ResearchArticle · 2019. 7. 30. · Analysis and Design of Associative Memories for Memristive...

Research ArticleAnalysis and Design of Associative Memories for MemristiveNeural Networks with Deviating Argument

Jin-E Zhang

Hubei Normal University Hubei 435002 China

Correspondence should be addressed to Jin-E Zhang zhang86021205163com

Received 19 April 2017 Revised 20 June 2017 Accepted 2 July 2017 Published 9 August 2017

Academic Editor Radek Matusu

Copyright copy 2017 Jin-E Zhang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate associativememories formemristive neural networks with deviating argument Firstly the existence and uniquenessof the solution formemristive neural networks with deviating argument are discussed Next some sufficient conditions for this classof neural networks to possess invariant manifolds are obtained In addition a global exponential stability criterion is presentedThen analysis and design of autoassociative memories and heteroassociative memories for memristive neural networks withdeviating argument are formulated respectively Finally several numerical examples are given to demonstrate the effectivenessof the obtained results

1 Introduction

In recent decades analysis and design of neurodynamic sys-tems have receivedmuch attention [1ndash16] Specifically neuro-dynamics of associative memories is a hot research issue [9ndash16] Associative memories refer to brain-inspired computingdesigned to store a set of prototype patterns such that thestored patterns can be retrieved with the recalling probescontaining sufficient information about the contents of pat-terns In associativememories for any given probe (eg noisyor corrupted version of a prototype pattern) the retrievaldynamics can converge to an ideal equilibrium point rep-resenting the prototype pattern At present system analystshave two main theories for explaining strange dynamicalbehaviors in recalling processes autoassociative memoriesand heteroassociative memories [12 13] Two types of meth-ods are usually used for solving problems in regard to analysisand synthesis of associative memories The first is whenneurodynamic systems aremultistable and then system statesmay converge to locally stable equilibrium points and theseequilibrium points are encoded as the memorized patternsThe second is when neurodynamic systems are globallymonostable and then the memorized patterns are associatedwith the external input patterns

As we know the human brain is believed to be the mostpowerful information processor because of its structure of

synapses Actually there is always a high demand for a singledevice to emulate artificial synaptic functions Using a mem-ristor essential synaptic plasticity and learning behaviors canbe mimicked By integrating memristors into a large-scaleneuromorphic circuit memristive neural networks basedneuromorphic circuits demonstrate spike-timing-dependentplasticity (a form of the Hebbian learning rule) [17ndash21] Fromthe viewpoint of theory and experiment a memristive neuralnetwork model is a state-dependent switching system Onthe other hand this new dynamical system shows many ofthe characteristic features of magnetic tunnel junction [1920] For these reasons system analysis and integration formemristive neural networks will be extremely tricky Forthe moment there are two major methods for qualitativeanalysis ofmemristive neural networksOne is the differentialinclusions approach [17ndash20] and the other is the fuzzy uncer-tainty approach [21] For the differential inclusions approachthe core idea is to integrate a switched network cluster Forthe fuzzy uncertainty approach the central idea is to dividemultijump network flows into a continuous subsystem andbounded subsystem The two kinds of analysis frameworksare just the type and level of points not the merits of goodpoints of difference From the perspective of cyberneticsmany reported analytical skills are merely a breakthrough ofsystem theory more than substance How to develop more

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1057909 16 pageshttpsdoiorg10115520171057909

2 Mathematical Problems in Engineering

effective analysis methods for memristive neural networks isworth studying

For the past few years hybrid dynamic systems haveremained one of the most active fields of research in thecontrol community [22ndash30] For instance to describe the sta-tionary distribution of temperature along the length of a wirethat is bended the nonlinear dynamic model with deviatingargument is often used The right-hand side in nonlinearsystems with deviating argument features a combination ofcontinuous and discrete systems Thus nonlinear systemswith deviating argument unify the advanced and retardedsystems Because of this property relatedworks on the controlstrategies for nonlinear systems with deviating argument arerelatively rare Viewed from systems involving the interplaydifferential equations and difference equations are includedin the nonlinear systems with deviating argument [28ndash30]However it is still obvious thatmany basic issues onnonlinearsystems with deviating argument remain to be addressedsuch as nonlinear dynamics systems design and analy-sis

Inspired by the above discussions the goal of this paper isto formulate the analysis and design of associative memoriesfor a class of memristive neural networks with deviatingargument On the whole the highlights of this paper can beoutlined as follows

(1) Sufficient conditions are derived to ascertain the exis-tence uniqueness and global exponential stabilityof the solution for memristive neural networks withdeviating argument

(2) The syntheticmechanismof the analysis and design ofassociativememories for a general class ofmemristiveneural networks with deviating argument is revealed

(3) A uniform associative memories model which unitesautoassociative memories and heteroassociativememories is proposed for memristive neural net-works with deviating argument

In addition it should be noted thatwhen special and strictconditions are exerted associative memoriesrsquo performance ofneural networks is usually limited Therefore the methodsapplicable to conventional stability analysis of neural net-works cannot be directly employed to investigate the asso-ciative memories for memristive neural networks Moreoverthe study of dynamics for memristive neural networks withdeviating argument is not an easy work since the conditionsfor dynamics of neural networks cannot be simply utilized toanalyze hybrid dynamic systems with deviating argument Inthis paper according to the fuzzy uncertainty approach incombination with the theories of set-valued maps and differ-ential inclusions analysis and design of associative memoriesfor memristive neural networks with deviating argument aredescribed in detail

The rest of the paper is arranged as follows Design prob-lem and preliminaries are given in Section 2 Main results arestated in Section 3 In Section 4 several numerical examplesare presented Finally concluding remarks are given inSection 5

2 Design Problem and Preliminaries

21 Problem Description Let Ω = 120573 = (1205731 1205732 120573119899)119879 |120573119894 = minus119901 119901 119894 = 1 2 119899 be the set of 119899-dimensionalbipolar vectors where 119901 is a positive constant and 119879 denotesthe transpose of a vector or matrix The problem consideredis described as follows

Given 119897 (119897 le 2119899) pairs of bipolar vectors (119906(1) 119902(1))(119906(2) 119902(2)) (119906(119897) 119902(119897)) where 119906(119903) 119902(119903) isin Ω for 119903 =1 2 119897 design an associative memory model which satis-fies the notion that the output of the model will converge tothe corresponding pattern 119902(119903) when 119906(119903) is fed to the modelinput as a memory pattern

Definition 1 The neural network is said to be an autoassocia-tive memory if 119906(119903) = 119902(119903) and a heteroassociative memory if119906(119903) = 119902(119903) for 119903 = 1 2 119897

In this paper let 119873 be the natural number set the normof vector 119909 = (1199091 1199092 119909119899)119879 is defined as 119909 = sum119899119894=1 |119909119894|DenoteR119899 as the 119899-dimensional Euclidean space Fix two realnumber sequences 120578119896 120599119896 119896 isin 119873 satisfying 120578119896 lt 120578119896+1120578119896 le 120599119896 le 120578119896+1 and lim119896rarr+infin120578119896 = +infin

22 Model Consider the following neural network model

(119905) = minus119860119909 (119905) + 119861 (119909 (119905)) 119891 (119909 (119905))+ 119862 (119909 (119905)) 119891 (119909 (120574 (119905))) + E119906(119903)

119908 (119905) = 119891 (119909 (119905)) 119903 = 1 2 119897

(1)

where 119909(119905) = (1199091(119905) 1199092(119905) 119909119899(119905))119879 isin R119899 is the neuronstate 119860 = diag(1198861 1198862 119886119899) which is a constant diagonalmatrix with 119886119894 gt 0 119894 isin 1 2 119899 indicates the self-inhibition matrix 119861(119909(119905)) = (119887119894119895(119909119895(119905)))119899times119899 and 119862(119909(119905)) =(119888119894119895(119909119895(119905)))119899times119899 are connection weight matrices at time 119905 and120574(119905) respectively 119887i119895(119909119895(119905)) and 119888119894119895(119909119895(119905)) are defined by

119887119894119895 (119909119895 (119905)) = 119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895

119888119894119895 (119909119895 (119905)) = 119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895

(2)

for 119894 119895 = 1 2 119899 and 119887119894119895(plusmn119879119895) = 119894119895 or 119894119895 119888119894119895(plusmn119879119895) =119888119894119895 or 119888119894119895 where 119879119895 gt 0 represents the switching jump and119894119895 119894119895 119888119894119895 and 119888119894119895 are all constants 119891(119909(sdot)) = (1198911(1199091(sdot)1198912(1199092(sdot) 119891119899(119909119899(sdot))119879 is the activation function The devi-ating function 120574(119905) = 120599119896 when 119905 isin [120578119896 120578119896+1) for any 119896 isin 119873 Estands for a transform matrix of order 119899 satisfying E119906(119903) =119902(119903) for 119903 = 1 2 119897 119906(119903) and 119902(119903) indicate the externalinput pattern and the corresponding memorized patternrespectively 119908(119905) = (1199081(119905) 1199082(119905) 119908119899(119905))119879 is the output of(1) corresponding to 119906(119903)

Mathematical Problems in Engineering 3

Neural networkmodel (1) is called an associativememoryif the output 119908(119905) converges to 119902(119903)

Obviously neural network model (1) is of hybrid typeswitching and mixed When |119909119895(119905)| gt 119879119895 119887119894119895(119909119895(119905)) = 119894119895119888119894119895(119909119895(119905)) = 119888119894119895 When |119909119895(119905)| lt 119879119895 119887119894119895(119909119895(119905)) = 119894119895 119888119894119895(119909119895(119905)) =119888119894119895 From this perspective (1) is a switching system On theother hand for any fixed interval [120578119896 120578119896+1) when 120574(119905) = 120599119896 lt119905 lt 120578119896+1 neural networkmodel (1) is a retarded systemWhen120578119896 le 119905 lt 120574(119905) = 120599119896 neural network model (1) is an advancedsystem Then from this point of view (1) is a mixed system

For neural networkmodel (1) the conventional definitionof solution for differential equations cannot apply here Totackle this problem the solution concept for differential equa-tionswith deviating argument is introduced [24ndash30] Accord-ing to this theory a solution 119909(119905) = (1199091(119905) 1199092(119905) 119909119899(119905))119879of (1) is a consecutive function such that (1) (119905) exists on[0 +infin) except at the points 120578119896 119896 isin 119873 where a one-sidedderivative exists and (2) 119909(119905) satisfies (1) within each interval(120578119896 120578119896+1) 119896 isin 119873

In this paper the activation functions are selected as

119891119894 (120594) = 2sumℓ=1

(minus1)ℓ 1199012119889 (10038161003816100381610038161003816120594 + 119911(119894)ℓ 10038161003816100381610038161003816 minus 10038161003816100381610038161003816120594 minus 119911(119894)ℓ 10038161003816100381610038161003816) 119894 = 1 2 119899

(3)

where 119901 gt 0 and 119889 gt 0 are used to adjust the slope and cornerpoint of activation functions and 119911(119894)1 = 119879119894 119911(119894)2 = 119879119894 + 119889

It is easy to see that 119891119894(sdot) satisfies 119891119894(plusmn119879119894) = 0 and1003816100381610038161003816119891119894 (1198971) minus 119891119894 (1198972)1003816100381610038161003816 le 120572 10038161003816100381610038161198971 minus 11989721003816100381610038161003816 119894 = 1 2 119899 (4)

for any 1198971 1198972 isin (minusinfin +infin) where 120572 = 119901119889For technical convenience denote 119887119894119895 = max|119894119895| |119894119895|1198870119894119895 = (12)(119894119895 + 119894119895) 119887+119894119895 = (12)|119894119895 + 119894119895| 119887minus119894119895 = (12)|119894119895 minus 119894119895|119888119894119895 = max|119888119894119895| | 119888119894119895| 1198880119894119895 = (12)(119888119894119895 + 119888119894119895) 119888+119894119895 = (12)|119888119894119895 + 119888119894119895|119888minus119894119895 = (12)|119888119894119895 minus 119888119894119895| and 119878119894 = max|119894119894119901 minus 119886119894119889| |119894119894119901 minus 119886119894119889| for119894 119895 = 1 2 119899 Let 119878119887 = max1le119894le119899(119886119894 +sum119899119895=1 120572(119887+119895119894 +119887minus119895119894)) 119878minus119887 =

min1le119894le119899(119886119894 minussum119899j=1(119887+119895119894 + 119887minus119895119894)120572) and 119878119888 = max1le119894le119899(sum119899119895=1 120572(119888+119895119894 +119888minus119895119894)) 119888119900M Mdenotes the convex closure of a set constitutedby real numbers M and M

23 AutoassociativeMemories FromDefinition 1matrixE isselected as an identity matrix when designing autoassociativememories (ie E119906(119903) = 119906(119903) = 119902(119903) for 119903 = 1 2 119897) For theconvenience of analysis and formulation in autoassociativememories neural network model (1) can be rewritten incomponent form

119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894

119908119894 (119905) = 119891119894 (119909119894 (119905)) 119894 = 1 2 119899

(5)

24 Properties For neural network model (1) we considerthe following set-valued maps

119870[119887119894119895 (119909119895 (119905))] =

119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119900 119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 = 119879119895119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895(6)

119870[119888119894119895 (119909119895 (119905))] =

119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119900 119888119894119895 119888 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 = 119879119895119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895(7)

for 119894 119895 = 1 2 119899Based on the theory of differential inclusions [17ndash20]

from (5) we can get

119894 (119905) isin minus119886119894119909119894 (119905) + 119899sum119895=1

119870[119887119894119895 (119909119895 (119905))] 119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[119888119894119895 (119909119895 (119905))] 119891119895 (119909119895 (120574 (119905))) + 119906119894(8)

or equivalently there exist 119887lowast119894119895 (119909119895(119905)) isin 119870[119887119894119895(119909119895(119905))] and119888lowast119894119895 (119909119895(119905)) isin 119870[119888119894119895(119909119895(119905))] such that

119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887lowast119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888lowast119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894119894 = 1 2 119899

(9)

for almost all 119905 isin [0 +infin)Employing the fuzzy uncertainty approach [21] neural

network model (5) can be expressed as follows

119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

120575119887119894119895 (119909119895 (119905)) 119887minus119894119895119891119895 (119909119895 (119905)) + 119899sum119895=1

120575119888119894119895 (119909119895 (119905))times 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(10)


where

120575119887119894119895 (119909119895 (119905)) = sgn (119894119895 minus 119894119895) 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895minus sgn (119894119895 minus 119894119895) 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895


(11)

and sgn(sdot) is the sign function

Definition 2 A constant vector 119909lowast = (119909lowast1 119909lowast2 119909lowast119899 )119879 is saidto be an equilibrium point of neural networkmodel (5) if andonly if

0 isin minus119886119894119909lowast119894 + 119899sum119895=1

119870[119887119894119895 (119909lowast119895 )] 119891119895 (119909lowast119895 )+ 119899sum119895=1

119870[119888119894119895 (119909lowast119895 )] 119891119895 (119909lowast119895 ) + 119906119894 119894 = 1 2 119899(12)

or equivalently there exist 119894119895(119909lowast119895 ) isin 119870[119887119894119895(119909lowast119895 )] and 119888119894119895(119909lowast119895 ) isin119870[119888119894119895(119909lowast119895 )] such that

minus 119886119894119909lowast119894 + 119899sum119895=1

119894119895 (119909lowast119895 ) 119891119895 (119909lowast119895 ) + 119899sum119895=1

119888119894119895 (119909lowast119895 ) 119891119895 (119909lowast119895 ) + 119906119894= 0

(13)

for 119894 = 1 2 119899Definition 3 The equilibrium point 119909lowast = (119909lowast1 119909lowast2 119909lowast119899 )119879 ofneural network model (5) is globally exponentially stable ifthere exist constants119872 gt 0 and 120598 gt 0 such that

1003817100381710038171003817119909 (119905) minus 119909lowast1003817100381710038171003817 le 119872100381710038171003817100381710038171199090 minus 119909lowast10038171003817100381710038171003817 exp minus120598119905 (14)

for any 119905 ge 0 where 119909(119905) is the state vector of neural networkmodel (5) with initial condition 1199090Lemma 4 For neural network model (5) one has

10038161003816100381610038161003816119870 [119887119894119895 (119909119895)] 119891119895 (119909119895) minus 119870 [119887119894119895 (119910119895)] 119891119895 (119910119895)10038161003816100381610038161003816le 119887119894119895120572 10038161003816100381610038161003816119909119895 minus 11991011989510038161003816100381610038161003816 10038161003816100381610038161003816119870 [119888119894119895 (119909119895)] 119891119895 (119909119895) minus 119870 [119888119894119895 (119910119895)] 119891119895 (119910119895)10038161003816100381610038161003816le 119888119894119895120572 10038161003816100381610038161003816119909119895 minus 11991011989510038161003816100381610038161003816

(15)

for 119894 119895 = 1 2 119899 119909119895 119910119895 isin (minusinfin +infin) where 119870[119887119894119895(sdot)] and119870[119888119894119895(sdot)] are defined as those in (6) and (7) respectively

Lemma 4 is a basic conclusion To get more specificplease see [Neural Networks vol 51 pp 1ndash8 2014] or[Information Sciences vol 279 pp 358ndash373 2014] or others

In the following we end this section with some basicassumptions

(A1) There exists a positive constant 120578 satisfying120578119896+1 minus 120578119896 lt 120578 119896 isin 119873 (16)

(A2) 120578(119878119887 + 2119878119888) exp119878119887120578 lt 1(A3) 120578[119878119888 + 119878119887(1 + 119878119888120578) exp119878119887120578] lt 1

3 Main Results

In this section we first present the conditions ensuring theexistence and uniqueness of solution for neural networkmodel (5) and then analyze its global exponential stabilityfinally we discuss how to design the procedures of autoasso-ciative memories and heteroassociative memories

31 Existence and Uniqueness of Solution

Theorem5 Let (A1) and (A2) holdThen there exists a uniquesolution 119909(119905) = 119909(119905 1199050 1199090) 119905 ge 1199050 of (5) for every (1199050 1199090) isin(R+R119899)Proof The proof of the assertion is divided into two steps

Firstly we discuss the existence of solutionFor a fixed 119896 isin 119873 assume that 120578119896 le 1199050 le 120599119896 = 120574(119905) lt 119905 lt120578119896+1 and the other case 120578119896 le 1199050 lt 119905 lt 120599119896 = 120574(119905) lt 120578119896+1 can be

discussed in a similar wayLet V(119905) = 119909(119905 1199050 1199090) and consider the following

equivalent equation

V119894 (119905) = 1199090119894 + int1199051199050

[[minus119886119894V119894 (119904) +

119899sum119895=1

1198870119894119895119891119895 (V119895 (119904)) + 119899sum119895=1

1198880119894119895times 119891119895 (V119895 (120599119896)) + 119906119894 + 119899sum

119895=1

120575119887119894119895 (V119895 (119905)) 119887minus119894119895119891119895 (V119895 (119904))

+ 119899sum119895=1

120575119888119894119895 (V119895 (119905)) 119888minus119894119895119891119895 (V119895 (120599119896))]] d119904119894 = 1 2 119899

(17)

Construct the following sequence V119898119894 (119905) 119898 isin 119873 119894 =1 2 119899 V119898+1119894 (119905) = 1199090119894 + int119905

1199050

[[minus119886119894V

119898119894 (119904) + 119899sum

119895=1

1198870119894119895119891119895 (V119898119895 (119904))+ 119899sum119895=1

1198880119894119895 times 119891119895 (V119898119895 (120599119896)) + 119906119894+ 119899sum119895=1

120575119887119894119895 (V119898119895 (119905)) 119887minus119894119895119891119895 (V119898119895 (119904))


+ 119899sum119895=1

120575119888119894119895 (V119898119895 (119905)) 119888minus119894119895119891119895 (V119898119895 (120599119896))]] d119904119894 = 1 2 119899

(18)

and V0119894 (119905) = 1199090119894 Define a norm V(119905)119872 = max[1199050 119905]V(119905) and we can get

10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 le [(119878119887 + 119878119888) 120578]119898 120589 (19)

where 120589 = 120578[sum119899119894=1 119886119894|1199090119894 | + sum119899119894=1sum119899119895=1(119887+119894119895 + 119887minus119894119895 + 119888+119894119895 + 119888minus119894119895 )119901 +sum119899119894=1 |119906119894|]From (A2) we have (119878119887 + 119878119888)120578 lt 1 and hence

lim119898rarr+infin

10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 = 0 (20)

Therefore there exists a solution V(119905) = 119909(119905 1199050 1199090) on theinterval [1199050 120599119896] for (17) From (4) the solution can beconsecutive to 120578119896+1 Utilizing a similar method 119909(119905) canbe consecutive to 120599119896+1 and then to 120578119896+2 Applying themathematical induction the proof of existence of solution for(5) is completed

Next we analyze the uniqueness of solutionFix 119896 isin 119873 and choose 1199050 isin [120578119896 120578119896+1] 1199091 isin R119899 1199092 isin

R119899 Denote 1199091(119905) = 119909(119905 1199050 1199091) and 1199092(119905) = 119909(119905 1199050 1199092) as twosolutions of (5) with different initial conditions (1199050 1199091) and(1199050 1199092) respectively

Then we get

100381710038171003817100381710038171199091 (119905) minus 1199092 (119905)10038171003817100381710038171003817le (100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817 + 119878119888120578 100381710038171003817100381710038171199091 (120599119896) minus 1199092 (120599119896)10038171003817100381710038171003817)+ int1199051199050

119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(21)

Based on the Gronwall-Bellman inequality

100381710038171003817100381710038171199091 (119905) minus 1199092 (119905)10038171003817100381710038171003817le (100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817 + 119878119888120578 100381710038171003817100381710038171199091 (120599119896) minus 1199092 (120599119896)10038171003817100381710038171003817)times exp 119878119887120578

(22)

and particularly


(23)

Hence

100381710038171003817100381710038171199091 (119905) minus 1199092 (119905)10038171003817100381710038171003817 le ( exp 1198781198871205781 minus 119878119888120578 exp 119878119887120578)100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817 (24)

Suppose that there exists some isin [120578119896 120578119896+1] satisfying1199091() = 1199092() then100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817 le 119878119888120578 100381710038171003817100381710038171199091 (120599119896) minus 1199092 (120599119896)10038171003817100381710038171003817

+ int1199050

119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(25)

From (A2) we get

0 lt 120578 exp 119878119887120578 (119878119887 + 119878119888) lt 1 minus 120578119878119888 exp 119878119887120578 (26)

that is

0 lt 120578 exp 119878119887120578 (119878119887 + 119878119888)1 minus 120578119878119888 exp 119878119887120578 lt 1 (27)

Substituting (24) into (25) from (27) we obtain

100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817 le 119878119888120578 exp 1198781198871205781 minus 119878119888120578 exp 119878119887120578100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

+ 119878119887120578 exp 1198781198871205781 minus 119878119888120578 exp 119878119887120578100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

le 120578 exp 119878119887120578 (119878119887 + 119878119888)1 minus 120578119878119888 exp 119878119887120578100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

lt 100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

(28)

This poses a contradiction Therefore the uniqueness ofsolution is true Taken together the proof of the existence anduniqueness of solution for (5) is completed

32 Invariant Manifolds

Lemma 6 The solution 119909(119905) of neural network model (5) willfall intoprod119899119894=1[119889 + 119879119894 +infin) ultimately if for 119894 = 1 2 119899 thefollowing conditions hold

119906119894 gt 119878119894 + 119886119894119879119894 + 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 (29)

Proof We distinguish four cases to prove the lemma

(1) If for some 119905 119909119894(119905) isin [119879119894 119889+119879119894) 119894 isin 1 2 119899 then119894 (119905) = minus119886119894119909119894 (119905) + 119899sum

119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) times 119891119895 (119909119895 (120574 (119905))) + 119906119894


ge minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) minus 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus119886119894 [119909119894 (119905) minus 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) minus 119879119894]minus 119886119894119879119894 minus 119899sum

119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) minus 119879119894] minus 119886119894119879119894minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 minus119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 minus 119886119894119879119894 minus 119899sum

119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus119878119894 minus 119886119894119879119894 minus 119899sum

119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 0(30)

That is to say 119894(119905) gt 0 when 119909119894(119905) isin [119879119894 119889 + 119879119894)(2) If for some 119905 119909119894(119905) isin (minus119879119894 119879119894) 119894 isin 1 2 119899 then119894 (119905) ge minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) minus 119899sum

119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus119886119894119879119894 minus 119899sum

119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 119878119894 gt 0(31)

That is to say 119894(119905) gt 0 when 119909119894(119905) isin (minus119879119894 119879119894)(3) If for some 119905 119909119894(119905) isin (minus119879119894 minus 119889 minus119879119894) 119894 isin 1 2 119899

then

119894 (119905) ge minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) minus 119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894

= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] minus

119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894ge minus 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 minus 119899sum

119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0(32)

That is to say 119894(119905) gt 0 when 119909119894(119905) isin (minus119879119894 minus 119889119894 minus119879119894)(4) If for some 119905 119909119894(119905) isin (minusinfin minus119879119894 minus 119889119894) 119894 isin 1 2 119899

then


119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894ge 119886119894 (119879119894 + 119889) minus 119887119894119894 (119909119894 (119905)) 119901 minus 119899sum

119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 minus 119899sum

119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0

(33)

That is to say 119894(119905) gt 0 when 119909119894(119905) isin (minusinfin minus119879119894 minus 119889119894)From the above analysis we can get that any component119909119894(119905) (119894 = 1 2 119899) of the solution 119909(119905) of neural network

model (5) will fall into [119889 + 119879119894 +infin) ultimately Hence theproof is completed

Lemma 7 The solution 119909(119905) of neural network model (5) willfall intoprod119899119894=1(minusinfin minus119889minus119879119894] ultimately if for 119894 = 1 2 119899 thefollowing conditions hold

119906119894 lt minus119878119894 minus 119886119894119879119894 minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 (34)



(1) If for some 119905 119909119894(119905) isin [119889 + 119879119894 +infin) 119894 isin 1 2 119899then

119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816119886119894119889 minus 119887119894119894 (119909119894 (119905)) 1199011003816100381610038161003816 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)

That is to say 119894(119905) lt 0 when 119909119894(119905) isin [119889 + 119879119894 +infin)(2) If for some 119905 119909119894(119905) isin (119879119894 119889 +119879119894) 119894 isin 1 2 119899 then119894 (119905) = minus119886119894119909119894 (119905) + 119899sum

119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894 [119909119894 (119905) minus 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) minus 119879119894]minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) minus 119879119894] minus 119886119894119879119894+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)

That is to say 119894(119905) lt 0 when 119909119894(119905) isin (119879119894 119889 + 119879119894)(3) If for some 119905 119909119894(119905) isin (minus119879119894 119879119894) 119894 isin 1 2 119899 then119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)

That is to say 119894(119905) lt 0 when 119909119894(119905) isin (minus119879119894 119879119894)(4) If for some 119905 119909119894(119905) isin (minus119879119894 minus 119889 minus119879119894) 119894 isin 1 2 119899

then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)

That is to say 119894(119905) lt 0 when 119909119894(119905) isin (minus119879119894 minus 119889119894 minus119879119894)


From the above analysis we can get that any component119909119894(119905) (119894 = 1 2 119899) of the solution 119909(119905) of neural networkmodel (5) will fall into (minusinfin minus119889 minus 119879119894] ultimately Hence theproof is completed

33 Global Exponential Stability Analysis Denote

Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)

In order to guarantee the existence of equilibrium pointfor neural network model (5) the following assumption isneeded

(A4) For any 119894 isin 1 2 119899 119894 isin Δ 1 cup Δ 2Theorem8 Let (A4) holdThen neural networkmodel (5) hasat least one equilibrium point 119909lowast isin Θ where

Θ = 119909 isin R119899 | 119909119894 le minus119879119894 minus 119889 or 119909119894 ge 119889 + 119879119894 119894

isin 1 2 119899 (40)

Proof Define a mapping

119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)

It is obvious that 119865(119909) is equicontinuousWhen 119894 isin Δ 1 there exists a small enough positive number1205761 satisfying

119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)

andwhen 119894 isin Δ 2 there exists a small enough positive number1205762 satisfying119906119894 lt minus119878119894 minus 119886119894119879119894 minus sum

119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)

Take 0 lt 120576 lt min(1205761 1205762) and denote

Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894

isin Δ 1 or 119909119894 isin [minus (119889 + 119879119894)120576 minus119889 minus 119879119894 minus 120576] 119894 isin Δ 2 (45)

which is a bounded and closed setWhen 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576] 119894 isin Δ 1119865119894 (119909) = 1119886119894 [[

119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)

Similarly we can obtain119865119894 (119909) lt minus119879119894 minus 119889 minus 120576 (47)

for 119909119894 isin [minus(119889 + 119879119894)120576 minus119889 minus 119879119894 minus 120576] 119894 isin Δ 2From the above discussion we can get 119865(119909) isin Θ for 119909 isinΘ Based on the generalized Brouwerrsquos fixed point theorem

there exists at least one equilibrium 119909lowast isin Θ This completesthe proof

Recalling (10) for 119894 = 1 2 119899 then119894 (119905) isin minus119886119894119909119894 (119905) + 119899sum

119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)

Let 119910119894(119905) = 119909119894(119905) minus 119909lowast119894 119894 = 1 2 119899119910119894 (119905) isin minus119886119894119910119894 (119905) + 119899sum

119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)


where 119892119895(119910119895(119905)) = 119891119895(119910119895(119905) + 119909lowast119895 ) minus 119891119895(119909lowast119895 ) 119860119887119894119895(119910119895(119905)) =119887minus119894119895 (119870[120575119887119894119895(119910119895(119905) + 119909lowast119895 )]119891119895(119910119895(119905) + 119909lowast119895 ) minus 119870[120575119887119894119895(119909lowast119895 )]119891119895(119909lowast119895 ))or 119860119888119894119895(119910119895(120574(119905))) = 119888minus119894119895 (119870[120575119888119894119895(119910119895(120574(119905)) + 119909lowast119895 )]119891119895(119910119895(120574(119905)) +119909lowast119895 ) minus 119870[120575119888119894119895(119909lowast119895 )]119891119895(119909lowast119895 )) or there exists 120593119887119894119895(119910119895(119905)) isin119860119887119894119895(119910119895(119905)) 120593119888119894119895(119910119895(119905)) isin 119860119888119894119895(119910119895(119905)) for 119894 119895 = 1 2 119899 suchthat

119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)

According to Lemma 410038161003816100381610038161003816119860119887119894119895 (119910119895 (119905))10038161003816100381610038161003816 le 119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 10038161003816100381610038161003816119860119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 le 119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 (51)

for 119894 119895 = 1 2 119899Lemma 9 Let (A1)ndash(A3) hold and 119910(119905) = (1199101(119905) 1199102(119905) 119910119899(119905))119879 be a solution of (49) or (50) Then1003817100381710038171003817119910 (120574 (119905))1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (52)

for 119905 ge 0 where 120583 = (1 minus 120578[119878119888 + 119878119887(1 + 119878119888120578) exp119878119887120578])minus1Proof For any 119905 gt 0 there exists a unique 119896 isin 119873 such that119905 isin [120578119896 120578119896+1) then

119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)

Taking the absolute value on both sides of (53)

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)

and then119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)

According to the Gronwall-Bellman inequality

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)


Exchanging the location of 119910119894(119905) and 119910119894(120599119896) in (53) we get1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)

holds for 119905 isin [120578119896 120578119896+1) so 119910(120574(119905)) le 120583119910(119905) where 120583 = (1minus120578[119878119888 +119878119887(1 + 119878119888120578) exp119878119887120578])minus1 This completes the proof

Remark 10 Neural network model (49) is also of a hybridtype The difficulty in investigating this class of neural net-works lies in the existence of a switching jump and deviatingargument For the switching jump we introduce the differen-tial inclusions and fuzzy uncertainty approach to compensatestate-jump uncertainty For the deviating argument by theaid of effective computational analysis we estimate the normof deviating state 119910(120574(119905)) by the corresponding state 119910(119905)

In the following the criterion to guarantee the globalexponential stability of neural network model (5) based onthe Lyapunov method is established

Theorem 11 Let (A1)ndash(A4) holdThe origin of neural networkmodel (49) is globally exponentially stable which implies thatthe equilibrium point 119909lowast = (119909lowast1 119909lowast2 119909lowast119899 )119879 of neural networkmodel (5) is globally exponentially stable if the followingcondition is satisfied

120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)

where 120582 = minus(12)(120583119878119888 minus 119878minus119887 ) gt 0 is a positive constant satisfy-ing

120582 + 120583119878119888 minus 119878minus119887 = 12 (120583119878119888 minus 119878minus119887 ) lt 0 (64)

Then for 119905 = 120578119896 119896 isin 119873 calculate the derivative of 119911119894(119905) alongthe trajectory of (49) or (50)

119894 (119905) = 120582 exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + exp 120582119905 sgn (119910119894 (119905)) 119910119894 (119905)= 120582 exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + exp 120582119905 sgn (119910119894 (119905))

sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905)))) le 120582 exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ exp 120582119905(minus119886119894 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum

119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816]] le exp 120582119905sdot [(120582 minus 119878minus119887 ) 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120574 (119905))1003817100381710038171003817]

(68)

Based on Lemma 9

(119905) le exp 120582119905 (120582 minus 119878minus119887 + 120583119878119888) 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (69)

From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then

(119905) le exp 120582119905 (120582 minus 119878minus119887 + 120583119878119888) 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 0 (71)

It is easy to see that

119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)

hence 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (0)1003817100381710038171003817 exp minus120582119905 (73)

that is 1003817100381710038171003817119909 (119905) minus 119909lowast1003817100381710038171003817 le 1003817100381710038171003817119909 (0) minus 119909lowast1003817100381710038171003817 exp minus120582119905 (74)

for 119905 ge 0 This completes the proof

34 Design Procedure of Autoassociative Memories Given119897 (119897 le 2119899) vectors 119906(1) 119906(2) 119906(119897) to be memorized where119906(119896) isin Ω for 119896 = 1 2 119897 the design procedure of autoas-sociative memories is stated as follows

(1) Select matrix E as an identity matrix(2) According to (29) and (34) choose appropriate 119878119894 119886119894119879119894 119887119894119895 and 119888119894119895(3) Based on 119878119894 = max(|119894119894119901 minus 119886119894119889| |119894119894119901 minus 119886119894119889|) 119887119894119895 =

max(|119894119895| |119894119895|) and 119888119894119895 = max(|119888119894119895| | 119888119894119895|) the values of119894119895 119894119895 119888119894119895 and 119888119894119895 can be selected

(4) Calculate 119878119887 119878minus119887 and 119878119888 Adjust the values of 119894119895 119894119895 119888119894119895and 119888119894119895 to make sure that 119878119888 minus 119878minus119887 lt 0

(5) Solving the following inequalities 120578(119878119887 +2119878119888) exp119878119887120578 lt 1 120578(119878119888 + 119878119887(1 + 119878119888120578) exp119878119887120578) lt 1we can get 0 lt 120578 lt 120578

(6) Take a proper 120578 isin (0 120578) such that 120583119878119888 minus 119878minus119887 lt 0

The output 119908(119905) will converge to the related memorypattern when matrices119860 119861 119862 and E and scalar 120578 are chosenas above

35 Design Procedure of Heteroassociative Memories Given119897 (119897 le 2119899) vectors to be memorized as 119902(1) 119902(2) 119902(119897) whichcorrespond to the external input patterns 119906(1) 119906(2) 119906(119897)where 119902(119896) isin Ω 119906(119896) isin Ω 119896 = 1 2 119897 set E119880 = 119876 where119880 = [119906(1) 119906(2) 119906(119897)]119876 = [119902(1) 119902(2) 119902(119897)] It is clear thatheteroassociative memories can be treated as autoassociativememories when transform matrix E is obtained Matrices119860 119861 and 119862 and scalar 120578 can be selected as the methodused in autoassociative memories The transform matrix E isobtained by the following steps

(1) When 119897 = 119899 that is 119876 and 119880 are square matricesutilizing Matlab Toolbox we can get inverse matrixof 119880 as 119880minus1 and E can be obtained as E = 119876119880minus1

(2) When 119897 lt 119899 add 119899 minus 119897 proper column vectors to 119880and 119876 which constructs new matrices 119880 and 119876respectively such that the ranks of matrices 119880 and 119876are full And the transform matrix E can be obtainedby E = 119876119880minus1

(3) When 119899 lt 119897 add 119897 minus 119899 proper row vectors to 119880and 119876 which constructs new matrices and 119876 suchthat the ranks of matrices and 119876 are full Andthe transform matrix E can be obtained by E =119876minus1 Note in particular in this circumstance thatthe dimensionality of each input and output patternincreases from 119899 to 119897 And the front 119899 values of eachexternal input pattern carry out associativememories


4 Experimental Verification

In this section in order to verify the effectiveness of theproposed results several numerical examples are provided

Example 12 The first example is state response of autoasso-ciative memory

Consider neural network model (5) with 119899 = 12 where119901 = 119889 = 1 120578 = 29 119879119894 = 01 119886119894 = 04 and119894119895 = 119888119894119895 =

0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)

with the activation functions

119891119894 (119909119894) = minus12 (1003816100381610038161003816119909119894 + 011003816100381610038161003816 minus 1003816100381610038161003816119909119894 minus 011003816100381610038161003816)+ 12 (1003816100381610038161003816119909119894 + 111003816100381610038161003816 minus 1003816100381610038161003816119909119894 minus 111003816100381610038161003816)

(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

Figure 1 State response in neural network model (5) with externalinput 119906

for 119894 119895 = 1 2 12 Denote 119906 = 119902 = (1 1 1 1 minus1 minus1 1minus1 minus1 1 1 1)119879 as the external input vector and the memo-rized vector respectively

It follows fromTheorem 11 that neural networkmodel (5)has a unique equilibrium point which is globally exponen-tially stable The simulation result is shown in Figure 1

It can be calculated easily that the equilibrium point inFigure 1 is

119909lowast = (271 271 271 271 minus231 minus231 271minus 231 minus231 271 271 271)119879 (77)

Clearly the output pattern is

119908lowast = (1 1 1 1 minus1 minus1 1 minus1 minus1 1 1 1)119879 (78)

which is equivalent to the memorized pattern 119902 Henceneural network model (5) can be implemented effectively asan autoassociative memory

Remark 13 Because the equilibrium point is globally expo-nentially stable the initial values can be selected randomlyand there is no influence on the performance of associativememories

Example 14 The second example includes influences of 119886119894 119889and 119901 on the location of equilibrium points

The influences are divided into two cases to discuss

Case 1 Let 119901 = 1 be a constant and denote 119906 = 119902 =(1 1 1 1 minus1 minus1 1 minus1 minus1 1 1 1)119879 as the external input vec-tor and thememorized vector respectively Consider 119886119894 and 119889

Table 1 The influences of 119886119894 and 119889 on the location of equilibriumpoints when 119901 = 1 is a constant119889 12 14 16 20 24 30

11988611989403 V V V V V mdash04 V V V V mdash mdash05 V V V mdash mdash mdash06 V V mdash mdash mdash mdash07 V mdash mdash mdash mdash mdash

ldquoVrdquo (ldquomdashrdquo) denotes the equilibrium points located (not located) in thesaturation regions

Table 2 The influences of 119886119894 and 120572 = 119901119889 on the location ofequilibrium points when 119889 = 025 is a constant120572 = 119901119889 12 23 34 1 12 15

119886119894040 mdash V V V V V045 mdash mdash V V V V060 mdash mdash mdash V V V070 mdash mdash mdash mdash V V090 mdash mdash mdash mdash mdash V

ldquoVrdquo (ldquomdashrdquo) denotes the equilibrium points located (not located) in thesaturation region

as variables in neural network model (5) taking 120578 = 19 andthe rest of the parameters are the same as those in Example 12The results are shown in Table 1 from which we can see thatthe smaller the value of 119886119894 the bigger the value range 119889 cantake for the sake of memorized pattern and vice versa

Case 2 Let 119889 = 025 be a constant and denote 119906 = 119902 =(119901 119901 119901 119901 minus119901 minus119901 119901 minus119901 minus119901 119901 119901 119901)119879 as the external inputvector and the memorized vector respectively where 119901 gt 0Consider 119886119894 and119901 (or 120572 = 119901119889) as variables in neural networkmodel (5) taking 120578 = 19 and the rest of the parametersare the same as those in Example 12 The results are shown inTable 2 from which we can see that 120572 (or 119901) should increasewhen 119886119894 increases for the sake of memorized pattern and viceversa

Remark 15 From Tables 1 and 2 it can be concluded that theassociative memories performance becomes better with theincrease of 120572 = 119901119889 when 119886119894 is fixed but not vice versaExample 16 The third example includes the influences of119886119894 and perturbed 119906 on the recall success rate of memorizedpatterns

Let 119906 = (1 1 1 1 minus1 minus1 1 minus1 minus1 1 1 1)119879 120578 = 19and consider 119886119894 (119894 = 1 2 12) as variables in neuralnetwork model (5) The other parameters are the same asthose in Example 12 In this example the influences of 119886119894and perturbed 119906 are analyzed on the recall success rate ofmemorized vectors Experimental results are illustrated inTable 3 which makes it clear that the robustness of neuralnetworkmodel (5) for perturbed input119906will become strongeras 119886119894 is small enough


Table 3 The influences of 119886119894 and perturbed 119906 on the recall successrate

119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0

Figure 2 Typical evolutions of output pattern with input vector 119906 =119902 under three random initial values

Remark 17 Table 3 illustrates that the robustness of associa-tive memories on perturbed 119906 becomes stronger with thedecrease of 119886119894Example 18 The fourth example includes the application ofautoassociative memory without input perturbation

According to neural network model (5) design anautoassociative memory to memorize pattern ldquoTrdquo which isrepresented by a 4 times 3-pixel image (black pixel = 1 whitepixel = minus1) and the corresponding input pattern is 119906 =(1 1 1 minus1 1 minus1 minus1 1 minus1 minus1 1 minus1)119879Theparameters are thesame as those in Example 12

Simulation results with three random initial values aredepicted in Figure 2 where the dark grey block indicates thatthe value of the related component of output 119908(119905) is in (0 1)and the light grey block indicates that the value of the relatedcomponent of output119908(119905) is in (minus1 0) Obviously the outputpattern 119908(119905) converges to the memorized pattern correctly

Example 19 The fifth example includes the application ofautoassociative memory with input perturbation

The aim of this experiment is to discuss the influence ofinput pattern perturbation on the output patternThe param-eters are the same as those in Example 12 Choose patternldquoTrdquo to be memorized and the perturbed vector is selected as119906 = 119896 lowast 119902 where 119902 = (1 1 1 minus1 1 minus1 minus1 1 minus1 minus1 1 minus1)119879 isthe corresponding memorized vector and 119896 is a perturbationcoefficient

Figure 3 Typical evolutions of output pattern with input vector 119906 =052 lowast 119902 under three random initial values


In what follows the perturbed vector 119906 is imposed onneural network model (5) as input vector with 119896 = 052 and119896 = 041 respectively Simulation results are shown in Figures3 and 4 In Figure 3 the output pattern is exactly the same asthe memorized pattern and in Figure 4 the output pattern isdifferent from the memorized pattern although they can bedistinguished from the outlines

Remark 20 From the analysis above we can see that theoutput pattern will converge to the memorized patternwhen the perturbation coefficient is not small enough Infact neural network model (5) is implemented well as anassociative memory if the coefficient satisfying 119896 ge 052 andthe other parameters are the same as those in Example 19

Example 21 The sixth example includes the application ofheteroassociative memory

Consider neural network model (1) and the parametersare the same as those in Example 12 except the transformmatrix E which will be given later The aim of this ex-periment is to design a heteroassociative memory to mem-orize pattern ldquoUrdquo when the input pattern is ldquoCrdquo It is clear thatthe input vector is 119906 = (1 1 1 1 minus1 minus1 1 minus1 minus1 1 1 1)119879and the corresponding output vector is 119902 = (1 minus1 1 1 minus1 11 minus1 1 1 1 1 )119879


According to the heteroassociative memories design pro-cedure the matrices 119880 and 119876 are constructed as

119880 =

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

1 0 0 0 0 0 0 0 0 0 0 01 1 0 0 0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0minus1 0 0 0 1 0 0 0 0 0 0 0minus1 0 0 0 0 1 0 0 0 0 0 01 0 0 0 0 0 1 0 0 0 0 0minus1 0 0 0 0 0 0 1 0 0 0 0minus1 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 1 0 01 0 0 0 0 0 0 0 0 0 1 01 0 0 0 0 0 0 0 0 0 0 1

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

119876 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus1 1 0 0 0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0minus1 0 0 0 1 0 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0 0 01 0 0 0 0 0 1 0 0 0 0 0minus1 0 0 0 0 0 0 1 0 0 0 01 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 1 0 01 0 0 0 0 0 0 0 0 0 1 01 0 0 0 0 0 0 0 0 0 0 1

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

(79)

respectivelyUsing the Matlab Toolbox we can have

E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)

Figure 5 Typical evolutions of pattern ldquoUrdquo with input pattern ldquoCrdquounder three random initial values

The simulation results are depicted in Figure 5 whichshows that the output pattern ldquoUrdquo is memorized when theinput pattern is ldquoCrdquo and the transform matrix E is selectedas above This indicates that neural network model (1) canperform well as a heteroassociative memory under a propertransformmatrixE which can be easily obtained by using theMatlab Toolbox

Remark 22 Simulation results in Figure 5 show that theassociative memories model presented in this paper canbe implemented well as autoassociative memories and het-eroassociative memoriesThis model possesses the resistancecapacity against external input perturbations and can beregarded as an extension and complement to the relevantresults [10 11 16]

5 Concluding Remarks

Analysis and design of associative memories for recurrentneural networks have drawn considerable attention whereasanalysis and design of associative memories for memristiveneural networks with deviating argument have not been wellinvestigatedThis paper presents a general class ofmemristiveneural networks with deviating argument and discussesthe analysis and design of autoassociative memories andheteroassociative memories Meanwhile the robustness ofthis class of neural networks is demonstrated when externalinput patterns are seriously perturbed To the best of theauthorrsquos knowledge this is the first theoretical result onassociative memories for memristive neural networks withdeviating argument Future works may aim at (1) studyingassociative memories based on multistability of memristiveneural networks with deviating argument (2) analyzingassociative memories based on discrete-time memristiveneural networks with deviating argument (3) exploringhigh-capacity associative memories for memristive neuralnetworks with deviating argument

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper


Acknowledgments

This work is supported by the Research Project of HubeiProvincial Department of Education of China under GrantT201412

References

[1] J Tian and S Zhong ldquoImproved delay-dependent stability cri-terion for neural networks with time-varying delayrdquo AppliedMathematics and Computation vol 217 no 24 pp 10278ndash102882011

[2] Y Du S Zhong and N Zhou ldquoGlobal asymptotic stability ofMarkovian jumping stochastic Cohen-Grossberg BAM neuralnetworks with discrete and distributed time-varying delaysrdquoApplied Mathematics and Computation vol 243 pp 624ndash6362014

[3] W-H Chen X Lu and W X Zheng ldquoImpulsive stabilizationand impulsive synchronization of discrete-time delayed neuralnetworksrdquo IEEE Transactions on Neural Networks and LearningSystems vol 26 no 4 pp 734ndash748 2015

[4] H Zhang F Yang X Liu and Q Zhang ldquoStability analysis forneural networks with time-varying delay based on quadraticconvex combinationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 24 no 4 pp 513ndash521 2013

[5] S Wen T Huang Z Zeng Y Chen and P Li ldquoCircuit designand exponential stabilization of memristive neural networksrdquoNeural Networks vol 63 pp 48ndash56 2015

[6] A L Wu L Liu T W Huang and Z G Zeng ldquoMittag-Lefflerstability of fractional-order neural networks in the presence ofgeneralized piecewise constant argumentsrdquo Neural Networksvol 85 pp 118ndash127 2017

[7] C Hu J Yu Z H Chen H J Jiang and T W Huang ldquoFixed-time stability of dynamical systems and fixed-time synchro-nization of coupled discontinuous neural networksrdquo NeuralNetworks vol 89 pp 74ndash83 2017

[8] T W Huang ldquoRobust stability of delayed fuzzy Cohen-Gross-berg neural networksrdquo Computers ampMathematics with Applica-tions vol 61 no 8 pp 2247ndash2250 2011

[9] G Grass ldquoOn discrete-time cellular neural networks for asso-ciative memoriesrdquo IEEE Transactions on Circuits and Systems IFundamentalTheory and Applications vol 48 no 1 pp 107ndash1112001

[10] Q Han X Liao T Huang J Peng C Li and H Huang ldquoAnal-ysis and design of associative memories based on stability ofcellular neural networksrdquoNeurocomputing vol 97 pp 192ndash2002012

[11] M S Tarkov ldquoOscillatory neural associative memories withsynapses based on memristor bridgesrdquo Optical Memory andNeural Networks vol 25 no 4 pp 219ndash227 2016

[12] M Baghelani A Ebrahimi and H B Ghavifekr ldquoDesign of RFMEMS based oscillatory neural network for ultra high speedassociative memoriesrdquo Neural Processing Letters vol 40 no 1pp 93ndash102 2014

[13] M Kobayashi ldquoGradient descent learning rule for complex-valued associative memories with large constant termsrdquo IEEJTransactions on Electrical and Electronic Engineering vol 11 no3 pp 357ndash363 2016

[14] A R Trivedi S Datta and S Mukhopadhyay ldquoApplicationof silicon-germanium source tunnel-fet to enable ultralowpower cellular neural network-based associativememoryrdquo IEEE

Transactions on Electron Devices vol 61 no 11 pp 3707ndash37152014

[15] Y Wu and S N Batalama ldquoImproved one-shot learning forfeedforward associative memories with application to compos-ite pattern associationrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 31 no 1 pp 119ndash125 2001

[16] C Zhou X Zeng J Yu and H Jiang ldquoA unified associativememory model based on external inputs of continuous recur-rent neural networksrdquo Neurocomputing vol 186 pp 44ndash532016

[17] R Zhang D Zeng S Zhong and Y Yu ldquoEvent-triggered sam-pling control for stability and stabilization of memristive neuralnetworks with communication delaysrdquo Applied Mathematicsand Computation vol 310 pp 57ndash74 2017

[18] G Zhang and Y Shen ldquoExponential stabilization of memristor-based chaotic neural networks with time-varying delays viaintermittent controlrdquo IEEE Transactions on Neural Networksand Learning Systems vol 26 no 7 pp 1431ndash1441 2015

[19] SWen Z Zeng T Huang and Y Zhang ldquoExponential adaptivelag synchronization of memristive neural networks via fuzzymethod and applications in pseudo random number genera-torsrdquo IEEE Transactions on Fuzzy Systems 2013

[20] AWu and Z Zhigang ldquoLagrange stability of memristive neuralnetworks with discrete and distributed delaysrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 25 no 4pp 690ndash703 2014

[21] S B Ding Z S Wang J D Wang and H G Zhang ldquoHinfin stateestimation for memristive neural networks with time-varyingdelaysThe discrete-time caserdquoNeural Networks vol 84 pp 47ndash56 2016

[22] Q Zhong J Cheng Y Zhao J Ma and B Huang ldquoFinite-timeHinfin filtering for a class of discrete-time Markovian jump sys-tems with switching transition probabilities subject to averagedwell time switchingrdquo Applied Mathematics and Computationvol 225 pp 278ndash294 2013

[23] K Shi X Liu H Zhu S Zhong Y Zeng and C Yin ldquoNoveldelay-dependent master-slave synchronization criteria of cha-otic Lurrsquoe systems with time-varying-delay feedback controlrdquoApplied Mathematics and Computation vol 282 pp 137ndash1542016

[24] M U Akhmet D Arugaslan and E Yilmaz ldquoStability analysisof recurrent neural networks with piecewise constant argumentof generalized typerdquoNeural Networks vol 23 no 7 pp 805ndash8112010

[25] MUAkhmet DArugaslan andE Yılmaz ldquoStability in cellularneural networks with a piecewise constant argumentrdquo Journalof Computational and Applied Mathematics vol 233 no 9 pp2365ndash2373 2010

[26] M U Akhmet and E Yılmaz ldquoImpulsive Hopfield-type neuralnetwork system with piecewise constant argumentrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 2584ndash25932010

[27] K-S Chiu M Pinto and J-C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae vol 133pp 133ndash152 2014

[28] L G Wan and A L Wu ldquoStabilization control of generalizedtype neural networks with piecewise constant argumentrdquo Jour-nal of Nonlinear Science and Applications vol 9 no 6 pp 3580ndash3599 2016


[29] L Liu A L Wu Z G Zeng and T W Huang ldquoGlobal meansquare exponential stability of stochastic neural networks withretarded and advanced argumentrdquoNeurocomputing vol 247 pp156ndash164 2017

[30] A Wu and Z Zeng ldquoOutput convergence of fuzzy neurody-namic system with piecewise constant argument of generalizedtype and time-varying inputrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1689ndash17022016

Submit your manuscripts athttpswwwhindawicom

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


Operations ResearchAdvances in

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

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


effective analysis methods for memristive neural networks isworth studying

For the past few years hybrid dynamic systems haveremained one of the most active fields of research in thecontrol community [22ndash30] For instance to describe the sta-tionary distribution of temperature along the length of a wirethat is bended the nonlinear dynamic model with deviatingargument is often used The right-hand side in nonlinearsystems with deviating argument features a combination ofcontinuous and discrete systems Thus nonlinear systemswith deviating argument unify the advanced and retardedsystems Because of this property relatedworks on the controlstrategies for nonlinear systems with deviating argument arerelatively rare Viewed from systems involving the interplaydifferential equations and difference equations are includedin the nonlinear systems with deviating argument [28ndash30]However it is still obvious thatmany basic issues onnonlinearsystems with deviating argument remain to be addressedsuch as nonlinear dynamics systems design and analy-sis

Inspired by the above discussions the goal of this paper isto formulate the analysis and design of associative memoriesfor a class of memristive neural networks with deviatingargument On the whole the highlights of this paper can beoutlined as follows

(1) Sufficient conditions are derived to ascertain the exis-tence uniqueness and global exponential stabilityof the solution for memristive neural networks withdeviating argument

(2) The syntheticmechanismof the analysis and design ofassociativememories for a general class ofmemristiveneural networks with deviating argument is revealed

(3) A uniform associative memories model which unitesautoassociative memories and heteroassociativememories is proposed for memristive neural net-works with deviating argument

In addition it should be noted thatwhen special and strictconditions are exerted associative memoriesrsquo performance ofneural networks is usually limited Therefore the methodsapplicable to conventional stability analysis of neural net-works cannot be directly employed to investigate the asso-ciative memories for memristive neural networks Moreoverthe study of dynamics for memristive neural networks withdeviating argument is not an easy work since the conditionsfor dynamics of neural networks cannot be simply utilized toanalyze hybrid dynamic systems with deviating argument Inthis paper according to the fuzzy uncertainty approach incombination with the theories of set-valued maps and differ-ential inclusions analysis and design of associative memoriesfor memristive neural networks with deviating argument aredescribed in detail

The rest of the paper is arranged as follows Design prob-lem and preliminaries are given in Section 2 Main results arestated in Section 3 In Section 4 several numerical examplesare presented Finally concluding remarks are given inSection 5

2 Design Problem and Preliminaries

21 Problem Description Let Ω = 120573 = (1205731 1205732 120573119899)119879 |120573119894 = minus119901 119901 119894 = 1 2 119899 be the set of 119899-dimensionalbipolar vectors where 119901 is a positive constant and 119879 denotesthe transpose of a vector or matrix The problem consideredis described as follows

Given 119897 (119897 le 2119899) pairs of bipolar vectors (119906(1) 119902(1))(119906(2) 119902(2)) (119906(119897) 119902(119897)) where 119906(119903) 119902(119903) isin Ω for 119903 =1 2 119897 design an associative memory model which satis-fies the notion that the output of the model will converge tothe corresponding pattern 119902(119903) when 119906(119903) is fed to the modelinput as a memory pattern

Definition 1 The neural network is said to be an autoassocia-tive memory if 119906(119903) = 119902(119903) and a heteroassociative memory if119906(119903) = 119902(119903) for 119903 = 1 2 119897

In this paper let 119873 be the natural number set the normof vector 119909 = (1199091 1199092 119909119899)119879 is defined as 119909 = sum119899119894=1 |119909119894|DenoteR119899 as the 119899-dimensional Euclidean space Fix two realnumber sequences 120578119896 120599119896 119896 isin 119873 satisfying 120578119896 lt 120578119896+1120578119896 le 120599119896 le 120578119896+1 and lim119896rarr+infin120578119896 = +infin

22 Model Consider the following neural network model

(119905) = minus119860119909 (119905) + 119861 (119909 (119905)) 119891 (119909 (119905))+ 119862 (119909 (119905)) 119891 (119909 (120574 (119905))) + E119906(119903)

119908 (119905) = 119891 (119909 (119905)) 119903 = 1 2 119897

(1)

where 119909(119905) = (1199091(119905) 1199092(119905) 119909119899(119905))119879 isin R119899 is the neuronstate 119860 = diag(1198861 1198862 119886119899) which is a constant diagonalmatrix with 119886119894 gt 0 119894 isin 1 2 119899 indicates the self-inhibition matrix 119861(119909(119905)) = (119887119894119895(119909119895(119905)))119899times119899 and 119862(119909(119905)) =(119888119894119895(119909119895(119905)))119899times119899 are connection weight matrices at time 119905 and120574(119905) respectively 119887i119895(119909119895(119905)) and 119888119894119895(119909119895(119905)) are defined by

119887119894119895 (119909119895 (119905)) = 119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895

119888119894119895 (119909119895 (119905)) = 119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895

(2)

for 119894 119895 = 1 2 119899 and 119887119894119895(plusmn119879119895) = 119894119895 or 119894119895 119888119894119895(plusmn119879119895) =119888119894119895 or 119888119894119895 where 119879119895 gt 0 represents the switching jump and119894119895 119894119895 119888119894119895 and 119888119894119895 are all constants 119891(119909(sdot)) = (1198911(1199091(sdot)1198912(1199092(sdot) 119891119899(119909119899(sdot))119879 is the activation function The devi-ating function 120574(119905) = 120599119896 when 119905 isin [120578119896 120578119896+1) for any 119896 isin 119873 Estands for a transform matrix of order 119899 satisfying E119906(119903) =119902(119903) for 119903 = 1 2 119897 119906(119903) and 119902(119903) indicate the externalinput pattern and the corresponding memorized patternrespectively 119908(119905) = (1199081(119905) 1199082(119905) 119908119899(119905))119879 is the output of(1) corresponding to 119906(119903)






119891119894 (120594) = 2sumℓ=1


(3)






119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894

119908119894 (119905) = 119891119894 (119909119894 (119905)) 119894 = 1 2 119899

(5)


119870[119887119894119895 (119909119895 (119905))] =

119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119900 119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 = 119879119895119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895(6)

119870[119888119894119895 (119909119895 (119905))] =

119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119900 119888119894119895 119888 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 = 119879119895119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895(7)


from (5) we can get

119894 (119905) isin minus119886119894119909119894 (119905) + 119899sum119895=1

119870[119887119894119895 (119909119895 (119905))] 119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[119888119894119895 (119909119895 (119905))] 119891119895 (119909119895 (120574 (119905))) + 119906119894(8)


119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887lowast119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888lowast119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894119894 = 1 2 119899

(9)



119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

120575119887119894119895 (119909119895 (119905)) 119887minus119894119895119891119895 (119909119895 (119905)) + 119899sum119895=1

120575119888119894119895 (119909119895 (119905))times 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(10)


where



(11)





119870[119888119894119895 (119909lowast119895 )] 119891119895 (119909lowast119895 ) + 119906119894 119894 = 1 2 119899(12)


minus 119886119894119909lowast119894 + 119899sum119895=1


119888119894119895 (119909lowast119895 ) 119891119895 (119909lowast119895 ) + 119906119894= 0

(13)





(15)





(A2) 120578(119878119887 + 2119878119888) exp119878119887120578 lt 1(A3) 120578[119878119888 + 119878119887(1 + 119878119888120578) exp119878119887120578] lt 1

3 Main Results






equivalent equation

V119894 (119905) = 1199090119894 + int1199051199050

[[minus119886119894V119894 (119904) +

119899sum119895=1

1198870119894119895119891119895 (V119895 (119904)) + 119899sum119895=1

1198880119894119895times 119891119895 (V119895 (120599119896)) + 119906119894 + 119899sum

119895=1

120575119887119894119895 (V119895 (119905)) 119887minus119894119895119891119895 (V119895 (119904))

+ 119899sum119895=1

120575119888119894119895 (V119895 (119905)) 119888minus119894119895119891119895 (V119895 (120599119896))]] d119904119894 = 1 2 119899

(17)


1199050

[[minus119886119894V

119898119894 (119904) + 119899sum

119895=1

1198870119894119895119891119895 (V119898119895 (119904))+ 119899sum119895=1

1198880119894119895 times 119891119895 (V119898119895 (120599119896)) + 119906119894+ 119899sum119895=1

120575119887119894119895 (V119898119895 (119905)) 119887minus119894119895119891119895 (V119898119895 (119904))


+ 119899sum119895=1

120575119888119894119895 (V119898119895 (119905)) 119888minus119894119895119891119895 (V119898119895 (120599119896))]] d119904119894 = 1 2 119899

(18)


10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 le [(119878119887 + 119878119888) 120578]119898 120589 (19)


lim119898rarr+infin

10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 = 0 (20)




Then we get


119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(21)



(22)

and particularly


(23)

Hence



+ int1199050

119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(25)

From (A2) we get


that is






lt 100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

(28)




119906119894 gt 119878119894 + 119886119894119879119894 + 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 (29)



119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) times 119891119895 (119909119895 (120574 (119905))) + 119906119894



119887119894119895119901 minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1


119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 minus119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 0(30)


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 119878119894 gt 0(31)


then


119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894

= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] minus

119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0(32)


then


119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0

(33)





119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 (34)




119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)


119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1


119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)


then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)





Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









119891119894 (120594) = 2sumℓ=1


(3)






119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894

119908119894 (119905) = 119891119894 (119909119894 (119905)) 119894 = 1 2 119899

(5)


119870[119887119894119895 (119909119895 (119905))] =

119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119900 119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 = 119879119895119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895(6)

119870[119888119894119895 (119909119895 (119905))] =

119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 gt 119879119895119888119900 119888119894119895 119888 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 = 119879119895119888119894119895 10038161003816100381610038161003816119909119895 (119905)10038161003816100381610038161003816 lt 119879119895(7)


from (5) we can get

119894 (119905) isin minus119886119894119909119894 (119905) + 119899sum119895=1

119870[119887119894119895 (119909119895 (119905))] 119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[119888119894119895 (119909119895 (119905))] 119891119895 (119909119895 (120574 (119905))) + 119906119894(8)


119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887lowast119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888lowast119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894119894 = 1 2 119899

(9)



119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

120575119887119894119895 (119909119895 (119905)) 119887minus119894119895119891119895 (119909119895 (119905)) + 119899sum119895=1

120575119888119894119895 (119909119895 (119905))times 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(10)


where



(11)





119870[119888119894119895 (119909lowast119895 )] 119891119895 (119909lowast119895 ) + 119906119894 119894 = 1 2 119899(12)


minus 119886119894119909lowast119894 + 119899sum119895=1


119888119894119895 (119909lowast119895 ) 119891119895 (119909lowast119895 ) + 119906119894= 0

(13)





(15)





(A2) 120578(119878119887 + 2119878119888) exp119878119887120578 lt 1(A3) 120578[119878119888 + 119878119887(1 + 119878119888120578) exp119878119887120578] lt 1

3 Main Results






equivalent equation

V119894 (119905) = 1199090119894 + int1199051199050

[[minus119886119894V119894 (119904) +

119899sum119895=1

1198870119894119895119891119895 (V119895 (119904)) + 119899sum119895=1

1198880119894119895times 119891119895 (V119895 (120599119896)) + 119906119894 + 119899sum

119895=1

120575119887119894119895 (V119895 (119905)) 119887minus119894119895119891119895 (V119895 (119904))

+ 119899sum119895=1

120575119888119894119895 (V119895 (119905)) 119888minus119894119895119891119895 (V119895 (120599119896))]] d119904119894 = 1 2 119899

(17)


1199050

[[minus119886119894V

119898119894 (119904) + 119899sum

119895=1

1198870119894119895119891119895 (V119898119895 (119904))+ 119899sum119895=1

1198880119894119895 times 119891119895 (V119898119895 (120599119896)) + 119906119894+ 119899sum119895=1

120575119887119894119895 (V119898119895 (119905)) 119887minus119894119895119891119895 (V119898119895 (119904))


+ 119899sum119895=1

120575119888119894119895 (V119898119895 (119905)) 119888minus119894119895119891119895 (V119898119895 (120599119896))]] d119904119894 = 1 2 119899

(18)


10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 le [(119878119887 + 119878119888) 120578]119898 120589 (19)


lim119898rarr+infin

10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 = 0 (20)




Then we get


119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(21)



(22)

and particularly


(23)

Hence



+ int1199050

119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(25)

From (A2) we get


that is






lt 100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

(28)




119906119894 gt 119878119894 + 119886119894119879119894 + 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 (29)



119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) times 119891119895 (119909119895 (120574 (119905))) + 119906119894



119887119894119895119901 minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1


119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 minus119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 0(30)


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 119878119894 gt 0(31)


then


119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894

= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] minus

119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0(32)


then


119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0

(33)





119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 (34)




119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)


119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1


119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)


then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)





Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





where



(11)





119870[119888119894119895 (119909lowast119895 )] 119891119895 (119909lowast119895 ) + 119906119894 119894 = 1 2 119899(12)


minus 119886119894119909lowast119894 + 119899sum119895=1


119888119894119895 (119909lowast119895 ) 119891119895 (119909lowast119895 ) + 119906119894= 0

(13)





(15)





(A2) 120578(119878119887 + 2119878119888) exp119878119887120578 lt 1(A3) 120578[119878119888 + 119878119887(1 + 119878119888120578) exp119878119887120578] lt 1

3 Main Results






equivalent equation

V119894 (119905) = 1199090119894 + int1199051199050

[[minus119886119894V119894 (119904) +

119899sum119895=1

1198870119894119895119891119895 (V119895 (119904)) + 119899sum119895=1

1198880119894119895times 119891119895 (V119895 (120599119896)) + 119906119894 + 119899sum

119895=1

120575119887119894119895 (V119895 (119905)) 119887minus119894119895119891119895 (V119895 (119904))

+ 119899sum119895=1

120575119888119894119895 (V119895 (119905)) 119888minus119894119895119891119895 (V119895 (120599119896))]] d119904119894 = 1 2 119899

(17)


1199050

[[minus119886119894V

119898119894 (119904) + 119899sum

119895=1

1198870119894119895119891119895 (V119898119895 (119904))+ 119899sum119895=1

1198880119894119895 times 119891119895 (V119898119895 (120599119896)) + 119906119894+ 119899sum119895=1

120575119887119894119895 (V119898119895 (119905)) 119887minus119894119895119891119895 (V119898119895 (119904))


+ 119899sum119895=1

120575119888119894119895 (V119898119895 (119905)) 119888minus119894119895119891119895 (V119898119895 (120599119896))]] d119904119894 = 1 2 119899

(18)


10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 le [(119878119887 + 119878119888) 120578]119898 120589 (19)


lim119898rarr+infin

10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 = 0 (20)




Then we get


119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(21)



(22)

and particularly


(23)

Hence



+ int1199050

119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(25)

From (A2) we get


that is






lt 100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

(28)




119906119894 gt 119878119894 + 119886119894119879119894 + 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 (29)



119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) times 119891119895 (119909119895 (120574 (119905))) + 119906119894



119887119894119895119901 minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1


119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 minus119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 0(30)


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 119878119894 gt 0(31)


then


119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894

= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] minus

119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0(32)


then


119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0

(33)





119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 (34)




119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)


119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1


119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)


then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)





Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





+ 119899sum119895=1

120575119888119894119895 (V119898119895 (119905)) 119888minus119894119895119891119895 (V119898119895 (120599119896))]] d119904119894 = 1 2 119899

(18)


10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 le [(119878119887 + 119878119888) 120578]119898 120589 (19)


lim119898rarr+infin

10038171003817100381710038171003817V119898+1 (119905) minus V119898 (119905)10038171003817100381710038171003817119872 = 0 (20)




Then we get


119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(21)



(22)

and particularly


(23)

Hence



+ int1199050

119878119887 100381710038171003817100381710038171199091 (119904) minus 1199092 (119904)10038171003817100381710038171003817 d119904(25)

From (A2) we get


that is






lt 100381710038171003817100381710038171199091 minus 119909210038171003817100381710038171003817

(28)




119906119894 gt 119878119894 + 119886119894119879119894 + 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 (29)



119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) times 119891119895 (119909119895 (120574 (119905))) + 119906119894



119887119894119895119901 minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1


119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 minus119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 0(30)


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 119878119894 gt 0(31)


then


119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894

= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] minus

119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0(32)


then


119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0

(33)





119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 (34)




119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)


119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1


119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)


then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)





Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119887119894119895119901 minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1


119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894ge minus 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 minus119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 0(30)


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 119878119894 gt 0(31)


then


119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894

= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]minus 119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] minus

119899sum119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0(32)


then


119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1


119895=1119895 =119894

119887119894119895119901minus 119899sum119895=1

119888119894119895119901 + 119906119894 gt 2119886119894119879119894 gt 0

(33)





119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 (34)




119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)


119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1


119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)


then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)





Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119894 (119905) = minus119886119894119909119894 (119905) + 119899sum119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1

119888119894119895 (119909119895 (119905)) 119891119895 (119909119895 (120574 (119905))) + 119906119894le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le minus119886119894 (119889 + 119879119894) + 119887119894119894 (119909119894 (119905)) 119901 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816 119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(35)


119895=1

119887119894119895 (119909119895 (119905)) 119891119895 (119909119895 (119905))+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1


119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894le 1003816100381610038161003816100381610038161003816minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889

1003816100381610038161003816100381610038161003816 119889 minus 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119878119894 minus 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus2119886119894119879119894lt 0

(36)


119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894le 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 lt minus119878119894 lt 0(37)


then

119894 (119905) le minus119886119894119909119894 (119905) + 119887119894119894 (119909119894 (119905)) 119891119894 (119909119894 (119905)) + 119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894= minus119886119894 [119909119894 (119905) + 119879119894] + 119887119894119894 (119909119894 (119905)) 119901119889 [119909119894 (119905) + 119879119894]+ 119899sum119895=1119895 =119894

119887119894119895119901 + 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894= [minus119886119894 + 119887119894119894 (119909119894 (119905)) 119901119889] [119909119894 (119905) + 119879119894] +

119899sum119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 + 119886119894119879119894le 1003816100381610038161003816119887119894119894 (119909119894 (119905)) 119901 minus 1198861198941198891003816100381610038161003816 + 119886119894119879119894 + 119899sum

119895=1119895 =119894

119887119894119895119901+ 119899sum119895=1

119888119894119895119901 + 119906119894 lt 0

(38)





Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Δ 1 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

Δ 2 = 119894 | 119894 isin 1 2 119899 119906119894 gt 119878119894 + 119886119894119879119894 +119899sum119895=1119895 =119894

119887119894119895119901

+ 119899sum119895=1

119888119894119895119901

(39)




isin 1 2 119899 (40)


119865 (119909) = (1198651 (119909) 1198652 (119909) 119865119899 (119909))119879 (41)

where

119865119894 (119909) = 1119886119894 [[119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]] 119894 = 1 2 119899(42)


119906119894 gt 119878119894 + 119886119894119879119894 + sum119895=1119895 =119894

119887119894119895119901 + sum119895=1

119888119894119895119901 + 1198861198941205761 (43)


119895=1119895 =119894

119887119894119895119901 minus sum119895=1

119888119894119895119901 minus 1198861198941205762 (44)


Θ = 119909 isin R119899 | 119909119894 isin [119889 + 119879119894 + 120576 (119889 + 119879119894)120576 ] 119894



119899sum119895=1

119870[119887119894119895 (119909119895)] 119891119895 (119909119895)

+ 119899sum119895=1

119870[119888119894119895 (119909119895)] 119891119895 (119909119895) + 119906119894]]ge 1119886119894 [[119870 [119887119894119894 (119909119894)] 119901 minus

119899sum119895=1119895 =119894

119887119894119895119901 minus 119899sum119895=1

119888119894119895119901 + 119906119894]]gt 1119886119894 [(119870 [119887119894119894 (119909119894)] 119901 minus 119886119894119889) + 119878119894 + 119886119894119879119894 + 119886119894119889 + 119886119894120576]ge 119879119894 + 119889 + 120576

(46)





119895=1

1198870119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

1198880119894119895119891119895 (119909119895 (120574 (119905))) + 119906119894+ 119899sum119895=1

119870[120575119887119894119895 (119909119895 (119905))] 119887minus119894119895119891119895 (119909119895 (119905))+ 119899sum119895=1

119870[120575119888119894119895 (119909119895 (119905))] 119888minus119894119895119891119895 (119909119895 (120574 (119905)))

(48)


119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

119860119887119894119895 (119910119895 (119905))+ 119899sum119895=1

119860119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(49)



119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119910119894 (119905) = minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))+ 119899sum119895=1

120593119888119894119895 (119910119895 (120574 (119905))) 119894 = 1 2 119899(50)




119910119894 (119905) = 119910119894 (120599119896) + int119905120599119896

[[minus119886119894119910119894 (119904) +

119899sum119895=1

1198870119894119895119892119895 (119910119895 (119904))

+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120599119896)) + 119899sum119895=1

120593119887119894119895 (119910119895 (119904))

+ 119899sum119895=1

120593119888119894119895 (119910119895 (120599119896))]] d119904 119894 = 1 2 119899

(53)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119886119894

1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904119894 = 1 2 119899

(54)


1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120599119896))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119904))10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120599119896))10038161003816100381610038161003816]] d119904

(55)

Applying Lemma 4

119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119904)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120599119896)10038161003816100381610038161003816]] d119904

= 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816 + int119905120599119896

[[119899sum119894=1

119886119894 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816

+ 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(56)

that is

1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119899sum119894=1

1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816+ int119905120599119896

[[119899sum119894=1

(119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119904)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1

120572 (119888+119895119894 + 119888minus119895119894)) 1003816100381610038161003816119910119894 (120599119896)1003816100381610038161003816]] d119904

(57)

then1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817] d119904le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 + int119905

120599119896

119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 d119904(58)


1003817100381710038171003817119910 (119905)1003817100381710038171003817 le (1 + 119878119888120578) 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 expint119905120599119896

119878119887d119904le (1 + 119878c120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(59)



120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






120599119896

[119878119887 1003817100381710038171003817119910 (119904)1003817100381710038171003817 + 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + int119905

120599119896

[119878119887 ((1 + 119878119888120578) exp 119878119887120578 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817)+ 119878119888 1003817100381710038171003817100381711991012059911989610038171003817100381710038171003817] d119904 le 1003817100381710038171003817119910 (119905)1003817100381710038171003817 + 120578 [119878119888 + 119878119887 (1+ 119878119888120578)] 1003817100381710038171003817119910 (120599119896)1003817100381710038171003817

(60)

and then

1003817100381710038171003817119910 (120599119896)1003817100381710038171003817 le 120583 1003817100381710038171003817119910 (119905)1003817100381710038171003817 (61)





120583119878119888 minus 119878minus119887 lt 0 (62)

Proof Denote

119911119894 (119905) = exp 120582119905 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 119894 = 1 2 119899 (63)





sdot (minus119886119894119910119894 (119905) + 119899sum119895=1

1198870119894119895119892119895 (119910119895 (119905))+ 119899sum119895=1

1198880119894119895119892119895 (119910119895 (120574 (119905))) + 119899sum119895=1

120593119887119894119895 (119910119895 (119905))

+ 119899sum119895=1


119895=1

119887+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895 10038161003816100381610038161003816119892119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816 +119899sum119895=1

10038161003816100381610038161003816120593119887119894119895 (119910119895 (119905))10038161003816100381610038161003816+ 119899sum119895=1

10038161003816100381610038161003816120593119888119894119895 (119910119895 (120574 (119905)))10038161003816100381610038161003816) 119894 = 1 2 119899(65)

Applying Lemma 4

119894 (119905) le exp 120582119905((120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816 +119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816+ 119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816) 119894 = 1 2 119899

(66)

Let

119881 (119905) = 119899sum119894=1

119911119894 (119905) (67)

then for 119905 = 120578119896 119896 isin 119873

(119905) = 119899sum119894=1

119894 (119905) le exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887+119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888+119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119887minus119894119895120572 10038161003816100381610038161003816119910119895 (119905)10038161003816100381610038161003816 +119899sum119894=1

119899sum119895=1

119888minus119894119895120572 10038161003816100381610038161003816119910119895 (120574 (119905))10038161003816100381610038161003816)

= exp 120582119905( 119899sum119894=1

(120582 minus 119886119894) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887+119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816+ 119899sum119894=1

119899sum119895=1

119888+119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816 + 119899sum119894=1

119899sum119895=1

119887minus119895119894120572 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816


+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





+ 119899sum119894=1

119899sum119895=1

119888minus119895119894120572 1003816100381610038161003816119910119894 (120574 (119905))1003816100381610038161003816) = exp 120582119905

sdot [[119899sum119894=1

(120582 minus 119886119894 + 119899sum119895=1

120572 (119887+119895119894 + 119887minus119895119894)) 1003816100381610038161003816119910119894 (119905)1003816100381610038161003816

+ 119899sum119894=1

( 119899sum119895=1


(68)

Based on Lemma 9


From (64)

120582 minus 119878minus119887 + 120583119878119888 lt 0 (70)

then



119881 (119905) = exp 120582119905 119899sum119894=1

1003816100381610038161003816119910119894 (119905)1003816100381610038161003816 = exp 120582119905 1003817100381710038171003817119910 (119905)1003817100381710038171003817 le 119881 (0)= 1003817100381710038171003817119910 (0)1003817100381710038171003817

(72)




















0012 119894 = 119895001 119894 = 119895

119894119895 = 119888119894119895 = 0008 119894 = 1198950006 119894 = 119895

(75)



(76)


minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus8

minus6

minus4

minus2

0

2

4

6

8

x

2 4 6 8 10 12 14 160t

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

























119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119886119894 03 04 05 06 0709u 100 100 100 100 100084u 100 100 100 100 6667063u 100 100 100 6667 0052u 100 100 6667 0041u 100 6667 0025u 6667 0024u 0















119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119880 =

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


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

119876 =

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


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

(79)


E = 119876119880minus1 =

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

1 0 0 0 0 0 0 0 0 0 0 0minus2 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 02 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 02 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

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

(80)









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




ResearchArticle · 2019. 7. 30. · Analysis and Design of Associative Memories for Memristive...

Documents

Transcript of ResearchArticle · 2019. 7. 30. · Analysis and Design of Associative Memories for Memristive...