20] P Li J Cao and Z Wang ldquoRobust impulsive synchronizationof coupled delayed neural networks with uncertaintiesrdquo PhysicaA vol 373 pp 261ndash272 2007

[21] X Yang and J Cao ldquoStochastic synchronization of coupledneural networks with intermittent controlrdquo Physics Letters Avol 373 no 36 pp 3259ndash3272 2009

[22] X Liu and T Chen ldquoBoundedness and synchronization of y-coupled Lorenz systems with or without controllersrdquo Physica Dvol 237 no 5 pp 630ndash639 2008

[23] J Zhou J Lu and J Lu ldquoAdaptive synchronization of anuncertain complex dynamical networkrdquo Institute of Electricaland Electronics Engineers vol 51 no 4 pp 652ndash656 2006

[24] Q Zhu and J Cao ldquoAdaptive synchronization under almostevery initial data for stochastic neural networks with time-varying delays and distributed delaysrdquo Communications inNonlinear Science and Numerical Simulation vol 16 no 4 pp2139ndash2159 2011


[25] M Chen and D Zhou ldquoSynchronization in uncertain complexnetworksrdquo Chaos vol 16 no 1 Article ID 013101 8 pages 2006

[26] J Cao Z Wang and Y Sun ldquoSynchronization in an arrayof linearly stochastically coupled networks with time delaysrdquoPhysica A vol 385 no 2 pp 718ndash728 2007

[27] J Zhou J Lu and J Lu ldquoPinning adaptive synchronization of ageneral complex dynamical networkrdquo Automatica vol 44 no4 pp 996ndash1003 2008

[28] J Zhou X Wu W Yu M Small and J-a Lu ldquoPinningsynchronization of delayed neural networksrdquo Chaos vol 18 no4 Article ID 043111 9 pages 2008

[29] W Guo F Austin S Chen and W Sun ldquoPinning synchroniza-tion of the complex networks with non-delayed and delayedcouplingrdquo Physics Letters A vol 373 no 17 pp 1565ndash1572 2009

[30] W Guo ldquoLag synchronization of complex networks via pinningcontrolrdquoNonlinear Analysis RealWorld Applications vol 12 no5 pp 2579ndash2585 2011

[31] X Wu and H Lu ldquoOuter synchronization of uncertain generalcomplex delayed networks with adaptive couplingrdquo Neurocom-puting vol 82 pp 157ndash166 2012

[32] Y Sun J Cao and Z Wang ldquoExponential synchronization ofstochastic perturbed chaotic delayed neural networksrdquo Neuro-computing vol 70 no 13 pp 2477ndash2485 2007

[33] Q Zhu and J Cao ldquopth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[34] J Liang Z Wang Y Liu and X Liu ldquoGlobal synchronizationcontrol of general delayed discrete-time networks with stochas-tic coupling and disturbancesrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 38 no 4 pp 1073ndash1083 2008

[35] W Yu G Chen and J Cao ldquoAdaptive synchronization ofuncertain coupled stochastic complex networksrdquo Asian Journalof Control vol 13 no 3 pp 418ndash429 2011

[36] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory Society forIndustrial and Applied Mathematics Philadelphia Pa USA1994

[37] K Q Gu V L Kharitonov and J Chen Stability of Time-DelaySystem Birkhauser Boston Mass USA 2003

[38] V Popov Hyperstability of Control System Springer BerlinGermany 1973

[39] X Wang X Li and G Chen Complex Network Theory and ItsApplication Tsinghua University Press Beijing China 2006

[40] W Yu G Chen and J Lu ldquoOn pinning synchronization ofcomplex dynamical networksrdquo Automatica vol 45 no 2 pp429ndash435 2009

[41] T Chen X Liu and W Lu ldquoPinning complex networks by asingle controllerrdquo IEEE Transactions on Circuits and Systems Ivol 54 no 6 pp 1317ndash1326 2007

[42] J Zhou X Wu W Yu M Small and J Lu ldquoPinning syn-chronization of delayed neural networksrdquo Chaos vol 18 no 4Article ID 043111 9 pages 2008

[43] Y Liu Z Wang and X Liu ldquoExponential synchronization ofcomplex networks with Markovian jump and mixed delaysrdquoPhysics Letters A vol 372 no 22 pp 3986ndash3998 2008

[44] Y Tang J A Fang and Q Miao ldquoOn the exponential synchro-nization of stochastic jumping chaotic neural networks withmixed delays and sector-bounded non-linearitiesrdquo Neurocom-puting vol 72 no 7ndash9 pp 1694ndash1701 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


impulsive control intermittent control and adaptive controlAdaptive control method receives particular attention ofresearchers in recently rears In [3] the authors studiedsynchronization in complex networks by using distributedadaptive control scheme By designing a simple adaptivecontroller authors of [23] investigated the locally and globallyadaptive synchronization of an uncertain complex dynamicalnetwork Authors in [24] investigated synchronization ofneural networks with time-varying delays and distributeddelays via adaptive control method By using the adaptivefeedback control scheme Chen and Zhou [25] studiedsynchronization of complex nondelayed networks and Caoet al [26] investigated the complete synchronization in anarray of linearly stochastically coupled identical networkswith delays By using adaptive pinning control method Zhouet al [27] studied local and global synchronization of complexnetworks without delays authors of [28 29] consideredthe global synchronization of the complex networks withnondelayed and delayed couplings and the authors of [30]investigated lag synchronization of complex networks viastate feedback pinning strategy Outer synchronization ofcomplex delayed networks with uncertain parameters wasconsidered by using adaptive coupling in [31] Howevermodels in the previous references are special that is eachof them does not consider general complex networks inwhich every dynamical node has mixed delays (discretedelay and distributed delay) and the complex networkshave nondelayed discrete-delayed and distributed-delayedcouplings

Complex networks are always affected by some unknownexternal perturbations due to environmental causes andhuman causes White noises brought by some random fluc-tuations in the course of transmission and other probabilitiescauses have received extensive attention in the literatures[21 24 32ndash35] However not all the external perturbationsare white noise and some of them may be nonlinear andnonstochastic perturbations When complex networks aredisturbed by nonlinear and nonstochastic perturbations thestates of the nodes will be changed dramatically whichwill affect the stability and synchronization of the complexnetworks Due to the fragility of complex networks if someimportant nodes are perturbed by such external perturba-tions whole states of the network will be affected or eventhe network cannot operate normally Hence how to realizesynchronization of all nodes for complex networks withuncertain nonlinear nonstochastic external perturbations isan urgent practical problem to be solved Obviously thecontrollers for stability and synchronization of stochasticperturbations are not applicable to the case of nonlinearnonstochastic perturbations especially when the functionsand bounds of the perturbations are unknown Thereforeto enhance antiperturbations capability and to realize syn-chronization of complex networks more effective controllershould be designed

Motivated by the previous analysis in this paper a classof more general complex networks is proposed The newmodel has nondelayed discrete-delayed and distributed-delayed couplings and every dynamical node has mixeddelays Unknown nonstochastic external perturbations to the

complex networks are also considered Then we study theglobal complete synchronization of the proposed model Anew simple but robust adaptive controller is designed toovercome the effects of such perturbations and synchronizethe complex networks even without knowing the exactfunctions and bounds of the perturbations Moreover theadaptive controller can also synchronize coupled systemswith stochastic perturbations since it includes existing adap-tive controller as special case Two cases are consideredall nodes or partial nodes are perturbed All nodes shouldbe controlled for the former case Pinning control schemecan also be used for the latter case Based on Lyapunovstability theory integral inequality Barbalat lemma andSchur Complement lemma rigorous proofs are given forsynchronization of the complex networks with unknownperturbations of the previous two cases It should be notedthat our new adaptive controllers can also prevent externalperturbations Therefore the new adaptive controllers arebetter than those in [23ndash29]Numerical simulations verify theeffectiveness of our theoretical results

Notations In the sequel if not explicitly stated matricesare assumed to have compatible dimensions 119868

119873denotes the

identity matrix of 119873 dimension The Euclidean norm in R119899

is denoted as sdot accordingly for vector 119909 isin R119899 119909 =

radic119909119879119909 where 119879 denotes transposition 119860 = (119886119894119895)119898times119898

denotes

a matrix of 119898 dimension 119860 = radic120582max(119860119879119860) and 119860

119904=

(12)(119860 + 119860119879) 119860 gt 0 or 119860 lt 0 denotes that the matrix 119860 is

symmetric and positive or negative definite matrix 120582min(119860119904)

is the minimum eigenvalues of the symmetric matrices 119860119904and 119860

119897denotes the matrix of the first 119897 row-column pairs of

119860 119860119888119897denotes the minor matrix of matrix 119860 by removing all

the first 119897 row-column elements of 119860The rest of this paper is organized as follows In Section 2

a class of general complex networks with mixed delaysand external perturbations is proposed Some necessaryassumptions and lemmas are also given in this section InSection 3 synchronization of the complex networks withall nodes perturbed is studied Synchronization with onlypartial nodes perturbed is considered in Section 4 Thenin Section 5 numerical simulations are given to show theeffectiveness of our results Finally in Section 6 conclusionsare given

2 Preliminaries

The general complex networks consisting of 119873 identicalnodeswith external perturbations andmixed-delay couplingsare described as

119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904


+ 120590119894(119905 119909119894 (119905) 119909119894 (119905 minus 120591 (119905)) int

119905

119905minus120579(119905)

119909119894 (119904) 119889119904)

+ 119877119894 119894 = 0 1 119873

(1)

where 119909119894(119905) = [119909

1198941(119905) 119909



119894isin R119899 is


119894119895)119873times119873

119881 =

(V119894119895)119873times119873

119882 = (119908119894119895)119873times119873


119906119894119895ge 0 (119894 = 119895) 119906

119894119894= minus

119873

sum

119895=1119895 = 119894

119906119894119895

V119894119895ge 0 (119894 = 119895) V

119894119894= minus

119873

sum

119895=1119895 = 119894

119906119894119895

119908119894119895ge 0 (119894 = 119895) 119908

119894119894= minus

119873

sum

119895=1119895 = 119894

119908119894119895

(2)


int119905




120579lt 1 120579(119905) is bounded



119909119894 (119904) = 120593119894 (119904) minus984858 le 119904 le 0 119894 = 0 1 2 119873 (3)


120590119894(119905 119909119894(119905) 119909119894(119905 minus 120591(119905)) int

119905

119905minus120579(119905)119909119895(119904)119889119904) with 120590

119894(119905)


(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(4)



120591(119905)) int119905





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

+ 120590119894(119905 119911 (119905) 119911 (119905 minus 120591 (119905)) int

119905

119905minus120579(119905)

119911 (119904) 119889119904)

(5)





119899 (6)


119872119894119896such that |120590

119894119896(119905 119906 V 119908)| le 119872


119906 V 119908 isin R119899 119894 = 1 2 119873 119896 = 1 2 119899







2) can


119894119896 119894 = 1 2 119873 119896 =

1 2 119899


1199091 (119905) = 1199092 (119905) = sdot sdot sdot = 119909119873 (119905) = 119911 (119905) (7)




119878 = [

[

11987811

11987812

119878119879

1211987822

]

]

lt 0 (8)


(L1) 11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0

(L2) 11987822lt 0 119878

11minus 11987812119878minus1

22S11987912lt 0

where 11987811= 119878119879

11 11987822= 119878119879

22


120590int

120590

0

120596119879(119904)119863120596 (119904) d119904 ge (int

120590

0

120596 (119904) d119904)119879

119863int

120590

0

120596 (119904) d119904 (9)



lim119905rarrinfin

int

119905

0

119891 (119904) d119904 (10)


119891(119905) = 0



Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890

119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120590119894 (119905) + 119877119894

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

(11)

where 119892(119890119894) = 119891(119909

119894(119905)) minus 119891(119911(119905)) 119894 = 1 2 119873

From (H1) and (H





2)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(12)

where 120596 gt 1 119901119894gt 0 and 120585


respectively 119894 = 1 2 119873


119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (13)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119873

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

119873

sum

119894=1

(119872119894minus 120573119894)2

120585119894

1198812 (119905) = int

119905

119905minus120591(119905)

120578119879(119904) 119876120578 (119904) 119889119904

1198813 (119905) = int

119905

119905minus120579(119905)

int

119905

120583

120578119879(119904) 119866120578 (119904) 119889119904 119889120583

(14)

120578(119905) = (1198901(119905) 119890

2(119905) 119890

119873(119905))119879 119872119894= max

1le119896le119899119872119894119896





(H1) and (H

2) we obtain

1 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119873

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

119873

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[

[

(119862 + 119860 ℎ minus 120572119896119894)1003817100381710038171003817119890119894(119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904

+ 120572

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120572120582min (Φ119904) 119906119894119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2


+ 120573

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817

+120574

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904]

]

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

le 120578119879(119905) ( (119862 + 119860 ℎ + 1) 119868119873

+120572 (Φ 119904

minus 119870)) 120578 (119905)

+ 120578119879(119905 minus 120591 (119905)) 119861

119879

119861120578 (119905 minus 120591 (119905))

+ (int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119863119879

119863int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(15)

where 119870 = diag (1198961 1198962 119896

119873) = (

119894119895)119873times119873

119894119895= 119906119894119895

119894 = 119895 119894119894= (120582min(Φ

119904)Φ)119906

119894119894 119861 = 119861ℎ119868

119873+ 120573Υ|119881| 119863 =

119863ℎ119868119873+ 120574Λ|119882| |119881| = (|V

119894119895|)119873times119873

|119882| = (|119908119894119895|)119873times119873


119873

sum

119894=1

[119890119879

119894(119905) 120590119894 (119905) minus 120596

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 minus

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816]

le

119873

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

le minus

119873

sum

119894=1

119899

sum

119896=1

(120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 le 0

(16)


2 (119905) = 120578

119879(119905) 119876120578 (119905) minus (1 minus (119905)) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

le 120578119879(119905) 119876120578 (119905) minus (1 minus ℎ120591) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

(17)


3 (119905) = 120579 (119905) 120578

119879(119905) 119866120578 (119905) minus (1 minus (119905)) int

119905

119905minus120579(119905)

120578119879(119904) 119866120578 (119904) 119889119904

le 120579max120578119879(119905) 119866120578 (119905)


120579min(int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119866int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(18)

Take119876 = (1(1minusℎ120591))119861119879

119861119866 = (120579min(1minusℎ120579))119863119879


(119905) le 120572120578119879(119905) [

1

120572(119862 + 119860 ℎ + 1) 119868119873

+ Φ 119904

+1

120572 (1 minus ℎ120591)119861119879

119861

+120579min120579max120572 (1 minus ℎ

120579)119863119879

119863 minus 119870]120578 (119905)

(19)

Let 119896119894= 120582max((1120572)(119862 + 119860ℎ + 1)119868

119873+ Φ

119904

+

(1120572(1 minus ℎ120591))119861119879

119861 + (120579min120579max(120572(1 minus ℎ120579)))119863119879

119863) + 1 where120582max((1120572)(119862 + 119860ℎ+ 1)119868119873 + Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1 minus ℎ120579))119863119879


119873+ Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1minusℎ120579))119863119879


(119905) le minus120572120578119879(119905) 120578 (119905) (20)


119881 (0) ge 119881 (119905) + 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 ge 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 (21)

Therefore


119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (22)


lim119905rarrinfin

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (23)

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (24)









1) nodes to


119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905)

+ 119877119894 119894 = 1 2 119897

1

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 119877119894

119894 = 1198971+ 1 1198971+ 2 119897

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(25)


minus20minus10

010

2030

minus20minus10

010

2030

minus30

60

50

40

30

20

10

0

1199111(119905)

1199112 (119905)

119911 3(119905)


Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 1 2 119897

1

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 119877119894 119894 = 119897

1+ 1 1198971+ 2 119897

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(26)



minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)


2(119905) (b) 120590

3(119905) (c)


1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +


120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868




119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)


2(119905)




1(119905) along the


1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 120590119894(119905 119909119894 (119905) 119909119894 (119905 minus 120591 (119905)) int

119905

119905minus120579(119905)

119909119894 (119904) 119889119904)

+ 119877119894 119894 = 0 1 119873

(1)

where 119909119894(119905) = [119909

1198941(119905) 119909



119894isin R119899 is


119894119895)119873times119873

119881 =

(V119894119895)119873times119873

119882 = (119908119894119895)119873times119873


119906119894119895ge 0 (119894 = 119895) 119906

119894119894= minus

119873

sum

119895=1119895 = 119894

119906119894119895

V119894119895ge 0 (119894 = 119895) V

119894119894= minus

119873

sum

119895=1119895 = 119894

119906119894119895

119908119894119895ge 0 (119894 = 119895) 119908

119894119894= minus

119873

sum

119895=1119895 = 119894

119908119894119895

(2)


int119905




120579lt 1 120579(119905) is bounded



119909119894 (119904) = 120593119894 (119904) minus984858 le 119904 le 0 119894 = 0 1 2 119873 (3)


120590119894(119905 119909119894(119905) 119909119894(119905 minus 120591(119905)) int

119905

119905minus120579(119905)119909119895(119904)119889119904) with 120590

119894(119905)


(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(4)



120591(119905)) int119905





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

+ 120590119894(119905 119911 (119905) 119911 (119905 minus 120591 (119905)) int

119905

119905minus120579(119905)

119911 (119904) 119889119904)

(5)





119899 (6)


119872119894119896such that |120590

119894119896(119905 119906 V 119908)| le 119872


119906 V 119908 isin R119899 119894 = 1 2 119873 119896 = 1 2 119899







2) can


119894119896 119894 = 1 2 119873 119896 =

1 2 119899


1199091 (119905) = 1199092 (119905) = sdot sdot sdot = 119909119873 (119905) = 119911 (119905) (7)




119878 = [

[

11987811

11987812

119878119879

1211987822

]

]

lt 0 (8)


(L1) 11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0

(L2) 11987822lt 0 119878

11minus 11987812119878minus1

22S11987912lt 0

where 11987811= 119878119879

11 11987822= 119878119879

22


120590int

120590

0

120596119879(119904)119863120596 (119904) d119904 ge (int

120590

0

120596 (119904) d119904)119879

119863int

120590

0

120596 (119904) d119904 (9)



lim119905rarrinfin

int

119905

0

119891 (119904) d119904 (10)


119891(119905) = 0



Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890

119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120590119894 (119905) + 119877119894

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

(11)

where 119892(119890119894) = 119891(119909

119894(119905)) minus 119891(119911(119905)) 119894 = 1 2 119873

From (H1) and (H





2)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(12)

where 120596 gt 1 119901119894gt 0 and 120585


respectively 119894 = 1 2 119873


119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (13)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119873

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

119873

sum

119894=1

(119872119894minus 120573119894)2

120585119894

1198812 (119905) = int

119905

119905minus120591(119905)

120578119879(119904) 119876120578 (119904) 119889119904

1198813 (119905) = int

119905

119905minus120579(119905)

int

119905

120583

120578119879(119904) 119866120578 (119904) 119889119904 119889120583

(14)

120578(119905) = (1198901(119905) 119890

2(119905) 119890

119873(119905))119879 119872119894= max

1le119896le119899119872119894119896





(H1) and (H

2) we obtain

1 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119873

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

119873

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[

[

(119862 + 119860 ℎ minus 120572119896119894)1003817100381710038171003817119890119894(119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904

+ 120572

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120572120582min (Φ119904) 119906119894119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2


+ 120573

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817

+120574

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904]

]

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

le 120578119879(119905) ( (119862 + 119860 ℎ + 1) 119868119873

+120572 (Φ 119904

minus 119870)) 120578 (119905)

+ 120578119879(119905 minus 120591 (119905)) 119861

119879

119861120578 (119905 minus 120591 (119905))

+ (int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119863119879

119863int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(15)

where 119870 = diag (1198961 1198962 119896

119873) = (

119894119895)119873times119873

119894119895= 119906119894119895

119894 = 119895 119894119894= (120582min(Φ

119904)Φ)119906

119894119894 119861 = 119861ℎ119868

119873+ 120573Υ|119881| 119863 =

119863ℎ119868119873+ 120574Λ|119882| |119881| = (|V

119894119895|)119873times119873

|119882| = (|119908119894119895|)119873times119873


119873

sum

119894=1

[119890119879

119894(119905) 120590119894 (119905) minus 120596

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 minus

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816]

le

119873

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

le minus

119873

sum

119894=1

119899

sum

119896=1

(120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 le 0

(16)


2 (119905) = 120578

119879(119905) 119876120578 (119905) minus (1 minus (119905)) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

le 120578119879(119905) 119876120578 (119905) minus (1 minus ℎ120591) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

(17)


3 (119905) = 120579 (119905) 120578

119879(119905) 119866120578 (119905) minus (1 minus (119905)) int

119905

119905minus120579(119905)

120578119879(119904) 119866120578 (119904) 119889119904

le 120579max120578119879(119905) 119866120578 (119905)


120579min(int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119866int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(18)

Take119876 = (1(1minusℎ120591))119861119879

119861119866 = (120579min(1minusℎ120579))119863119879


(119905) le 120572120578119879(119905) [

1

120572(119862 + 119860 ℎ + 1) 119868119873

+ Φ 119904

+1

120572 (1 minus ℎ120591)119861119879

119861

+120579min120579max120572 (1 minus ℎ

120579)119863119879

119863 minus 119870]120578 (119905)

(19)

Let 119896119894= 120582max((1120572)(119862 + 119860ℎ + 1)119868

119873+ Φ

119904

+

(1120572(1 minus ℎ120591))119861119879

119861 + (120579min120579max(120572(1 minus ℎ120579)))119863119879

119863) + 1 where120582max((1120572)(119862 + 119860ℎ+ 1)119868119873 + Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1 minus ℎ120579))119863119879


119873+ Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1minusℎ120579))119863119879


(119905) le minus120572120578119879(119905) 120578 (119905) (20)


119881 (0) ge 119881 (119905) + 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 ge 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 (21)

Therefore


119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (22)


lim119905rarrinfin

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (23)

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (24)









1) nodes to


119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905)

+ 119877119894 119894 = 1 2 119897

1

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 119877119894

119894 = 1198971+ 1 1198971+ 2 119897

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(25)


minus20minus10

010

2030

minus20minus10

010

2030

minus30

60

50

40

30

20

10

0

1199111(119905)

1199112 (119905)

119911 3(119905)


Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 1 2 119897

1

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 119877119894 119894 = 119897

1+ 1 1198971+ 2 119897

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(26)



minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)


2(119905) (b) 120590

3(119905) (c)


1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +


120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868




119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)


2(119905)




1(119905) along the


1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878 = [

[

11987811

11987812

119878119879

1211987822

]

]

lt 0 (8)


(L1) 11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0

(L2) 11987822lt 0 119878

11minus 11987812119878minus1

22S11987912lt 0

where 11987811= 119878119879

11 11987822= 119878119879

22


120590int

120590

0

120596119879(119904)119863120596 (119904) d119904 ge (int

120590

0

120596 (119904) d119904)119879

119863int

120590

0

120596 (119904) d119904 (9)



lim119905rarrinfin

int

119905

0

119891 (119904) d119904 (10)


119891(119905) = 0



Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890

119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120590119894 (119905) + 119877119894

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

(11)

where 119892(119890119894) = 119891(119909

119894(119905)) minus 119891(119911(119905)) 119894 = 1 2 119873

From (H1) and (H





2)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(12)

where 120596 gt 1 119901119894gt 0 and 120585


respectively 119894 = 1 2 119873


119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (13)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119873

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

119873

sum

119894=1

(119872119894minus 120573119894)2

120585119894

1198812 (119905) = int

119905

119905minus120591(119905)

120578119879(119904) 119876120578 (119904) 119889119904

1198813 (119905) = int

119905

119905minus120579(119905)

int

119905

120583

120578119879(119904) 119866120578 (119904) 119889119904 119889120583

(14)

120578(119905) = (1198901(119905) 119890

2(119905) 119890

119873(119905))119879 119872119894= max

1le119896le119899119872119894119896





(H1) and (H

2) we obtain

1 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119873

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

119873

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[

[

(119862 + 119860 ℎ minus 120572119896119894)1003817100381710038171003817119890119894(119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904

+ 120572

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120572120582min (Φ119904) 119906119894119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2


+ 120573

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817

+120574

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904]

]

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

le 120578119879(119905) ( (119862 + 119860 ℎ + 1) 119868119873

+120572 (Φ 119904

minus 119870)) 120578 (119905)

+ 120578119879(119905 minus 120591 (119905)) 119861

119879

119861120578 (119905 minus 120591 (119905))

+ (int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119863119879

119863int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(15)

where 119870 = diag (1198961 1198962 119896

119873) = (

119894119895)119873times119873

119894119895= 119906119894119895

119894 = 119895 119894119894= (120582min(Φ

119904)Φ)119906

119894119894 119861 = 119861ℎ119868

119873+ 120573Υ|119881| 119863 =

119863ℎ119868119873+ 120574Λ|119882| |119881| = (|V

119894119895|)119873times119873

|119882| = (|119908119894119895|)119873times119873


119873

sum

119894=1

[119890119879

119894(119905) 120590119894 (119905) minus 120596

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 minus

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816]

le

119873

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

le minus

119873

sum

119894=1

119899

sum

119896=1

(120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 le 0

(16)


2 (119905) = 120578

119879(119905) 119876120578 (119905) minus (1 minus (119905)) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

le 120578119879(119905) 119876120578 (119905) minus (1 minus ℎ120591) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

(17)


3 (119905) = 120579 (119905) 120578

119879(119905) 119866120578 (119905) minus (1 minus (119905)) int

119905

119905minus120579(119905)

120578119879(119904) 119866120578 (119904) 119889119904

le 120579max120578119879(119905) 119866120578 (119905)


120579min(int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119866int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(18)

Take119876 = (1(1minusℎ120591))119861119879

119861119866 = (120579min(1minusℎ120579))119863119879


(119905) le 120572120578119879(119905) [

1

120572(119862 + 119860 ℎ + 1) 119868119873

+ Φ 119904

+1

120572 (1 minus ℎ120591)119861119879

119861

+120579min120579max120572 (1 minus ℎ

120579)119863119879

119863 minus 119870]120578 (119905)

(19)

Let 119896119894= 120582max((1120572)(119862 + 119860ℎ + 1)119868

119873+ Φ

119904

+

(1120572(1 minus ℎ120591))119861119879

119861 + (120579min120579max(120572(1 minus ℎ120579)))119863119879

119863) + 1 where120582max((1120572)(119862 + 119860ℎ+ 1)119868119873 + Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1 minus ℎ120579))119863119879


119873+ Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1minusℎ120579))119863119879


(119905) le minus120572120578119879(119905) 120578 (119905) (20)


119881 (0) ge 119881 (119905) + 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 ge 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 (21)

Therefore


119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (22)


lim119905rarrinfin

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (23)

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (24)









1) nodes to


119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905)

+ 119877119894 119894 = 1 2 119897

1

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 119877119894

119894 = 1198971+ 1 1198971+ 2 119897

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(25)


minus20minus10

010

2030

minus20minus10

010

2030

minus30

60

50

40

30

20

10

0

1199111(119905)

1199112 (119905)

119911 3(119905)


Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 1 2 119897

1

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 119877119894 119894 = 119897

1+ 1 1198971+ 2 119897

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(26)



minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)


2(119905) (b) 120590

3(119905) (c)


1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +


120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868




119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)


2(119905)




1(119905) along the


1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 120573

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817

+120574

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904]

]

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

le 120578119879(119905) ( (119862 + 119860 ℎ + 1) 119868119873

+120572 (Φ 119904

minus 119870)) 120578 (119905)

+ 120578119879(119905 minus 120591 (119905)) 119861

119879

119861120578 (119905 minus 120591 (119905))

+ (int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119863119879

119863int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(15)

where 119870 = diag (1198961 1198962 119896

119873) = (

119894119895)119873times119873

119894119895= 119906119894119895

119894 = 119895 119894119894= (120582min(Φ

119904)Φ)119906

119894119894 119861 = 119861ℎ119868

119873+ 120573Υ|119881| 119863 =

119863ℎ119868119873+ 120574Λ|119882| |119881| = (|V

119894119895|)119873times119873

|119882| = (|119908119894119895|)119873times119873


119873

sum

119894=1

[119890119879

119894(119905) 120590119894 (119905) minus 120596

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 minus

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816]

le

119873

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

le minus

119873

sum

119894=1

119899

sum

119896=1

(120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 le 0

(16)


2 (119905) = 120578

119879(119905) 119876120578 (119905) minus (1 minus (119905)) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

le 120578119879(119905) 119876120578 (119905) minus (1 minus ℎ120591) 120578

119879(119905 minus 120591 (119905)) 119876120578 (119905 minus 120591 (119905))

(17)


3 (119905) = 120579 (119905) 120578

119879(119905) 119866120578 (119905) minus (1 minus (119905)) int

119905

119905minus120579(119905)

120578119879(119904) 119866120578 (119904) 119889119904

le 120579max120578119879(119905) 119866120578 (119905)


120579min(int

119905

119905minus120579(119905)

120578 (119904) 119889119904)

119879

119866int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(18)

Take119876 = (1(1minusℎ120591))119861119879

119861119866 = (120579min(1minusℎ120579))119863119879


(119905) le 120572120578119879(119905) [

1

120572(119862 + 119860 ℎ + 1) 119868119873

+ Φ 119904

+1

120572 (1 minus ℎ120591)119861119879

119861

+120579min120579max120572 (1 minus ℎ

120579)119863119879

119863 minus 119870]120578 (119905)

(19)

Let 119896119894= 120582max((1120572)(119862 + 119860ℎ + 1)119868

119873+ Φ

119904

+

(1120572(1 minus ℎ120591))119861119879

119861 + (120579min120579max(120572(1 minus ℎ120579)))119863119879

119863) + 1 where120582max((1120572)(119862 + 119860ℎ+ 1)119868119873 + Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1 minus ℎ120579))119863119879


119873+ Φ

119904

+ (1120572(1 minus ℎ120591))119861119879

119861 +

(120579min120579max120572(1minusℎ120579))119863119879


(119905) le minus120572120578119879(119905) 120578 (119905) (20)


119881 (0) ge 119881 (119905) + 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 ge 120572

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 (21)

Therefore


119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (22)


lim119905rarrinfin

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (23)

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (24)









1) nodes to


119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905)

+ 119877119894 119894 = 1 2 119897

1

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 119877119894

119894 = 1198971+ 1 1198971+ 2 119897

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(25)


minus20minus10

010

2030

minus20minus10

010

2030

minus30

60

50

40

30

20

10

0

1199111(119905)

1199112 (119905)

119911 3(119905)


Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 1 2 119897

1

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 119877119894 119894 = 119897

1+ 1 1198971+ 2 119897

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(26)



minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)


2(119905) (b) 120590

3(119905) (c)


1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +


120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868




119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)


2(119905)




1(119905) along the


1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1) nodes to


119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905)

+ 119877119894 119894 = 1 2 119897

1

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 119877119894

119894 = 1198971+ 1 1198971+ 2 119897

119894 (119905) = 119862119909119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 119868 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(25)


minus20minus10

010

2030

minus20minus10

010

2030

minus30

60

50

40

30

20

10

0

1199111(119905)

1199112 (119905)

119911 3(119905)


Let 119890119894(119905) = 119909



119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 1 2 119897

1

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

+ 119877119894 119894 = 119897

1+ 1 1198971+ 2 119897

119890119894 (119905) = 119862119890119894 (119905) + 119860119892 (119890119894 (119905)) + 119861119892 (119890119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119892 (119890119894 (119904)) 119889119904 + 120572

119873

sum

119895=1

119906119894119895Φ119890119895 (119905)

+ 120573

119873

sum

119895=1

V119894119895Υ119890119895 (119905 minus 120591 (119905)) + 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119890119895 (119904) 119889119904

119894 = 119897 + 1 119897 + 2 119873

(26)



minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)


2(119905) (b) 120590

3(119905) (c)


1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +


120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868




119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)


2(119905)




1(119905) along the


1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(a)

minus200 10

20 30

605040302010

040

200

minus20minus40

minus10

(b)

minus200 10

20 30

minus20minus10

01020

30

minus30

605040302010

0

minus10

(c)


2(119905) (b) 120590

3(119905) (c)


1) and (H

2) hold If

2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

lt 0 (27)


119877119894= minus120572120576

119894119890119894 (119905) minus 120596120573119894 sign (119890119894 (119905))

120576119894= 119901119894119890119894(119905)119879119890119894 (119905)

119894= 120585119894

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

(28)

where 119894 = 1 2 119897 Σ = 2(119862 + 119860ℎ) + ((120579max120579min +


120579))119863ℎ119868

119873+ 120574Λ|119882| + ((2 minus ℎ

120591)(1 minus

ℎ120591))119861ℎ119868




119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) (29)

where

1198811 (119905) =

1

2

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905)

+1

2

119897

sum

119894=1

120572(120576119894minus 119896119894)2

119901119894

+1

2

1198971

sum

119894=1

(119872119894minus 120573119894)2

120585119894

(30)


2(119905)




1(119905) along the


1198811 (119905) =

119873

sum

119894=1

119890119879

119894(119905) 119890119894 (119905) + 120572

119897

sum

119894=1

(120576119894minus 119896119894) 119890119879

119894(119905) 119890119894 (119905)

minus

1198971

sum

119894=1

(119872119894minus 120573119894)

119899

sum

119896=1

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

119873

sum

119894=1

[ (119862 + 119860 ℎ)1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 119861 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

1003817100381710038171003817119890119894 (119905 minus 120591 (119905))1003817100381710038171003817

+ 119863 ℎ1003817100381710038171003817119890119894 (119905)

1003817100381710038171003817 int

119905

119905minus120579(119905)

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817 119889119904]

minus 120572

119897

sum

119894=1

119896119894

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2+ 120572

119873

sum

119894=1

120582Φ119904

min1199061198941198941003817100381710038171003817119890119894 (119905)

1003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119873

sum

119895=1119895 = 119894

119906119894119895 Φ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905)

10038171003817100381710038171003817

+ 120573

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816V119894119895

10038161003816100381610038161003816Υ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

10038171003817100381710038171003817119890119895 (119905 minus 120591 (119905))

10038171003817100381710038171003817


119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119905

0 05 1 15 2 25 3 35 4 45 5

4

2

0

minus2

minus4

minus6

minus8

119890 1198941(119905)

1le119894le

3

(a)119905

10

5

0

minus5

minus10

minus150 05 1 15 2 25 3 35 4 45 5

119890 1198942(119905)

1le119894le

3

(b)

0 05 1 15 2 25 3 35 4 45 5

119890 1198943(119905)

1le119894le

3

10

5

0

minus5

minus10

minus15

119905

(c)


1198942(119905) (c) 119890

1198943(119905) 119894 = 1 2 3

0 05 1 15 2 25 3 35 4 45 5119905

30

25

20

15

10

5

0

120576 119894(119905)

1le119894le

3

(a)

0 05 1 15 2 25 3 35 4 45 5119905

55

5

45

4

35

3

25

2

120573119894(119905)

1le119894le

3

(b)


119894 1 le 119894 le 3


+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 120574

119873

sum

119894=1

119873

sum

119895=1

10038161003816100381610038161003816119908119894119895

10038161003816100381610038161003816Λ

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 int

119905

119905minus120579(119905)

10038171003817100381710038171003817119890119895 (119904)

10038171003817100381710038171003817119889119904

= 120578119879(119905) ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ

119904

minus 119870)) 120578 (119905)

+ 120578119879(119905) (119861 ℎ119868119873 + 120573 Υ |119881|) 120578 (119905 minus 120591 (119905))

+ 120578119879(119905) (119863 ℎ119868119873 + 120574 Λ |119882|) int

119905

119905minus120579(119905)

120578 (119904) 119889119904

(31)

where119870 = diag(1198961 119896

119897 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873minus119897


tion is used1198971

sum

119894=1

119890119879

119894(119905) 120590119894 (119905) minus 120596

1198971

sum

119894=1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

minus

1198971

sum

119894=1

119899

sum

119896=1

(119872119894minus 120573119894)1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le

1198971

sum

119894=1

119899

sum

119896=1

[1003816100381610038161003816119890119894119896 (119905)

1003816100381610038161003816119872119894119896 minus1198721198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus (120596 minus 1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816]

minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816

le minus

1198971

sum

119894=1

119899

sum

119896=1

(120596minus1) 1205731198941003816100381610038161003816119890119894119896

1003816100381610038161003816 minus 120596

119897

sum

119894=1198971+1

119899

sum

119896=1

120573119894

1003816100381610038161003816119890119894119896 (119905)1003816100381610038161003816 le 0

(32)


119881 (119905) le

1

2120577119879(119905)Π120577 (119905) (33)


120578(119904)119889119904)119879)119879 and

Π =

[[[[[

[

Π11

Π12

Π13

Π119879


120591) 119876 0

Π119879

130 minus

2 (1 minus ℎ120579)

120579min119866

]]]]]

]

(34)

withΠ11= 2((119862+ 119860ℎ)119868

119873+120572(Φ

119904

minus119870) + 119876 + 120579max119866)Π12= 119861ℎ119868

119873+ 120573Υ|119881| Π

13= 119863ℎ119868

119873+ 120574Λ|119882|


Δ = 2 ((119862 + 119860 ℎ) 119868119873 + 120572 (Φ 119904

minus 119870) + 119876 + 120579max119866)

+1

2 (1 minus ℎ120591)(119861 ℎ119868119873 + 120573 Υ |119881|)119876

minus1

times (119861 ℎ119868119873 + 120573 Υ |119881|119879)

+120579min

2 (1 minus ℎ120579)(119863 ℎ119868119873 + 120574 Λ |119882|) 119866

minus1

times (119863 ℎ119868119873 + 120574 Λ |119882|119879) lt 0

(35)


15

10

5

0

minus5

minus10

119911 2(119905)



10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119911 2(119905)


Let119876 = (12(1minusℎ120591))119861ℎ119868

119873+120573Υ|119881|119868

119873119866 = (120579min2(1minus

ℎ120579))119863ℎ119868

119873+ 120574Λ|119882|119868

119873 We have

Δ le 2 (119862 + 119860 ℎ) 119868119873 + 2120572 (Φ 119904

minus 119870)

+120579max120579min + 1 minus ℎ120579

1 minus ℎ120579

1003817100381710038171003817119863 ℎ119868119873 + 120574 Λ |119882|1003817100381710038171003817 119868119873

+2 minus ℎ120591

1 minus ℎ120591

1003817100381710038171003817119861 ℎ119868119873 + 120573 Υ |119881|1003817100381710038171003817 119868119873minus1

= [

Δ11

2120572 Φ (119904

)lowast

2120572 Φ (119904

)119879

lowast2120572 Φ (

119904

)119888

119897+ Σ119868119873minus119897

]

(36)

where Δ11= 2120572Φ(

119904

)119897minus 2120572119870

119897+ Σ119868119897 2120572Φ(119904)

lowastis matrix



(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)


0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198941(119905)

1le119894le

10

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

119890 1198942(119905)

1le119894le

10

(b)


1198942 1 le 119894 le 10

Since 2120572Φ(119904)119888119897+ Σ119868119873minus119897


1 1198962 119896

119897such that

2120572 Φ (119904

)119897minus 2120572119870

119897+ Σ119868119897minus (2120572 Φ)

2(119904

)lowast

times (2120572 Φ (119904

)119888

119897+ Σ119868119873minus119897

)

minus1

(119904

)119879

lowastlt 0

(37)


119881 (119905) le minus120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2le 0 (38)


119881 (0) ge 119881 (119905) + 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

ge 120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904

(39)

Therefore

lim119905rarrinfin

120582min

119873

sum

119894=1

int

119905

0

1003817100381710038171003817119890119894 (119904)1003817100381710038171003817

2119889119904 le 119881 (0) (40)


lim119905rarrinfin

120582min

119873

sum

119894=1

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817

2= 0 (41)


0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10 11119905

55

5

45

4

35

3

25

2

15

1

05

0

120576 1(119905)120576 2(119905)

(a)

0 1 2 3 4 5 6 7 8 9 10 11119905

45

4

35

3

25

2

15

1

05

0

1205731(119905)1205732(119905)

(b)


120573119894 119894 = 1 2

which in turn means

lim119905rarrinfin

1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 0 119894 = 1 2 119873 (42)















119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891120591 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891120579(119909119894 (119904)) 119889119904 + 119868 (119905)

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120573

119873

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905))

+ 120574

119873

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904 + 120590119894 (119905) + 119877119894

119894 = 0 1 119873

(43)





(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) (44)


where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119862 = [

[

minus10 10 0

28 minus1 0

0 0 83

]

]

119860 = [

[

0 0 0

0 1 0

0 0 1

]

]

(45)


taken as 1199111(0) = 08 119911

2(0) = 2 119911



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 120590119894 (119905) 119894 = 1 2 3 (46)

where 1205901(119905) = (01119911

2

1(119905) 02119911

2(119905) 02119911

3(119905))119879 1205902(119905) =

(011199111(119905) 005119911

2

2(119905) sin 119911

3(119905))119879 1205903(119905) = (01119911

1(119905) cos 119911

2(119905)



2(0) = 2 119911

3(0) = 25


119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905))

+ 120572

119873

sum

119895=1

119906119894119895Φ119909119895 (119905) + 120590119894 (119905) + 119877119894 119894 = 0 1 3

(47)

where 120572 = 05 Φ = 1198683and

119880 = [

[

minus1 1 0

1 minus1 0

1 1 minus2

]

]

(48)






1198791199092(0) = (1 2 3)

119879 1199093(0) = (4 5 6)

119879 120576119894(0) = 1 120573

119894(0) = 2 119901

119894=


errors 119890119894119895(119905) = 119909

119894119895(119905) minus 119911

119895(119905) 119894 119895 = 1 2 3 Figure 4 shows



(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905)

(49)

where 119911(119905) = (1199111(119905) 1199112(119905))119879 120591(119905) = 1 120590(119905) = 03 119891(119911(119905)) =

(tanh(1199111(119905) tanh(119911

2(119905))119879

119862 = [minus12 0

0 minus1] 119860 = [

3 minus03

8 5]

119868 = [0

2] 119861 = [

minus14 01

03 minus8] 119863 = [

minus12 01

minus28 minus1]

(50)




(119905) = 119862119911 (119905) + 119860119891 (119911 (119905)) + 119861119891 (119911 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120590(119905)

119891 (119911 (119904)) 119889119904 + 119868 (119905) + 120590 (119905)

(51)

where 120590(119905) = (021199111(119905 minus 1) 02 int

119905

119905minus031199112(119904)119889119904)


1(119905) = 04



119894 (119905) = 119862119909

119894 (119905) + 119860119891 (119909119894 (119905)) + 119861119891 (119909119894 (119905 minus 120591 (119905)))

+ 119863int

119905

119905minus120579(119905)

119891 (119909119894 (119904)) 119889119904 + 119868 (119905) + 120572

10

sum

119895=1

119906119894119895Φ119909119895 (119905)

+ 120573

10

sum

119895=1

V119894119895Υ119909119895 (119905 minus 120591 (119905)) + 120574

10

sum

119895=1

119908119894119895Λint

119905

119905minus120579(119905)

119909119895 (119904) 119889119904

+ 120590119894 (119905) + 119877119894 119894 = 0 1 10

(52)

where 120572 = 3 120573 = 120574 = 1 1205902(119905) = 120590(119905) else 120590

119894(119905) = 0


119880 =

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

[

minus4 1 1 0 0 0 0 0 1 1

1 minus5 1 1 0 0 0 0 1 1

1 1 minus4 1 1 0 0 0 0 0

0 1 1 minus4 1 1 0 0 0 0

0 0 1 1 minus4 1 1 0 0 0

0 0 0 1 1 minus4 1 1 0 0

0 0 0 0 1 1 minus4 1 1 0

0 0 0 0 0 1 1 minus4 1 1

1 1 0 0 0 0 1 1 minus4 0

1 1 0 0 0 0 0 1 0 minus3

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

]

(53)

119881 = 119882 =

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

[

minus2 1 0 0 0 1 0 0 0 0

1 minus2 1 0 0 0 0 0 0 0

0 1 minus2 0 0 0 0 0 1 0

0 0 0 minus1 1 0 0 0 0 0

0 0 0 1 minus3 0 1 0 0 1

1 0 0 0 0 minus1 0 0 0 0

0 0 0 0 1 0 minus2 1 0 0

0 0 0 0 0 0 1 minus2 1 0

0 0 1 0 0 0 0 1 minus3 1

0 0 0 0 1 0 0 0 1 minus2

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

]

(54)




119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894(0) = (minus11 + 2119894 minus10 +

2119894)119879 119894 = 1 2 10 120576

119894(0) = 120573

119894(0) = 119901

119894= 120585119894= 1 119894 =


119909119894119895(119905) minus 119911



6 Conclusions


Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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