Research Article Hopf Bifurcation and Global Periodic ...downloads.hindawi.com/journals/aaa/2014/835310.pdf · Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System

Research ArticleHopf Bifurcation and Global Periodic Solutions in aPredator-Prey System with Michaelis-Menten Type FunctionalResponse and Two Delays

Yunxian Dai1 Yiping Lin1 and Huitao Zhao12

1 Department of Applied Mathematics Kunming University of Science and Technology Kunming Yunnan 650093 China2Department of Mathematics and Information Science Zhoukou Normal University Zhoukou Henan 466001 China

Correspondence should be addressed to Yiping Lin linyiping689163com

Received 27 November 2013 Accepted 31 March 2014 Published 11 May 2014

Academic Editor Yuming Chen

Copyright copy 2014 Yunxian Dai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider a predator-prey systemwithMichaelis-Menten type functional response and two delaysWe focus on the case with twounequal and non-zero delays present in the model study the local stability of the equilibria and the existence of Hopf bifurcationand then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and centermanifold theorem Special attention is paid to the global continuation of local Hopf bifurcation when the delays 120591

1= 1205912

1 Introduction

In [1] Xu and Chaplain studied the following delayedpredator-prey model with Michaelis-Menten type functionalresponse

1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905 minus 12059111) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905 minus 12059121)

1198981+ 1199091(119905 minus 12059121)

minus119886221199092(119905 minus 12059122) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 12059132)

1198982+ 1199092(119905 minus 12059132)

minus 119886331199093(119905 minus 12059133)]

(1)

with initial conditions

119909119894(119905) = 120601

119894(119905) 119905 isin [minus120591 0] 120601

119894(0) gt 0 119894 = 1 2 3 (2)

where 1199091(119905) 119909

2(119905) and 119909

3(119905) denote the densities of the

prey predator and top predator population respectively

119886119894 119886119894119895(119894 119895 = 1 2 3) are positive constants 120591

11 12059121 12059122 12059132

and 12059133

are nonnegative constants 12059111 12059122 12059133

denote thedelay in the negative feedback of the prey predator andtop predator crowding respectively 120591

21 12059132 are constant

delays due to gestation that is mature adult predators canonly contribute to the production of predator biomass 120591 =

max12059111 12059121 12059122 12059132 12059133 120601119894(119905) (119894 = 1 2 3) are continuous

bounded functions in the interval [minus120591 0] The authorsproved that the system is uniformly persistent under someappropriate conditions By means of constructing suitableLyapunov functional sufficient conditions are derived for theglobal asymptotic stability of the positive equilibrium of thesystem

Time delays of one type or another have been incorpo-rated into systems by many researchers since a time delaycould cause a stable equilibrium to become unstable andfluctuation In [2ndash12] authors showed effects of two delayson dynamical behaviors of system

It is well known that periodic solutions can arise throughtheHopf bifurcation in delay differential equations Howeverthese periodic solutions bifurcating from Hopf bifurcationsare generally local Under some circumstances periodicsolutions exist when the parameter is far away from thecritical value Therefore global existence of Hopf bifurcation

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 835310 16 pageshttpdxdoiorg1011552014835310

2 Abstract and Applied Analysis

is a more interesting and difficult topic A great deal ofresearch has been devoted to the topics [13ndash21] In this paperlet 12059111

= 12059122

= 12059133

= 0 12059121

= 1205911 12059132

= 1205912in (1) we

considerHopf bifurcation and global periodic solutions of thefollowing system with two unequal and nonzero delays

1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905 minus 1205911)

1198981+ 1199091(119905 minus 1205911)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 1205912)

1198982+ 1199092(119905 minus 1205912)

minus 119886331199093(119905)]

(3)


119909119894(119905) = 120601

119894(119905) 119905 isin [minus120591 0] 120601

119894(0) gt 0

119894 = 1 2 3 120591 = max 1205911 1205912

(4)

Our goal is to investigate the possible stability switches of thepositive equilibrium and stability of periodic orbits arisingdue to a Hopf bifurcation when one of the delays is treatedas a bifurcation parameter Special attention is paid to theglobal continuation of local Hopf bifurcation when the delays1205911

= 1205912

This paper is organized as follows In Section 2 byanalyzing the characteristic equation of the linearized systemof system (3) at positive equilibrium the sufficient conditionsensuring the local stability of the positive equilibrium andthe existence of Hopf bifurcation are obtained [22] Someexplicit formulas determining the direction and stabilityof periodic solutions bifurcating from Hopf bifurcationsare demonstrated by applying the normal form methodand center manifold theory due to Hassard et al [23] inSection 3 In Section 4 we consider the global existence ofthese bifurcating periodic solutions [24] with two differentdelays Some numerical simulation results are included inSection 5

2 Stability of the Positive Equilibrium andLocal Hopf Bifurcations

In this section we first study the existence and local stabilityof the positive equilibrium and then investigate the effect ofdelay and the conditions for existence of Hopf bifurcations

There are at most four nonnegative equilibria for system(3)

1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864

lowast= (119909lowast

1 119909lowast

2 119909lowast

3)

(5)

where (1199091 1199092 0) and (119909

lowast

1 119909lowast

2 119909lowast

3) satisfy

1198861minus 119886111199091minus

119886121199092

1198981+ 1199091

= 0

minus1198862+

119886211199091

1198981+ 1199091

minus 119886221199092= 0

(6)

1198861minus 11988611119909lowast

1(119905) minus

11988612119909lowast

2(119905)

1198981+ 119909lowast

1(119905)

= 0

minus1198862+

11988621119909lowast

1(119905)

1198981+ 119909lowast

1(119905)

minus 11988622119909lowast

2(119905) minus

11988623119909lowast

3(119905)

1198982+ 119909lowast

2(119905)

= 0

minus1198863+

11988632119909lowast

2(119905)

1198982+ 119909lowast

2(119905)


3(119905) = 0

(7)

where 1198643is a nonnegative equilibrium point if there is a

positive solution of (6) and 119864lowastis a nonnegative equilibrium

point if there is a positive solution of (7)Let

(H1) 1198861(11988621

minus 1198862) gt 119898

1119886211988611

(H2) (11988632minus1198863)[1198861(11988621minus1198862)minus11989811198862]minus1198982119886311988622(1198861+119898111988611) gt 0

(H3) 1199092(11988632

minus 1198863) minus 11989821198863gt 0

(H4) 11988622(11988611

minus (11988611198981)) minus (119886

12119886211198982

1) gt 0

From [1 25] we know that if (1198671) (1198672) (1198673) and (119867

4)

hold 1198643and 119864

lowastalways exist as nonnegative equilibria

Let119864 = (11990910 11990920 11990930) be the arbitrary equilibrium point

and let 1199091(119905) = 119909

1(119905) minus 119909

10 1199092(119905) = 119909

2(119905) minus 119909

20 1199093(119905) =

1199093(119905)minus119909

30 still denote 119909

1(119905) 1199092(119905) 1199093(119905) by 119909

1(119905) 1199092(119905) 1199093(119905)

respectively then the linearized system of the correspondingequations at 119864 is as follows

(119905) = 119861119906 (119905) + 119862119906 (119905 minus 1205911) + 119863119906 (119905 minus 120591

2) (8)

where

119906 (119905) = (1199091(119905) 1199092(119905) 1199093(119905))119879

119861 = (119887119894119895)3times3

119862 = (119888119894119895)3times3

119863 = (119889119894119895)3times3

11988711

= 1198861minus 21198861111990910

minus

11988612119898111990920

(1198981+ 11990910)2 119887

12= minus

1198861211990910

1198981+ 11990910

11988722

= minus1198862+

1198862111990910

1198981+ 11990910

minus 21198862211990920

minus

11988623119898211990930

(1198982+ 11990920)2

11988723

= minus

1198862311990920

1198982+ 11990920

11988733

= minus1198863+

1198863211990920

1198982+ 11990920

minus 21198863311990930

11988821

=

11988621119898111990920

(1198981+ 11990910)2 119889

32=

11988632119898211990930

(1198982+ 11990920)2

(9)

all the others of 119887119894119895 119888119894119895 and 119889

119894119895are 0

The characteristic equation for system (8) is

1205823+11990121205822+1199011120582 + 1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582 + 1199030) 119890minus1205821205912= 0

(10)

Abstract and Applied Analysis 3

where

1199012= minus (119887

11+ 11988722

+ 11988733)

1199011= 1198871111988722

+ 1198872211988733

+ 1198871111988733

1199010= minus119887111198872211988733

1199021= minus1198871211988821 1199020= 119887121198882111988733

1199031= minus1198872311988932 1199030= 119887111198872311988932

(11)

We consider the following cases

(1) 119864 = 1198641 The characteristic equation reduces to

(120582 minus 1198861) (120582 + 119886

2) (120582 + 119886

3) = 0 (12)

There are always a positive root 1198861and two negative roots

1198862 1198863of (12) hence 119864

1is a saddle point

(2) 119864 = 1198642 Equation (10) takes the form

(120582 + 1198861) (120582 + 119886

2minus

119886111988612

119898111988611

+ 1198861

) (120582 + 1198863) = 0 (13)

There is a positive root 120582 = (119886111988612(119898111988611

+ 1198861)) minus 119886

2if

119886111988612(119898111988611

+ 1198861) gt 119886

2 hence 119864

2is a saddle point If

119886111988612(119898111988611

+ 1198861) lt 119886

2 1198642is locally asymptotically stable

(3) 119864 = 1198643 The characteristic equation is

(120582 minus 11988733) [1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821119890minus1205821205911] = 0 (14)

We will analyse the distribution of the characteristic root of(14) from Ruan and Wei [26] which is stated as follows

Lemma 1 Consider the exponential polynomial

119875 (120582 119890minus1205821205911 119890

minus120582120591119898)

= 120582119899+ 119901(0)

1120582119899minus1

+ sdot sdot sdot + 119901(0)

119899minus1120582 + 119901(0)

119899

+ [119901(1)

1120582119899minus1


119899minus1120582 + 119901(1)

119899] 119890minus1205821205911

+ sdot sdot sdot + [119901(119898)

1120582119899minus1

+ sdot sdot sdot + 119901(119898)

119899minus1120582 + 119901(119898)

119899] 119890minus120582120591119898

(15)

where 120591119894⩾ 0 (119894 = 1 2 119898) and 119901

(119894)

119895(119894 = 0 1 119898 119895 =

1 2 119899) are constants As (1205911 1205912 120591

119898) vary the sum of

the order of the zeros of 119875(120582 119890minus1205821205911 119890minus120582120591119898) on the open righthalf plane can change only if a zero appears on or crosses theimaginary axis

By using Lemma 1 we can easily obtain the followingresults

Lemma 2 If 1198643is a nonnegative equilibrium point then

(1) 1198643is unstable if 119887

33gt 0

(2) 1198643is locally asymptotically stable if 119887

33lt 0 11988711

+ 11988722

lt

0 1198871111988722

minus 1198871211988821

gt 0 and 1198871111988722

+ 1198871211988821

gt 0

Proof (1) 120582 = 11988733

is a root of (14) if 11988733

gt 0 then 1198643is

unstable(2) Clearly 120582 = 0 is not a root of (14) we should discuss

the following equation instead of (14)

1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821119890minus1205821205911= 0 (16)

Assume that 119894120596 with 120596 gt 0 is a solution of (16) Substituting120582 = 119894120596 into (16) and separating the real and imaginary partsyield

minus1205962+ 1198871111988722

= 1198871211988821cos120596120591

1

120596 (11988711

+ 11988722) = 1198871211988821sin120596120591

1

(17)

which implies

1205964+ (1198872

11+ 1198872

22) 1205962+ 1198872

111198872

22minus 1198872

121198882

21= 0 (18)

If 1198872111198872

22minus 1198872

121198882

21gt 0 that is (119887

1111988722

+ 1198871211988821)(1198871111988722

minus 1198871211988821) gt

0 there is no real root of (16) Hence there is no purelyimaginary root of (18) When 120591

1= 0 (16) reduces to

1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821

= 0 (19)

If 11988711+ 11988722

lt 0 and 1198871111988722minus 1198871211988821

gt 0 both roots of (19) havenegative real parts Thus by using Lemma 1 when 119887

33lt 0

11988711

+ 11988722

lt 0 1198871111988722

minus 1198871211988821

gt 0 and 1198871111988722

+ 1198871211988821

gt 0 1198643is

locally asymptotically stable

(4)119864 = 119864lowastThe characteristic equation about119864

lowastis (10) In the

following we will analyse the distribution of roots of (10)Weconsider four cases

Case a Consider1205911= 1205912= 0

The associated characteristic equation of system (3) is

1205823+ 11990121205822+ (1199011+ 1199021+ 1199031) 120582 + (119901

0+ 1199020+ 1199030) = 0 (20)

Let

(H5) 1199012gt 0 119901

2(1199011+1199021+1199031)minus(1199010+1199020+1199030) gt 0 119901

0+1199020+1199030gt

0

By Routh-Hurwitz criterion we have the following

Theorem 3 For 1205911= 1205912= 0 assume that (119867

1)ndash(1198675) hold

Thenwhen 1205911= 1205912= 0 the positive equilibrium119864

lowast(119909lowast

1 119909lowast

2 119909lowast

3)

of system (3) is locally asymptotically stable

Case b Consider1205911= 0 1205912gt 0


1205823+ 11990121205822+ (1199011+ 1199021) 120582 + (119901

0+ 1199020) + (1199031120582 + 1199030) 119890minus1205821205912= 0

(21)


We want to determine if the real part of some rootincreases to reach zero and eventually becomes positive as 120591varies Let 120582 = 119894120596 (120596 gt 0) be a root of (21) then we have

minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901

0+ 1199020)

+ (1199031120596119894 + 119903

0) (cos120596120591

2minus 119894 sin120596120591

2) = 0

(22)

Separating the real and imaginary parts we have

minus1205963+ (1199011+ 1199021) 120596 = 119903

0sin120596120591

2minus 1199031120596 cos120596120591

2

minus11990121205962+ (1199010+ 1199020) = minus119903

1120596 sin120596120591

2minus 1199030cos120596120591

2

(23)

It follows that

1205966+ 119898121205964+ 119898111205962+ 11989810

= 0 (24)

where 11989812

= 1199012

2minus 2(119901

1+ 1199021) 11989811

= (1199011+ 1199021)2minus 21199012(1199010+

1199020) minus 1199032

1 11989810

= (1199010+ 1199020)2minus 1199032

0

Denoting 119911 = 1205962 (24) becomes

1199113+ 119898121199112+ 11989811119911 + 119898

10= 0 (25)

Let

ℎ1(119911) = 119911

3+ 119898121199112+ 11989811119911 + 119898

10 (26)

we have

119889ℎ1(119911)

119889119911

= 31199112+ 211989812119911 + 119898

11 (27)

If 11989810

= (1199010+ 1199020)2minus 1199032

0lt 0 then ℎ

1(0) lt

0 lim119911rarr+infin

ℎ1(119911) = +infin We can know that (25) has at least

one positive rootIf 11989810

= (1199010+ 1199020)2minus 1199032

0ge 0 we obtain that when Δ =

1198982

12minus 311989811

le 0 (25) has no positive roots for 119911 isin [0 +infin)On the other hand when Δ = 119898

2

12minus 311989811

gt 0 the followingequation

31199112+ 211989812119911 + 119898

11= 0 (28)

has two real roots 119911lowast11

= (minus11989812

+ radicΔ)3 119911lowast

12= (minus119898

12minus

radicΔ)3 Because of ℎ101584010158401(119911lowast

11) = 2radicΔ gt 0 ℎ

10158401015840

1(119911lowast

12) = minus2radicΔ lt

0 119911lowast

11and 119911lowast12are the local minimum and the localmaximum

of ℎ1(119911) respectively By the above analysis we immediately

obtain the following

Lemma 4 (1) If 11989810

ge 0 and Δ = 1198982

12minus 311989811

le 0 (25) hasno positive root for 119911 isin [0 +infin)

(2) If 11989810

ge 0 and Δ = 1198982

12minus 311989811

gt 0 (25) has at leastone positive root if and only if 119911lowast

11= (minus119898

12+ radicΔ)3 gt 0 and

ℎ1(119911lowast

11) le 0

(3) If 11989810

lt 0 (25) has at least one positive root

Without loss of generality we assume that (25) has threepositive roots defined by 119911

11 11991112 11991113 respectively Then

(24) has three positive roots

12059611

= radic11991111 120596

12= radic11991112 120596

13= radic11991113 (29)

From (23) we have

cos120596111989612059121119896

=

11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896minus 1199030(1199020+ 1199010)

1199032

0+ 1199032

11205962

1119896

(30)

Thus if we denote

120591(119895)

21119896

=

1

1205961119896

times arccos ((11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896

minus 1199030(1199020+ 1199010))

times (1199032

0+ 1199032

11205962

1119896)

minus1

) + 2119895120587

(31)

where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205961119896is a pair of purely

imaginary roots of (21) corresponding to 120591(119895)

21119896

Define

120591210

= 120591(0)

211198960

= min119896=123

120591(0)

21119896

12059610

= 12059611198960

(32)

Let 120582(1205912) = 120572(120591

2)+119894120596(120591

2) be the root of (21) near 120591

2= 120591(119895)

21119896

satisfying

120572 (120591(119895)

21119896

) = 0 120596 (120591(119895)

21119896

) = 1205961119896 (33)

Substituting 120582(1205912) into (21) and taking the derivative with

respect to 1205912 we have

31205822+ 21199012120582 + (119901

1+ 1199021) + 1199031119890minus1205821205912minus 1205912(1199031120582 + 1199030) 119890minus1205821205912

119889120582

1198891205912

= 120582 (1199031120582 + 1199030) 119890minus1205821205912

(34)

Therefore

[

119889120582

1198891205912

]

minus1

=

[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

+

1199031

120582 (1199031120582 + 1199030)

minus

1205912

120582

(35)

When 1205912

= 120591(119895)

21119896

120582(120591(119895)

21119896

) = 1198941205961119896

(119896 = 1 2 3) 120582(1199031120582 +

1199030)|1205912=120591(119895)

21119896

= minus11990311205962

1119896+ 11989411990301205961119896 [31205822

+ 21199012120582 + (119901

1+

1199021)]1198901205821205912|1205912=120591(119895)

21119896

= [minus31205962

1119896+ (1199011+ 1199021)] cos(120596

1119896120591(119895)

21119896

) minus

211990121205961119896sin(1205961119896120591(119895)

21119896

) + 119894211990121205961119896cos(120596

1119896120591(119895)

21119896

) + [minus31205962

1119896+ (1199011+

1199021)] sin(120596

1119896120591(119895)

21119896

)


According to (35) we have

[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

= Re[[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

+ Re[ 1199031

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

=

1

Λ1

minus 11990311205962

1119896[minus31205962

1119896+ (1199011+ 1199021)] cos (120596

1119896120591(119895)

21119896

)

+ 2119903111990121205963

1119896sin (120596

1119896120591(119895)

21119896

) minus 1199032

11205962

1119896

+ 2119903011990121205962

1119896cos (120596

1119896120591(119895)

21119896

)

+ 1199030[minus31205962

1119896+ (1199011+ 1199021)] 1205961119896sin (120596

1119896120591(119895)

21119896

)

=

1

Λ1

31205966

1119896+ 2 [119901

2

2minus 2 (119901

1+ 1199021)] 1205964

1119896

+ [(1199011+ 1199021)2

minus 21199012(1199010+ 1199020) minus 1199032

1] 1205962

1119896

=

1

Λ1

1199111119896(31199112

1119896+ 2119898121199111119896

+ 11989811)

=

1

Λ1

1199111119896ℎ1015840

1(1199111119896)

(36)

where Λ1= 1199032

11205964

1119896+ 1199032

01205962

1119896gt 0 Notice that Λ

1gt 0 1199111119896

gt 0

sign

[

Re 119889 (120582 (1205912))

1198891205912

]

1205912=120591(119895)

21119896

= sign

[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

(37)

then we have the following lemma

Lemma 5 Suppose that 1199111119896

= 1205962

1119896and ℎ

1015840

1(1199111119896) = 0 where

ℎ1(119911) is defined by (26) then 119889(Re 120582(120591(119895)

21119896

))1198891205912has the same

sign with ℎ1015840

1(1199111119896)

From Lemmas 1 4 and 5 and Theorem 3 we can easilyobtain the following theorem

Theorem6 For 1205911= 0 120591

2gt 0 suppose that (119867


(i) If 11989810

ge 0 and Δ = 1198982

12minus 311989811

le 0 then all rootsof (10) have negative real parts for all 120591

2ge 0 and the

positive equilibrium 119864lowastis locally asymptotically stable

for all 1205912ge 0

(ii) If either 11989810

lt 0 or 11989810

ge 0 Δ = 1198982

12minus 311989811

gt

0 119911lowast

11gt 0 and ℎ

1(119911lowast

11) le 0 then ℎ

1(119911) has at least one

positive roots and all roots of (23) have negative realparts for 120591

2isin [0 120591

210

) and the positive equilibrium 119864lowast

is locally asymptotically stable for 1205912isin [0 120591

210

)

(iii) If (ii) holds and ℎ1015840

1(1199111119896) = 0 then system (3)undergoes

Hopf bifurcations at the positive equilibrium119864lowastfor 1205912=

120591(119895)

21119896

(119896 = 1 2 3 119895 = 0 1 2 )

Case c Consider1205911gt 0 1205912= 0


1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901

0+ 1199030) + (119902

1120582 + 1199020) 119890minus1205821205911= 0

(38)

Similar to the analysis of Case 119887 we get the followingtheorem

Theorem7 For 1205911gt 0 120591

2= 0 suppose that (119867


(i) If 11989820

ge 0 and Δ = 1198982

22minus 311989821


1ge 0 and the


for all 1205911ge 0


lt 0 or 11989820

ge 0 Δ = 1198982

22minus 311989821

gt 0119911lowast

21gt 0 and ℎ

2(119911lowast

21) le 0 then ℎ


positive root 1199112119896 and all roots of (38) have negative real

parts for 1205911isin [0 120591

120



120

)


2(1199112119896) = 0 then system (3) undergoes


120591(119895)

12119896

(119896 = 1 2 3 119895 = 0 1 2 )

where

11989822

= 1199012

2minus 2 (119901

1+ 1199031)

11989821

= (1199011+ 1199031)2

minus 21199012(1199010+ 1199030) minus 1199022

1

11989820

= (1199010+ 1199030)2

minus 1199022

0

ℎ2(119911) = 119911

3+ 119898221199112+ 11989821119911 + 119898

20 119911lowast

21=

minus11989822

+ radicΔ

3

120591(119895)

12119896

=

1

1205962119896

times arccos ((11990211205964

2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962

2119896

minus 1199020(1199030+ 1199010) )

times (1199022

0+ 1199022

11205962

2119896)

minus1

) + 2119895120587

(39)



12119896

Define

120591120

= 120591(0)

121198960

= min119896=123

120591(0)

12119896

12059610

= 12059611198960

(40)


Case d Consider1205911gt 0 1205912gt 0 1205911

= 1205912


1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0

(41)

We consider (41) with 1205912= 120591lowast

2in its stable interval [0 120591

210

)Regard 120591

1as a parameter

Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have

minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591

1minus 119894 sin120596120591

1)

+ (1199030+ 1198941199031120596) (cos120596120591lowast

2minus 119894 sin120596120591

lowast

2) = 0

(42)


1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast

2+ 1199030sin120596120591

lowast

2

= 1199021120596 cos120596120591

1minus 1199020sin120596120591

1

11990121205962minus 1199010minus 1199030cos120596120591lowast

2minus 1199031120596 sin120596120591

lowast

2

= 1199020cos120596120591

1+ 1199021120596 sin120596120591

1

(43)

It follows that

1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830

= 0 (44)

where

11989833

= 1199012


2

11989832

= 2 (1199030minus 11990121199031) sin120596120591

lowast

2

11989831

= 1199012

1minus 211990101199012minus 2 (119901

21199030minus 11990111199031) cos120596120591lowast

2+ 1199032

1minus 1199022

1

11989830

= 1199012

0+ 211990101199030cos120596120591lowast

2+ 1199032

0minus 1199022

0

(45)

Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If

11989830

lt 0 then

119865 (0) lt 0 lim120596rarr+infin

119865 (120596) = +infin (46)

We can obtain that (44) has at most six positive roots1205961 1205962 120596

6 For every fixed 120596

119896 119896 = 1 2 6 there exists

a sequence 120591(119895)1119896

| 119895 = 0 1 2 3 such that (43) holdsLet

12059110

= min 120591(119895)

1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)

When 1205911

= 120591(119895)

1119896 (41) has a pair of purely imaginary roots

plusmn119894120596(119895)

1119896for 120591lowast2isin [0 120591

210

)In the following we assume that

(H6) ((119889Re(120582))119889120591

1)|120582=plusmn119894120596

(119895)

1119896

= 0

Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)

Theorem 8 For 1205911

gt 0 1205912

gt 0 1205911

= 1205912 suppose that

(1198671)ndash(1198676) is satisfied If 119898

30lt 0 and 120591

lowast

2isin [0 120591

210

] thenthe positive equilibrium 119864

lowastis locally asymptotically stable for

1205911

isin [0 12059110) System (3) undergoes Hopf bifurcations at the

positive equilibrium 119864lowastfor 1205911= 120591(119895)

1119896

3 Direction and Stability ofthe Hopf Bifurcation

In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591

2= 120591lowast

2isin [0 120591

210

)

and regard 1205911as a parameter We will derive the explicit

formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864

lowastat

the critical values 1205911by using the normal form and the center

manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591

1=

120591(119895)

1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591

1 at which (43) has a

pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864

lowast

Throughout this section we always assume that 120591lowast2lt 12059110

Let 1199061= 1199091minus 119909lowast

1 1199062= 1199091minus 119909lowast

2 1199063= 1199092minus 119909lowast

3 119905 = 120591

1119905

and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation

value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)

(119905) = 119871120583(119906119905) + 119891 (120583 119906

119905) (48)

where 119906 = (1199061 1199062 1199063)119879isin R3 and

119871120583(120601) = (120591

1+ 120583) 119861[

[

1206011(0)

1206012(0)

1206013(0)

]

]

+ (1205911+ 120583)119862[

[

1206011(minus1)

1206012(minus1)

1206013(minus1)

]

]

+ (1205911+ 120583)119863

[[[[[[[[[[

[

1206011(minus

120591lowast

2

1205911

)

1206012(minus

120591lowast

2

1205911

)

1206013(minus

120591lowast

2

1205911

)

]]]]]]]]]]

]

(49)

119891 (120583 120601) = (1205911+ 120583)[

[

1198911

1198912

1198913

]

]

(50)


where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and

119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012

1(minus1) minus 119897

81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911

)1206013(0) + sdot sdot sdot

1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840

1(119909lowast) 1199101lowast

1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)

Obviously 119871120583(120601) is a continuous linear function mapping

C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that

119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)

In fact we can choose

119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)

where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define

119860 (120583) 120601 =

119889120601 (120579)

119889120579

120579 isin [minus1 0)

int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)

Then when 120579 = 0 the system is equivalent to

119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)

where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)

lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)

and a bilinear inner product

⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)

where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast

are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591

1are eigenvalues of 119860 Thus they are also

eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860

lowast corresponding to 1198941205961205911and minus119894120596120591

1 respectively

Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860

corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591

1119902(120579) From the

definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =

(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)

and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572

lowast 120573lowast)1198901198941199041205961205911

be the eigenvector of 119860lowast corresponding to minus119894120596120591

1 By the

definition of 119860lowast we can compute

120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)

From (57) we have⟨119902lowast(119904) 119902 (120579)⟩

= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)

such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0

In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862

0

at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define

119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)

On the center manifold 1198620 we have

119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)

where 119911 and 119911 are local coordinates for center manifold 1198620in

the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We

consider only real solutions For the solution 119909119905isin 1198620of (48)

since 120583 = 0 we have

= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)


It follows together with (50) that

119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + sdot sdot sdot ]

+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911

) + sdot sdot sdot ]

(67)

Comparing the coefficients with (65) we have

11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]

+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]

+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)

minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]

+ 120572lowast[minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)

1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897

7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore

we can determine each 119892119894119895by the parameters and delay in (3)

Thus we can compute the following values

1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

1 Suppose

Re1205821015840(1205911) gt 0 120583

2determines the directions of the Hopf

bifurcation if 1205832

gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591

1(lt 1205911) 1205732determines the stability of the bifurcation

periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573

2lt 0(gt 0) and119879

2determines the period

of the bifurcating periodic solutions the period increases(decreases) if 119879

2gt 0(lt 0)

4 Numerical Simulation

We consider system (3) by taking the following coefficients1198861

= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633

= 12 We have the unique positiveequilibrium 119864

lowast= (00451 00357 00551)

By computation we get 11989810

= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20

= 32348 FromTheorem 6 we know that when 120591

1= 0 the positive

equilibrium 119864lowastis locally asymptotically stable for 120591

2isin

[0 32348) When 1205912crosses 120591

20

the equilibrium 119864lowastloses its

stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583

2= 56646 120573

2= minus31583 119879

2= 5445

whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2

Regarding 1205911as a parameter and let 120591

2= 29 isin

[0 32348) we can observe that with 1205911increasing the pos-

itive equilibrium 119864lowastloses its stability and Hopf bifurcation

occurs (see Figures 3 and 4)

5 Global Continuationof Local Hopf Bifurcations

In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)

1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this

section we follow closely the notations in [24] and assumethat 120591

2= 120591lowast

2isin [0 120591

210

) regarding 1205911as a parameter For

simplification of notations setting 119911119905(119905) = (119909

1119905 1199092119905 1199093119905)119879 we

may rewrite system (3) as the following functional differentialequation

(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))

119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909

1(119905) 1199092(119905)

and 1199093(119905) denote the densities of the prey the predator and

the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773

+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious

that (71) has a unique positive equilibrium 119864lowast(119909lowast

1 119909lowast

2 119909lowast

3) in

1198773

+under the assumption (119867

1)ndash(1198674) Following the work of

[24] we need to define

119883 = 119862([minus1205911 0] 119877

3

+)

Γ = 119862119897 (119911 1205911 119901) isin X times R times R+

119911 is a 119901-periodic solution of system (71)

N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)

1119896)denote the connected component pass-

ing through (119864lowast 120591(119895)

1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)

1119896is defined by

(43) We know that ℓ(119864lowast120591(119895)

11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)

is nonemptyFor the benefit of readers we first state the global Hopf

bifurcation theory due to Wu [24] for functional differentialequations

Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-

fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)

theconnected component of (119911

lowast 120591 119901) in Γ Then either

(i) ℓ(119911lowast120591119901)

is unbounded or(ii) ℓ(119911lowast120591119901)

is bounded ℓ(119911lowast120591119901)

cap Γ is finite and

sum

(119911120591119901)isinℓ(119911lowast120591119901)capN

120574119898(119911lowast 120591 119901) = 0 (73)

for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing

number of (119911lowast 120591 119901) if119898 isin 119869(119911

lowast 120591 119901) or it is zero if otherwise

Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)

isunbounded Thus if the projections of ℓ

(119911lowast120591119901)

onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ

(119911lowast120591119901)

onto 120591-space is unbounded Further if we can show that the


0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)

Figure 1 The trajectories and the phase graph with 1205911= 0 120591

2= 29 lt 120591

20= 32348 119864

lowastis locally asymptotically stable

0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591

20= 32348 a periodic orbit bifurcate from 119864

lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast


projection of ℓ(119911lowast120591119901)

onto 120591-space is away from zero thenthe projection of ℓ

(119911lowast120591119901)

onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation

Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-

ial periodic solutions of system (71) with initial conditions

1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)

are uniformly bounded

Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-

odic solutions of system (3) and define

1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)

It follows from system (3) that

1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)

which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits

must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909

1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0

From the first equation of system (3) we can get

0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)

From the second equation of (3) we obtain

0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)

therefore one can get

1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)

Applying the third equation of system (3) we know

0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)

This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete

Lemma 11 If the conditions (1198671)ndash(1198674) and

(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0

hold then system (3) has no nontrivial 1205911-periodic solution

Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591

1 Then the following

system (83) of ordinary differential equations has nontrivialperiodic solution

1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)

which has the same equilibria to system (3) that is

1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)

Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of

system (83) and the orbits of system (83) do not intersect each


other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864

1 1198642 1198643are located on the coordinate

axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867

7) holds it is well known

that the positive equilibrium 119864lowastis globally asymptotically

stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed

Theorem 12 Suppose the conditions of Theorem 8 and (1198677)

hold let120596119896and 120591(119895)1119896

be defined in Section 2 then when 1205911gt 120591(119895)

1119896

system (3) has at least 119895 minus 1 periodic solutions

Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)

11198962120587120596

119896)onto 120591

1-space is [120591

1 +infin) for each 119895 ge 1 where

1205911le 120591(119895)

1119896

In following we prove that the hypotheses (A1)ndash(A4) in[24] hold

(1) From system (3) we know easily that the followingconditions hold

(A1) 119865 isin 1198622(1198773

+times119877+times119877+) where 119865 = 119865|

1198773

+times119877+times119877+

rarr

1198773

+

(A3) 119865(120601 1205911 119901) is differential with respect to 120601

(2) It follows from system (3) that

119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)

Then under the assumption (1198671)ndash(1198674) we have

det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[

minus11988611119909lowast

1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862

21199101lowast1199102lowast

(1 + 11988721199101lowast)3[minus119909lowast+

11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)

From (86) we know that the hypothesis (A2) in [24]is satisfied

(3) The characteristic matrix of (71) at a stationarysolution (119911 120591

0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3

takes the following form

Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)

Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864

lowast 120591(119895)

1119896 2120587120596

119896) is an isolated center and there exist 120598 gt

0 120575 gt 0 and a smooth curve 120582 (120591(119895)

1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)

It is easy to see that on [120591(119895)

1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864

lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0

1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2

Therefore the hypothesis (A4) in [24] is satisfiedIf we define

119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)

then we have the crossing number of isolated center(119864lowast 120591(119895)

1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)

where (119911 1205911 119901) has all or parts of the form

(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from

Lemma 9 that the connected component ℓ(119864lowast120591(119895)

11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596

119896) is unbounded for each center

(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2

we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)

where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)

1119896

for 119895 ge 1Now we prove that the projection of ℓ

(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591

1 +infin) where 120591

1le 120591(119895)

1119896 Clearly it follows

from the proof of Lemma 11 that system (3) with 1205911

= 0

has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)

11198962120587120596

119896)onto 120591

1-space is away from zero

For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)

11198962120587120596

119896)onto 120591

1-space is bounded this means that the

projection of ℓ(119864lowast120591(119895)

11198962120587120596

119896)onto 120591

1-space is included in a

interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895

1119896and applying Lemma 11

we have 119901 lt 120591lowast for (119911(119905) 120591

1 119901) belonging to ℓ

(119864lowast120591(119895)

11198962120587120596

119896)

This implies that the projection of ℓ(119864lowast120591(119895)

11198962120587120596

119896)onto 119901-

space is bounded Then applying Lemma 10 we get thatthe connected component ℓ

(119864lowast120591(119895)

11198962120587120596

119896)is bounded This

contradiction completes the proof

6 Conclusion

In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591

1= 1205912 we derived the explicit

formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)


References

[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002

[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008

[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009

[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012

[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012

[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012

[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013

[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013

[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013

[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013

[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013

[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011

[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012

[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013

[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012

[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011

[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004

[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005

[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004

[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997

[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008

[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981

[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998

[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990

[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is a more interesting and difficult topic A great deal ofresearch has been devoted to the topics [13ndash21] In this paperlet 12059111

= 12059122

= 12059133

= 0 12059121

= 1205911 12059132

= 1205912in (1) we

considerHopf bifurcation and global periodic solutions of thefollowing system with two unequal and nonzero delays

1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905 minus 1205911)

1198981+ 1199091(119905 minus 1205911)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 1205912)

1198982+ 1199092(119905 minus 1205912)

minus 119886331199093(119905)]

(3)


119909119894(119905) = 120601

119894(119905) 119905 isin [minus120591 0] 120601

119894(0) gt 0

119894 = 1 2 3 120591 = max 1205911 1205912

(4)

Our goal is to investigate the possible stability switches of thepositive equilibrium and stability of periodic orbits arisingdue to a Hopf bifurcation when one of the delays is treatedas a bifurcation parameter Special attention is paid to theglobal continuation of local Hopf bifurcation when the delays1205911

= 1205912

This paper is organized as follows In Section 2 byanalyzing the characteristic equation of the linearized systemof system (3) at positive equilibrium the sufficient conditionsensuring the local stability of the positive equilibrium andthe existence of Hopf bifurcation are obtained [22] Someexplicit formulas determining the direction and stabilityof periodic solutions bifurcating from Hopf bifurcationsare demonstrated by applying the normal form methodand center manifold theory due to Hassard et al [23] inSection 3 In Section 4 we consider the global existence ofthese bifurcating periodic solutions [24] with two differentdelays Some numerical simulation results are included inSection 5

2 Stability of the Positive Equilibrium andLocal Hopf Bifurcations

In this section we first study the existence and local stabilityof the positive equilibrium and then investigate the effect ofdelay and the conditions for existence of Hopf bifurcations

There are at most four nonnegative equilibria for system(3)

1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(5)

where (1199091 1199092 0) and (119909

lowast

1 119909lowast

2 119909lowast

3) satisfy

1198861minus 119886111199091minus

119886121199092

1198981+ 1199091

= 0

minus1198862+

119886211199091

1198981+ 1199091

minus 119886221199092= 0

(6)

1198861minus 11988611119909lowast

1(119905) minus

11988612119909lowast

2(119905)

1198981+ 119909lowast

1(119905)

= 0

minus1198862+

11988621119909lowast

1(119905)

1198981+ 119909lowast

1(119905)


2(119905) minus

11988623119909lowast

3(119905)

1198982+ 119909lowast

2(119905)

= 0

minus1198863+

11988632119909lowast

2(119905)

1198982+ 119909lowast

2(119905)


3(119905) = 0

(7)

where 1198643is a nonnegative equilibrium point if there is a

positive solution of (6) and 119864lowastis a nonnegative equilibrium

point if there is a positive solution of (7)Let

(H1) 1198861(11988621

minus 1198862) gt 119898

1119886211988611

(H2) (11988632minus1198863)[1198861(11988621minus1198862)minus11989811198862]minus1198982119886311988622(1198861+119898111988611) gt 0

(H3) 1199092(11988632

minus 1198863) minus 11989821198863gt 0

(H4) 11988622(11988611

minus (11988611198981)) minus (119886

12119886211198982

1) gt 0

From [1 25] we know that if (1198671) (1198672) (1198673) and (119867

4)

hold 1198643and 119864

lowastalways exist as nonnegative equilibria

Let119864 = (11990910 11990920 11990930) be the arbitrary equilibrium point

and let 1199091(119905) = 119909

1(119905) minus 119909

10 1199092(119905) = 119909

2(119905) minus 119909

20 1199093(119905) =

1199093(119905)minus119909

30 still denote 119909

1(119905) 1199092(119905) 1199093(119905) by 119909

1(119905) 1199092(119905) 1199093(119905)

respectively then the linearized system of the correspondingequations at 119864 is as follows

(119905) = 119861119906 (119905) + 119862119906 (119905 minus 1205911) + 119863119906 (119905 minus 120591

2) (8)

where

119906 (119905) = (1199091(119905) 1199092(119905) 1199093(119905))119879

119861 = (119887119894119895)3times3

119862 = (119888119894119895)3times3

119863 = (119889119894119895)3times3

11988711

= 1198861minus 21198861111990910

minus

11988612119898111990920

(1198981+ 11990910)2 119887

12= minus

1198861211990910

1198981+ 11990910

11988722

= minus1198862+

1198862111990910

1198981+ 11990910

minus 21198862211990920

minus

11988623119898211990930

(1198982+ 11990920)2

11988723

= minus

1198862311990920

1198982+ 11990920

11988733

= minus1198863+

1198863211990920

1198982+ 11990920

minus 21198863311990930

11988821

=

11988621119898111990920

(1198981+ 11990910)2 119889

32=

11988632119898211990930

(1198982+ 11990920)2

(9)

all the others of 119887119894119895 119888119894119895 and 119889

119894119895are 0

The characteristic equation for system (8) is

1205823+11990121205822+1199011120582 + 1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582 + 1199030) 119890minus1205821205912= 0

(10)


where

1199012= minus (119887

11+ 11988722

+ 11988733)

1199011= 1198871111988722

+ 1198872211988733

+ 1198871111988733

1199010= minus119887111198872211988733

1199021= minus1198871211988821 1199020= 119887121198882111988733

1199031= minus1198872311988932 1199030= 119887111198872311988932

(11)



(120582 minus 1198861) (120582 + 119886

2) (120582 + 119886

3) = 0 (12)


1198862 1198863of (12) hence 119864

1is a saddle point


(120582 + 1198861) (120582 + 119886

2minus

119886111988612

119898111988611

+ 1198861

) (120582 + 1198863) = 0 (13)


+ 1198861)) minus 119886

2if

119886111988612(119898111988611

+ 1198861) gt 119886

2 hence 119864


119886111988612(119898111988611

+ 1198861) lt 119886



(120582 minus 11988733) [1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821119890minus1205821205911] = 0 (14)



119875 (120582 119890minus1205821205911 119890

minus120582120591119898)

= 120582119899+ 119901(0)

1120582119899minus1


119899minus1120582 + 119901(0)

119899

+ [119901(1)

1120582119899minus1


119899minus1120582 + 119901(1)

119899] 119890minus1205821205911

+ sdot sdot sdot + [119901(119898)

1120582119899minus1

+ sdot sdot sdot + 119901(119898)

119899minus1120582 + 119901(119898)

119899] 119890minus120582120591119898

(15)

where 120591119894⩾ 0 (119894 = 1 2 119898) and 119901

(119894)

119895(119894 = 0 1 119898 119895 =

1 2 119899) are constants As (1205911 1205912 120591





(1) 1198643is unstable if 119887

33gt 0


33lt 0 11988711

+ 11988722

lt

0 1198871111988722

minus 1198871211988821

gt 0 and 1198871111988722

+ 1198871211988821

gt 0

Proof (1) 120582 = 11988733

is a root of (14) if 11988733

gt 0 then 1198643is



1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821119890minus1205821205911= 0 (16)


minus1205962+ 1198871111988722

= 1198871211988821cos120596120591

1

120596 (11988711

+ 11988722) = 1198871211988821sin120596120591

1

(17)

which implies

1205964+ (1198872

11+ 1198872

22) 1205962+ 1198872

111198872

22minus 1198872

121198882

21= 0 (18)

If 1198872111198872

22minus 1198872

121198882

21gt 0 that is (119887

1111988722

+ 1198871211988821)(1198871111988722

minus 1198871211988821) gt



1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821

= 0 (19)

If 11988711+ 11988722

lt 0 and 1198871111988722minus 1198871211988821


33lt 0

11988711

+ 11988722

lt 0 1198871111988722

minus 1198871211988821

gt 0 and 1198871111988722

+ 1198871211988821

gt 0 1198643is





Case a Consider1205911= 1205912= 0


1205823+ 11990121205822+ (1199011+ 1199021+ 1199031) 120582 + (119901

0+ 1199020+ 1199030) = 0 (20)

Let

(H5) 1199012gt 0 119901

2(1199011+1199021+1199031)minus(1199010+1199020+1199030) gt 0 119901

0+1199020+1199030gt

0





lowast(119909lowast

1 119909lowast

2 119909lowast

3)




1205823+ 11990121205822+ (1199011+ 1199021) 120582 + (119901

0+ 1199020) + (1199031120582 + 1199030) 119890minus1205821205912= 0

(21)



minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901

0+ 1199020)

+ (1199031120596119894 + 119903

0) (cos120596120591

2minus 119894 sin120596120591

2) = 0

(22)


minus1205963+ (1199011+ 1199021) 120596 = 119903

0sin120596120591

2minus 1199031120596 cos120596120591

2

minus11990121205962+ (1199010+ 1199020) = minus119903

1120596 sin120596120591

2minus 1199030cos120596120591

2

(23)

It follows that

1205966+ 119898121205964+ 119898111205962+ 11989810

= 0 (24)

where 11989812

= 1199012

2minus 2(119901

1+ 1199021) 11989811

= (1199011+ 1199021)2minus 21199012(1199010+

1199020) minus 1199032

1 11989810

= (1199010+ 1199020)2minus 1199032

0


1199113+ 119898121199112+ 11989811119911 + 119898

10= 0 (25)

Let

ℎ1(119911) = 119911

3+ 119898121199112+ 11989811119911 + 119898

10 (26)

we have

119889ℎ1(119911)

119889119911

= 31199112+ 211989812119911 + 119898

11 (27)

If 11989810

= (1199010+ 1199020)2minus 1199032

0lt 0 then ℎ

1(0) lt




= (1199010+ 1199020)2minus 1199032


1198982

12minus 311989811


2

12minus 311989811


31199112+ 211989812119911 + 119898

11= 0 (28)


= (minus11989812


12= (minus119898

12minus



10158401015840

1(119911lowast


0 119911lowast




Lemma 4 (1) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


(2) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


11= (minus119898


ℎ1(119911lowast

11) le 0

(3) If 11989810





12059611

= radic11991111 120596

12= radic11991112 120596

13= radic11991113 (29)

From (23) we have

cos120596111989612059121119896

=

11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896minus 1199030(1199020+ 1199010)

1199032

0+ 1199032

11205962

1119896

(30)

Thus if we denote

120591(119895)

21119896

=

1

1205961119896

times arccos ((11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896

minus 1199030(1199020+ 1199010))

times (1199032

0+ 1199032

11205962

1119896)

minus1

) + 2119895120587

(31)



21119896

Define

120591210

= 120591(0)

211198960

= min119896=123

120591(0)

21119896

12059610

= 12059611198960

(32)

Let 120582(1205912) = 120572(120591

2)+119894120596(120591


2= 120591(119895)

21119896

satisfying

120572 (120591(119895)

21119896

) = 0 120596 (120591(119895)

21119896

) = 1205961119896 (33)



31205822+ 21199012120582 + (119901


119889120582

1198891205912

= 120582 (1199031120582 + 1199030) 119890minus1205821205912

(34)

Therefore

[

119889120582

1198891205912

]

minus1

=

[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

+

1199031

120582 (1199031120582 + 1199030)

minus

1205912

120582

(35)

When 1205912

= 120591(119895)

21119896

120582(120591(119895)

21119896

) = 1198941205961119896

(119896 = 1 2 3) 120582(1199031120582 +

1199030)|1205912=120591(119895)

21119896

= minus11990311205962

1119896+ 11989411990301205961119896 [31205822

+ 21199012120582 + (119901

1+

1199021)]1198901205821205912|1205912=120591(119895)

21119896

= [minus31205962

1119896+ (1199011+ 1199021)] cos(120596

1119896120591(119895)

21119896

) minus

211990121205961119896sin(1205961119896120591(119895)

21119896

) + 119894211990121205961119896cos(120596

1119896120591(119895)

21119896

) + [minus31205962

1119896+ (1199011+

1199021)] sin(120596

1119896120591(119895)

21119896

)



[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

= Re[[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

+ Re[ 1199031

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

=

1

Λ1

minus 11990311205962

1119896[minus31205962

1119896+ (1199011+ 1199021)] cos (120596

1119896120591(119895)

21119896

)

+ 2119903111990121205963

1119896sin (120596

1119896120591(119895)

21119896

) minus 1199032

11205962

1119896

+ 2119903011990121205962

1119896cos (120596

1119896120591(119895)

21119896

)

+ 1199030[minus31205962

1119896+ (1199011+ 1199021)] 1205961119896sin (120596

1119896120591(119895)

21119896

)

=

1

Λ1

31205966

1119896+ 2 [119901

2

2minus 2 (119901

1+ 1199021)] 1205964

1119896

+ [(1199011+ 1199021)2

minus 21199012(1199010+ 1199020) minus 1199032

1] 1205962

1119896

=

1

Λ1

1199111119896(31199112

1119896+ 2119898121199111119896

+ 11989811)

=

1

Λ1

1199111119896ℎ1015840

1(1199111119896)

(36)

where Λ1= 1199032

11205964

1119896+ 1199032

01205962


1gt 0 1199111119896

gt 0

sign

[

Re 119889 (120582 (1205912))

1198891205912

]

1205912=120591(119895)

21119896

= sign

[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

(37)



= 1205962

1119896and ℎ

1015840

1(1199111119896) = 0 where


21119896

))1198891205912has the same


1(1199111119896)


Theorem6 For 1205911= 0 120591



(i) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


2ge 0 and the


for all 1205912ge 0


lt 0 or 11989810

ge 0 Δ = 1198982

12minus 311989811

gt

0 119911lowast

11gt 0 and ℎ

1(119911lowast

11) le 0 then ℎ



2isin [0 120591

210



210

)




120591(119895)

21119896

(119896 = 1 2 3 119895 = 0 1 2 )



1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901

0+ 1199030) + (119902

1120582 + 1199020) 119890minus1205821205911= 0

(38)


Theorem7 For 1205911gt 0 120591



(i) If 11989820

ge 0 and Δ = 1198982

22minus 311989821


1ge 0 and the


for all 1205911ge 0


lt 0 or 11989820

ge 0 Δ = 1198982

22minus 311989821

gt 0119911lowast

21gt 0 and ℎ

2(119911lowast

21) le 0 then ℎ



parts for 1205911isin [0 120591

120



120

)




120591(119895)

12119896

(119896 = 1 2 3 119895 = 0 1 2 )

where

11989822

= 1199012

2minus 2 (119901

1+ 1199031)

11989821

= (1199011+ 1199031)2

minus 21199012(1199010+ 1199030) minus 1199022

1

11989820

= (1199010+ 1199030)2

minus 1199022

0

ℎ2(119911) = 119911

3+ 119898221199112+ 11989821119911 + 119898

20 119911lowast

21=

minus11989822

+ radicΔ

3

120591(119895)

12119896

=

1

1205962119896

times arccos ((11990211205964

2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962

2119896

minus 1199020(1199030+ 1199010) )

times (1199022

0+ 1199022

11205962

2119896)

minus1

) + 2119895120587

(39)



12119896

Define

120591120

= 120591(0)

121198960

= min119896=123

120591(0)

12119896

12059610

= 12059611198960

(40)



= 1205912


1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0

(41)



210

)Regard 120591

1as a parameter


minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591

1minus 119894 sin120596120591

1)

+ (1199030+ 1198941199031120596) (cos120596120591lowast

2minus 119894 sin120596120591

lowast

2) = 0

(42)



2+ 1199030sin120596120591

lowast

2

= 1199021120596 cos120596120591

1minus 1199020sin120596120591

1


2minus 1199031120596 sin120596120591

lowast

2

= 1199020cos120596120591

1+ 1199021120596 sin120596120591

1

(43)

It follows that

1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830

= 0 (44)

where

11989833

= 1199012


2

11989832

= 2 (1199030minus 11990121199031) sin120596120591

lowast

2

11989831

= 1199012

1minus 211990101199012minus 2 (119901

21199030minus 11990111199031) cos120596120591lowast

2+ 1199032

1minus 1199022

1

11989830

= 1199012

0+ 211990101199030cos120596120591lowast

2+ 1199032

0minus 1199022

0

(45)

Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If

11989830

lt 0 then


119865 (120596) = +infin (46)



119896 119896 = 1 2 6 there exists

a sequence 120591(119895)1119896


12059110

= min 120591(119895)

1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)

When 1205911

= 120591(119895)


plusmn119894120596(119895)

1119896for 120591lowast2isin [0 120591

210


(H6) ((119889Re(120582))119889120591

1)|120582=plusmn119894120596

(119895)

1119896

= 0



gt 0 1205912

gt 0 1205911



30lt 0 and 120591

lowast

2isin [0 120591

210



1205911



1119896



2= 120591lowast

2isin [0 120591

210

)



lowastat



1=

120591(119895)

1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591



lowast


Let 1199061= 1199091minus 119909lowast

1 1199062= 1199091minus 119909lowast

2 1199063= 1199092minus 119909lowast

3 119905 = 120591

1119905



(119905) = 119871120583(119906119905) + 119891 (120583 119906

119905) (48)

where 119906 = (1199061 1199062 1199063)119879isin R3 and

119871120583(120601) = (120591

1+ 120583) 119861[

[

1206011(0)

1206012(0)

1206013(0)

]

]

+ (1205911+ 120583)119862[

[

1206011(minus1)

1206012(minus1)

1206013(minus1)

]

]

+ (1205911+ 120583)119863

[[[[[[[[[[

[

1206011(minus

120591lowast

2

1205911

)

1206012(minus

120591lowast

2

1205911

)

1206013(minus

120591lowast

2

1205911

)

]]]]]]]]]]

]

(49)

119891 (120583 120601) = (1205911+ 120583)[

[

1198911

1198912

1198913

]

]

(50)



119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911


1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840


1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)



119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)


119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)


119860 (120583) 120601 =

119889120601 (120579)

119889120579


int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)


119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)


lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)


⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)






1 respectively



1119902(120579) From the


(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)


lowast 120573lowast)1198901198941199041205961205911


1 By the


120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)


= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1199012= minus (119887

11+ 11988722

+ 11988733)

1199011= 1198871111988722

+ 1198872211988733

+ 1198871111988733

1199010= minus119887111198872211988733

1199021= minus1198871211988821 1199020= 119887121198882111988733

1199031= minus1198872311988932 1199030= 119887111198872311988932

(11)



(120582 minus 1198861) (120582 + 119886

2) (120582 + 119886

3) = 0 (12)


1198862 1198863of (12) hence 119864

1is a saddle point


(120582 + 1198861) (120582 + 119886

2minus

119886111988612

119898111988611

+ 1198861

) (120582 + 1198863) = 0 (13)


+ 1198861)) minus 119886

2if

119886111988612(119898111988611

+ 1198861) gt 119886

2 hence 119864


119886111988612(119898111988611

+ 1198861) lt 119886



(120582 minus 11988733) [1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821119890minus1205821205911] = 0 (14)



119875 (120582 119890minus1205821205911 119890

minus120582120591119898)

= 120582119899+ 119901(0)

1120582119899minus1


119899minus1120582 + 119901(0)

119899

+ [119901(1)

1120582119899minus1


119899minus1120582 + 119901(1)

119899] 119890minus1205821205911

+ sdot sdot sdot + [119901(119898)

1120582119899minus1

+ sdot sdot sdot + 119901(119898)

119899minus1120582 + 119901(119898)

119899] 119890minus120582120591119898

(15)

where 120591119894⩾ 0 (119894 = 1 2 119898) and 119901

(119894)

119895(119894 = 0 1 119898 119895 =

1 2 119899) are constants As (1205911 1205912 120591





(1) 1198643is unstable if 119887

33gt 0


33lt 0 11988711

+ 11988722

lt

0 1198871111988722

minus 1198871211988821

gt 0 and 1198871111988722

+ 1198871211988821

gt 0

Proof (1) 120582 = 11988733

is a root of (14) if 11988733

gt 0 then 1198643is



1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821119890minus1205821205911= 0 (16)


minus1205962+ 1198871111988722

= 1198871211988821cos120596120591

1

120596 (11988711

+ 11988722) = 1198871211988821sin120596120591

1

(17)

which implies

1205964+ (1198872

11+ 1198872

22) 1205962+ 1198872

111198872

22minus 1198872

121198882

21= 0 (18)

If 1198872111198872

22minus 1198872

121198882

21gt 0 that is (119887

1111988722

+ 1198871211988821)(1198871111988722

minus 1198871211988821) gt



1205822minus (11988711

+ 11988722) 120582 + 119887

1111988722

minus 1198871211988821

= 0 (19)

If 11988711+ 11988722

lt 0 and 1198871111988722minus 1198871211988821


33lt 0

11988711

+ 11988722

lt 0 1198871111988722

minus 1198871211988821

gt 0 and 1198871111988722

+ 1198871211988821

gt 0 1198643is





Case a Consider1205911= 1205912= 0


1205823+ 11990121205822+ (1199011+ 1199021+ 1199031) 120582 + (119901

0+ 1199020+ 1199030) = 0 (20)

Let

(H5) 1199012gt 0 119901

2(1199011+1199021+1199031)minus(1199010+1199020+1199030) gt 0 119901

0+1199020+1199030gt

0





lowast(119909lowast

1 119909lowast

2 119909lowast

3)




1205823+ 11990121205822+ (1199011+ 1199021) 120582 + (119901

0+ 1199020) + (1199031120582 + 1199030) 119890minus1205821205912= 0

(21)



minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901

0+ 1199020)

+ (1199031120596119894 + 119903

0) (cos120596120591

2minus 119894 sin120596120591

2) = 0

(22)


minus1205963+ (1199011+ 1199021) 120596 = 119903

0sin120596120591

2minus 1199031120596 cos120596120591

2

minus11990121205962+ (1199010+ 1199020) = minus119903

1120596 sin120596120591

2minus 1199030cos120596120591

2

(23)

It follows that

1205966+ 119898121205964+ 119898111205962+ 11989810

= 0 (24)

where 11989812

= 1199012

2minus 2(119901

1+ 1199021) 11989811

= (1199011+ 1199021)2minus 21199012(1199010+

1199020) minus 1199032

1 11989810

= (1199010+ 1199020)2minus 1199032

0


1199113+ 119898121199112+ 11989811119911 + 119898

10= 0 (25)

Let

ℎ1(119911) = 119911

3+ 119898121199112+ 11989811119911 + 119898

10 (26)

we have

119889ℎ1(119911)

119889119911

= 31199112+ 211989812119911 + 119898

11 (27)

If 11989810

= (1199010+ 1199020)2minus 1199032

0lt 0 then ℎ

1(0) lt




= (1199010+ 1199020)2minus 1199032


1198982

12minus 311989811


2

12minus 311989811


31199112+ 211989812119911 + 119898

11= 0 (28)


= (minus11989812


12= (minus119898

12minus



10158401015840

1(119911lowast


0 119911lowast




Lemma 4 (1) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


(2) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


11= (minus119898


ℎ1(119911lowast

11) le 0

(3) If 11989810





12059611

= radic11991111 120596

12= radic11991112 120596

13= radic11991113 (29)

From (23) we have

cos120596111989612059121119896

=

11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896minus 1199030(1199020+ 1199010)

1199032

0+ 1199032

11205962

1119896

(30)

Thus if we denote

120591(119895)

21119896

=

1

1205961119896

times arccos ((11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896

minus 1199030(1199020+ 1199010))

times (1199032

0+ 1199032

11205962

1119896)

minus1

) + 2119895120587

(31)



21119896

Define

120591210

= 120591(0)

211198960

= min119896=123

120591(0)

21119896

12059610

= 12059611198960

(32)

Let 120582(1205912) = 120572(120591

2)+119894120596(120591


2= 120591(119895)

21119896

satisfying

120572 (120591(119895)

21119896

) = 0 120596 (120591(119895)

21119896

) = 1205961119896 (33)



31205822+ 21199012120582 + (119901


119889120582

1198891205912

= 120582 (1199031120582 + 1199030) 119890minus1205821205912

(34)

Therefore

[

119889120582

1198891205912

]

minus1

=

[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

+

1199031

120582 (1199031120582 + 1199030)

minus

1205912

120582

(35)

When 1205912

= 120591(119895)

21119896

120582(120591(119895)

21119896

) = 1198941205961119896

(119896 = 1 2 3) 120582(1199031120582 +

1199030)|1205912=120591(119895)

21119896

= minus11990311205962

1119896+ 11989411990301205961119896 [31205822

+ 21199012120582 + (119901

1+

1199021)]1198901205821205912|1205912=120591(119895)

21119896

= [minus31205962

1119896+ (1199011+ 1199021)] cos(120596

1119896120591(119895)

21119896

) minus

211990121205961119896sin(1205961119896120591(119895)

21119896

) + 119894211990121205961119896cos(120596

1119896120591(119895)

21119896

) + [minus31205962

1119896+ (1199011+

1199021)] sin(120596

1119896120591(119895)

21119896

)



[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

= Re[[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

+ Re[ 1199031

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

=

1

Λ1

minus 11990311205962

1119896[minus31205962

1119896+ (1199011+ 1199021)] cos (120596

1119896120591(119895)

21119896

)

+ 2119903111990121205963

1119896sin (120596

1119896120591(119895)

21119896

) minus 1199032

11205962

1119896

+ 2119903011990121205962

1119896cos (120596

1119896120591(119895)

21119896

)

+ 1199030[minus31205962

1119896+ (1199011+ 1199021)] 1205961119896sin (120596

1119896120591(119895)

21119896

)

=

1

Λ1

31205966

1119896+ 2 [119901

2

2minus 2 (119901

1+ 1199021)] 1205964

1119896

+ [(1199011+ 1199021)2

minus 21199012(1199010+ 1199020) minus 1199032

1] 1205962

1119896

=

1

Λ1

1199111119896(31199112

1119896+ 2119898121199111119896

+ 11989811)

=

1

Λ1

1199111119896ℎ1015840

1(1199111119896)

(36)

where Λ1= 1199032

11205964

1119896+ 1199032

01205962


1gt 0 1199111119896

gt 0

sign

[

Re 119889 (120582 (1205912))

1198891205912

]

1205912=120591(119895)

21119896

= sign

[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

(37)



= 1205962

1119896and ℎ

1015840

1(1199111119896) = 0 where


21119896

))1198891205912has the same


1(1199111119896)


Theorem6 For 1205911= 0 120591



(i) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


2ge 0 and the


for all 1205912ge 0


lt 0 or 11989810

ge 0 Δ = 1198982

12minus 311989811

gt

0 119911lowast

11gt 0 and ℎ

1(119911lowast

11) le 0 then ℎ



2isin [0 120591

210



210

)




120591(119895)

21119896

(119896 = 1 2 3 119895 = 0 1 2 )



1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901

0+ 1199030) + (119902

1120582 + 1199020) 119890minus1205821205911= 0

(38)


Theorem7 For 1205911gt 0 120591



(i) If 11989820

ge 0 and Δ = 1198982

22minus 311989821


1ge 0 and the


for all 1205911ge 0


lt 0 or 11989820

ge 0 Δ = 1198982

22minus 311989821

gt 0119911lowast

21gt 0 and ℎ

2(119911lowast

21) le 0 then ℎ



parts for 1205911isin [0 120591

120



120

)




120591(119895)

12119896

(119896 = 1 2 3 119895 = 0 1 2 )

where

11989822

= 1199012

2minus 2 (119901

1+ 1199031)

11989821

= (1199011+ 1199031)2

minus 21199012(1199010+ 1199030) minus 1199022

1

11989820

= (1199010+ 1199030)2

minus 1199022

0

ℎ2(119911) = 119911

3+ 119898221199112+ 11989821119911 + 119898

20 119911lowast

21=

minus11989822

+ radicΔ

3

120591(119895)

12119896

=

1

1205962119896

times arccos ((11990211205964

2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962

2119896

minus 1199020(1199030+ 1199010) )

times (1199022

0+ 1199022

11205962

2119896)

minus1

) + 2119895120587

(39)



12119896

Define

120591120

= 120591(0)

121198960

= min119896=123

120591(0)

12119896

12059610

= 12059611198960

(40)



= 1205912


1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0

(41)



210

)Regard 120591

1as a parameter


minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591

1minus 119894 sin120596120591

1)

+ (1199030+ 1198941199031120596) (cos120596120591lowast

2minus 119894 sin120596120591

lowast

2) = 0

(42)



2+ 1199030sin120596120591

lowast

2

= 1199021120596 cos120596120591

1minus 1199020sin120596120591

1


2minus 1199031120596 sin120596120591

lowast

2

= 1199020cos120596120591

1+ 1199021120596 sin120596120591

1

(43)

It follows that

1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830

= 0 (44)

where

11989833

= 1199012


2

11989832

= 2 (1199030minus 11990121199031) sin120596120591

lowast

2

11989831

= 1199012

1minus 211990101199012minus 2 (119901

21199030minus 11990111199031) cos120596120591lowast

2+ 1199032

1minus 1199022

1

11989830

= 1199012

0+ 211990101199030cos120596120591lowast

2+ 1199032

0minus 1199022

0

(45)

Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If

11989830

lt 0 then


119865 (120596) = +infin (46)



119896 119896 = 1 2 6 there exists

a sequence 120591(119895)1119896


12059110

= min 120591(119895)

1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)

When 1205911

= 120591(119895)


plusmn119894120596(119895)

1119896for 120591lowast2isin [0 120591

210


(H6) ((119889Re(120582))119889120591

1)|120582=plusmn119894120596

(119895)

1119896

= 0



gt 0 1205912

gt 0 1205911



30lt 0 and 120591

lowast

2isin [0 120591

210



1205911



1119896



2= 120591lowast

2isin [0 120591

210

)



lowastat



1=

120591(119895)

1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591



lowast


Let 1199061= 1199091minus 119909lowast

1 1199062= 1199091minus 119909lowast

2 1199063= 1199092minus 119909lowast

3 119905 = 120591

1119905



(119905) = 119871120583(119906119905) + 119891 (120583 119906

119905) (48)

where 119906 = (1199061 1199062 1199063)119879isin R3 and

119871120583(120601) = (120591

1+ 120583) 119861[

[

1206011(0)

1206012(0)

1206013(0)

]

]

+ (1205911+ 120583)119862[

[

1206011(minus1)

1206012(minus1)

1206013(minus1)

]

]

+ (1205911+ 120583)119863

[[[[[[[[[[

[

1206011(minus

120591lowast

2

1205911

)

1206012(minus

120591lowast

2

1205911

)

1206013(minus

120591lowast

2

1205911

)

]]]]]]]]]]

]

(49)

119891 (120583 120601) = (1205911+ 120583)[

[

1198911

1198912

1198913

]

]

(50)



119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911


1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840


1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)



119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)


119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)


119860 (120583) 120601 =

119889120601 (120579)

119889120579


int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)


119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)


lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)


⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)






1 respectively



1119902(120579) From the


(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)


lowast 120573lowast)1198901198941199041205961205911


1 By the


120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)


= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901

0+ 1199020)

+ (1199031120596119894 + 119903

0) (cos120596120591

2minus 119894 sin120596120591

2) = 0

(22)


minus1205963+ (1199011+ 1199021) 120596 = 119903

0sin120596120591

2minus 1199031120596 cos120596120591

2

minus11990121205962+ (1199010+ 1199020) = minus119903

1120596 sin120596120591

2minus 1199030cos120596120591

2

(23)

It follows that

1205966+ 119898121205964+ 119898111205962+ 11989810

= 0 (24)

where 11989812

= 1199012

2minus 2(119901

1+ 1199021) 11989811

= (1199011+ 1199021)2minus 21199012(1199010+

1199020) minus 1199032

1 11989810

= (1199010+ 1199020)2minus 1199032

0


1199113+ 119898121199112+ 11989811119911 + 119898

10= 0 (25)

Let

ℎ1(119911) = 119911

3+ 119898121199112+ 11989811119911 + 119898

10 (26)

we have

119889ℎ1(119911)

119889119911

= 31199112+ 211989812119911 + 119898

11 (27)

If 11989810

= (1199010+ 1199020)2minus 1199032

0lt 0 then ℎ

1(0) lt




= (1199010+ 1199020)2minus 1199032


1198982

12minus 311989811


2

12minus 311989811


31199112+ 211989812119911 + 119898

11= 0 (28)


= (minus11989812


12= (minus119898

12minus



10158401015840

1(119911lowast


0 119911lowast




Lemma 4 (1) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


(2) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


11= (minus119898


ℎ1(119911lowast

11) le 0

(3) If 11989810





12059611

= radic11991111 120596

12= radic11991112 120596

13= radic11991113 (29)

From (23) we have

cos120596111989612059121119896

=

11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896minus 1199030(1199020+ 1199010)

1199032

0+ 1199032

11205962

1119896

(30)

Thus if we denote

120591(119895)

21119896

=

1

1205961119896

times arccos ((11990311205964

1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962

1119896

minus 1199030(1199020+ 1199010))

times (1199032

0+ 1199032

11205962

1119896)

minus1

) + 2119895120587

(31)



21119896

Define

120591210

= 120591(0)

211198960

= min119896=123

120591(0)

21119896

12059610

= 12059611198960

(32)

Let 120582(1205912) = 120572(120591

2)+119894120596(120591


2= 120591(119895)

21119896

satisfying

120572 (120591(119895)

21119896

) = 0 120596 (120591(119895)

21119896

) = 1205961119896 (33)



31205822+ 21199012120582 + (119901


119889120582

1198891205912

= 120582 (1199031120582 + 1199030) 119890minus1205821205912

(34)

Therefore

[

119889120582

1198891205912

]

minus1

=

[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

+

1199031

120582 (1199031120582 + 1199030)

minus

1205912

120582

(35)

When 1205912

= 120591(119895)

21119896

120582(120591(119895)

21119896

) = 1198941205961119896

(119896 = 1 2 3) 120582(1199031120582 +

1199030)|1205912=120591(119895)

21119896

= minus11990311205962

1119896+ 11989411990301205961119896 [31205822

+ 21199012120582 + (119901

1+

1199021)]1198901205821205912|1205912=120591(119895)

21119896

= [minus31205962

1119896+ (1199011+ 1199021)] cos(120596

1119896120591(119895)

21119896

) minus

211990121205961119896sin(1205961119896120591(119895)

21119896

) + 119894211990121205961119896cos(120596

1119896120591(119895)

21119896

) + [minus31205962

1119896+ (1199011+

1199021)] sin(120596

1119896120591(119895)

21119896

)



[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

= Re[[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

+ Re[ 1199031

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

=

1

Λ1

minus 11990311205962

1119896[minus31205962

1119896+ (1199011+ 1199021)] cos (120596

1119896120591(119895)

21119896

)

+ 2119903111990121205963

1119896sin (120596

1119896120591(119895)

21119896

) minus 1199032

11205962

1119896

+ 2119903011990121205962

1119896cos (120596

1119896120591(119895)

21119896

)

+ 1199030[minus31205962

1119896+ (1199011+ 1199021)] 1205961119896sin (120596

1119896120591(119895)

21119896

)

=

1

Λ1

31205966

1119896+ 2 [119901

2

2minus 2 (119901

1+ 1199021)] 1205964

1119896

+ [(1199011+ 1199021)2

minus 21199012(1199010+ 1199020) minus 1199032

1] 1205962

1119896

=

1

Λ1

1199111119896(31199112

1119896+ 2119898121199111119896

+ 11989811)

=

1

Λ1

1199111119896ℎ1015840

1(1199111119896)

(36)

where Λ1= 1199032

11205964

1119896+ 1199032

01205962


1gt 0 1199111119896

gt 0

sign

[

Re 119889 (120582 (1205912))

1198891205912

]

1205912=120591(119895)

21119896

= sign

[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

(37)



= 1205962

1119896and ℎ

1015840

1(1199111119896) = 0 where


21119896

))1198891205912has the same


1(1199111119896)


Theorem6 For 1205911= 0 120591



(i) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


2ge 0 and the


for all 1205912ge 0


lt 0 or 11989810

ge 0 Δ = 1198982

12minus 311989811

gt

0 119911lowast

11gt 0 and ℎ

1(119911lowast

11) le 0 then ℎ



2isin [0 120591

210



210

)




120591(119895)

21119896

(119896 = 1 2 3 119895 = 0 1 2 )



1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901

0+ 1199030) + (119902

1120582 + 1199020) 119890minus1205821205911= 0

(38)


Theorem7 For 1205911gt 0 120591



(i) If 11989820

ge 0 and Δ = 1198982

22minus 311989821


1ge 0 and the


for all 1205911ge 0


lt 0 or 11989820

ge 0 Δ = 1198982

22minus 311989821

gt 0119911lowast

21gt 0 and ℎ

2(119911lowast

21) le 0 then ℎ



parts for 1205911isin [0 120591

120



120

)




120591(119895)

12119896

(119896 = 1 2 3 119895 = 0 1 2 )

where

11989822

= 1199012

2minus 2 (119901

1+ 1199031)

11989821

= (1199011+ 1199031)2

minus 21199012(1199010+ 1199030) minus 1199022

1

11989820

= (1199010+ 1199030)2

minus 1199022

0

ℎ2(119911) = 119911

3+ 119898221199112+ 11989821119911 + 119898

20 119911lowast

21=

minus11989822

+ radicΔ

3

120591(119895)

12119896

=

1

1205962119896

times arccos ((11990211205964

2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962

2119896

minus 1199020(1199030+ 1199010) )

times (1199022

0+ 1199022

11205962

2119896)

minus1

) + 2119895120587

(39)



12119896

Define

120591120

= 120591(0)

121198960

= min119896=123

120591(0)

12119896

12059610

= 12059611198960

(40)



= 1205912


1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0

(41)



210

)Regard 120591

1as a parameter


minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591

1minus 119894 sin120596120591

1)

+ (1199030+ 1198941199031120596) (cos120596120591lowast

2minus 119894 sin120596120591

lowast

2) = 0

(42)



2+ 1199030sin120596120591

lowast

2

= 1199021120596 cos120596120591

1minus 1199020sin120596120591

1


2minus 1199031120596 sin120596120591

lowast

2

= 1199020cos120596120591

1+ 1199021120596 sin120596120591

1

(43)

It follows that

1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830

= 0 (44)

where

11989833

= 1199012


2

11989832

= 2 (1199030minus 11990121199031) sin120596120591

lowast

2

11989831

= 1199012

1minus 211990101199012minus 2 (119901

21199030minus 11990111199031) cos120596120591lowast

2+ 1199032

1minus 1199022

1

11989830

= 1199012

0+ 211990101199030cos120596120591lowast

2+ 1199032

0minus 1199022

0

(45)

Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If

11989830

lt 0 then


119865 (120596) = +infin (46)



119896 119896 = 1 2 6 there exists

a sequence 120591(119895)1119896


12059110

= min 120591(119895)

1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)

When 1205911

= 120591(119895)


plusmn119894120596(119895)

1119896for 120591lowast2isin [0 120591

210


(H6) ((119889Re(120582))119889120591

1)|120582=plusmn119894120596

(119895)

1119896

= 0



gt 0 1205912

gt 0 1205911



30lt 0 and 120591

lowast

2isin [0 120591

210



1205911



1119896



2= 120591lowast

2isin [0 120591

210

)



lowastat



1=

120591(119895)

1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591



lowast


Let 1199061= 1199091minus 119909lowast

1 1199062= 1199091minus 119909lowast

2 1199063= 1199092minus 119909lowast

3 119905 = 120591

1119905



(119905) = 119871120583(119906119905) + 119891 (120583 119906

119905) (48)

where 119906 = (1199061 1199062 1199063)119879isin R3 and

119871120583(120601) = (120591

1+ 120583) 119861[

[

1206011(0)

1206012(0)

1206013(0)

]

]

+ (1205911+ 120583)119862[

[

1206011(minus1)

1206012(minus1)

1206013(minus1)

]

]

+ (1205911+ 120583)119863

[[[[[[[[[[

[

1206011(minus

120591lowast

2

1205911

)

1206012(minus

120591lowast

2

1205911

)

1206013(minus

120591lowast

2

1205911

)

]]]]]]]]]]

]

(49)

119891 (120583 120601) = (1205911+ 120583)[

[

1198911

1198912

1198913

]

]

(50)



119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911


1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840


1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)



119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)


119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)


119860 (120583) 120601 =

119889120601 (120579)

119889120579


int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)


119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)


lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)


⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)






1 respectively



1119902(120579) From the


(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)


lowast 120573lowast)1198901198941199041205961205911


1 By the


120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)


= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

= Re[[31205822+ 21199012120582 + (119901

1+ 1199021)] 1198901205821205912

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

+ Re[ 1199031

120582 (1199031120582 + 1199030)

]

1205912=120591(119895)

21119896

=

1

Λ1

minus 11990311205962

1119896[minus31205962

1119896+ (1199011+ 1199021)] cos (120596

1119896120591(119895)

21119896

)

+ 2119903111990121205963

1119896sin (120596

1119896120591(119895)

21119896

) minus 1199032

11205962

1119896

+ 2119903011990121205962

1119896cos (120596

1119896120591(119895)

21119896

)

+ 1199030[minus31205962

1119896+ (1199011+ 1199021)] 1205961119896sin (120596

1119896120591(119895)

21119896

)

=

1

Λ1

31205966

1119896+ 2 [119901

2

2minus 2 (119901

1+ 1199021)] 1205964

1119896

+ [(1199011+ 1199021)2

minus 21199012(1199010+ 1199020) minus 1199032

1] 1205962

1119896

=

1

Λ1

1199111119896(31199112

1119896+ 2119898121199111119896

+ 11989811)

=

1

Λ1

1199111119896ℎ1015840

1(1199111119896)

(36)

where Λ1= 1199032

11205964

1119896+ 1199032

01205962


1gt 0 1199111119896

gt 0

sign

[

Re 119889 (120582 (1205912))

1198891205912

]

1205912=120591(119895)

21119896

= sign

[

Re 119889 (120582 (1205912))

1198891205912

]

minus1

1205912=120591(119895)

21119896

(37)



= 1205962

1119896and ℎ

1015840

1(1199111119896) = 0 where


21119896

))1198891205912has the same


1(1199111119896)


Theorem6 For 1205911= 0 120591



(i) If 11989810

ge 0 and Δ = 1198982

12minus 311989811


2ge 0 and the


for all 1205912ge 0


lt 0 or 11989810

ge 0 Δ = 1198982

12minus 311989811

gt

0 119911lowast

11gt 0 and ℎ

1(119911lowast

11) le 0 then ℎ



2isin [0 120591

210



210

)




120591(119895)

21119896

(119896 = 1 2 3 119895 = 0 1 2 )



1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901

0+ 1199030) + (119902

1120582 + 1199020) 119890minus1205821205911= 0

(38)


Theorem7 For 1205911gt 0 120591



(i) If 11989820

ge 0 and Δ = 1198982

22minus 311989821


1ge 0 and the


for all 1205911ge 0


lt 0 or 11989820

ge 0 Δ = 1198982

22minus 311989821

gt 0119911lowast

21gt 0 and ℎ

2(119911lowast

21) le 0 then ℎ



parts for 1205911isin [0 120591

120



120

)




120591(119895)

12119896

(119896 = 1 2 3 119895 = 0 1 2 )

where

11989822

= 1199012

2minus 2 (119901

1+ 1199031)

11989821

= (1199011+ 1199031)2

minus 21199012(1199010+ 1199030) minus 1199022

1

11989820

= (1199010+ 1199030)2

minus 1199022

0

ℎ2(119911) = 119911

3+ 119898221199112+ 11989821119911 + 119898

20 119911lowast

21=

minus11989822

+ radicΔ

3

120591(119895)

12119896

=

1

1205962119896

times arccos ((11990211205964

2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962

2119896

minus 1199020(1199030+ 1199010) )

times (1199022

0+ 1199022

11205962

2119896)

minus1

) + 2119895120587

(39)



12119896

Define

120591120

= 120591(0)

121198960

= min119896=123

120591(0)

12119896

12059610

= 12059611198960

(40)



= 1205912


1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0

(41)



210

)Regard 120591

1as a parameter


minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591

1minus 119894 sin120596120591

1)

+ (1199030+ 1198941199031120596) (cos120596120591lowast

2minus 119894 sin120596120591

lowast

2) = 0

(42)



2+ 1199030sin120596120591

lowast

2

= 1199021120596 cos120596120591

1minus 1199020sin120596120591

1


2minus 1199031120596 sin120596120591

lowast

2

= 1199020cos120596120591

1+ 1199021120596 sin120596120591

1

(43)

It follows that

1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830

= 0 (44)

where

11989833

= 1199012


2

11989832

= 2 (1199030minus 11990121199031) sin120596120591

lowast

2

11989831

= 1199012

1minus 211990101199012minus 2 (119901

21199030minus 11990111199031) cos120596120591lowast

2+ 1199032

1minus 1199022

1

11989830

= 1199012

0+ 211990101199030cos120596120591lowast

2+ 1199032

0minus 1199022

0

(45)

Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If

11989830

lt 0 then


119865 (120596) = +infin (46)



119896 119896 = 1 2 6 there exists

a sequence 120591(119895)1119896


12059110

= min 120591(119895)

1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)

When 1205911

= 120591(119895)


plusmn119894120596(119895)

1119896for 120591lowast2isin [0 120591

210


(H6) ((119889Re(120582))119889120591

1)|120582=plusmn119894120596

(119895)

1119896

= 0



gt 0 1205912

gt 0 1205911



30lt 0 and 120591

lowast

2isin [0 120591

210



1205911



1119896



2= 120591lowast

2isin [0 120591

210

)



lowastat



1=

120591(119895)

1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591



lowast


Let 1199061= 1199091minus 119909lowast

1 1199062= 1199091minus 119909lowast

2 1199063= 1199092minus 119909lowast

3 119905 = 120591

1119905



(119905) = 119871120583(119906119905) + 119891 (120583 119906

119905) (48)

where 119906 = (1199061 1199062 1199063)119879isin R3 and

119871120583(120601) = (120591

1+ 120583) 119861[

[

1206011(0)

1206012(0)

1206013(0)

]

]

+ (1205911+ 120583)119862[

[

1206011(minus1)

1206012(minus1)

1206013(minus1)

]

]

+ (1205911+ 120583)119863

[[[[[[[[[[

[

1206011(minus

120591lowast

2

1205911

)

1206012(minus

120591lowast

2

1205911

)

1206013(minus

120591lowast

2

1205911

)

]]]]]]]]]]

]

(49)

119891 (120583 120601) = (1205911+ 120583)[

[

1198911

1198912

1198913

]

]

(50)



119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911


1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840


1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)



119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)


119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)


119860 (120583) 120601 =

119889120601 (120579)

119889120579


int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)


119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)


lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)


⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)






1 respectively



1119902(120579) From the


(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)


lowast 120573lowast)1198901198941199041205961205911


1 By the


120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)


= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= 1205912


1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0

(41)



210

)Regard 120591

1as a parameter


minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591

1minus 119894 sin120596120591

1)

+ (1199030+ 1198941199031120596) (cos120596120591lowast

2minus 119894 sin120596120591

lowast

2) = 0

(42)



2+ 1199030sin120596120591

lowast

2

= 1199021120596 cos120596120591

1minus 1199020sin120596120591

1


2minus 1199031120596 sin120596120591

lowast

2

= 1199020cos120596120591

1+ 1199021120596 sin120596120591

1

(43)

It follows that

1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830

= 0 (44)

where

11989833

= 1199012


2

11989832

= 2 (1199030minus 11990121199031) sin120596120591

lowast

2

11989831

= 1199012

1minus 211990101199012minus 2 (119901

21199030minus 11990111199031) cos120596120591lowast

2+ 1199032

1minus 1199022

1

11989830

= 1199012

0+ 211990101199030cos120596120591lowast

2+ 1199032

0minus 1199022

0

(45)

Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If

11989830

lt 0 then


119865 (120596) = +infin (46)



119896 119896 = 1 2 6 there exists

a sequence 120591(119895)1119896


12059110

= min 120591(119895)

1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)

When 1205911

= 120591(119895)


plusmn119894120596(119895)

1119896for 120591lowast2isin [0 120591

210


(H6) ((119889Re(120582))119889120591

1)|120582=plusmn119894120596

(119895)

1119896

= 0



gt 0 1205912

gt 0 1205911



30lt 0 and 120591

lowast

2isin [0 120591

210



1205911



1119896



2= 120591lowast

2isin [0 120591

210

)



lowastat



1=

120591(119895)

1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591



lowast


Let 1199061= 1199091minus 119909lowast

1 1199062= 1199091minus 119909lowast

2 1199063= 1199092minus 119909lowast

3 119905 = 120591

1119905



(119905) = 119871120583(119906119905) + 119891 (120583 119906

119905) (48)

where 119906 = (1199061 1199062 1199063)119879isin R3 and

119871120583(120601) = (120591

1+ 120583) 119861[

[

1206011(0)

1206012(0)

1206013(0)

]

]

+ (1205911+ 120583)119862[

[

1206011(minus1)

1206012(minus1)

1206013(minus1)

]

]

+ (1205911+ 120583)119863

[[[[[[[[[[

[

1206011(minus

120591lowast

2

1205911

)

1206012(minus

120591lowast

2

1205911

)

1206013(minus

120591lowast

2

1205911

)

]]]]]]]]]]

]

(49)

119891 (120583 120601) = (1205911+ 120583)[

[

1198911

1198912

1198913

]

]

(50)



119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911


1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840


1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)



119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)


119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)


119860 (120583) 120601 =

119889120601 (120579)

119889120579


int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)


119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)


lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)


⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)






1 respectively



1119902(120579) From the


(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)


lowast 120573lowast)1198901198941199041205961205911


1 By the


120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)


= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119861 = [

[

11988711

11988712

0

0 11988722

11988723

0 0 11988733

]

]

119862 = [

[

0 0 0

11988821

0 0

0 0 0

]

]

119863 = [

[

0 0 0

0 0 0

0 11988932

0

]

]

1198911= minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0) minus 119897

21206012

1(0) 1206012(0)

minus 11989751206013

1(0) minus 119897

41206013

1(0) 1206012(0) + sdot sdot sdot

1198912= minus11989761206012(0) 1206013(0) minus (119897

7+ 11988622) 1206012

2(0) + 119897

11206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013

1(minus1) + 119897

41206013

1(minus1) 120601

2(0)

minus 119897101206013

2(0) 1206013(0) + sdot sdot sdot

1198913= 11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

) minus 119886331206012

3(0)

+ 11989791206012

2(minus

120591lowast

2

1205911

)1206013(0) + 119897

81206013

2(minus

120591lowast

2

1205911

)

+ 119897101206013

2(minus

120591lowast

2

1205911


1199011(119909) =

1198861119909

1 + 1198871119909

1199012(119909) =

1198862119909

1 + 1198872119909

1198971= 1199011015840

1(119909lowast)

1198972=

1

2

11990110158401015840

1(119909lowast) 1198973=

1

2

11990110158401015840


1198974=

1

3

119901101584010158401015840

1(119909lowast) 1198975=

1

3

119901101584010158401015840


1198976= 1199011015840

2(1199101lowast) 1198977=

1

2

11990110158401015840


1198978=

1

3

119901101584010158401015840

2(1199101lowast) 1199102lowast 1198979=

1

2

11990110158401015840

2(1199101lowast)

11989710

=

1

3

119901101584010158401015840

2(1199101lowast)

(51)



119871120583120601 = int

0

minus1

119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)


119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +

120591lowast

2

1205911

)]

(53)


119860 (120583) 120601 =

119889120601 (120579)

119889120579


int

0

minus1

119889120578 (119904 120583) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(54)


119905= 119860 (120583) 119909

119905+ 119877 (120583) 119909

119905 (55)


lowast)

define

119860lowast120595 (119904) =

minus

119889120595 (119904)

119889119904

119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(56)


⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint

0

minus1

int

120579

120585=0

120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585

(57)






1 respectively



1119902(120579) From the


(1 120572 120573)1198791198901198941205791205961205911 where

120572 =

119894120596 minus 11988711

11988712

120573 =

11988932(119894120596 minus 119887

11)

11988712(119894120596 minus 119887

33) 119890119894120596120591lowast

2

(58)


lowast 120573lowast)1198901198941199041205961205911


1 By the


120572lowast=

minus119894120596 minus 11988711

119888211198901198941205961205911

120573lowast=

11988723(minus119894120596 minus 119887

11)

11988821(119894120596 minus 119887

33) 1198901198941205961205911

(59)


= 119863(1 120572lowast 120573

lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573

lowast

) 119890minus1198941205961205911(120585minus120579)

119889120578 (120579)

times (1 120572 120573)119879

1198901198941205961205911120585119889120585

= 1198631 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

(60)


Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus we can choose

119863 = 1 + 120572120572lowast+ 120573120573

lowast

+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573

lowast

120591lowast

2119890minus119894120596120591lowast

2

minus1

(61)



0


119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(62)


119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2

+11988211(120579) 119911119911 +119882

02(120579)

1199112

2

+11988230(120579)

1199113

6

+ sdot sdot sdot

(63)





= 1198941205961205911119911 + ⟨119902

lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)

+ 2Re 119911 (119905) 119902 (120579))⟩

= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))

= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591

1119911 + 119892 (119911 119911)

(64)

where

119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892

20

1199112

2

+ 11989211119911119911

+ 11989202

1199112

2

+ 11989221

1199112119911

2

+ sdot sdot sdot

(65)

By (62) we have 119909119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199091119905(0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2

+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2

+119882(2)

11(0) 119911119911 +119882

(2)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2

+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus1) = 119911119890

minus1198941205961205911+ 1199111198901198941205961205911+119882(1)

20(minus1)

1199112

2

+119882(1)

11(minus1) 119911119911 +119882

(1)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus1) = 119911120572119890

minus1198941205961205911+ 119911 120572119890

1198941205961205911+119882(2)

20(minus1)

1199112

2

+119882(2)

11(minus1) 119911119911 +119882

(2)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus1) = 119911120573119890

minus1198941205961205911+ 119911120573119890

1205961205911+119882(3)

20(minus1)

1199112

2

+119882(3)

11(minus1) 119911119911 +119882

(3)

02(minus1)

1199112

2

+ 119900 (|(119911 119911)|3)

1199091119905(minus

120591lowast

2

12059110

) = 119911119890minus119894120596120591lowast

2+ 119911119890119894120596120591lowast

2+119882(1)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(1)

11(minus

120591lowast

2

1205911

)119911119911 +119882(1)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199092119905(minus

120591lowast

2

1205911

) = 119911120572119890minus119894120596120591lowast

2+ 119911 120572119890

119894120596120591lowast

2+119882(2)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(2)

11(minus

120591lowast

2

1205911

)119911119911 +119882(2)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

1199093119905(minus

120591lowast

2

1205911

) = 119911120573119890minus119894120596120591lowast

2+ 119911120573119890

120596120591lowast

2+119882(3)

20(minus

120591lowast

2

1205911

)

1199112

2

+119882(3)

11(minus

120591lowast

2

1205911

)119911119911 +119882(3)

02(minus

120591lowast

2

1205911

)

1199112

2

+ 119900 (|(119911 119911)|3)

(66)



119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591

1(1 120572lowast 120573

lowast

) (119891(0)

1119891(0)

2119891(0)

3)

119879

= 1198631205911[minus (119886

11+ 1198973) 1206012

1(0) minus 119897

11206011(0) 1206012(0)

minus 11989721206012

1(0) 1206012(0) minus 119897

51206013

1(0) minus 119897

41206013

1(0) 1206012(0)

+ sdot sdot sdot ]

+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012

2(0)

+ 11989711206011(minus1) 120601

2(0)

+ 11989731206012


81206013

2(0) minus 119897

91206012

2(0) 1206013(0)

+ 11989721206012

1(minus1) 120601

2(0) + 119897

31206013


+ 120573

lowast

[11989761206012(minus

120591lowast

2

1205911

)1206013(0) + 119897

71206012

2(minus

120591lowast

2

1205911

)

minus 119886331206012

3(0) + 119897

81206013

2(minus

120591lowast

2

1205911


(67)


11989220

= 1198631205911[minus2 (119886

11+ 1198973) minus 2120572119897

1]


1

minus21198976120572120573 minus 2 (119897

7+ 11988622) 1205722]

+ 120573

lowast

[21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2

minus2119886331205732]

11989211

= 1198631205911[minus2 (119886

11+ 1198973) minus 1198971(120572 + 120572)]


minus2 (1198977+ 11988622) 120572120572 + 2119897

3]

+ 120573

lowast

[1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2)

+ 21198977120572120572 minus 119886

33120573120573]

11989202

= 21198631205911[minus2 (119886

11+ 1198973) minus 21198971120572]


7+ 11988622) 1205722+ 211989711205721198901198941205961205911

+2119897311989021198941205961205911]

+ 120573

lowast

[21198976120572120573119890119894120596120591lowast

2+ 2119897712057221198902119894120596120591lowast

2minus 11988633120573

2

]

11989221

= 1198631205911[minus (119886

11+ 1198973) (2119882

(1)

20(0) + 4119882

(1)

11(0))

minus 1198971(2120572119882

(1)

11(0) + 120572119882

(1)

20(0) + 119882

(2)

20(0)

+ 2119882(2)

11(0))]

+ 120572lowast[minus1198976(2120573119882

(2)

11(0) + 120572119882

(3)

20(0) + 120573119882

(2)

20(0)

+ 2120572119882(3)

11(0)) minus (119897

7+ 11988622)

times (4120572119882(2)

11(0) + 2120572119882

(2)

20(0))

+ 1198971(2120572119882

(1)

11(minus1) + 120572119882

(1)

20(minus1)

+ 119882(2)

20(0) 1198901198941205961205911+ 2119882

(2)

11(0) 119890minus1198941205961205911)

+ 1198973(4119882(1)

11(minus1) 119890

minus1198941205961205911+2119882(1)

20(minus1) 119890

1198941205961205911)]

+ 120573

lowast

[1198976(2120573119882

(2)

11(minus

120591lowast

2

1205911

) + 120573119882(2)

20(minus

120591lowast

2

1205911

)

+ 120572119882(3)

20(0) 119890119894120596120591lowast

2+ 2120572119882

(3)

11(0) 119890minus119894120596120591lowast

2)

+ 1198977(4120572119882

(2)

11(minus

120591lowast

2

1205911

) 119890minus119894120596120591lowast

2

+2120572119882(2)

20(minus

120591lowast

2

1205911

) 119890119894120596120591lowast

2)

minus 11988633(4120573119882

(3)

11(0) + 2120573119882

(3)

20(0))]

(68)

where

11988220(120579) =

11989411989220

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989202

31205961205911

119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579

11988211(120579) = minus

11989411989211

1205961205911

119902 (0) 1198901198941205961205911120579+

11989411989211

1205961205911

119902 (0) 119890minus1198941205961205911120579+ 1198642

1198641= 2

[[[[[

[

2119894120596 minus 11988711

minus11988712

0

minus11988821119890minus2119894120596120591

12119894120596 minus 119887

22minus11988723

0 minus11988932119890minus2119894120596120591

lowast

22119894120596 minus 119887

33

]]]]]

]

minus1

times

[[[[[

[

minus2 (11988611

+ 1198973) minus 2120572119897

1

21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591

1minus 21198976120572120573 minus 2 (119897

7+ 11988622) 1205722

21198976120572120573119890minus119894120596120591lowast

2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591

lowast

2minus 2119886331205732

]]]]]

]


1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198642= 2

[[[[

[

minus11988711

minus11988712

0

minus11988821

minus11988722

minus11988723

0 minus11988932

minus11988733

]]]]

]

minus1

times

[[[[

[

minus2 (11988611

+ 1198973) minus 1198971(120572 + 120572)


7+ 11988622) 120572120572 + 2119897

3

1198976(120573120572119890119894120596120591lowast

2+ 120572120573119890

minus119894120596120591lowast

2) + 21198977120572120572 minus 119886

33120573120573

]]]]

]

(69)





1198881(0) =

119894

21205961205911

(1198922011989211

minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205911)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(1205911)

1205961205911

1205732= 2Re 119888

1(0)

(70)


1 Suppose

Re1205821015840(1205911) gt 0 120583






2lt 0(gt 0) and119879



2gt 0(lt 0)



= 03 11988611

= 58889 11988612

= 1 1198981

= 1 1198862

=

01 11988621

= 27 11988622

= 12 11988623

= 12 1198982

= 1 1198863

=

02 11988632

= 25 11988633


lowast= (00451 00357 00551)


= minus00104 12059611

= 0516411991111

= 02666 ℎ10158401(11991111) = 03402 120591

20


1= 0 the positive


2isin

[0 32348) When 1205912crosses 120591

20



2= 56646 120573

2= minus31583 119879

2= 5445



2= 29 isin






1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this


2= 120591lowast

2isin [0 120591

210



1119905 1199092119905 1199093119905)119879 we


(119905) = 119865 (119911119905 1205911 119901) (71)

where 119911119905(120579) = (119909

1119905(120579) 1199092119905(120579) 1199093119905(120579))119879

= (1199091(119905 + 120579) 119909

2(119905 +

120579) 1199093(119905 + 120579))


1(119905) 1199092(119905)



+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909

2gt 0 119909

3gt 0 It is obvious


1 119909lowast

2 119909lowast

3) in

1198773




119883 = 119862([minus1205911 0] 119877

3

+)



N = (119911 1205911 119901) 119865 (119911 120591

1 119901) = 0

(72)

Let ℓ(119864lowast120591(119895)

11198962120587120596

(119895)



1119896 2120587120596

(119895)

1119896) in Γ where 120591

(119895)



11198962120587120596

(119895)

1119896)through (119864

lowast 120591(119895)

1119896 2120587120596

(119895)

1119896)







(i) ℓ(119911lowast120591119901)




sum


120574119898(119911lowast 120591 119901) = 0 (73)






(119911lowast120591119901)


(119911lowast120591119901)



0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 60000451

00451

00452

t

x1(t)

t

0 200 400 60000356

00357

00358

x2(t)

t

0 200 400 6000055

00551

00552

x3(t)

00451

00451

00452

00356

00357

00357

003570055

0055

00551

00551

00552

x 1(t)

x2 (t)

x3(t)


2= 29 lt 120591

20= 32348 119864


0 500 1000004

0045

005

0 500 1000002

004

006

0 500 10000

005

01

t

t

t

x1(t)

x2(t)

x3(t)

0044

0046

0048

002

003

004

005003

004

005

006

007

008

x 1(t)

x2 (t)

x3(t)


2= 33 gt 120591


lowast


0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500

00451

00452

0 500 1000 15000035

0036

0037

0 500 1000 15000054

0055

0056

t

t

t

x1(t)

x2(t)

x3(t)

0045100451

0045200452

0035

00355

003600546

00548

0055

00552

00554

00556

00558

x1(t)

x2 (t)

x3(t)


2= 29 119864


0 500 1000 15000044

0045

0046

0 500 1000 1500002

003

004

0 500 1000 1500004

006

008

t

t

t

x1(t)

x2(t)

x3(t)

00445

0045

00455

0032

0034

0036

00380052

0054

0056

0058

006

0062

0064

x 1(t)

x2 (t)

x3(t)



lowast




(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119911lowast120591119901)




1199091(120579) = 120593 (120579) ge 0 119909

2(120579) = 120595 (120579) ge 0

1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591

1 0)

120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0

(74)




1199091(1205851) = min 119909

1(119905) 119909

1(1205781) = max 119909

1(119905)

1199092(1205852) = min 119909

2(119905) 119909

2(1205782) = max 119909

2(119905)

1199093(1205853) = min 119909

3(119905) 119909

3(1205783) = max 119909

3(119905)

(75)


1199091(119905) = 119909

1(0) expint

119905

0

[1198861minus 119886111199091(119904) minus

119886121199092(119904)

1198981+ 1199091(119904)

] 119889119904

1199092(119905) = 119909

2(0) expint

119905

0

[minus1198862+

119886211199091(119904 minus 1205911)

1198981+ 1199091(119904 minus 1205911)

minus 119886221199092(119904) minus

119886231199093(119904)

1198982+ 1199092(119904)

] 119889119904

1199093(119905) = 119909

3(0) expint

119905

0

[minus1198863+

119886321199092(119904 minus 120591

lowast

2)

1198982+ 1199092(119904 minus 120591

lowast

2)

minus 119886331199093(119904)] 119889119904

(76)



1(119905) gt 0 119909

2(119905) gt

0 1199093(119905) gt 0 for 119905 ge 0


0 = 1198861minus 119886111199091(1205781) minus

119886121199092(1205781)

1198981+ 1199091(1205781)

le 1198861minus 119886111199091(1205781) (77)

thus we have

1199091(1205781) le

1198861

11988611

(78)


0 = minus 1198862+

119886211199091(1205782minus 1205911)

1198981+ 1199091(1205782minus 1205911)

minus 119886221199092(1205782)

minus

119886231199093(1205782)

1198982+ 1199092(1205782)

le minus1198862+

11988621(119886111988611)

1198981+ (119886111988611)

minus 119886221199092(1205782)

(79)


1199092(1205782) le

minus1198862(119886111198981+ 1198861) + 119886111988621

11988622(119886111198981+ 1198861)

≜ 1198721 (80)


0 = minus1198863+

119886321199092(1205783minus 120591lowast

2)

1198982+ 1199092(1205783minus 120591lowast

2)

minus 119886331199093(1205783)

le minus1198863+

119886321198721

1198982+1198721

minus 119886331199093(1205783)

(81)

It follows that

1199093(1205783) le

minus1198863(1198982+1198721) + 119886321198721

11988633(1198982+1198721)

≜ 1198722 (82)



(H7) 11988622

minus (119886211198982) gt 0 [(119886

11minus 11988611198981)11988622

minus

(11988612119886211198982

1)]11988633

minus (11988611

minus (11988611198981))[(119886211198982)11988633

+

(11988623119886321198982

2)] gt 0





1198891199091

119889119905

= 1199091(119905) [1198861minus 119886111199091(119905) minus

119886121199092(119905)

1198981+ 1199091(119905)

]

1198891199092

119889119905

= 1199092(119905) [minus119886

2+

119886211199091(119905)

1198981+ 1199091(119905)

minus 119886221199092(119905) minus

119886231199093(119905)

1198982+ 1199092(119905)

]

1198891199093

119889119905

= 1199093(119905) [minus119886

3+

119886321199092(119905 minus 120591

lowast

2)

1198982+ 1199092(119905 minus 120591

lowast

2)

minus 119886331199093(119905)]

(83)


1198641= (0 0 0) 119864

2= (

1198861

11988611

0 0)

1198643= (1199091 1199092 0) 119864


1 119909lowast

2 119909lowast

3)

(84)











hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















hold let120596119896and 120591(119895)1119896


1119896



11198962120587120596

119896)onto 120591

1-space is [120591


1205911le 120591(119895)

1119896



(A1) 119865 isin 1198622(1198773


1198773


rarr

1198773

+



119863119911119865 (119911 120591

1 119901) =

[[[[[[[[[[[

[

1198861minus 2119886111199091minus

1198861211989811199092

(1198981+ 1199091)2

minus

119886121199091

1198981+ 1199091

0

1198862111989811199092

(1198981+ 1199091)2

minus1198862+

119886211199091

1198981+ 1199091

minus 2119886221199092minus

1198862311989821199093

(1198982+ 1199092)2

minus

119886231199092

1198982+ 1199092

0

1198863211989821199093

(1198982+ 1199092)2

minus1198863+

119886321199092

1198982+ 1199092

minus 2119886331199093

]]]]]]]]]]]

]

(85)


det119863119911119865 (119911lowast 1205911 119901)

= det

[[[[[[[[

[


1+

11988612119909lowast

1119909lowast

2

(1198981+ 119909lowast

1)2

11988612119909lowast

1

1198981+ 119909lowast

1

0

119886211198981119909lowast

2

(1198981+ 119909lowast

1)2


2+

11988623119909lowast

2119909lowast

3

(1198982+ 119909lowast

2)2

minus

11988623119909lowast

2

1198982+ 119909lowast

2

0

119886321198982119909lowast

3

(1198982+ 119909lowast

2)2


3

]]]]]]]]

]

= minus

1198862



11988611198871119909lowast1199101lowast

(1 + 1198871119909lowast)2] = 0

(86)



0 1199010) where 119911 = (119911

(1) 119911(2) 119911(3)) isin 119877

3


Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863

120601119865 (119911 120591

1 119901) (119890

120582119868) (87)

that is

Δ (119911 1205911 119901) (120582)

=

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

[

120582 minus 1198861+ 211988611119911(1)

+

119886121198981119911(2)

(1198981+ 119911(1))

2

11988612119911(1)

1198981+ 119911(1)

0

minus

119886211198981119911(2)

(1198981+ 119911(1))

2119890minus1205821205911

120582 + 1198862minus

11988621119911(1)

1198981+ 119911(1)

+ 211988622119911(2)

+

119886231198982119911(3)

(1198982+ 119911(2))

2

11988623119911(2)

1198982+ 119911(2)

0 minus

119886321198982119911(3)

(1198982+ 119911(2))

2119890minus120582120591lowast

2120582 + 1198863minus

11988632119911(2)

1198982+ 119911(2)

+ 211988633119911(3)

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

]

(88)


From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










From (88) we have

det (Δ (119864lowast 1205911 119901) (120582))

= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902

0+ 1199030)] 119890minus1205821205911

(89)


lowast 120591(119895)

1119896 2120587120596



1119896minus 120575 120591

(119895)

1119896+ 120575) rarr C

such that det(Δ(120582(1205911))) = 0 |120582(120591

1) minus 120596119896| lt 120598 for all 120591

1isin

[120591(119895)

1119896minus 120575 120591(119895)

1119896+ 120575] and

120582 (120591(119895)

1119896) = 120596119896119894

119889Re 120582 (1205911)

1198891205911

1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)

1119896

gt 0 (90)

Let

Ω1205982120587120596

119896

= (120578 119901) 0 lt 120578 lt 120598

10038161003816100381610038161003816100381610038161003816

119901 minus

2120587

120596119896

10038161003816100381610038161003816100381610038161003816

lt 120598 (91)


1119896minus 120575 120591

(119895)

1119896+ 120575] times 120597Ω

1205982120587120596119896

det(Δ(119864


1205911= 120591(119895)

1119896 119901 = 2120587120596

119896 119896 = 1 2 3 119895 = 0 1 2


119867plusmn(119864lowast 120591(119895)

1119896

2120587

120596119896

) (120578 119901)

= det(Δ (119864lowast 120591(119895)

1119896plusmn 120575 119901) (120578 +

2120587

119901

119894))

(92)


1119896 2120587120596

119896) as follows

120574(119864lowast 120591(119895)

1119896

2120587

120596119896

) = deg119861(119867minus(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

minus deg119861(119867+(119864lowast 120591(119895)

1119896

2120587

120596119896

) Ω1205982120587120596

119896

)

= minus1

(93)

Thus we have

sum

(1199111205911119901)isinC

(119864lowast120591(119895)

11198962120587120596119896)

120574 (119911 1205911 119901) lt 0

(94)


(119864lowast 120591(119896)

1119895 2120587120596

119896) (119895 = 0 1 ) It follows from


11198962120587120596

119896)

through (119864lowast 120591(119895)

1119896 2120587120596



we have

120591(119895)

1119896=

1

120596119896

times arccos(((11990121205962

119896minus1199010) (1199030+1199020)

+ 1205962(1205962minus1199011) (1199031+1199021))

times ((1199030+1199020)2

+ (1199031+1199021)2

1205962

119896)

minus1

) +2119895120587

(95)


1119896


(119864lowast120591(119895)

11198962120587120596

119896)onto

1205911-space is [120591


1le 120591(119895)



= 0


11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591



11198962120587120596

119896)onto 120591






(119864lowast120591(119895)

11198962120587120596

119896)


11198962120587120596

119896)onto 119901-


(119864lowast120591(119895)

11198962120587120596



6 Conclusion






Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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