research article complexity analysis of a master-slave...

Research ArticleComplexity Analysis of a Master-Slave Oligopoly Model andChaos Control

Junhai Ma1 Fang Zhang12 and Yanyan He2

1 College of Management and Economics Tianjin University Tianjin 300072 China2Department of Mathematics Tianjin Polytechnic University Tianjin 300387 China

Correspondence should be addressed to Fang Zhang zhangfangsx163com

Received 11 April 2014 Revised 11 June 2014 Accepted 18 June 2014 Published 13 August 2014

Academic Editor Simone Marsiglio

Copyright copy 2014 Junhai Ma 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 establish a master-slave oligopoly game model with an upstream monopoly whose output is considered and two downstreamoligopolies whose prices are consideredThe existence and the local stable region of theNash equilibriumpoint are investigatedThecomplex dynamic properties such as bifurcation and chaos are analyzed using bifurcation diagrams the largest Lyapunov exponentdiagrams and the strange attractor graphWe further analyze the long-run average profit of the three firms and find that they are alloptimal in the stable region In addition delay feedback control method and limiter control method are used in nondelayed modelto control chaos Furthermore a delayed master-slave oligopoly game model is considered and the three firmsrsquo profit in variousconditions is analyzed We find that suitable delayed parameters are important for eliminating chaos and maximizing the profit ofthe players

1 Introduction

Oligopoly is a market structure between monopoly andperfect competition It is characterized by a domination ofseveral firms which completely control trade These firmsmanufacture the same or homogeneous products They haveto consider both the market demand that is the behaviorof consumers and the strategies of their competitors thatis they form expectations concerning how their rivals willact The most widely used and simultaneously the firstformalmodel of oligopolymarket were proposed byCournotCournot model assumed that each company adjusts itsquantity of production to that of its rivals and there is noretaliation at all so that in every step the player perceivesthe latest move made by the competitors to remain his lastBesides Cournot model there is another important modelthe Bertrand model The former is under the assumptionthat producers in an oligopoly decide their policy assumingthat other producers will maintain their output at its existinglevel while Bertrand model is based on the assumption thatproducers act on the belief that competitors will maintaintheir price to maximize profits rather than their output

Works on Cournot or Bertrand model showed that it hasan ample dynamical behavior under different expectations[1ndash6] A large number of literatures have been publishedPuu considered a triopoly Cournot competition model [7]Agiza and Elsadany studied the dynamics of a Cournotduopoly with heterogeneous players [8] Zhang and Maanalyzed Bertrand competition model of four oligopolistswith heterogeneous expectations [9] And there are somearticles about Cournot-Bertrand competition [10 11] Forinstance Ma and Pu studied a Cournot-Bertrand duopolymodel with bounded rational expectations [12] The master-slave Cournot or Bertrand model exists in realistic economyXin and Chen studied a master-slave duopoly Bertrand gamemodel in the setting that the upstream firm might be themonopolistic supplier of fresh water and the downstreamfirm might be the monopolistic supplier of pure distilledwater [13] And the upstream monopolistic firmrsquos output isused as the main factor of production by the downstreammonopolistic firm who is a negligible purchaser of theupstream monopolistrsquos output

In general situations a system with nonlinear term willnot always be stable and sometimes can even be chaotic

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 970205 13 pageshttpdxdoiorg1011552014970205

2 Abstract and Applied Analysis

However the appearance of chaos in the economic systemis not expected and even is harmful Thus people hope tofind some methods to control the chaos of economic systemBy controlling the chaotic phenomenon occurring on themarket bifurcation and chaos are delayed or eliminatedand the system is stabilized to the Nash equilibrium thatis the market goes back to orderly competition In recentyears scholars put forwardmany controlmethods to differentchaotic systems such as OGY control adaptive control andfeedback control

As the game player makes decisions at time 119905 that dependon past observed variables by means of a prediction feedbackand the functional relationships describing the dynamics ofthe model may depend on both the current state and thepast states a delayed structure in economicsmodels emergesYassen andAgiza considered a delayed duopoly game and gotsome important conclusions [14] Since then many expertsand scholars also extensively studied delay oligopoly modelssuch as Peng et al [15] and Ma and Wu [16] These studiesfocus on the changes of stability domain of system in thecase of delay or the bifurcation of system with the parameterschanging but the research to the playersrsquo profits is less

Based on the research of experts and scholars on thesemodels this paper builds a master-slave oligopoly gamemodel in which upstream monopolistic firm competes inoutput and two downstream oligopolistic firms compete inprice We use two control methods the delayed feedbackcontrol method and the limiter control method to controlchaos In addition we study the delayed game model Andwe have given thorough discussion on profits of players

The paper is organized as follows In Section 2 a non-delayed oligopoly model with bounded rational expectationsis presented Equilibrium points and stability are analyzedAnd numerical simulations which confirm analytical resultsare given Two chaos control methods the delayed feedbackcontrol method and the limiter control method are shownin Section 3 Delayed system is investigated in Section 4Section 5 gives the conclusion

2 The Nondelayed Master-Slave Model

21 Model In an area there are three firms in themarket andfirm 119894 produces goods 119909

119894 119894 = 1 2 3 Firm 1 represents the

upstream monopolistic firm and firms 2 and 3 represent twodownstream oligopolistic firms The output and price of firm119894 are represented respectively as 119902

119894and 119901

119894

This model is based on the following assumptions

(1) The upstream monopolist supplies output (eg freshwater) to two downstream firms that compete in afinal goods market (eg distilled bottled water) Theupstream market is monopoly (a single firm firm 1)and the downstream market is duopoly (two firms 2and 3) In addition the two downstream oligopolisticfirms do not cooperate Then we have 119902

1gt 1199022+ 1199023

1199011lt 1199012and 119901

1lt 1199013

(2) The cost of the two upstream monopolistic firms is aquadratic function Because firm 1rsquos price will affectthe cost of the downstream firms (assuming 1 unit of

fresh water is used to make 1 unit spring water) weassume the two firmsrsquo marginal cost is 119901

1

(3) In the vertically connected market the three firmsmake decisions at the same time The upstreammonopolistic firmrsquos decision variable is its output 119902

1

while the two downstream firmsrsquo decision variablesare their prices 119901

2and 119901

3

Assume that the inverse demand function of firm 1 anddemand functions of firms 2 and 3 respectively are

1199011 (119905) = 119886

1minus 11988711199021 (119905)

1199022 (119905) = 119886

2minus 11988721199012 (119905) + 11988921199013 (119905)

1199023 (119905) = 119886

3minus 11988731199013 (119905) + 11988931199012 (119905)

(1)

where 119886119894 119887119894gt 0 119894 = 1 2 3 and 119889

119894gt 0 119894 = 2 3 119889

2and 119889

3

mean the competition parameters between firms 2 and 3The cost functions of the first firm has the following form

[17]

1198621 (119905) = 119888

11199022

1(119905) (2)

where 21198881ismarginal cost and 119888

1is positive for any 119905 gt 0 Firm

1rsquos price will affect the cost of the two downstream firmsThatis 119862119895= 119891(119901

1) 119895 = 2 3 We assume firm 119895rsquos cost function is

119862119895 (119905) = 119901

1 (119905) 119902119895 (119905) (3)

where 119895 = 2 3The profit functions of the three firms are

Π1 (119905) = 119902

1 (119905) 1199011 (119905) minus 1198621 (119905)

= 1199021 (119905) (1198861 minus 11988711199021 (119905)) minus 1198881119902

2

1(119905)

= minus11988711199022

1(119905) minus 1198881119902

2

1(119905) + 11988611199021 (119905)

Π2 (119905) = 119902

2 (119905) 1199012 (119905) minus 1198622 (119905)

= (1198862minus 11988721199012 (119905) + 11988921199013 (119905)) 1199012 (119905) minus 1199011 (119905) 1199022 (119905)

= 11988621199012 (119905) minus 1198872119901

2

2(119905) + 11988921199012 (119905) 1199013 (119905)

minus (1198861minus 11988711199021 (119905)) (1198862 minus 11988721199012 (119905) + 11988921199013 (119905))

Π3 (119905) = 119902

3 (119905) 1199013 (119905) minus 1198623 (119905)

= (1198863minus 11988731199013 (119905) + 11988931199012 (119905)) 1199013 (119905) minus 1199011 (119905) 1199023 (119905)

= 11988631199013 (119905) minus 1198873119901

2

3(119905) + 11988931199012 (119905) 1199013 (119905)


(4)

We assume that the three firms do not have a completeknowledge of the market and the other players In gamesplayers behave adaptively following a bounded rationaladjustment process and they build decisions on the basis ofthe expected marginal profit that is if the marginal profit ispositive (negative) they increase (decrease) their production

Abstract and Applied Analysis 3

or price in the next period The marginal profit functions ofthe three firms are as follows120597Π1 (119905)

1205971199021 (119905)

= minus 211988711199021 (119905) minus 211988811199021 (119905) + 1198861

120597Π2 (119905)

1205971199012 (119905)

= 1198862minus 211988721199012 (119905) + 11988921199013 (119905) + 1198872 (1198861 minus 11988711199021 (119905))

= 1198862+ 11988611198872minus 119887111988721199021 (119905) minus 211988721199012 (119905) + 11988921199013 (119905)

120597Π3 (119905)

1205971199013 (119905)

= 1198863minus 211988731199013 (119905) + 11988931199012 (119905) + 1198873 (1198861 minus 11988711199021 (119905))

= 1198863+ 11988611198873minus 119887111988731199021 (119905) + 11988931199012 (119905) minus 211988731199013 (119905)

(5)

Under the above assumptions the dynamic adjustment equa-tion of the master-slave game is

1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905)120597Π1 (119905)

1205971199021 (119905)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)120597Π2 (119905)

1205971199012 (119905)

1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)120597Π3 (119905)

1205971199013 (119905)

(6)

where 120572 120573 120574 are the adjustment speeds of the three firmsrespectively And 120572 120573 120574 gt 0

Substituting (5) into (6) then themodel has the followingform1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905)

times (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)

times [1198862+ 11988611198872minus 119887111988721199021 (119905) minus 211988721199012 (119905) + 11988921199013 (119905)]

1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)

times [1198863+ 11988611198873minus 119887111988731199021 (119905) + 11988931199012 (119905) minus 211988731199013 (119905)]

(7)

22 EquilibriumPoints and Local Stability Thesystem (7) haseight equilibrium points

1198640= (0 0 0)

1198641= (0 0

(1198863+ 11988611198873)

21198873

)

1198642= (0

(1198862+ 11988611198872)

21198872

0)

1198643= (

1198861

2 (1198871+ 1198881) 0 0)

1198644= (0

(211988621198873+ 2119886111988721198873+ 11988631198892+ 119886111988731198892)

(411988721198873minus 11988921198893)

(211988631198872+ 2119886111988721198873+ 11988621198893+ 119886111988721198893)

(411988721198873minus 11988921198893)

)

1198645= (

1198861

2 (1198871+ 1198881) 0

(211988621198871+ 211988631198881+ 119886111988711198873+ 2119886111988731198881)

41198873(1198871+ 1198881)

)

1198646= (

1198861

2 (1198871+ 1198881)(211988621198871+ 211988621198881+ 119886111988711198872+ 2119886111988721198881)

41198872(1198871+ 1198881)

0)

119864lowast= (

1198861

2 (1198871+ 1198881)ℎ2

119892ℎ3

119892) = (119902

lowast

1 119901lowast

2 119901lowast

3)

(8)

where

ℎ2= 4119886211988711198873+ 4119886211988731198881+ 21198861119887111988721198873+ 41198861119887211988731198881

+ 2119886311988711198892+ 2119886311988811198892+ 1198861119887111988731198892+ 21198861119887311988811198892

ℎ3= 4119886311988711198872+ 4119886311988721198881+ 2119886211988711198893+ 2119886211988811198893

+ 21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988721198893+ 21198861119887211988811198893

119892 = 2 (1198871+ 1198881) (411988721198873minus 11988921198893)

(9)

Since all the equilibrium points should be nonnegative theparameters satisfy119889

11198892lt 411988721198873 Since 119902lowast

1gt 119902lowast

2+119902lowast

3 we should

have

2 [211988721198873(1198862+ 1198863) + (119886

211988731198893+ 119886311988721198893)] (1198871+ 1198881)

+ 119886111988721198873[(1198892+ 1198893) (1198871+ 21198881) minus 2]

lt 1198861[1 + (119887

1+ 21198881) (1198872+ 1198873)] (211988721198873minus 11988921198893)

(10)

The local stability of equilibrium points can be deter-mined by the nature of the eigenvalues of the Jacobianmatrix evaluated at the corresponding equilibrium pointsThe Jacobian matrix of the system (7) corresponding to thestate variables (119902

1 1199012 1199013) is

119869 (119864) = (

11990811

0 0

11990821

11990822

11990823

11990831

11990832

11990833

) (11)

where

11990811= 1 + 120572 (minus4119887

11199021+ 1198861minus 411988811199021)

11990821= minus119887111988721205731199012

11990822= 1 + 120573 (119886

2+ 11988611198872minus 119887111988721199021minus 411988721199012+ 11988921199013)

11990823= 11988921205731199012

11990831= minus119887111988731205741199013

11990832= 11988931205741199013

11990833= 1 + 120574 (119886

3+ 11988611198873minus 119887111988731199021+ 11988931199012minus 411988731199013)

(12)

Theorem 1 All the boundary equilibrium points 1198640 1198641 1198646

are unstable


Proof 1198640 1198641 1198642 and 119864

4all have the eigenvalue 120582

1= 1 +

1198861120572 Since 119886

1 120572 gt 0 then 120582

1gt 1 Hence the equilibrium

points 1198640 1198641 1198642 and 119864

4are unstable equilibrium points [18

19] Similarly 1198643has one eigenvalue 120582 = 1 + 119886

2120573 + 11988611198872120573(1198871+

21198881)2(1198871+ 1198881) gt 1 120582 = 1 + 119886

2120573 + 120573(2119886

311988711198891+ 2119886311988811198892+

21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988731198892+ 21198861119887311988811198892)41198873(1198871+ 1198881) gt 1

for 1198645and 120582 = 1 + 119886

3120574 + 120574(2119886

211988711198893+ 2119886311988811198893+ 21198862119887111988721198873+

41198861119887211988731198881+1198861119887111988721198893+21198861119887211988811198893)41198872(1198871+1198881) gt 1 for 119864

6 Then

all the boundary equilibrium points are unstable

Nowwe investigate the local stability of Nash equilibriumpoint 119864lowast The Jacobian matrix 119869(119864lowast) is

119869 (119864lowast) = (

119908lowast

110 0

119908lowast

21119908lowast

22119908lowast

23

119908lowast

31119908lowast

32119908lowast

33

) (13)

where119908lowast

11= 1 + 120572 (minus4119887

1119902lowast

1+ 1198861minus 41198881119902lowast

1)

119908lowast

21= minus11988711198872120573119901lowast

2

119908lowast

22= 1 + 120573 (119886

2+ 11988611198872minus 11988711198872119902lowast

1minus 41198872119901lowast

2+ 1198892119901lowast

3)

119908lowast

23= 1198892120573119901lowast

2

119908lowast

31= minus11988711198873120574119901lowast

3

119908lowast

32= 1198893120574119901lowast

3

119908lowast

33= 1 + 120574 (119886

3+ 11988611198873minus 11988711198873119902lowast

1+ 1198893119901lowast


3)

(14)

The characteristic polynomial of the Jacobian matrix 119869(119864lowast) is

119891 (120582) = 1205823+ 1198601205822+ 119861120582 + 119862 (15)

And its local stability is given by the Jury conditions [20]

(i) 119891 (1) = 119860 + 119861 + 119862 + 1 gt 0

(ii) 119891 (minus1) = 119860 minus 119861 + 119862 minus 1 lt 0

(iii) 1198622 minus 1 lt 0

(iv) (1 minus 1198622)2

minus (119861 minus 119860119862)2gt 0

(16)

In order to analyze the stability ofNash equilibriumpointwe perform some numerical simulations

23 Numerical Simulations In this section we will show thecomplex behaviors of the system (7) including bifurcationand strange attractor In order to further analyze long-runprofit of the three firms with parameters changing the long-run average profit figures are given It is convenient to takethe parameters values as follows 119886

1= 2 119886

2= 4 119886

3= 3

1198871= 02 119887

2= 2 119887

3= 15 119888

1= 03 119889

2= 05 and 119889

3=

06 The initial values are chosen as (1199021(0) 1199012(0) 1199013(0)) =

(18 21 22) Through (7) the Nash equilibrium point is(2 20769 22154) Then its Jacobian matrix is

119869 (119864lowast) = (

1 minus 2120572 0 0

minus08308120573 1 minus 83077120573 10385120573

minus06646120574 13292120574 1 minus 66462120574

) (17)

00

05

10

15

00

01

02

03

04

04

03

02

01

000

0 5

Figure 1 The stable region of the Nash equilibrium point 119864lowast

where 119860 = 2120572 + 83077120573 + 66462120574 minus 3 119861 = (66462120574 minus

1)(2120572 + 83077120573 minus 2) minus 13804120573120574 + (2120572 minus 1)(83077120573 minus 1) and119862 = minus13804120573120574(2120572 + 83077120573minus2) + 13804120573120574(83077120573minus1)+(2120572 minus 1)(83077120573 minus 1)(66462120574 minus 1)

Figure 1 gives the stable region of the Nash equilibriumpoint 119864lowast We can see that the stable region is 120572 lt 1 120573 lt 024120574 lt 03 approximately From the figure we can conclude thatthe stability region is asymmetric and the higher adjustmentspeeds will push the system out of the stable region

Figure 2 displays the bifurcation diagram and the largestLyapunov exponent with respect to the parameter 120572 whichis the adjustment speed of the upstream monopoly when120573 = 015 and 120574 = 02 By comparing the largest Lyapunovexponent diagram one can have a better understanding of theparticular properties of the system In Figure 2 the system (7)converges to the Nash equilibrium point for 0 lt 120572 lt 099 If120572 increases that is 120572 gt 099 the system turns unstable andcomplex dynamic behavior is observed At 120572 = 099 a flipbifurcation arises which is followed by further flips and thelargest Lyapunov exponent increases to zero for the first timehence the system enters a period doubling routes to chaosWhen 120572 ge 129 the largest Lyapunov exponent is positiveand chaos emerges

Figure 3 is the bifurcation diagram with respect to theparameter 120573 when given 120572 = 05 and 120574 = 02 In Figure 3 theoutput 119902

1(119905) of upstreamfirm is always stablewhich illustrates

that the adjustment speed 120573 has little effect on the output1199021(119905) of the upstream firm while the prices 119901

2(119905) 1199013(119905) of

the two downstream firms generate bifurcation behaviors at120573 = 02312 And when 120573 ge 0306 the largest Lyapunovexponent is positive then the system is in a state of chaos

Similar to Figure 3 Figure 4 gives the bifurcation diagramwith respect to the parameter 120574 when given 120572 = 05 and 120573 =

015 In Figure 4 the output 1199021(119905) is also stable And when


0 05 1 15

3

25

2

15

1

05

0

minus05

120572

q1 p2

p3

q1 p2

p3

Figure 2 Bifurcation diagram and the Largest Lyapunov exponentof the discrete dynamic system (7) for 120573 = 015 120574 = 02 and 120572

varying from 0 to 15

0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

q1

p3

q1

p3

p2

Figure 3 Bifurcation diagram and the Largest Lyapunov exponentof the discrete dynamic system (7) for120572 = 05 120574 = 02 and120573 varyingfrom 0 to 034

120574 gt 02856 the system turns unstable and enters chaos when120574 ge 03822

From the above analysis it can be seen that the adjust-ment parameter 120572 has an important influence on system (7)That is to say the behavior of upstream monopolist has adecisive influence on the market in economics And it isharmful for the development of the two downstream firmsif the changes of adjustment parameters 120572 120573 120574 are too big

Figure 5 represents the graph of a strange attractors of thedynamical system (7) for the adjustment parameter values120573 = 03 120574 = 035 and 120572 = 135 which exhibits fractalstructure of the system

Then we analyse the long-run average profit of the threefirms The results are shown in Figures 6 7 and 8

From these figures we can see that the long-run averageprofit of firm 1 is larger than the other firms Figure 6 shows

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

q1

p2p3

q1

p2p3



01

23

3

2

1

0

3

25

2

15

1

05

0

q1

p2

p3

Figure 5 Strange attractors for 120572 = 135 120573 = 03 and 120574 = 035

0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873

Figure 6 The long-run average profits of the players with 120572 forsystem (7)


0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

1205871

1205872

1205873


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

1205871

1205872

1205873


that the long-run average profit of the three firms fluctuatesand is lower than that of the stable state with the productionadjustment speed increasing when the system (7) enters intobifurcation and chaotic states When 120572 ge 099 the profitof the three firms begins to decrease From Figures 7 and 8we can see that the average profit of the upstream firm is aconstant value so we deduce that the adjustment speeds 120573and 120574 have little effect on the profit of firm 1 In Figure 7the average profit of firms 1 and 3 is always higher than firm2 and the long-run average profit of the downstream firmsdecreases when the bifurcation begins When 120573 = 0255the profit of firm 2 is negative In Figure 8 the long-runaverage profit of the downstream firms decreases when thebifurcation begins and when 120574 = 03024 they are the sameFrom then on the profit of firm 2 is higher than firm 3 andwhen 120574 = 0336 the profit of firm 3 turns to negative

Our study finds that the overall profit of the system willdecrease in bifurcation and chaotic states with adjustmentspeeds increasing

3 Chaos Control of NondelayedMaster-Slave Model

31 Delay Feedback Control Method In this part we first takethe delay feedback control method [21 22] to control chaosThe method is based on the difference between the 119879-timedelayed state and the current state where 119879 denotes a periodof the stabilized orbits The controlled system is

119909 (119905 + 1) = 119891 (119909 (119905) 119906 (119905)) (18)

where 119906(119905) is the input signal and 119909(119905) is the state Consider

119906 (119905) = 119896 (119909 (119905 + 1 minus 119879) minus 119909 (119905 + 1)) 119905 gt 119879 (19)

where 119879 is the time delay and 119896 is the controlling factorBecause the behavior of firm 1 has great effect on the

system we add the function in the first equation from thesystem (7) Then the controlled system is

1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

+ 119896 (1199021 (119905 + 1 minus 119879) minus 1199021 (119905 + 1))

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(20)

By choosing 119879 = 1 the system becomes

1199021 (119905 + 1) = 119902

1 (119905) +1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

(1 + 119896)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(21)

It can be seen from Figure 9 (120572 = 135 120573 = 015120574 = 02) that the chaotic system was gradually controlledwith controlling parameter 119896 increasing When 119896 = 0 itturns uncontrolled system (7) which is at chaotic state Andwhen 119896 ge 036 the system is controlled Taking 119896 = 08we can see that the stable region of 120572 expands to 1775 fromFigure 10 which indicates that chaos is delayed or eliminatedcompletely Figure 11 gives the long-run average profit of theplayers for the controlled system (21) Comparing Figure 11with Figure 6 we see that in Figure 6 the three firmsrsquo profitstarts to fall in 120572 = 099 while in Figure 11 this phenomenondoes not happen

32 Limiter Control Method We use limiter control method[23 24] which is better for firms 2 and 3 This method only


0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3

Figure 9 The trend of system with 119896 increasing for 120572 = 135 120573 =

015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3

Figure 10The bifurcation diagram of the controlled system (21) for119896 = 08 120573 = 015 120574 = 02 and 120572 varying from 0 to 25

0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873

Figure 11 The long-run average profits of the players with 120572 for thecontrolled system (21)

0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3

Figure 12The bifurcation diagram of the controlled system (22) for120573 = 015 and 120574 = 02

requires the player who wants to improve his performanceto take measures without the other playersrsquo cooperation Weimpose lower limiters on the price of firms 2 and 3 noted119901min2

119901min3

and it has no effect on the behavior of firm 1The controlled system is

1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)

By choosing 119901min2

= 20907 and 119901min3

= 22264 Figures12 and 13 give bifurcation diagram and long-run averageprofit of the three firms with the 120572 changing They showthat the behaviors of firm 1 are the same with the originalsystem (7) Figure 12 shows that the bifurcation and chaoticbehaviors of firms 2 and 3 have been controlled In Figure 13the decreasing speed of 120587

2and 120587

3is under control

4 The Delayed Master-Slave Model

41 Model The primary reason for the occurrence of sucha delayed structure in economic models is that (a) decisions


0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


made by economic agents at time 119905 depend on past observedvariables by means of a prediction feedback and (b) the func-tional relationships describing the dynamics of the modelmay not only depend on the current state of the firm butalso in a nontrivial manner on past states Considering thesereasons we introduce the delayed model and compare thethree firmsrsquo profits in various cases

Then the bounded rationality dynamical model evolvedfrom system (7) with one step delayed is given by

1199021 (119905 + 1)

= 1199021 (119905) + 1205721199021 (119905) (minus21198871 ((1 minus 1199081) 1199021 (119905) + 11990811199021 (119905 minus 1))

minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 11990811199021 (119905 minus 1))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)

119908119894represent the weights given to previous production and

prices and 0 le 1199081 1199082 1199083le 1

It is convenient to take 119909(119905) = 1199021(119905 minus 1) 119910(119905) = 119901

2(119905 minus 1)

and 119911(119905) = 1199013(119905 minus 1) then system (24) becomes

1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)

We can get eight equilibrium points denoted by 119864119894=

(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894

3) (119894 = 0 1 7) have the same values as

equilibrium points of system (7) The Jacobi matrix for thesystem (25) is

119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0

Figure 14 The bifurcation diagram and the corresponding largest Lyapunov exponent of the delayed system (25)

1198640 119864

6are boundary equilibrium points 119864

7is the

unique Nash equilibrium point The Jury conditions are

(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0

(ii) 119865 (minus1) = 1205750minus 1205751+ 1205752minus 1205753+ 1205754minus 1205755+ 1205756gt 0

(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)

where 119865(120582) = 1205750+1205751120582+12057521205822+ sdot sdot sdot + 120575

61205826 is the characteristic

polynomial and 119891119894 120593119894(119894 = 1 2 3 4) are the functions of 120575

119894

(119894 = 1 2 3 4 5 6)

42 Numerical Simulations In order to discuss the complex-ity of delayed system (25) we first take the 120572 120573 120574 as constantvalues and then analyze behaviors with 119908

1 1199082 1199083changing

Figure 14 gives the bifurcation diagrams with respect tothe parameters 119908

119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =

02 in (a) (b) and (c) In the nondelayed system when 120572 120573 120574take the values as above as we see from Figure 2 system (7) isat the state of chaos while in Figure 14(a) when119908

2= 1199083= 0

system (25) from chaotic state changes to stable state thenfrom stable state to chaotic state with the changing of119908

1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the

system (25) is always in the chaotic stateWe conclude that1199081

plays an important role in changing system (7) from chaos tostable state with the rest parameters fixed

Similarly taking 120572 = 05 120573 = 032 and 120574 = 02 we knowfrom Figure 4 that the system is in chaotic state In Figure 15the bifurcation diagram (b) shows that when 119908

1= 1199083= 0

the system (25) is also from the chaotic state to stable statethen from stable state to unstable state with 119908

2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


Figure 16 shows the bifurcation diagrams and the largestLyapunov exponent diagram taking 120572 = 05 120573 = 015 120574 =

041 When 1199081= 1199082= 0 system (25) also experiences the

process from chaos to stability then again to the chaos as thechanges of the 119908

3

From the analysis of the Figures 14ndash16 we conclude thatthe appropriate delay parameters play an important role forcontrolling the system from chaos to stability in system (7)

Tables 1 2 and 3 show three firmsrsquo profit and the totalprofit when the systems are nondelayed delay feedbackcontrolled limiter controlled and delayed We conclude thefollowing

(a) The profit of upstream firm is apparently higher thanthe other two downstream firmsrsquo

(b) In Table 1 although the value of 120572 goes beyond thestable region of system (7) when 119908

1= 02 the three

firms achieve maximal profit 1205871= 60 120587

2= 136473

1205873= 170414 total profit sum120587 = 906888 In Table 2

when the value of 120573 goes beyond the stable regionthe three firms achieve maximal profit with 119908

2= 02

While in Table 3 when the value of 120574 goes beyond thestable region the three firms achieve maximal profitwith119908

3= 02 It can be seen that the effect of selecting

appropriate delay parameters is the same as applyinga control on the system which can make profit of thesystem maximization

(c) Taking the appropriate value of 119896 can reach the bestcontrol effect for this case where 120572 goes beyond thestable region This proves that the delay feedbackcontrol method is advantageous to the upstream firm

(d) According to the comparison of the three tableslimiter control method is also effective in controllingchaos And it has great significance when requiring


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0


the player who wants to improve his performance totake measures without the other playersrsquo cooperation

5 Conclusions

In this paper we investigate the dynamics of a boundedrational oligopoly model which contains three players oneis the upstream monopoly who chooses his production tocompete and the other two downstream oligopolies adjusttheir prices to maximize their expected profit The existenceand stability of equilibria bifurcation and chaotic behaviorare analyzed in this game In addition the largest Lyapunovexponent and strange attractors are also applied to display thebifurcation and chaotic behavior of this systemWe show thatif the adjustment speeds of players are too high then theywill change the stability of equilibrium and cause a marketstructure to behave chaotically Furthermore we give long-run average profit of the three firms which demonstrates thatthe equilibrium state is satisfactory to the three firms

We adopt two kinds of control methods and consider thedelayed systemThen the following conclusions are obtained

(1) Delay feedback control method and limiter controlmethod both can control chaos and make profitincrease

(2) From the perspective of profits the delay feedbackcontrol method is advantageous to the upstream firmThe limiter control method is effective in preventingand controlling the two downstream firmsrsquo profitdecline

(3) The effect of selecting appropriate delay parameters isthe same as applying a control on the system


Table 1 Comparative analysis of oligarchsrsquo profits when 120572 = 136 120573 = 015 and 120574 = 02

Parameters Nondelayed Delay feedback control Limiter control Delayed Delayed Delayed1199081

0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146


Parameters Nondelayed Limiter control Delayed Delayed Delayed1199081

0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

minus167885 99633 minus156247 136473 minus130483

1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873

minus132909 139479 minus192747 minus103930 170414

sum120587119894

561364 866260 495898 598387 906888

Conflict of Interests

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

Acknowledgments

The authors would like to thank the reviewers for theircareful reading and for providing some pertinent suggestionsThe research was supported by the National Natural ScienceFoundation of China (no 61273231) and Doctoral Fund ofMinistry of Education of China (Grant no 20130032110073)and it was supported by Tianjin University Innovation Fund

References

[1] H N Agiza ldquoOn the analysis of stability bifurcation chaos andchaos control of Kopel maprdquo Chaos Solitons amp Fractals vol 10no 11 pp 1909ndash1916 1999

[2] G I Bischi and A Naimzada ldquoGlobal analysis of a dynamicduopoly game with bounded rationalityrdquo in Advances inDynamicGames andApplication vol 5 pp 361ndash385 BirkhauserBasel Switzerland 2000

[3] HNAgiza A SHegazi andAA Elsadany ldquoComplex dynam-ics and synchronization of a duopoly game with boundedrationalityrdquoMathematics and Computers in Simulation vol 58no 2 pp 133ndash146 2002


[4] F Tramontana ldquoHeterogeneous duopoly with isoelasticdemand functionrdquo Economic Modelling vol 27 no 1 pp350ndash357 2010

[5] Y Fan T Xie and J Du ldquoComplex dynamics of duopoly gamewith heterogeneous players a further analysis of the outputmodelrdquo Applied Mathematics and Computation vol 218 no 15pp 7829ndash7838 2012

[6] L Fanti and L Gori ldquoThe dynamics of a differentiated duopolywith quantity competitionrdquo Economic Modelling vol 29 no 2pp 421ndash427 2012

[7] T Puu ldquoComplex dynamics with three oligopolistsrdquo ChaosSolitons and Fractals vol 7 no 12 pp 2075ndash2081 1996

[8] HN Agiza andAA Elsadany ldquoChaotic dynamics in nonlinearduopoly game with heterogeneous playersrdquo Applied Mathemat-ics and Computation vol 149 no 3 pp 843ndash860 2004

[9] J Zhang and J Ma ldquoResearch on the price game model for fouroligarchs with different decision rules and its chaos controlrdquoNonlinear Dynamics vol 70 no 1 pp 323ndash334 2012

[10] A K Naimzada and F Tramontana ldquoDynamic properties of aCournot-Bertrand duopoly game with differentiated productsrdquoEconomic Modelling vol 29 no 4 pp 1436ndash1439 2012

[11] C H Tremblay and V J Tremblay ldquoThe Cournot-Bertrandmodel and the degree of product differentiationrdquo EconomicsLetters vol 111 no 3 pp 233ndash235 2011

[12] J Ma and X Pu ldquoThe research on Cournot-Bertrand duopolymodel with heterogeneous goods and its complex characteris-ticsrdquo Nonlinear Dynamics vol 72 no 4 pp 895ndash903 2013

[13] B Xin and T Chen ldquoOn a master-slave Bertrand game modelrdquoEconomic Modelling vol 28 no 4 pp 1864ndash1870 2011

[14] M T Yassen and H N Agiza ldquoAnalysis of a duopoly gamewith delayed bounded rationalityrdquo Applied Mathematics andComputation vol 138 no 2-3 pp 387ndash402 2003

[15] J Peng Z Miao and F Peng ldquoStudy on a 3-dimensional gamemodel with delayed bounded rationalityrdquo Applied Mathematicsand Computation vol 218 no 5 pp 1568ndash1576 2011

[16] J Ma and K Wu ldquoComplex system and influence of delayeddecision on the stability of a triopoly price game modelrdquoNonlinear Dynamics vol 73 no 3 pp 1741ndash1751 2013

[17] T Dubiel-Teleszynski ldquoNonlinear dynamics in a heterogeneousduopoly game with adjusting players and diseconomies ofscalerdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 1 pp 296ndash308 2011

[18] B Xin T Chen and J Ma ldquoNeimark-Sacker bifurcation ina discrete-time financial systemrdquo Discrete Dynamics in Natureand Society vol 2010 Article ID 405639 12 pages 2010

[19] A A Elsadany ldquoCompetition analysis of a triopoly game withbounded rationalityrdquoChaos Solitons and Fractals vol 45 no 11pp 1343ndash1348 2012

[20] S ElaydiAn Introduction toDifference Equations SpringerNewYork NY USA 2005

[21] K Pyragas ldquoContinuous control of chaos by self-controllingfeedbackrdquo Physics Letters A vol 170 no 6 pp 421ndash428 1992

[22] J A Holyst and K Urbanowicz ldquoChaos control in economicalmodel by time-delayed feedback methodrdquo Physica A StatisticalMechanics and its Applications vol 287 no 3-4 pp 587ndash5982000

[23] J Du T Huang Z Sheng and H Zhang ldquoA new method tocontrol chaos in an economic systemrdquoAppliedMathematics andComputation vol 217 no 6 pp 2370ndash2380 2010

[24] J Du Y Fan Z Sheng and Y Hou ldquoDynamics analysis andchaos control of a duopoly game with heterogeneous playersand output limiterrdquo Economic Modelling vol 33 pp 507ndash5162013

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

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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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


However the appearance of chaos in the economic systemis not expected and even is harmful Thus people hope tofind some methods to control the chaos of economic systemBy controlling the chaotic phenomenon occurring on themarket bifurcation and chaos are delayed or eliminatedand the system is stabilized to the Nash equilibrium thatis the market goes back to orderly competition In recentyears scholars put forwardmany controlmethods to differentchaotic systems such as OGY control adaptive control andfeedback control

As the game player makes decisions at time 119905 that dependon past observed variables by means of a prediction feedbackand the functional relationships describing the dynamics ofthe model may depend on both the current state and thepast states a delayed structure in economicsmodels emergesYassen andAgiza considered a delayed duopoly game and gotsome important conclusions [14] Since then many expertsand scholars also extensively studied delay oligopoly modelssuch as Peng et al [15] and Ma and Wu [16] These studiesfocus on the changes of stability domain of system in thecase of delay or the bifurcation of system with the parameterschanging but the research to the playersrsquo profits is less

Based on the research of experts and scholars on thesemodels this paper builds a master-slave oligopoly gamemodel in which upstream monopolistic firm competes inoutput and two downstream oligopolistic firms compete inprice We use two control methods the delayed feedbackcontrol method and the limiter control method to controlchaos In addition we study the delayed game model Andwe have given thorough discussion on profits of players

The paper is organized as follows In Section 2 a non-delayed oligopoly model with bounded rational expectationsis presented Equilibrium points and stability are analyzedAnd numerical simulations which confirm analytical resultsare given Two chaos control methods the delayed feedbackcontrol method and the limiter control method are shownin Section 3 Delayed system is investigated in Section 4Section 5 gives the conclusion

2 The Nondelayed Master-Slave Model

21 Model In an area there are three firms in themarket andfirm 119894 produces goods 119909

119894 119894 = 1 2 3 Firm 1 represents the

upstream monopolistic firm and firms 2 and 3 represent twodownstream oligopolistic firms The output and price of firm119894 are represented respectively as 119902

119894and 119901

119894

This model is based on the following assumptions

(1) The upstream monopolist supplies output (eg freshwater) to two downstream firms that compete in afinal goods market (eg distilled bottled water) Theupstream market is monopoly (a single firm firm 1)and the downstream market is duopoly (two firms 2and 3) In addition the two downstream oligopolisticfirms do not cooperate Then we have 119902

1gt 1199022+ 1199023

1199011lt 1199012and 119901

1lt 1199013

(2) The cost of the two upstream monopolistic firms is aquadratic function Because firm 1rsquos price will affectthe cost of the downstream firms (assuming 1 unit of

fresh water is used to make 1 unit spring water) weassume the two firmsrsquo marginal cost is 119901

1

(3) In the vertically connected market the three firmsmake decisions at the same time The upstreammonopolistic firmrsquos decision variable is its output 119902

1

while the two downstream firmsrsquo decision variablesare their prices 119901

2and 119901

3

Assume that the inverse demand function of firm 1 anddemand functions of firms 2 and 3 respectively are

1199011 (119905) = 119886

1minus 11988711199021 (119905)

1199022 (119905) = 119886

2minus 11988721199012 (119905) + 11988921199013 (119905)

1199023 (119905) = 119886

3minus 11988731199013 (119905) + 11988931199012 (119905)

(1)

where 119886119894 119887119894gt 0 119894 = 1 2 3 and 119889

119894gt 0 119894 = 2 3 119889

2and 119889

3

mean the competition parameters between firms 2 and 3The cost functions of the first firm has the following form

[17]

1198621 (119905) = 119888

11199022

1(119905) (2)

where 21198881ismarginal cost and 119888

1is positive for any 119905 gt 0 Firm

1rsquos price will affect the cost of the two downstream firmsThatis 119862119895= 119891(119901

1) 119895 = 2 3 We assume firm 119895rsquos cost function is

119862119895 (119905) = 119901

1 (119905) 119902119895 (119905) (3)

where 119895 = 2 3The profit functions of the three firms are

Π1 (119905) = 119902

1 (119905) 1199011 (119905) minus 1198621 (119905)

= 1199021 (119905) (1198861 minus 11988711199021 (119905)) minus 1198881119902

2

1(119905)

= minus11988711199022

1(119905) minus 1198881119902

2

1(119905) + 11988611199021 (119905)

Π2 (119905) = 119902

2 (119905) 1199012 (119905) minus 1198622 (119905)

= (1198862minus 11988721199012 (119905) + 11988921199013 (119905)) 1199012 (119905) minus 1199011 (119905) 1199022 (119905)

= 11988621199012 (119905) minus 1198872119901

2

2(119905) + 11988921199012 (119905) 1199013 (119905)


Π3 (119905) = 119902

3 (119905) 1199013 (119905) minus 1198623 (119905)

= (1198863minus 11988731199013 (119905) + 11988931199012 (119905)) 1199013 (119905) minus 1199011 (119905) 1199023 (119905)

= 11988631199013 (119905) minus 1198873119901

2

3(119905) + 11988931199012 (119905) 1199013 (119905)


(4)

We assume that the three firms do not have a completeknowledge of the market and the other players In gamesplayers behave adaptively following a bounded rationaladjustment process and they build decisions on the basis ofthe expected marginal profit that is if the marginal profit ispositive (negative) they increase (decrease) their production



1205971199021 (119905)

= minus 211988711199021 (119905) minus 211988811199021 (119905) + 1198861

120597Π2 (119905)

1205971199012 (119905)

= 1198862minus 211988721199012 (119905) + 11988921199013 (119905) + 1198872 (1198861 minus 11988711199021 (119905))

= 1198862+ 11988611198872minus 119887111988721199021 (119905) minus 211988721199012 (119905) + 11988921199013 (119905)

120597Π3 (119905)

1205971199013 (119905)

= 1198863minus 211988731199013 (119905) + 11988931199012 (119905) + 1198873 (1198861 minus 11988711199021 (119905))

= 1198863+ 11988611198873minus 119887111988731199021 (119905) + 11988931199012 (119905) minus 211988731199013 (119905)

(5)


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905)120597Π1 (119905)

1205971199021 (119905)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)120597Π2 (119905)

1205971199012 (119905)

1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)120597Π3 (119905)

1205971199013 (119905)

(6)



1 (119905) + 1205721199021 (119905)


1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(7)


1198640= (0 0 0)

1198641= (0 0

(1198863+ 11988611198873)

21198873

)

1198642= (0

(1198862+ 11988611198872)

21198872

0)

1198643= (

1198861

2 (1198871+ 1198881) 0 0)

1198644= (0

(211988621198873+ 2119886111988721198873+ 11988631198892+ 119886111988731198892)

(411988721198873minus 11988921198893)

(211988631198872+ 2119886111988721198873+ 11988621198893+ 119886111988721198893)

(411988721198873minus 11988921198893)

)

1198645= (

1198861

2 (1198871+ 1198881) 0

(211988621198871+ 211988631198881+ 119886111988711198873+ 2119886111988731198881)

41198873(1198871+ 1198881)

)

1198646= (

1198861

2 (1198871+ 1198881)(211988621198871+ 211988621198881+ 119886111988711198872+ 2119886111988721198881)

41198872(1198871+ 1198881)

0)

119864lowast= (

1198861

2 (1198871+ 1198881)ℎ2

119892ℎ3

119892) = (119902

lowast

1 119901lowast

2 119901lowast

3)

(8)

where

ℎ2= 4119886211988711198873+ 4119886211988731198881+ 21198861119887111988721198873+ 41198861119887211988731198881

+ 2119886311988711198892+ 2119886311988811198892+ 1198861119887111988731198892+ 21198861119887311988811198892

ℎ3= 4119886311988711198872+ 4119886311988721198881+ 2119886211988711198893+ 2119886211988811198893

+ 21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988721198893+ 21198861119887211988811198893

119892 = 2 (1198871+ 1198881) (411988721198873minus 11988921198893)

(9)


11198892lt 411988721198873 Since 119902lowast

1gt 119902lowast

2+119902lowast

3 we should

have

2 [211988721198873(1198862+ 1198863) + (119886

211988731198893+ 119886311988721198893)] (1198871+ 1198881)

+ 119886111988721198873[(1198892+ 1198893) (1198871+ 21198881) minus 2]

lt 1198861[1 + (119887

1+ 21198881) (1198872+ 1198873)] (211988721198873minus 11988921198893)

(10)


1 1199012 1199013) is

119869 (119864) = (

11990811

0 0

11990821

11990822

11990823

11990831

11990832

11990833

) (11)

where

11990811= 1 + 120572 (minus4119887

11199021+ 1198861minus 411988811199021)

11990821= minus119887111988721205731199012

11990822= 1 + 120573 (119886

2+ 11988611198872minus 119887111988721199021minus 411988721199012+ 11988921199013)

11990823= 11988921205731199012

11990831= minus119887111988731205741199013

11990832= 11988931205741199013

11990833= 1 + 120574 (119886

3+ 11988611198873minus 119887111988731199021+ 11988931199012minus 411988731199013)

(12)


are unstable


Proof 1198640 1198641 1198642 and 119864


1= 1 +

1198861120572 Since 119886

1 120572 gt 0 then 120582


points 1198640 1198641 1198642 and 119864



2120573 + 11988611198872120573(1198871+

21198881)2(1198871+ 1198881) gt 1 120582 = 1 + 119886

2120573 + 120573(2119886

311988711198891+ 2119886311988811198892+

21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988731198892+ 21198861119887311988811198892)41198873(1198871+ 1198881) gt 1

for 1198645and 120582 = 1 + 119886

3120574 + 120574(2119886

211988711198893+ 2119886311988811198893+ 21198862119887111988721198873+

41198861119887211988731198881+1198861119887111988721198893+21198861119887211988811198893)41198872(1198871+1198881) gt 1 for 119864

6 Then



119869 (119864lowast) = (

119908lowast

110 0

119908lowast

21119908lowast

22119908lowast

23

119908lowast

31119908lowast

32119908lowast

33

) (13)

where119908lowast

11= 1 + 120572 (minus4119887

1119902lowast

1+ 1198861minus 41198881119902lowast

1)

119908lowast

21= minus11988711198872120573119901lowast

2

119908lowast

22= 1 + 120573 (119886

2+ 11988611198872minus 11988711198872119902lowast


2+ 1198892119901lowast

3)

119908lowast

23= 1198892120573119901lowast

2

119908lowast

31= minus11988711198873120574119901lowast

3

119908lowast

32= 1198893120574119901lowast

3

119908lowast

33= 1 + 120574 (119886

3+ 11988611198873minus 11988711198873119902lowast

1+ 1198893119901lowast


3)

(14)


119891 (120582) = 1205823+ 1198601205822+ 119861120582 + 119862 (15)


(i) 119891 (1) = 119860 + 119861 + 119862 + 1 gt 0


(iii) 1198622 minus 1 lt 0

(iv) (1 minus 1198622)2

minus (119861 minus 119860119862)2gt 0

(16)



1= 2 119886

2= 4 119886

3= 3

1198871= 02 119887

2= 2 119887

3= 15 119888

1= 03 119889

2= 05 and 119889

3=



119869 (119864lowast) = (

1 minus 2120572 0 0

minus08308120573 1 minus 83077120573 10385120573

minus06646120574 13292120574 1 minus 66462120574

) (17)

00

05

10

15

00

01

02

03

04

04

03

02

01

000

0 5









2(119905) 1199013(119905) of





0 05 1 15

3

25

2

15

1

05

0

minus05

120572

q1 p2

p3

q1 p2

p3



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

q1

p3

q1

p3

p2







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

q1

p2p3

q1

p2p3



01

23

3

2

1

0

3

25

2

15

1

05

0

q1

p2

p3


0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

1205871

1205872

1205873


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

1205871

1205872

1205873






119909 (119905 + 1) = 119891 (119909 (119905) 119906 (119905)) (18)


119906 (119905) = 119896 (119909 (119905 + 1 minus 119879) minus 119909 (119905 + 1)) 119905 gt 119879 (19)



1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

+ 119896 (1199021 (119905 + 1 minus 119879) minus 1199021 (119905 + 1))

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(20)


1199021 (119905 + 1) = 119902

1 (119905) +1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

(1 + 119896)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(21)




0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3


015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3


0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3



119901min3


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)


= 20907 and 119901min3


2and 120587

3is under control




0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205971199021 (119905)

= minus 211988711199021 (119905) minus 211988811199021 (119905) + 1198861

120597Π2 (119905)

1205971199012 (119905)

= 1198862minus 211988721199012 (119905) + 11988921199013 (119905) + 1198872 (1198861 minus 11988711199021 (119905))

= 1198862+ 11988611198872minus 119887111988721199021 (119905) minus 211988721199012 (119905) + 11988921199013 (119905)

120597Π3 (119905)

1205971199013 (119905)

= 1198863minus 211988731199013 (119905) + 11988931199012 (119905) + 1198873 (1198861 minus 11988711199021 (119905))

= 1198863+ 11988611198873minus 119887111988731199021 (119905) + 11988931199012 (119905) minus 211988731199013 (119905)

(5)


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905)120597Π1 (119905)

1205971199021 (119905)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)120597Π2 (119905)

1205971199012 (119905)

1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)120597Π3 (119905)

1205971199013 (119905)

(6)



1 (119905) + 1205721199021 (119905)


1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(7)


1198640= (0 0 0)

1198641= (0 0

(1198863+ 11988611198873)

21198873

)

1198642= (0

(1198862+ 11988611198872)

21198872

0)

1198643= (

1198861

2 (1198871+ 1198881) 0 0)

1198644= (0

(211988621198873+ 2119886111988721198873+ 11988631198892+ 119886111988731198892)

(411988721198873minus 11988921198893)

(211988631198872+ 2119886111988721198873+ 11988621198893+ 119886111988721198893)

(411988721198873minus 11988921198893)

)

1198645= (

1198861

2 (1198871+ 1198881) 0

(211988621198871+ 211988631198881+ 119886111988711198873+ 2119886111988731198881)

41198873(1198871+ 1198881)

)

1198646= (

1198861

2 (1198871+ 1198881)(211988621198871+ 211988621198881+ 119886111988711198872+ 2119886111988721198881)

41198872(1198871+ 1198881)

0)

119864lowast= (

1198861

2 (1198871+ 1198881)ℎ2

119892ℎ3

119892) = (119902

lowast

1 119901lowast

2 119901lowast

3)

(8)

where

ℎ2= 4119886211988711198873+ 4119886211988731198881+ 21198861119887111988721198873+ 41198861119887211988731198881

+ 2119886311988711198892+ 2119886311988811198892+ 1198861119887111988731198892+ 21198861119887311988811198892

ℎ3= 4119886311988711198872+ 4119886311988721198881+ 2119886211988711198893+ 2119886211988811198893

+ 21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988721198893+ 21198861119887211988811198893

119892 = 2 (1198871+ 1198881) (411988721198873minus 11988921198893)

(9)


11198892lt 411988721198873 Since 119902lowast

1gt 119902lowast

2+119902lowast

3 we should

have

2 [211988721198873(1198862+ 1198863) + (119886

211988731198893+ 119886311988721198893)] (1198871+ 1198881)

+ 119886111988721198873[(1198892+ 1198893) (1198871+ 21198881) minus 2]

lt 1198861[1 + (119887

1+ 21198881) (1198872+ 1198873)] (211988721198873minus 11988921198893)

(10)


1 1199012 1199013) is

119869 (119864) = (

11990811

0 0

11990821

11990822

11990823

11990831

11990832

11990833

) (11)

where

11990811= 1 + 120572 (minus4119887

11199021+ 1198861minus 411988811199021)

11990821= minus119887111988721205731199012

11990822= 1 + 120573 (119886

2+ 11988611198872minus 119887111988721199021minus 411988721199012+ 11988921199013)

11990823= 11988921205731199012

11990831= minus119887111988731205741199013

11990832= 11988931205741199013

11990833= 1 + 120574 (119886

3+ 11988611198873minus 119887111988731199021+ 11988931199012minus 411988731199013)

(12)


are unstable


Proof 1198640 1198641 1198642 and 119864


1= 1 +

1198861120572 Since 119886

1 120572 gt 0 then 120582


points 1198640 1198641 1198642 and 119864



2120573 + 11988611198872120573(1198871+

21198881)2(1198871+ 1198881) gt 1 120582 = 1 + 119886

2120573 + 120573(2119886

311988711198891+ 2119886311988811198892+

21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988731198892+ 21198861119887311988811198892)41198873(1198871+ 1198881) gt 1

for 1198645and 120582 = 1 + 119886

3120574 + 120574(2119886

211988711198893+ 2119886311988811198893+ 21198862119887111988721198873+

41198861119887211988731198881+1198861119887111988721198893+21198861119887211988811198893)41198872(1198871+1198881) gt 1 for 119864

6 Then



119869 (119864lowast) = (

119908lowast

110 0

119908lowast

21119908lowast

22119908lowast

23

119908lowast

31119908lowast

32119908lowast

33

) (13)

where119908lowast

11= 1 + 120572 (minus4119887

1119902lowast

1+ 1198861minus 41198881119902lowast

1)

119908lowast

21= minus11988711198872120573119901lowast

2

119908lowast

22= 1 + 120573 (119886

2+ 11988611198872minus 11988711198872119902lowast


2+ 1198892119901lowast

3)

119908lowast

23= 1198892120573119901lowast

2

119908lowast

31= minus11988711198873120574119901lowast

3

119908lowast

32= 1198893120574119901lowast

3

119908lowast

33= 1 + 120574 (119886

3+ 11988611198873minus 11988711198873119902lowast

1+ 1198893119901lowast


3)

(14)


119891 (120582) = 1205823+ 1198601205822+ 119861120582 + 119862 (15)


(i) 119891 (1) = 119860 + 119861 + 119862 + 1 gt 0


(iii) 1198622 minus 1 lt 0

(iv) (1 minus 1198622)2

minus (119861 minus 119860119862)2gt 0

(16)



1= 2 119886

2= 4 119886

3= 3

1198871= 02 119887

2= 2 119887

3= 15 119888

1= 03 119889

2= 05 and 119889

3=



119869 (119864lowast) = (

1 minus 2120572 0 0

minus08308120573 1 minus 83077120573 10385120573

minus06646120574 13292120574 1 minus 66462120574

) (17)

00

05

10

15

00

01

02

03

04

04

03

02

01

000

0 5









2(119905) 1199013(119905) of





0 05 1 15

3

25

2

15

1

05

0

minus05

120572

q1 p2

p3

q1 p2

p3



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

q1

p3

q1

p3

p2







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

q1

p2p3

q1

p2p3



01

23

3

2

1

0

3

25

2

15

1

05

0

q1

p2

p3


0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

1205871

1205872

1205873


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

1205871

1205872

1205873






119909 (119905 + 1) = 119891 (119909 (119905) 119906 (119905)) (18)


119906 (119905) = 119896 (119909 (119905 + 1 minus 119879) minus 119909 (119905 + 1)) 119905 gt 119879 (19)



1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

+ 119896 (1199021 (119905 + 1 minus 119879) minus 1199021 (119905 + 1))

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(20)


1199021 (119905 + 1) = 119902

1 (119905) +1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

(1 + 119896)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(21)




0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3


015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3


0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3



119901min3


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)


= 20907 and 119901min3


2and 120587

3is under control




0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Proof 1198640 1198641 1198642 and 119864


1= 1 +

1198861120572 Since 119886

1 120572 gt 0 then 120582


points 1198640 1198641 1198642 and 119864



2120573 + 11988611198872120573(1198871+

21198881)2(1198871+ 1198881) gt 1 120582 = 1 + 119886

2120573 + 120573(2119886

311988711198891+ 2119886311988811198892+

21198861119887111988721198873+ 41198861119887211988731198881+ 1198861119887111988731198892+ 21198861119887311988811198892)41198873(1198871+ 1198881) gt 1

for 1198645and 120582 = 1 + 119886

3120574 + 120574(2119886

211988711198893+ 2119886311988811198893+ 21198862119887111988721198873+

41198861119887211988731198881+1198861119887111988721198893+21198861119887211988811198893)41198872(1198871+1198881) gt 1 for 119864

6 Then



119869 (119864lowast) = (

119908lowast

110 0

119908lowast

21119908lowast

22119908lowast

23

119908lowast

31119908lowast

32119908lowast

33

) (13)

where119908lowast

11= 1 + 120572 (minus4119887

1119902lowast

1+ 1198861minus 41198881119902lowast

1)

119908lowast

21= minus11988711198872120573119901lowast

2

119908lowast

22= 1 + 120573 (119886

2+ 11988611198872minus 11988711198872119902lowast


2+ 1198892119901lowast

3)

119908lowast

23= 1198892120573119901lowast

2

119908lowast

31= minus11988711198873120574119901lowast

3

119908lowast

32= 1198893120574119901lowast

3

119908lowast

33= 1 + 120574 (119886

3+ 11988611198873minus 11988711198873119902lowast

1+ 1198893119901lowast


3)

(14)


119891 (120582) = 1205823+ 1198601205822+ 119861120582 + 119862 (15)


(i) 119891 (1) = 119860 + 119861 + 119862 + 1 gt 0


(iii) 1198622 minus 1 lt 0

(iv) (1 minus 1198622)2

minus (119861 minus 119860119862)2gt 0

(16)



1= 2 119886

2= 4 119886

3= 3

1198871= 02 119887

2= 2 119887

3= 15 119888

1= 03 119889

2= 05 and 119889

3=



119869 (119864lowast) = (

1 minus 2120572 0 0

minus08308120573 1 minus 83077120573 10385120573

minus06646120574 13292120574 1 minus 66462120574

) (17)

00

05

10

15

00

01

02

03

04

04

03

02

01

000

0 5









2(119905) 1199013(119905) of





0 05 1 15

3

25

2

15

1

05

0

minus05

120572

q1 p2

p3

q1 p2

p3



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

q1

p3

q1

p3

p2







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

q1

p2p3

q1

p2p3



01

23

3

2

1

0

3

25

2

15

1

05

0

q1

p2

p3


0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

1205871

1205872

1205873


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

1205871

1205872

1205873






119909 (119905 + 1) = 119891 (119909 (119905) 119906 (119905)) (18)


119906 (119905) = 119896 (119909 (119905 + 1 minus 119879) minus 119909 (119905 + 1)) 119905 gt 119879 (19)



1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

+ 119896 (1199021 (119905 + 1 minus 119879) minus 1199021 (119905 + 1))

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(20)


1199021 (119905 + 1) = 119902

1 (119905) +1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

(1 + 119896)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(21)




0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3


015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3


0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3



119901min3


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)


= 20907 and 119901min3


2and 120587

3is under control




0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15

3

25

2

15

1

05

0

minus05

120572

q1 p2

p3

q1 p2

p3



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

q1

p3

q1

p3

p2







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

q1

p2p3

q1

p2p3



01

23

3

2

1

0

3

25

2

15

1

05

0

q1

p2

p3


0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873



0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

1205871

1205872

1205873


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

1205871

1205872

1205873






119909 (119905 + 1) = 119891 (119909 (119905) 119906 (119905)) (18)


119906 (119905) = 119896 (119909 (119905 + 1 minus 119879) minus 119909 (119905 + 1)) 119905 gt 119879 (19)



1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

+ 119896 (1199021 (119905 + 1 minus 119879) minus 1199021 (119905 + 1))

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(20)


1199021 (119905 + 1) = 119902

1 (119905) +1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

(1 + 119896)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(21)




0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3


015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3


0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3



119901min3


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)


= 20907 and 119901min3


2and 120587

3is under control




0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 02 025 03

3

25

2

15

1

05

0

minus05

minus1

120573

1205871

1205872

1205873


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

120574

1205871

1205872

1205873






119909 (119905 + 1) = 119891 (119909 (119905) 119906 (119905)) (18)


119906 (119905) = 119896 (119909 (119905 + 1 minus 119879) minus 119909 (119905 + 1)) 119905 gt 119879 (19)



1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

+ 119896 (1199021 (119905 + 1 minus 119879) minus 1199021 (119905 + 1))

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(20)


1199021 (119905 + 1) = 119902

1 (119905) +1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

(1 + 119896)

1199012 (119905 + 1) = 119901

2 (119905) + 1205731199012 (119905)


1199013 (119905 + 1) = 119901

3 (119905) + 1205741199013 (119905)


(21)




0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3


015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3


0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3



119901min3


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)


= 20907 and 119901min3


2and 120587

3is under control




0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15

3

25

2

15

1

05

0

k

q1p2

p3

q1p2

p3


015 and 120574 = 02

0 05 2 251 15

3

25

2

15

1

05

0

minus05

minus1

120572

q1 p2

p3

q1 p2

p3


0 05 1 2 2515

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873


0 05 1 15

3

25

2

15

1

05

0

120572

q1p2

p3

q1p2

p3



119901min3


1199021 (119905 + 1) = 119902

1 (119905) + 1205721199021 (119905) (minus211988711199021 (119905) minus 211988811199021 (119905) + 1198861)

1199012 (119905 + 1) = Max [119891

2(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min2

]

1199013 (119905 + 1) = Max [119891

3(1199021 (119905) 1199012 (119905) 1199013 (119905)) 119901

min3

]

(22)

where 119901min2

119901min3

gt 0 and

1198912(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199012 (119905) + 1205731199012 (119905)


1198913(1199021 (119905) 1199012 (119905) 1199013 (119905))

= 1199013 (119905) + 1205741199013 (119905)


(23)


= 20907 and 119901min3


2and 120587

3is under control




0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15

3

25

2

15

1

05

0

minus05

120572

1205871

1205872

1205873




1199021 (119905 + 1)


minus 21198881((1 minus 119908

1) 1199021 (119905) +11990811199021 (119905 minus 1))+ 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872


minus 21198872((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873


+ 1198893((1 minus 119908

2) 1199012 (119905) + 11990821199012 (119905 minus 1))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 11990831199013 (119905 minus 1))]

(24)


prices and 0 le 1199081 1199082 1199083le 1


2(119905 minus 1)


1199021 (119905 + 1)

= 1199021(119905) + 1205721199021 (119905) ( minus 21198871 ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198881((1 minus 119908

1) 1199021 (119905) + 1199081119909 (119905)) + 1198861)

1199012 (119905 + 1)

= 1199012 (119905) + 1205731199012 (119905) [1198862 + 11988611198872 minus 11988711198872

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

minus 21198872((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

+ 1198892((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

1199013 (119905 + 1)

= 1199013 (119905) + 1205741199013 (119905) [1198863 + 11988611198873 minus 11988711198873

times ((1 minus 1199081) 1199021 (119905) + 1199081119909 (119905))

+ 1198893((1 minus 119908

2) 1199012 (119905) + 1199082119910 (119905))

minus 21198873((1 minus 119908

3) 1199013 (119905) + 1199083119911 (119905))]

119909 (119905 + 1) = 1199021 (119905)

119910 (119905 + 1) = 1199012 (119905)

119911 (119905 + 1) = 1199013 (119905)

(25)


(119902119894

1 119901119894

2 119901119894

3 119909119894 119910119894 119911119894) Consider (119902

119894

1 119901119894

2 119901119894

3) = (119909

119894 119910119894 119911119894)

and (119902119894

1 119901119894

2 119901119894



119869 (119864) = (

(

11986911

0 0 11986914

0 0

11986921

11986922

11986923

11986924

11986925

11986926

11986931

11986932

11986933

11986934

11986935

11986936

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

)

)

(26)

where 11986911

= 11990811+ 2120572119902

11199081(1198871+ 1198881) 11986914

= minus211990811205721199021(1198871+ 1198881)

11986921= 11990831(1 minus 119908

1) 11986922= 11990822+ 2120573119908

211990121198872 11986923= 11990823(1 minus 119908

3)

11986924= 119908111990821 11986925= minus2119887

211990821205731199012 11986926= 119908311990823 11986931= 11990831(1minus1199081)

11986932= 11990832(1 minus 119908

2) 11986933= 11990833+ 2120574119908

311990131198873 11986934= 119908211990832 11986935=

119908211990832 and 119869

36= minus2119887

312057411990131199083


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

q1p2

p3

w1

q1p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0


1198640 119864


7is the


(i) 119865 (1) = 1205750+ 1205751+ 1205752+ 1205753+ 1205754+ 1205755+ 1205756gt 0


(iii) 100381610038161003816100381612057501003816100381610038161003816lt 1205756

(iv) 10038161003816100381610038161198911 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205931(1205750 120575

6)1003816100381610038161003816

(v) 10038161003816100381610038161198912 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205932(1205750 120575

6)1003816100381610038161003816

(vi) 10038161003816100381610038161198913 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205933(1205750 120575

6)1003816100381610038161003816

(vii) 10038161003816100381610038161198914 (1205750 1205756)1003816100381610038161003816gt10038161003816100381610038161205934(1205750 120575

6)1003816100381610038161003816

(27)




119894

(119894 = 1 2 3 4 5 6)


1 1199082 1199083changing


119894 (119894 = 1 2 3) when 120572 = 136 120573 = 015 120574 =


2= 1199083= 0


1 And

when 1199081= 0 119908

119894= 0 as 119908

119895(119894 119895 = 2 3 119894 = 119895) changing the




1= 1199083= 0


2changing


0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

minus05

minus1

w2

q1

p2

p3

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2

p3

(c) 1199081 = 0 1199082 = 0





3





1= 02 the three


2= 136473



2= 02







0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w1

q1

p2

p3

(a) 1199082 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w2

q1

p2

p3

(b) 1199081 = 0 1199083 = 0

0 005 01 015 02 025 03 035 04

3

25

2

15

1

05

0

w3

q1

p2 p3

minus05

q1

p2 p3

(c) 1199081 = 0 1199082 = 0



5 Conclusions









0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 0 0 02 0 0

1199082

0 0 0 0 02 0

1199083

0 0 0 0 0 02

119896 0 08 0 0 0 0

119901min2

0 0 20842 0 0 0

119901min3

0 0 22213 0 0 0

1205871

539601 60 539601 60 539601 539601

1205872

121514 136473 126415 136473 124463 127958

1205873

158208 170414 163254 170414 155874 153587

sum120587119894

819322 906888 829271 906888 829938 821146



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872


1205873

109234 165478 106893 170414 124387

sum120587119894

541349 865111 550646 906888 593904



0 0 02 0 0

1199082

0 0 0 02 0

1199083

0 0 0 0 02

119901min2

0 20842 0 0 0

119901min3

0 22213 0 0 0

1205871

60 60 60 60 60

1205872

94274 126782 88646 102317 136473

1205873


sum120587119894

561364 866260 495898 598387 906888



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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