Event,cause, space timeand quantum memory register
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The International Institute for Science, Technology and Education (IISTE) Journals Call for paper http://www.iiste.org/Journals

The International Institute for Science, Technology and Education (IISTE) Journals Call for paper http://www.iiste.org/Journals

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    Event,cause, space timeand quantum memory register Event,cause, space timeand quantum memory register Document Transcript

    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 EVENT,CAUSE, SPACE-TIMEAND QUANTUM MEMORY REGISTER- AN AUGMENTATION-ARRONDISSEMENT MODEL 1 DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3 PROF C S BAGEWADIABSTRACT: We study a consolidated system of event; cause and n Qubit register which makescomputation with n Qubits. Model extensively dilates upon systemic properties and analyses the systemicbehaviour of the equations together with other concomitant properties. Inclusion of event and cause ,wefeel enhances the “Quantum ness” of the system holistically and brings out a relevance in the QuantumComputation on par with the classical system, in so far as the analysis is concerned. AdditionalVARIABLES OF Space Time provide bastion for the quantum space time studies.INTRODUCTION:EVENT AND ITS VINDICATION:There definitely is a sense of compunction, contrition, hesitation, regret, remorse, hesitation andreservation to the acknowledgement of the fact that there is a personal relation to what happens tooneself. Louis de Broglie said that the events have already happened and it shall disclose to the peoplebased on their level of consciousness. So there is destiny to start with! Say I am undergoing someseemingly insurmountable problem, which has hurt my sensibilities, susceptibilities and sentimentalitiesthat I refuse to accept that that event was waiting for me to happen. In fact this is the statement of stoicphilosophy which is referred to almost as bookish or abstract. Wound is there; it had to happen to me.So I was wounded. Stoics tell us that the wound existed before me; I was born to embody it. It is thequestion of consummation, consolidation, concretization, consubstantiation, that of this, that creates an"event" in us; thus you have become a quasi cause for this wound. For instance, my feeling to becomean actor made me to behave with such perfectionism everywhere, that people’s expectations rose andwhen I did not come up to them I fell; thus the wound was waiting for me and "I was waiting for thewound! One fellow professor used to say like you are searching for ides, ideas also searching for you.Thus the wound possesses in itself a nature which is "impersonal and preindividual" in character, beyondgeneral and particular, the collective and the private. It is the question of becoming universalistic andholistic in your outlook. Unless this fate had not befallen you, the "grand design" would not have takenplace in its entire entirety. It had to happen. And the concomitant ramifications and pernicious or positiveimplications. Everything is in order because the fate befell you. It is not as if the wound had to getsomething that is best from me or that I am a chosen by God to face the event. As said earlier ‘the granddesign" would have been altered. And it cannot alter. You got to play your part and go; there is just noother way. The legacy must go on. You shall be torch bearer and you shall hand over the torch tosomebody. This is the name of the game in totalistic and holistic way.When it comes to ethics, I would say it makes no sense if any obstreperous, obstreperous, ululations,serenading, tintinnabulations are made for the event has happened to me. It means to say that you areunworthy of the fate that has befallen you. To feel that what happened to you was unwarranted and notautonomous, telling the world that you are aggressively iconoclastic, veritably resentful, and volitionallyresentient, is choosing the cast of allegation aspersions and accusations at the Grand Design. What isimmoral is to invoke the name of god, because some event has happened to you. Cursing him isimmoral. Realize that it is all "grand design" and you are playing a part. Resignation, renunciation,revocation is only one form of resentience. Willing the event is primarily to release the eternal truth; infact you cannot release an event despite the fact everyone tries all ways and means they pray god; theyprostrate for others destitution, poverty, penury, misery. But releasing an event is something like an"action at a distance" which only super natural power can do.Here we are face to face with volitional intuition and repetitive transmutation. Like a premeditatedskirmisher, one quarrel with one self, with others, with god, and finally the accuser leaves this world indespair. Now look at this sentence which was quoted by I think Bousquet "if there is a failure of will", "Iwill substitute a longing for death" for that shall be apotheosis, a perpetual and progressive glorificationof the will. 108
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012EVENT AND SINGULARITIES IN QUANTUM SYSTEMS:What is an event? Or for that matter an ideal event? An event is a singularity or rather a set ofsingularities or set of singular points characterizing a mathematical curve, a physical state of affairs, apsychological person or a moral person. Singularities are turning points and points of inflection: they arebottle necks, foyers and centers; they are points of fusion; condensation and boiling; points of tears andjoy; sickness and health; hope and anxiety; they are so to say “sensitive" points; such singularities shouldnot be confused or confounded, aggravated or exacerbated with personality of a system expressing itself;or the individuality and idiosyncrasies of a system which is designated with a proposition. They shouldalso not be fused with the generalizational concept or universalistic axiomatic predications andpostulation alcovishness, or the dipsomaniac flageolet dirge of a concept. Possible a concept could besignified by a figurative representation or a schematic configuration. "Singularity is essentially, preindividual, and has no personalized bias in it, or for that matter a prejudice or pre circumspection of aconceptual scheme. It is in this sense we can define a "singularity" as being neither affirmative nor nonaffirmative. It can be positive or negative; it can create or destroy. On the other hand it must be notedthat singularity is different both in its thematic discursive from the run of the mill day to day musings andmundane drooling. They are in that sense "extra-ordinary".Each singularity is a source and resource, the origin, reason and raison d’être of a mathematical series, itcould be any series any type, and that is interpolated or extrapolated to the structural location of thedestination of another singularity. This according to this standpoint, there are different. It can be positiveor negative; it can create or destroy. On the other hand it must be noted that singularity is different both inits thematic discursive from the run of the mill day to day musings and mundane drooling. There are inthat sense "extra-ordinary". This according to the widely held standpoint, there are different, multifarious, myriad, series IN A structure.In the eventuality of the fact that we conduct an unbiased and prudent examination of the series belongingto different "singularities" we can come to indubitable conclusions that the "singularity" of one system isdifferent from the "other system" in the subterranean realm and ceratoid dualism of comparison andcontrastEPR experiment derived that there exists a communications between two particles. We go a further stepto say that there exists a channel of communication however slovenly, inept, clumpy, between the twosingularities. It is also possible the communication exchange could be one of belligerence,cantankerousness, tempestuousness, astutely truculent, with ensorcelled frenzy. That does not matter. Allwe are telling is that singularities communicate with each other.Now, how do find the reaction of systems to these singularities. You do the same thing a boss does foryou. "Problematize" the events and see how you behave. I will resort to "pressure tactics”. “intimidationof deriding report", or “cut in the increment" to make you undergo trials, travails and tribulations. I amhappy to see if you improve your work; but may or may not be sad if you succumb to it and hangyourself! We do the same thing with systems. systems show conducive response, felicitous reciprocationor behave erratically with inner roil, eponymous radicalism without and with blitzy conviction say like asolipsist nature of bellicose and blustering particles, or for that matter coruscation, trepidiational motionin fluid flows, or seemingly perfidious incendiaries in gormandizing fellow elementary particles,abnormal ebullitions, surcharges calumniations and unwarranted(you think so but the system does not!)unrighteous fulminations. So the point that is made here is “like we problematize the "events" to understand the human behaviourwe have to "problematize" the events of systems to understand their behaviour.This statement is made in connection to the fact that there shall be creation or destruction of particles orcomplete obliteration of the system (blackhole evaporation) or obfuscation of results. Some systems arelike “inside traders" they will not put signature at all! How do you find they did it! Anyway, there arepossibilities of a CIA finding out as they recently did! So we can do the same thing with systems to. Thisis accentuation, corroboration, fortification, .fomentatory notes to explain the various coefficients wehave used in the model as also the dissipations called forIn the Bank example we have clarified that various systems are individually conservative, and theirconservativeness extends holisticallytoo.that one law is universal does not mean there is completeadjudication of nonexistence of totality or global or holistic figure. Total always exists and “individual”systems always exist, if we do not bring Kant in to picture! For the time being let us not! Equationswould become more eneuretic and frenzied...Various, myriad, series in a structure. In the eventuality of the fact that we conduct an unbiased andprudent examination of the series belonging to different "singularities" we can come to indubitable 109
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012conclusions that the "singularity" of one system is different from the "other system" in the subterraneanrealm and ceratoid dualism of comparison and contrast. .CONSERVATION LAWS:Conservation laws bears ample testimony ,infallible observatory, and impeccable demonstration to thefact that the essential predications, character constitutions, ontological consonances remain unchangedwith evolution despite the system’s astute truculence, serenading whimsicality,assymetric disposition oron the other hand anachronistic dispensation ,eponymous radicality,entropic entrepotishness or thesubdued ,relationally contributive, diverse parametrisizational,conducive reciprocity to environment,unconventional behaviour,eneuretic nonlinear frenetic ness ,ensorcelled frenzy, abnormalebulliations,surcharged fulminations , or the inner roil. And that holds well with the evolution with time.We present a model of the generalizational conservation of the theories. A theory of all the conservationtheories. That all conservation laws hold and there is no relationship between them is bê noir. We shall teon this premise build a 36 storey model that deliberates on various issues, structural, dependent, thematicand discursive,Note THAT The classification is executed on systemic properties and parameters. And everything that isknown to us measurable. We do not know”intangible”.Nor we accept or acknowledge that. All laws ofconservation must holds. Hence the holistic laws must hold. Towards that end, interrelationships mustexist. All science like law wants evidence and here we shall provide one under the premise that for allconservations laws to hold each must be interrelated to the other, lest the very conception is a fricativecontretemps. And we live in “Measurement” world.QUANTUM REGISTER:Devices that harness and explore the fundamental axiomatic predications of Physics has wide rangingamplitidunial ramification with its essence of locus and focus on information processing thatoutperforms their classical counterparts, and for unconditionally secure communication. However, inparticular, implementations based on condensed-matter systems face the challenge of short coherencetimes. Carbon materials, particularly diamond, however, are suitable for hosting robust solid-statequantum registers, owing to their spin-free lattice and weak spin–orbit coupling. Studies with thestructurally notched criticism and schizoid fragments of manifestations of historical perspective ofdiamond hosting quantum register have borne ample testimony and, and at differential and determinatelevels have articulated the generalized significations and manifestations of quantum logic elements canbe realized by exploring long-range magnetic dipolar coupling between individually addressable singleelectron spins associated with separate colour centres in diamond. The strong distance dependence ofthis coupling was used to characterize the separation of single qubits (98± 3 Å) with accuracy close to thevalue of the crystal-lattice spacing. Coherent control over electron spins, conditional dynamics,selective readout as well as switchable interaction should rip open glittering faç for a prosperous and adescintillating irreducible affirmation of open the way towards a viable room-temperature solid-statequantum register. As both electron spins are optically addressable, this solid-state quantum deviceoperating at ambient conditions provides a degree of control that is at present available only for a fewsystems at low temperature (See for instance P. Neumann, R. Kolesov, B. Naydenov, J. Bec F. Rempp,M. Steiner, V. Jacques,, G. Balasubramanian,M, M. L. Markham,, D. J. Twitchen,, S. Pezzagna,, J.Meijer, J. Twamley, F. Jelezko & J. Wrachtrup) CAUSE AND EVENT: MODULE NUMBERED ONE 110
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012NOTATION : : CATEGORY ONE OF CAUSE : CATEGORY TWO OF CAUSE : CATEGORY THREE OF CAUSE : CATEGORY ONE OF EVENT : CATEGORY TWO OF EVENT :CATEGORY THREE OFEVENT FIRST TWO CATEGORIES OF QUBITS COMPUTATION: MODULE NUMBERED TWO:========================================================================== === : CATEGORY ONE OF FIRST SET OF QUBITS : CATEGORY TWO OF FIRST SET OF QUBITS : CATEGORY THREE OF FIRST SET OF QUBITS :CATEGORY ONE OF SECOND SET OF QUBITS : CATEGORY TWO OF SECOND SET OF QUBITS : CATEGORY THREE OF SECOND SET OF QUBITS THIRD SET OF QUBITS AND FOURTH SET OF QUBITS: MODULE NUMBERED THREE:============================================================================= : CATEGORY ONE OF THIRD SET OF QUBITS :CATEGORY TWO OF THIRD SET OF QUBITS : CATEGORY THREE OF THIRD SET OF QUBITS : CATEGORY ONE OF FOURTH SET OF QUBITS :CATEGORY TWO OF FOURTH SET OF QUBITS : CATEGORY THREE OF FOURTH SET OF QUBITS FIFTH SET OF QUBITS AND SIXTH SET OF QUBITS : MODULE NUMBERED FOUR: 111
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012============================================================================ : CATEGORY ONE OF FIFTH SET OF QUBITS : CATEGORY TWO OF FIFTH SET OF QUBITS : CATEGORY THREE OF FIFTH SET OF QUBITS :CATEGORY ONE OF SIXTH SET OF QUBITS :CATEGORY TWO OF SIXTH SET OF QUBITS : CATEGORY THREE OF SIXTH SET OF QUBITS SEVENTH SET OF QUBITS AND EIGHTH SET OF QUBITS: MODULE NUMBERED FIVE:============================================================================= : CATEGORY ONE OF SEVENTH SET OF QUBITS : CATEGORY TWO OFSEVENTH SET OF QUBITS :CATEGORY THREE OF SEVENTH SET OF QUBITS :CATEGORY ONE OF EIGHTH SET OF QUBITS :CATEGORY TWO OF EIGHTH SET OF QUBITS :CATEGORY THREE OF EIGHTH SET OF QUBITS (n-1)TH SET OF QUBITS AND nTH SET OF QUBITS : MODULE NUMBERED SIX:============================================================================= : CATEGORY ONE OF(n-1)TH SET OF QUBITS : CATEGORY TWO OF(n-1)TH SET OF QUBITS : CATEGORY THREE OF (N-1)TH SET OF QUBITS : CATEGORY ONE OF n TH SET OF QUBITS : CATEGORY TWO OF n TH SET OF QUBITS : CATEGORY THREE OF n TH SET OF QUBITS 112
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012GLOSSARY OF MODULE NUMBERED SEVEN========================================================================== : CATEGORY ONE OF TIME : CATEGORY TWO OF TIME : CATEGORY THREE OF TIME : CATEGORY ONE OF SPACE : CATEGORY TWO OF SPACE : CATEGORY THREE OF SPACE===============================================================================( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) : ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Accentuation coefficients( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) , ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Dissipation coefficients CAUSE AND EVENT: 1 MODULE NUMBERED ONEThe differential system of this model is now (Module Numbered one) ( )( ) [( )( ) ( )( ) ( )] 2 ( )( ) [( )( ) ( )( ) ( )] 3 ( )( ) [( )( ) ( )( ) ( )] 4 ( )( ) [( )( ) ( )( ) ( )] 5 ( )( ) [( )( ) ( )( ) ( )] 6 ( )( ) [( )( ) ( )( ) ( )] 7 ( )( ) ( ) First augmentation factor 8 113
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( ) First detritions factor FIRST TWO CATEGORIES OF QUBITS COMPUTATION: 9 MODULE NUMBERED TWO:The differential system of this model is now ( Module numbered two) ( )( ) [( )( ) ( )( ) ( )] 10 ( )( ) [( )( ) ( )( ) ( )] 11 ( )( ) [( )( ) ( )( ) ( )] 12 ( )( ) [( )( ) ( )( ) (( ) )] 13 ( )( ) [( )( ) ( )( ) (( ) )] 14 ( )( ) [( )( ) ( )( ) (( ) )] 15 ( )( ) ( ) First augmentation factor 16 ( )( ) (( ) ) First detritions factor 17 THIRD SET OF QUBITS AND FOURTH SET OF QUBITS: 18 MODULE NUMBERED THREEThe differential system of this model is now (Module numbered three) ( )( ) [( )( ) ( )( ) ( )] 19 ( )( ) [( )( ) ( )( ) ( )] 20 ( )( ) [( )( ) ( )( ) ( )] 21 ( )( ) [( )( ) ( )( ) ( )] 22 ( )( ) [( )( ) ( )( ) ( )] 23 ( )( ) [( )( ) ( )( ) ( )] 24 ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factor 25 FIFTH SET OF QUBITS AND SIXTH SET OF QUBITS 26 : MODULE NUMBERED FOUR 114
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012The differential system of this model is now (Module numbered Four) ( )( ) [( )( ) ( )( ) ( )] 27 ( )( ) [( )( ) ( )( ) ( )] 28 ( )( ) [( )( ) ( )( ) ( )] 29 ( )( ) [( )( ) ( )( ) (( ) )] 30 ( )( ) [( )( ) ( )( ) (( ) )] 31 ( )( ) [( )( ) ( )( ) (( ) )] 32 ( )( ) ( ) First augmentation factor 33 ( )( ) (( ) ) First detritions factor 34 SEVENTH SET OF QUBITS AND EIGHTH SET OF QUBITS: 35 MODULE NUMBERED FIVEThe differential system of this model is now (Module number five) ( )( ) [( )( ) ( )( ) ( )] 36 ( )( ) [( )( ) ( )( ) ( )] 37 ( )( ) [( )( ) ( )( ) ( )] 38 ( )( ) [( )( ) ( )( ) (( ) )] 39 ( )( ) [( )( ) ( )( ) (( ) )] 40 ( )( ) [( )( ) ( )( ) (( ) )] 41 ( )( ) ( ) First augmentation factor 42 ( )( ) (( ) ) First detritions factor 43 44 n-1)TH SET OF QUBITS AND nTH SET OF QUBITS : 45 MODULE NUMBERED SIX: 115
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012The differential system of this model is now (Module numbered Six) ( )( ) [( )( ) ( )( ) ( )] 46 ( )( ) [( )( ) ( )( ) ( )] 47 ( )( ) [( )( ) ( )( ) ( )] 48 ( )( ) [( )( ) ( )( ) (( ) )] 49 ( )( ) [( )( ) ( )( ) (( ) )] 50 ( )( ) [( )( ) ( )( ) (( ) )] 51 ( )( ) ( ) First augmentation factor 52 53 GOVERNING EQUATIONS:The differential system of this model is now (SEVENTH MODULE) ( )( ) [( )( ) ( )( ) ( )] 54 ( )( ) [( )( ) ( )( ) ( )] 55 ( )( ) [( )( ) ( )( ) ( )] 56 ( )( ) [( )( ) ( )( ) (( ) )] 57 ( )( ) [( )( ) ( )( ) (( ) )] 58 59 116
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) [( )( ) ( )( ) (( ) )] 60 ( )( ) ( ) First augmentation factor 61 ( )( ) (( ) ) First detritions factor 62FIRST MODULE CONCATENATION: ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) [ ] Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fourth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARESEVENTHAUGMENTATION COEFFICIENTS ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] 117
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) –( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SEVENTH DETRITIONCOEFFICIENTS 118
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 63 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 64 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1, 2 and 3SECOND MODULE CONCATENATION: 65 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 66 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 67 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 68 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ]Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 69 119
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficient for category 1, 2 and3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are fifth augmentation coefficient for category 1, 2 and3 70 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentation coefficient for category 1, 2 and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SEVENTH DETRITION 71COEFFICIENTS ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 72 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 73 ( )( ) –( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 74 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 75 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1,2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( )THIRD MODULE CONCATENATION: 76 120
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 77 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 78 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 79 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1,2 and 3 80 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are seventh augmentation coefficient 81 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 82 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 83 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 84 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detritions coefficients for category 1, 2 and 3 85 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 121
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detritions coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth detritions coefficients for category 1, 2and 3–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) are seventh detritions coefficients 86====================================================================================FOURTH MODULE CONCATENATION: ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 87 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 88 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 89 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ] 90 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 91 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1, 2,and3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2,and 3 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are sixth augmentation coefficients for category 1, 2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SEVENTH augmentationcoefficients 122
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 92 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 93 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 94 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 95 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 96 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( )–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) SEVENTH DETRITIONCOEFFICIENTS 97FIFTH MODULE CONCATENATION: 98 99 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 100 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 123
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 101 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 102 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1,2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1,2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1,2, 3 103 104 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 105 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) [ ( ) ( ) ] ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 106 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ ( )( ) ( ) ] –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 107 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1,2, and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1,2, and 3–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detrition coefficients for category 1,2, and 3SIXTH MODULE CONCATENATION 108 124
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 109 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 110 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] 111 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 112 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - are fourth augmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - fifth augmentation coefficients ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) sixth augmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ARE SVENTHAUGMENTATION COEFFICIENTS 113 114 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 115 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 116 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) [ –( )( ) ( ) ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 117 125
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1, 2, and 3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1, 2, and3–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detrition coefficients for category 1, 2, and3–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ARE SEVENTH DETRITIONCOEFFICIENTS 118 119SEVENTH MODULE CONCATENATION: ( )( ) [( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 120 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )] 121 ( )( ) [( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 122 ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ] ( ) 123 ( )( ) [( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ] 124 125 ( )( ) [( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) (( ) ) 126 ] ( )( ) [( )( ) ( )( ) (( ) )] 127 ( )( ) [( )( ) ( )( ) (( ) )] 128 129 126
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 130 131 132 ( )( ) ( ) First augmentation factor 134(A) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 135(B) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 136Definition of ( )( ) ( )( ) : 137 ( ) ( )( ) ( ) ( )( ) ( ̂ ) 138 ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) 139(C) ( )( ) ( ) ( )( ) 140 ( )( ) (( ) ) ( )( ) 141Definition of ( ̂ )( ) ( ̂ )( ) : 142Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 143 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 144 )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ 145With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 146and( )( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It isto be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) ( ) then the function ( ) ( ) , the SECOND augmentation coefficient would be absolutelycontinuous.Definition of ( ̂ )( ) (̂ )( ) : 147(D) ( ̂ )( ) (̂ )( ) are positive constants 148 ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 149There exists two constants ( ̂ )( ) and ( ̂ )( ) which togetherwith ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) 150(̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) 151( ̂ )( ) 127
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 Where we suppose 152 (E) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 153 The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) 154 ( )( ) ( ) ( )( ) 155 Definition of ( ̂ )( ) ( ̂ )( ) : 156 Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 157 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 158 )( ) 159 ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 160 and( ) ( ( ) ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , the THIRD augmentation coefficient, would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) : 161 (F) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with 162 ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) 163 satisfy the inequalities 164 ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) 165 ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) 166 ( ̂ )( ) 167 Where we suppose 168(G) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 169 (H) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : 128
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) 170 (I) ( )( ) ( ( )( ) ) ( )( ) (( ) ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 171 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 172 and( )( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ ( ) ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , the FOURTH augmentation coefficient WOULD be absolutely continuous. 173 Defi174nition of ( ̂ )( ) (̂ )( ) : 174(J) ( ̂ ) ( ) (̂ )( ) are positive constants(K) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 175 (L) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose 176(M) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 177 (N) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) 129
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) 178 (O) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 179 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 180 and( ) ( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ ) ( ) then the function ( )( ) ( ) , theFIFTH augmentation coefficient attributable would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) : 181(P) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 182(Q) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose 183 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 184 (R) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) (̂ )( ) 185 (S) ( )( ) ( ) ( )( ) 130
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 186 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 187and( ) (( ) ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It isto be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ )( ) then the function ( )( ) ( ) , the SIXTH augmentation coefficient would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 188( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 189There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( ) 190Theorem 1: if the conditions IN THE FOREGOING above are fulfilled, there exists a solution 191satisfying the conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 192 193Definition of ( ) ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 131
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 194 195 ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Definition of ( ) ( ): 196 ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 197Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 198Definition of ( ) ( ): 199 ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Proof: Consider operator ( ) defined on the space of sextuples of continuous functions 200 which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 201 ) ( ̂ )( ) ( ) ( ̂ )( 202 ) ( ̂ )( ) ( ) ( ̂ )( 203By 204 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 205 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 206̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 207 132
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 208̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 209Where ( ) is the integrand that is integrated over an interval ( ) 210Proof: 211 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 212 ) ( ̂ )( ) ( ) ( ̂ )( 213 ) ( ̂ )( ) ( ) ( ̂ )( 214By 215 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 216 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 217̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 218̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 219̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 220Where ( ) is the integrand that is integrated over an interval ( )Proof: 221 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 222 ) ( ̂ )( ) ( ) ( ̂ )( 223 ) ( ̂ )( ) ( ) ( ̂ )( 224By 225 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 226 133
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 227̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 228̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 229̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 230Where ( ) is the integrand that is integrated over an interval ( ) ( ) 231Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 232 ) ( ̂ )( ) ( ) ( ̂ )( 233 ) ( ̂ )( ) ( ) ( ̂ )( 234By 235 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 236 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 237̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 238̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 239̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 240Where ( ) is the integrand that is integrated over an interval ( ) ( ) 241Consider operator defined on the space of sextuples of continuous functionswhich satisfy 242 ( ) ( ) ( ̂ )( ) ( ̂ )( ) 243 ) ( ̂ )( ) ( ) ( ̂ )( 244 ) ( ̂ )( ) ( ) ( ̂ )( 245By 246 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 247 134
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 248̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 249̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 250̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 251Where ( ) is the integrand that is integrated over an interval ( ) 252 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 253 ) ( ̂ )( ) ( ) ( ̂ )( 254 ) ( ̂ )( ) ( ) ( ̂ )( 255By 256 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 257 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 258̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 259̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 260̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 261Where ( ) is the integrand that is integrated over an interval ( ) 262(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into 263 itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 264 135
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )(( ( ) ) ) [(( ) ) ( ̂ )( ) ] ( ̂ )(( ) is as defined in the statement of theorem 1Analogous inequalities hold also for 265(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into 266 itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( ( )( ) ) 267( )( ) ( ̂ )( ) )( ) ( (̂ ) ( ̂ )( )From which it follows that 268 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )(( ( ) ) ) [(( ) ) ( ̂ )( ) ] ( ̂ )(Analogous inequalities hold also for 269(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into 270 itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 271 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( )Analogous inequalities hold also for 272(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed 273 it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 274 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ) ( ) is as defined in the statement of theorem 1(c) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed 275 it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) 136
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 276 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) (̂ )( ) ] ( ̂ )(( ) is as defined in the statement of theorem 1(d) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed 277 it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 278 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ) ( ) is as defined in the statement of theorem 6Analogous inequalities hold also for 279 280 ( )( ) ( )( ) 281It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have 282 (̂ )( ) 283 ( ) ( )( ) [( ̂ ) ( ) (( ̂ ) ( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 284 ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( )In order that the operator ( ) transforms the space of sextuples of functions satisfying 285GLOBAL EQUATIONS into itself ( ) 286The operator is a contraction with respect to the metric ( ) ( ) ( ) ( ) (( )( )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | | 137
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012Indeed if we denote 287Definition of ̃ ̃ : ( ̃ ̃) ( ) ( )It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( ) 288| | (( ) ( ) ( ) ( ) (̂ ) ( ) (̂ ) ( ) (̂ ) ( ) ) (( ( ) ( ) ( ) ( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 289not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact then itsuffices to consider that ( )( ) ( )( ) depend only on and respectively on ( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 290From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] 291 ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) : 292Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively. 138
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012Remark 4: If bounded, from below, the same property holds for The proof is 293analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) ( ( ) )) ( )( ) then 294Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) ( ( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 295 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) ( ( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions 296 ( )( ) ( )( ) 297It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( ̂ )( ) large to have (̂ )( ) 298 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) 299 (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 300In order that the operator transforms the space of sextuples of functions satisfying ( ) 301The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 302Definition of ̃ ̃ : ( ̃ ̃ ) ( ) ( )It results 303 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) 139
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the interval 304From the hypotheses it follows )( ) ( )( ) | (̂ )( ) 305|( (( )( ) ( ) ( ) (̂ ) ( )(̂ )( )(̂ ) ( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result follows 306Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 307not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by ̂ )( ) ̂ )( )( ̂ )( ) ( ( ̂ )( ) ( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact then itsuffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where () () 308From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] () ( () ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 309Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( ) 310If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 311analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 312Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) () ( )( ) 140
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012Then ( )( ) ( ) ( ) which leads to 313 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) 314 ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions 315 ( )( ) ( )( ) 316It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 317 ( ) ( )( ) [( ̂ ( ) ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 318 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ ) ( )( ̂ )( ) ( ) 319In order that the operator transforms the space of sextuples of functions into itself ( ) 320The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 321Definition of ̃ ̃ :( (̃) ( ) ) ̃ ( ) (( )( ))It results 322 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) 323 ( ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( ) 324| | (( ) ( ) ( ) ( ) (̂ ) ( )(̂ )( ) ( ) ̂(̂ ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) )) 141
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 325not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact then it ( ) ( )suffices to consider that ( ) ( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 326From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 327Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 328analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 329Definition of ( )( ) : 330Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to 331 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions 332 142
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) 333It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 334 ( ) ( )( ) [( ̂ )( ) (( ̂ ) ( ) ) ] ( ̂ ) ( )( ̂ )( ) (̂ )( ) 335 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ ) ( ) ] ( ̂ ) ( )( ̂ )( ) ( ) 336In order that the operator transforms the space of sextuples of functions satisfying IN toitself ( ) 337The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows 338 )( ) ( )( ) | (̂ )( ) 339|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 340not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by 143
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ) (̂ )( ) ) (̂ )( )( ̂ )( ( ̂ )( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 341From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 342Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 343analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 344Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 345 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUSinequalities hold also for 346 144
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) 347 It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 348 ( ) ( )( ) [( ̂ ) ( ) (( ̂ )( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 349 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ ) ( )( ̂ )( ) ( ) 350In order that the operator transforms the space of sextuples of functions into itself ( ) 351The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows 352 )( ) ( )( ) | (̂ )( ) 353|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis (35,35,36) the resultfollowsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 354 145
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by ( ) ( )( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 355From GLOBAL EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 356Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 357analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 358Definition of ( )( ) :Indeed let be so that for 359 ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to 360 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( ) 146
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012We now state a more precise theorem about the behaviors at infinity of the solutionsAnalogous inequalities hold also for 361 ( )( ) ( )( ) 362It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 363 ( ) ( )( ) [( ̂ ) ( ) (( ̂ )( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 364 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ ) ( )( ̂ )( ) ( ) 365In order that the operator transforms the space of sextuples of functions into itself ( ) 366The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the interval 367From the hypotheses it follows )( ) ( )( ) | (̂ )( ) 368|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result follows 147
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 369not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by ( ) ( )( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact then ( ) ( )it suffices to consider that ( ) ( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 370From 69 to 32 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 371Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 372analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 373Definition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( ) 374Then ( )( ) ( )( ) which leads to 375 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is 148
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012unbounded. The same property holds for if ( )( ) (( )( ) ( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions 376Behavior of the solutions 377If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 378(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( )Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 379 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( ) ( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :- 380(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 381 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( ) 382are defined respectivelyThen the solution satisfies the inequalities 383 (( )( ) ( )( ) ) ( )( ) ( ) 149
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 384( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 385 ( ) ( )( ) (( )( ) ( )( ) ) 386 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 387 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 388 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutions 389If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) : 390(d) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 391 ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) 392Definition of ( )( ) ( )( ) ( )( ) ( )( ) : 393By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots 394(e) of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 395 and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and 396Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 397By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the 398roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 399 150
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 400Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 401(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 402( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 403( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) 404and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 405and analogously 406( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 407Then the solution satisfies the inequalities 408 (( )( ) ( )( ) ) ( ) ( )( )( )( ) is defined 409 (( )( ) ( )( ) ) ( )( ) 410 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 411( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 412 ( ) ( )( ) (( )( ) ( )( ) ) 413 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 414 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) ) Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 415 Where ( )( ) ( )( ) ( )( ) ( )( ) 416 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 417 ( )( ) ( )( ) ( )( ) 151
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 418Behavior of the solutions 419If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) : 420(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 421(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 422 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )( )( ) is defined 423 (( )( ) ( )( ) ) ( )( ) 424 ( ) ( )( ) ( )( ) 152
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 425( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 426 ( ) ( )( ) (( )( ) ( )( ) ) 427 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 428 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 429 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 430 431Behavior of the solutions 432If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(d) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 433(e) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 434 435 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) 436Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 153
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 437 438 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined by 59 and 64 respectivelyThen the solution satisfies the inequalities 439 440 (( )( ) ( )( ) ) ( ) ( )( ) 441 442 where ( )( ) is defined 443 444 445 (( )( ) ( )( ) ) ( ) ( )( ) 446 ( )( ) ( )( ) 447 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 448( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) ( ) (( )( ) ( )( ) ) 449 ( )( ) (( )( ) ( )( ) ) 450 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 451 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 452 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 453Behavior of the solutions 454 154
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(g) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 455(h) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 456 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(i) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 457 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 458 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) 459 ( ) ( )( ) ( )( ) 460 155
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 461( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 462 ( ) ( )( ) (( )( ) ( )( ) ) 463 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 464 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 465 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutions 466If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(j) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 467(k) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 468 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(l) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 470 156
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 471 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 472 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) 473 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 474( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 475 ( ) ( )( ) (( )( ) ( )( ) ) 476 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 477 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 478 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 479Proof : From GLOBAL EQUATIONS we obtain 480 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) 157
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 Definition of ( ) :- ( ) 481 It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 482 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 483 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 484 (c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain 158
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL E486QUATIONS we get easily the result stated in the theorem. Particular case : 485 If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) ( ) ( ) then ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also ( ) defines ( ) for the special case Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( )( ) and ( ̅ )( ) and definition of ( )( ) 486 we obtain 487 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 488 Definition of :- It follows 489 ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains 490 Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( )(( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( )(( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 491 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( ) ((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( ) 492(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 493 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) 159
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(f) If ( )( ) ( ̅ )( ) ( )( ) , we obtain 494 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( )And so with the notation of the first part of condition (c) , we have ( )Definition of ( ) :- 495 ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtain 496 ( )Definition of ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ). 497Particular case : 498If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( )Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an importantconsequence of the relation between ( )( ) and ( ̅ )( ) 499From GLOBAL EQUATIONS we obtain 500 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( )Definition of ( ) :- ( ) 501It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) 502 From which one obtains(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) 160
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 ( )( ) ( ) ( ) ( )( ) In the same manner , we get 503 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) Definition of ( ̅ )( ) :- From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 504 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain 505 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( )( ) and ( ̅ )( ) 506 : From GLOBAL EQUATIONS we obtain 507 161
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- 508 It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( , ( )( ) [ ( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 509 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) ( ̅ )( ) [ ( ( )( ) (̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 510 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 511 512 (f) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ) ( ) (̅ )( ) ( ̅ )( ) (̅ )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) 162
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also 513 ( ) defines ( ) for the special case . Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( ) and ( ̅ ) and definition of ( )( ) ( ) ( ) 514 From GLOBAL EQUATIONS we obtain 515 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(g) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 516 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(h) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 517 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 163
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 518 (i) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ) ( ) (̅ )( ) ( ̅ )( ) (̅ )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) 519 And so with the notation of the first part of condition (c) , we have Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) ( ) ( ) then ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also ( ) defines ( ) for the special case . Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( ) and ( ̅ ) and definition of ( )( ) ( ) ( ) 520 we obtain 521 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(j) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) 164
    • Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 ( )( ) ( ) ( ) ( )( ) In the same manner , we get 522 [ ( )( )((̅ )( ) (̅ )( ) ) ] 523 (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(k) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 524 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ̅ )( ) ( ̅ )( ) 525 (l) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) ( ) ( ) then ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also ( ) defines ( ) for the special case . Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( ) ( ) ( ) ( ) ( ̅ ) if in addition ( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( ) and ( ̅ ) and definition of ( )( ) ( ) ( ) 526 527 527 We can prove the following 528 Theorem 3: If ( )( ) ( )( ) are independent on , and the conditions ( )( ) ( )( ) ( )( ) ( )( ) 165
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined, then the system 529If ( )( ) ( )( ) are independent on , and the conditions 530.( )( ) ( )( ) ( )( ) ( )( ) 531( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 532( )( ) ( )( ) ( )( ) ( )( ) , 533( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 534 ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 535( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 536( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 537( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined satisfied , then the system 166
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012If ( )( ) ( )( ) are independent on , and the conditions 538( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) , 539 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) as defined are satisfied , then the system( )( ) [( )( ) ( )( ) ( )] 540( )( ) [( )( ) ( )( ) ( )] 541( )( ) [( )( ) ( )( ) ( )] 542( )( ) ( )( ) ( )( ) ( ) 543( )( ) ( )( ) ( )( ) ( ) 544( )( ) ( )( ) ( )( ) ( ) 545has a unique positive solution , which is an equilibrium solution for the system 546( )( ) [( )( ) ( )( ) ( )] 547( )( ) [( )( ) ( )( ) ( )] 548( )( ) [( )( ) ( )( ) ( )] 549( )( ) ( )( ) ( )( ) ( ) 550( )( ) ( )( ) ( )( ) ( ) 551( )( ) ( )( ) ( )( ) ( ) 552has a unique positive solution , which is an equilibrium solution for 553( )( ) [( )( ) ( )( ) ( )] 554( )( ) [( )( ) ( )( ) ( )] 555( )( ) [( )( ) ( )( ) ( )] 556( )( ) ( )( ) ( )( ) ( ) 557( )( ) ( )( ) ( )( ) ( ) 558( )( ) ( )( ) ( )( ) ( ) 559has a unique positive solution , which is an equilibrium solution 560( )( ) [( )( ) ( )( ) ( )] 561 167
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012( )( ) [( )( ) ( )( ) ( )] 563( )( ) [( )( ) ( )( ) ( )] 564( )( ) ( )( ) ( )( ) (( )) 565( )( ) ( )( ) ( )( ) (( )) 566( )( ) ( )( ) ( )( ) (( )) 567has a unique positive solution , which is an equilibrium solution for the system 568( )( ) [( )( ) ( )( ) ( )] 569( )( ) [( )( ) ( )( ) ( )] 570( )( ) [( )( ) ( )( ) ( )] 571( )( ) ( )( ) ( )( ) ( ) 572( )( ) ( )( ) ( )( ) ( ) 573( )( ) ( )( ) ( )( ) ( ) 574has a unique positive solution , which is an equilibrium solution for the system 575( )( ) [( )( ) ( )( ) ( )] 576( )( ) [( )( ) ( )( ) ( )] 577( )( ) [( )( ) ( )( ) ( )] 578( )( ) ( )( ) ( )( ) ( ) 579( )( ) ( )( ) ( )( ) ( ) 580( )( ) ( )( ) ( )( ) ( ) 584has a unique positive solution , which is an equilibrium solution for the system 582 583 584(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) 168
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012( )( ) ( )( )( ) ( ) 585(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( ) 586 587(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( ) 588(a) Indeed the first two equations have a nontrivial solution if ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) 589(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( ( )( ) ) 560(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( )Definition and uniqueness of :- 561After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 562After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations 169
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) 563 , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 564After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] 565Definition and uniqueness of :- 566After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 567After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 568After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )](e) By the same argument, the equations 92,93 admit solutions if 569 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) Where in ( ) must be replaced by their values from 96. It is easy to see thatis a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that ( )(f) By the same argument, the equations 92,93 admit solutions if 570 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) 170
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012Where in ( )( ) must be replaced by their values from 96. It is easy to see that 571 is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ) )(g) By the same argument, the concatenated equations admit solutions if 572 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( ) must be replaced by their values from 96. It is easy to see thatis a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ) ) 573(h) By the same argument, the equations of modules admit solutions if 574 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to seethat is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that (( ) )(i) By the same argument, the equations (modules) admit solutions if 575 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to seethat is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that (( ) )(j) By the same argument, the equations (modules) admit solutions if 578 579 ( ) ( )( ) ( )( ) ( )( ) ( )( ) 580[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) 581Where in ( )( ) must be replaced by their values It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows thatthere exists a unique such that ( )Finally we obtain the unique solution of 89 to 94 582 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] 171
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( )( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 583 (( )) , ( ) and 584 ( )( ) ( )( ) 585 , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 586 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )] Obviously, these values represent an equilibrium solution 587Finally we obtain the unique solution 588 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 589 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 590 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 591 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 592 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 593 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] 172
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ( )( ) ( )( ) 594 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionASYMPTOTIC STABILITY ANALYSIS 595Theorem 4: If the conditions of the previous theorem are satisfied and if the functions( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stable.Proof: DenoteDefinition of :- , 596 ( )( ) ( )( ) ( ) ( )( ) , ( )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 597 (( )( ) ( )( ) ) ( )( ) ( )( ) 598 (( )( ) ( )( ) ) ( )( ) ( )( ) 599 (( )( ) ( )( ) ) ( )( ) ( )( ) 600 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 601 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 602 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 603If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 604Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenote 605Definition of :- , 606 ( )( ) ( )( ) 607 ( ) ( )( ) , (( ) )taking into account equations (global)and neglecting the terms of power 2, we obtain 608 (( )( ) ( )( ) ) ( )( ) ( )( ) 609 (( )( ) ( )( ) ) ( )( ) ( )( ) 610 (( )( ) ( )( ) ) ( )( ) ( )( ) 611 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 612 173
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 613 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 614If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 615Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) ) 616Then taking into account equations (global) and neglecting the terms of power 2, we obtain 617 (( )( ) ( )( ) ) ( )( ) ( )( ) 618 (( )( ) ( )( ) ) ( )( ) ( )( ) 619 (( )( ) ( )( ) ) ( )( ) ( )( ) 6120 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 621 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 622 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 623If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 624Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- 625 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 626 (( )( ) ( )( ) ) ( )( ) ( )( ) 627 (( )( ) ( )( ) ) ( )( ) ( )( ) 628 (( )( ) ( )( ) ) ( )( ) ( )( ) 629 174
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 630 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 631 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 632 633 If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( )Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- 634 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 635 (( )( ) ( )( ) ) ( )( ) ( )( ) 636 (( )( ) ( )( ) ) ( )( ) ( )( ) 637 (( )( ) ( )( ) ) ( )( ) ( )( ) 638 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 639 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 640 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 641If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 642Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- 643 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations(global) and neglecting the terms of power 2, we obtain 644 (( )( ) ( )( ) ) ( )( ) ( )( ) 645 (( )( ) ( )( ) ) ( )( ) ( )( ) 646 175
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 (( )( ) ( )( ) ) ( )( ) ( )( ) 647 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 648 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 649 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 650 651The characteristic equation of this system is 652(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )] 653((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) 176
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) 177
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ) 178
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )And as one sees, all the coefficients are positive. It follows that all the roots have negative real part,and this proves the theorem.Acknowledgments:The introduction is a collection of information from various articles, Books, NewsPaper reports, Home Pages Of authors, Journal Reviews, Nature ‘s L:etters,ArticleAbstracts, Research papers, Abstracts Of Research Papers, Stanford Encyclopedia,Web Pages, Ask a Physicist Column, Deliberations with Professors, the internetincluding Wikipedia. We acknowledge all authors who have contributed to the same.In the eventuality of the fact that there has been any act of omission on the part ofthe authors, we regret with great deal of compunction, contrition, regret,trepidiation and remorse. As Newton said, it is only because erudite and eminentpeople allowed one to piggy ride on their backs; probably an attempt has been madeto look slightly further. Once again, it is stated that the references are onlyillustrative and not comprehensive REFERENCES1. Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi - MEASUREMENTDISTURBS EXPLANATION OF QUANTUM MECHANICAL STATES-A HIDDEN VARIABLETHEORY - published at: "International Journal of Scientific and Research Publications, Volume 2,Issue 5, May 2012 Edition".2. DR K N PRASANNA KUMAR, PROF B S KIRANAGI and PROF C S BAGEWADI -CLASSIC 2 FLAVOUR COLOR SUPERCONDUCTIVITY AND ORDINARY NUCLEARMATTER-A NEW PARADIGM STATEMENT - published at: "International Journal of Scientificand Research Publications, Volume 2, Issue 5, May 2012 Edition".3. A HAIMOVICI: “On the growth of a two species ecological system divided on age groups”. Tensor, Vol 37 (1982),Commemoration volume dedicated to Professor Akitsugu Kawaguchi on his 80th birthday4. FRTJOF CAPRA: “The web of life” Flamingo, Harper Collins See "Dissipative structures” pages 172-1885. HEYLIGHEN F. (2001): "The Science of Self-organization and Adaptivity", in L. D. Kiel, (ed) . Knowledge Management, Organizational Intelligence and Learning, and Complexity, in: The Encyclopedia of Life Support Systems ((EOLSS), (Eolss Publishers, Oxford) [http://www.eolss.net6. MATSUI, T, H. Masunaga, S. M. Kreidenweis, R. A. Pielke Sr., W.-K. Tao, M. Chin, and Y. J Kaufman (2006), “Satellite-based assessment of marine low cloud variability associated with aerosol, atmospheric stability, and the diurnal cycle”, J. Geophys. Res., 111, D17204, doi:10.1029/2005JD0060977. STEVENS, B, G. Feingold, W.R. Cotton and R.L. Walko, “Elements of the microphysical 179
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012 structure of numerically simulated nonprecipitating stratocumulus” J. Atmos. Sci., 53, 980-10068. FEINGOLD, G, Koren, I; Wang, HL; Xue, HW; Brewer, WA (2010), “Precipitation-generated oscillations in open cellular cloud fields” Nature, 466 (7308) 849-852, doi: 10.1038/nature09314, Published 12-Aug 2010 180
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012(9)^ a b c Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhaltabhängig?", Annalen der Physik 18:639 Bibcode 1905AnP...323..639E,DOI:10.1002/andp.19053231314. See also the English translation.(10)^ a b Paul Allen Tipler, Ralph A. Llewellyn (2003-01), Modern Physics, W. H. Freeman andCompany, pp. 87–88, ISBN 0-7167-4345-0(11)^ a b Rainville, S. et al. World Year of Physics: A direct test of E=mc2. Nature 438, 1096-1097(22 December 2005) | doi: 10.1038/4381096a; Published online 21 December 2005.(12)^ In F. Fernflores. The Equivalence of Mass and Energy. Stanford Encyclopedia of Philosophy(13)^ Note that the relativistic mass, in contrast to the rest mass m0, is not a relativistic invariant, andthat the velocity is not a Minkowski four-vector, in contrast to the quantity , where is the differentialof the proper time. However, the energy-momentum four-vector is a genuine Minkowski four-vector,and the intrinsic origin of the square-root in the definition of the relativistic mass is the distinctionbetween dτ and dt.(14)^ Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0(15)^ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8(16)^ Hans, H. S.; Puri, S. P. (2003). Mechanics (2 ed.). Tata McGraw-Hill. p. 433. ISBN 0-07-047360-9., Chapter 12 page 433(17)^ E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. Freeman and Co., NY. 1992.ISBN 0-7167-2327-1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclearbombs, until heat is allowed to escape.(18)^ Mould, Richard A. (2002). Basic relativity (2 ed.). Springer. p. 126. ISBN 0-387-95210-1., Chapter 5 page 126(19)^ Chow, Tail L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones &Bartlett Learning. p. 392. ISBN 0-7637-3827-1., Chapter 10 page 392(20)^ [2] Cockcroft-Walton experiment(21)^ a b c Conversions used: 1956 International (Steam) Table (IT) values where one calorie≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers conversion value of one gram TNT 181
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012≡ 1000 calories used.(22)^ Assuming the dam is generating at its peak capacity of 6,809 MW.(23)^ Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominatedaverage Cp of 25.8, 5.134 moles of metal, and 132 J.K-1 for the prototype. A variation of± picograms is of course, much smaller than the actual uncertainty in the mass of the international 1.5prototype, which are ± micrograms. 2(24)^ [3] Article on Earth rotation energy. Divided by c^2.(25)^ a b Earths gravitational self-energy is 4.6 × 10-10 that of Earths total mass, or 2.7 trillion metrictons. Citation: The Apache Point Observatory Lunar Laser-Ranging Operation (APOLLO), T. W.Murphy, Jr. et al. University of Washington, Dept. of Physics (132 kB PDF, here.).(26)^ There is usually more than one possible way to define a field energy, because any field can bemade to couple to gravity in many different ways. By general scaling arguments, the correct answer ateveryday distances, which are long compared to the quantum gravity scale, should be minimalcoupling, which means that no powers of the curvature tensor appear. Any non-minimal couplings,along with other higher order terms, are presumably only determined by a theory of quantum gravity,and within string theory, they only start to contribute to experiments at the string scale.(27)^ G. t Hooft, "Computation of the quantum effects due to a four-dimensional pseudoparticle",Physical Review D14:3432–3450 (1976).(28)^ A. Belavin, A. M. Polyakov, A. Schwarz, Yu. Tyupkin, "Pseudoparticle Solutions to YangMills Equations", Physics Letters 59B:85 (1975).(29)^ F. Klinkhammer, N. Manton, "A Saddle Point Solution in the Weinberg Salam Theory",Physical Review D 30:2212.(30)^ Rubakov V. A. "Monopole Catalysis of Proton Decay", Reports on Progress in Physics 51:189–241 (1988).(31)^ S.W. Hawking "Black Holes Explosions?" Nature 248:30 (1974).(32)^ Einstein, A. (1905), "Zur Elektrodynamik bewegter Körper." (PDF), Annalen der Physik 17:891–921, Bibcode 1905AnP...322...891E,DOI:10.1002/andp.19053221004. English translation. 182
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012(33)^ See e.g. Lev B.Okun, The concept of Mass, Physics Today 42 (6), June 1969, p. 31–36, http://www.physicstoday.org/vol-42/iss-6/vol42no6p31_36.pdf(34)^ Max Jammer (1999), Concepts of mass in contemporary physics and philosophy, PrincetonUniversity Press, p. 51, ISBN 0-691-01017-X(35)^ Eriksen, Erik; Vøyenli, Kjell (1976), "The classical and relativistic concepts ofmass",Foundations of Physics (Springer) 6: 115–124, Bibcode 1976FoPh....6..115E,DOI:10.1007/BF00708670(36)^ a b Jannsen, M., Mecklenburg, M. (2007), From classical to relativistic mechanics:Electromagnetic models of the electron., in V. F. Hendricks, et al., , Interactions: Mathematics,Physics and Philosophy (Dordrecht: Springer): 65–134(37)^ a b Whittaker, E.T. (1951–1953), 2. Edition: A History of the theories of aether and electricity,vol. 1: The classical theories / vol. 2: The modern theories 1900–1926, London: Nelson(38)^ Miller, Arthur I. (1981), Albert Einsteins special theory of relativity. Emergence (1905) andearly interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2(39)^ a b Darrigol, O. (2005), "The Genesis of the theory of relativity." (PDF), Séminaire Poincaré1:1–22(40)^ Philip Ball (Aug 23, 2011). "Did Einstein discover E = mc2?” Physics World.(41)^ Ives, Herbert E. (1952), "Derivation of the mass-energy relation", Journal of the Optical Societyof America 42 (8): 540–543, DOI:10.1364/JOSA.42.000540(42)^ Jammer, Max (1961/1997). Concepts of Mass in Classical and Modern Physics. New York:Dover. ISBN 0-486-29998-8.(43)^ Stachel, John; Torretti, Roberto (1982), "Einsteins first derivation of mass-energyequivalence", American Journal of Physics 50 (8): 760–763, Bibcode1982AmJPh..50..760S, DOI:10.1119/1.12764(44)^ Ohanian, Hans (2008), "Did Einstein prove E=mc2?", Studies In History and Philosophy ofScience Part B 40 (2): 167–173, arXiv:0805.1400,DOI:10.1016/j.shpsb.2009.03.002(45)^ Hecht, Eugene (2011), "How Einstein confirmed E0=mc2", American Journal ofPhysics 79 (6): 591–600, Bibcode 2011AmJPh..79..591H, DOI:10.1119/1.3549223 183
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012(46)^ Rohrlich, Fritz (1990), "An elementary derivation of E=mc2", American Journal ofPhysics 58 (4): 348–349, Bibcode 1990AmJPh..58..348R, DOI:10.1119/1.16168(47) (1996). Lise Meitner: A Life in Physics. California Studies in the History of Science. 13.Berkeley: University of California Press. pp. 236–237. ISBN 0-520-20860-(48)^ UIBK.ac.at(49)^ J. J. L. Morton; et al. (2008). "Solid-state quantum memory using the 31P nuclearspin". Nature 455 (7216): 1085–1088. Bibcode 2008Natur.455.1085M.DOI:10.1038/nature07295.(50)^ S. Weisner (1983). "Conjugate coding". Association of Computing Machinery, Special InterestGroup in Algorithms and Computation Theory 15: 78–88.(51)^ A. Zelinger, Dance of the Photons: From Einstein to Quantum Teleportation, Farrar, Straus &Giroux, New York, 2010, pp. 189, 192, ISBN 0374239665(52)^ B. Schumacher (1995). "Quantum coding". Physical Review A 51 (4): 2738–2747. Bibcode 1995PhRvA..51.2738S. DOI:10.1103/PhysRevA.51.2738. 184
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.7, 2012First Author: 1Mr. K. N.Prasanna Kumar has three doctorates one each in Mathematics, Economics, PoliticalScience. Thesis was based on Mathematical Modeling. He was recently awarded D.litt. for his work on‘Mathematical Models in Political Science’--- Department of studies in Mathematics, Kuvempu University,Shimoga, Karnataka, India Corresponding Author:drknpkumar@gmail.comSecond Author: 2Prof. B.S Kiranagi is the Former Chairman of the Department of Studies in Mathematics,Manasa Gangotri and present Professor Emeritus of UGC in the Department. Professor Kiranagi has guided over25 students and he has received many encomiums and laurels for his contribution to Co homology Groups andMathematical Sciences. Known for his prolific writing, and one of the senior most Professors of the country, hehas over 150 publications to his credit. A prolific writer and a prodigious thinker, he has to his credit severalbooks on Lie Groups, Co Homology Groups, and other mathematical application topics, and excellentpublication history.-- UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri,University of Mysore, Karnataka, IndiaThird Author: 3Prof. C.S. Bagewadi is the present Chairman of Department of Mathematics and Departmentof Studies in Computer Science and has guided over 25 students. He has published articles in both national andinternational journals. Professor Bagewadi specializes in Differential Geometry and its wide-rangingramifications. He has to his credit more than 159 research papers. Several Books on Differential Geometry,Differential Equations are coauthored by him--- Chairman, Department of studies in Mathematics and Computerscience, Jnanasahyadri Kuvempu University, Shankarghatta, Shimoga district, Karnataka, India============================================================================== 185
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