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The theory of zero point energy of vacuum

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  • 1. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 The Theory of Zero Point Energy Of Vacuum, Cosmological Constant Variability, Dark Matter Super Symmetry, Dark Energy, Expanding Universe, Microwave Sky ,Motion OfOrientation Of The Solar System, Mass Of Quantum Vacuum, Deceleration Of Acceleration Of Expansionary Universe,Discrete Structure Of Space And Time And GTR---A “Fricative Contretemps” And “Deus Ex Machina” Model. *1 Dr K N Prasanna Kumar, 2Prof B S Kiranagi And 3Prof C S Bagewadi *1 Dr K N Prasanna Kumar, Post doctoral researcher, Dr KNP Kumar has three PhD’s, one each in Mathematics, Economics and Political science and a D.Litt. in Political Science, Department of studies in Mathematics, Kuvempu University, Shimoga, Karnataka, India Correspondence Mail id : drknpkumar@gmail.com 2 Prof B S Kiranagi, UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri, University of Mysore, Karnataka, India 3 Prof C S Bagewadi, Chairman , Department of studies in Mathematics and Computer science, Jnanasahyadri Kuvempu university, Shankarghatta, Shimoga district, Karnataka, IndiaAbstract:Laws bears ample testimony ,infallible observatory, and impeccable demonstration to the fact that theessential predications, character constitutions, ontological consonances remain unchanged with evolution,despite the system’s astute truculence, serenading whimsicality,assymetric dispensation or on the other handanachronistic dispensation ,eponymous radicality,entropic entrepotishness or the subdued behaviour,relationally contributive, diverse parametrisizational,conducive reciprocity to environment, unconventionalbehavior, enuretic nonlinear freneticness ,ensorcelled frenzy, abnormal ebulliations,surcharged fulminationsor the inner roil. And that holds well with the evolution with time. We present a model of thegeneralizational conservation of the theories. A theory of all the theories. That all conservation laws hold andthere is no relationship between them is bê noir. We shall on this premise build a 36 storey model that tedeliberates on various issues, structural, dependent, thematic and discursive, discursive. Paper throws lighton at least six unsolved problems in physics, if not completely solve them, for which we are putting allconcerted efforts and protracted endeavors.Key words Zero point energy of vacuum, Dark matter, Dark energyIntroduction:What is an event? Or for that matter an ideal event? An event is a singularity or rather a set of singularities ora set of singular points characterizing a mathematical curve, a physical state of affairs, a psychologicalperson or a moral person. Singularities are turning points and points of inflection : they are bottle necks,foyers and centers ;they are points of fusion; condensation and boiling points or tears and joy; sickness andhealth; hope and anxiety; they are so to say “sensitive" points; such singularities should not be confused orconfounded, aggravated or exacerbated with personality of a system expressing itself; or the individualityand idiosyncrasies,penchance,predilections,proclivities,propensities of a system which is designated with aproposition. They should also not be fused with the generalizational concept or universalistic axiomaticpredications and postulation alcovishness, or the dipsomaniac flageolet dirge of a concept. Possibly aconcept could be signified by a figurative representation or a schematic configuration. "Singularity” is 173
  • 2. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012essentially, pre individual, and has no personalized bias in it, nor for that matter a prejudice orprocircumspection of a conceptual scheme. It is in this sense we can define a "singularity" as being neitheraffirmative nor non affirmative. It can be positive or negative; it can create or destroy. On the other hand itmust be noted that singularity is different both in its thematic discursive forms from the” run of the mill dayto day musings and mundane drooling. There are in that sense "extra-ordinary”. Each singularity is a sourceand resouce,the origin, reason and raison d’être of a mathematical series, it could be any series any type, andthat is interpolated or extrapolated to the structural location of the destination of another singularity. Thisaccording to this standpoint, there are different, multifarious, myriad, series in a structure. In the eventualityof the fact that we conduct an unbiased and prudent examination of the series belonging to different"singularities" we can come to indubitable conclusion that the "singularity" of one system is different fromthe "other system" in the subterranean realm and ceratoid dualism of comparison and contrast of systems.EPR experiment derived that there exists a communications between two particles. We go a further step tosay 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. All we are telling is thatsingularities communicate with each other.Now, how do find the reaction of systems to these singularities?. You do the same thing a boss does for you."Problematize" the events and see how you behave. I will resort to "pressure tactics”. “intimidation,terrorization of deriding report", or “cut in the increment" to make you undergo trials, travails andtribulations. I am happy to see if you improve your work; but may or may not be sad if you succumb to it andhang yourself! We do the same thing with systems. Systems show conducive response, felicitousreciprocation or behave erratically with inner roil, eponymous radicalism without and with glitzy convictionsay like a solipsist nature of bellicose and blustering particles, or for that matter coruscation, trepidiationalmotion in 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 behaviour wehave to "problematize" the events of systems to understand their behaviour.This statement in 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 are like“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 too. This isaccentuation, corroboration, fortificational,.fomentatory note to explain the various coefficients we haveused in the model as also the dissipations called for.In the bank example we have clarified that various systems are individually conservative, and theirconservativeness extends holistically too. 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! Equations wouldbecome more energetic and frenzied..We take in to consideration the following variables:1. Zero Point Energy Of the Vacuum2. Cosmological constant variability3. Dark Matter4. Super symmetry5. Dark Energy6. Expanding Universe7. Microwave Sky8. Motion and orientation of the Solar System.9. Mass of Quantum Vacuum10. Deceleration of the accelerated expansion of the universe 174
  • 3. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 201211. Discrete Structure of Space and Time12. GTR and Quantum Mechanics(We have given the linkage model of these two in a separate paper)Classification methodologies. Bank’s example of equality of Assets and Liabilities, and interconnectedinherent inter accountal transaction holds good for these systems too.ZERO POINT ENERGY OF THE VACUUM AND COSMOLOGICAL CONSTANTVARAIBILITY:MODULE NUMBERED ONENotation : : Category One Ofzero Point Energy : Category Two Of The Zero Point Energy(There Are Many Vacuums) : Category Three Of The Zero Point Energy : Category One Of The Variability Of Cosmological Constant(Note That There Exists DifferentVacuums And Constantancy Does Not Hinder The Production And Dissipation Of Zero Point Energy) : Category Two Of The Variability Of Cosmological Constant :Category Three Of The Variability Of The Cosmological Constant(We Repeat Assets=LiabilitiesDoes Not Mean Inter Account Transfers ,Production Of, Or Closure Of The Accounts Or In This CaseSystems)Dark Matter And Super Symmetry---Module Numbered Two: : Category One Of Super Symmetry : Category Two Of Super Symmetry : Category Three Of Super Symmetry :Category One Ofdark Matter : Category Two Of Dark Matter : Category Three Of Dark MatterExpanding Universe And Dark Energy:Module Numbered Three: : Category One Of Dark Energy :Category Two Of Dark Energy : Category Three Of Dark Energy 175
  • 4. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 : Category One Of Expanding Universe :Category Two Of Expanding Universe : Category Three Of Expanding UniverseMotion And Orientation Of The Solar system And Microwave Sky(Note Both Change From Time ToTime): Module Numbered Four: : Category One Of Motion And Orientation Of The Solar System : Category Two Of Motion And Orientation Of The Solar System : Category Three Of Motion And Orientation Of The Solar System :Category One Of Microwave Sky :Category Two Ofmicrowave Sky : Category Three Of Microwave SkyMass Of Quantum Vacuum And Deceleration Of The Expanding Universe:Module Numbered Five: : Category One Of Deceleration Of The Accelerated Expansion Of The Universe(Rate Is NotConstant) : Category Two Of Deceleration Of The Accelerated Expansion Of The Universe :Category Three Of deceleration Of The Accelerated Expansion Of The Universe :Category One Ofmass Of Quantum Vacuum :Category Two Of Mass Of Quantum Vacuum :Category Three Of mass Of Quantum VacuumGtr And Quantum Mechanics And Discrete Nature Of Space And Time:Module Numbered Six: : Category One Of Gtr And Quantum Mechanics(There Are Many Quantum Systems And There AreMany Systems To Which Gtr Would Hold Classification Is Based On Those Systems) : Category Two Of Gtr And Qm : Category Three Ofgtr And Qm : Category One Of Discrete Natute Of St : Category Two Of Discrete Nature Of St : Category Three Of Discrete Nature Of ST( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 176
  • 5. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( )( ) : ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ),( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Accentuation coefficients( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) , ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Dissipation coefficientsZERO POINT ENERGY OF THE VACUUM AND COSMOLOGICAL CONSTANTVARAIBILITY:MODULE NUMBERED ONEThe differential system of this model is now (Module Numbered one) ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factor DARK MATTER AND SUPER SYMMETRY--- MODULE NUMBERED TWO A theory in physics proposing a type of symmetry that would apply to all elementary particles (Note again that there are various systems of elementary particles. Bank example stands in good stead every time)The differential system of this model is now ( Module numbered two) ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] 177
  • 6. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factor EXPANDING UNIVERSE AND DARK ENERGY: MODULE NUMBERED THREEThe differential system of this model is now (Module numbered three) ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factor MOTION AND ORIENTATION OF THE SOLARSYSTEM AND MICROWAVE SKY(NOTE BOTH CHANGE FROM TIME TO TIME) : MODULE NUMBERED FOUR:The differential system of this model is now (Module numbered Four) ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] 178
  • 7. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factor MASS OF QUANTUM VACUUM AND DECELERATION OF THE EXPANDING UNIVERSE :MODULE NUMBERED FIVEThe differential system of this model is now (Module number five) ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factor GTR AND QUANTUM MECHANICS AND DISCRETE NATURE OF SPACE AND TIME :MODULE NUMBERED SIXThe differential system of this model is now (Module numbered Six) ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) ( )] ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) [( )( ) ( )( ) (( ) )] 179
  • 8. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) [( )( ) ( )( ) (( ) )] ( )( ) ( ) First augmentation factor ( )( ) (( ) ) First detritions factorHOLISTIC CONCATENATE SYTEMAL EQUATIONS HENCEFORTH REFERRED TO AS “GLOBALEQUATIONS” (1) Zero Point Energy Of the Vacuum (2) Cosmological constant variability (3) Dark Matter (4) Super symmetry (5) Dark Energy (6) Expanding Universe (7) Microwave Sky (8) Motion and orientation of the Solar System. (9) Mass of Quantum Vacuum (10) Deceleration of the accelerated expansion of the universe (11) Discrete Structure of Space and Time (12) GTR and Quantum Mechanics(We have given the linkage model of these two in a separate paper ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )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 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ] ( ) ( ) ( ) ( ) ( ) ( ) 180
  • 9. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detrition coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 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 3 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )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 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 181
  • 10. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 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 3 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) 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, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1, 2 and3 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 182
  • 11. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second detrition coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1, 2 and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1, 2 and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth detrition coefficients for category 1, 2 and 3 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 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,and 3 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) –( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) –( ) ( ) 183
  • 12. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( )–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 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 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 184
  • 13. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 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 3 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - are fourth augmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - fifth augmentation coefficients ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) sixth augmentation coefficients ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) –( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 185
  • 14. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 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 3Where we suppose(A) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )(B) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(C) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions( )( ) ( ) and( )( ) ( ) ( ) and ( ) are points belonging to the interval[( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of thefact, that if ( ̂ )( ) then the function ( ) ( ( ) ) , the first augmentation coefficient WOULD beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) :(D) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 186
  • 15. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012(E) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and ( ̂ )( ) and the constants ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( )Where we suppose(F) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )(G) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( ) ( )( ) ( ) ( )( ) ( ̂ ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(H) ( )( ) ( ) ( )( ) ( )( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) :Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is tobe noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) ( )then the function ( ) ( ) , the SECOND augmentation coefficient would be absolutely continuous.Definition of ( ̂ )( ) (̂ )( ) :(I) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) :There exists two constants ( ̂ )( ) and ( ̂ )( ) which togetherwith ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) satisfy the inequalities 187
  • 16. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose (J) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 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 ( ̂ )( ) (̂ )( ) : (K) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose(L) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 188
  • 17. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 (M) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) (N) ( )( ) ( ( )( ) ) ( )( ) (( ) ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 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. Definition of ( ̂ )( ) (̂ )( ) :(O) ( ̂ )( ) (̂ )( ) are positive constants(P) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : (Q) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose(R) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (S) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : 189
  • 18. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) (T) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 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 ( ̂ )( ) (̂ )( ) :(U) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) :(V) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (W) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) (̂ )( ) 190
  • 19. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012(X) ( )( ) ( ) ( )( ) ( ) ( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( )and( ) (( ) ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is tobe noted that ( ) ( ( ) ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( )then the function ( )( ) ( ) , the SIXTH augmentation coefficient would be absolutely continuous.Definition of ( ̂ )( ) (̂ )( ) :( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) :There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( )Theorem 1: if the conditions IN THE FOREGOING above are fulfilled, there exists a solution satisfyingthe conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Definition of ( ) ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 191
  • 20. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ( )Proof: Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 192
  • 21. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( )Proof: ( )Consider operator defined on the space of sextuples of continuous functions whichsatisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( )Proof: ( )Consider operator defined on the space of sextuples of continuous functions whichsatisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 193
  • 22. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( )Consider operator defined on the space of sextuples of continuous functions whichsatisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 194
  • 23. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( )Consider operator defined on the space of sextuples of continuous functions whichsatisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) ) ( ̂ )( ) ( ) ( ̂ )( ) ( ̂ )( ) ( ) ( ̂ )(By ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( )Where ( ) is the integrand that is integrated over an interval ( ) ( )(a) The operator maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 195
  • 24. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )( ( ) ) [(( ( ) ) ( ̂ )( ) ] ( ̂ )( )( ) is as defined in the statement of theorem 1Analogous inequalities hold also for ( )(b) The operator maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( ) ( )( ) ( ̂ )( ) ( ̂ )( )( ( ) ) ( ) ( ̂ )( )From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )(( ( ) ) ) [(( ) ) ( ̂ )( ) ] ( ̂ )(Analogous inequalities hold also for ( )(a) The operator maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )(Analogous inequalities hold also for(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ( ) is as defined in the statement of theorem 1(c) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) 196
  • 25. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) (̂ )( ) ] ( ̂ )(( ) is as defined in the statement of theorem 1 ( )(d) The operator maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ( ) is as defined in the statement of theorem 6Analogous inequalities hold also for ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) ( )( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )(̂ )( ) ( )In order that the operator transforms the space of sextuples of functions satisfying GLOBALEQUATIONS into itself ( )The operator is a contraction with respect to the metric ( ) ( ) ( ) ( ) (( )( )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | | 197
  • 26. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Indeed if we denoteDefinition of ̃ ̃ : ( ̃ ̃) ( ) ( )It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( )| | (( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ) (( ( ) ( ) ( ) ( ) ))(̂ )( )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 notconformal with the reality, however we have put this hypothesis ,in order that we can postulate condition ( ) ( )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 ( ) ( )From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 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. 198
  • 27. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Remark 4: If bounded, from below, the same property holds for The proof isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) ( ( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for( )( ) ( )( ) ( ( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is unbounded. ( ) ( )The same property holds for if ( ) ( ( ) ) ( )We now state a more precise theorem about the behaviors at infinity of the solutions ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( ̂ )( ) large to have (̂ )( ) ( ) ( )( ) [( ̂ )( ) (( ̂ ) ( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions satisfying ( )The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of ̃ ̃ : ( ̃ ̃ ) ( ) ( )It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) 199
  • 28. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows )( ) ( )( ) | (̂ )( )|( (( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )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 notconformal with the reality, however we have put this hypothesis ,in order that we can postulate condition ̂ )( ) ̂ )( )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 () ()From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] () ( () ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 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 isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) () ( )( )Then ( )( ) ( ) ( ) which leads to 200
  • 29. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is unbounded. The ( ) ( )same property holds for if ( ) (( )( ) ) ( )We now state a more precise theorem about the behaviors at infinity of the solutions ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) ( ) ( )( ) ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )(̂ )( (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( )The 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 ( ) ( ) (̂ )( )| | (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result follows 201
  • 30. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as notconformal with the reality, however we have put this hypothesis ,in order that we can postulate condition ( ) ( )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 ( ) ( )From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 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 isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is unbounded.The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( ) 202
  • 31. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012( ̂ )( ) (̂ )( ) large to have (̂ )( ) ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) ( )( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )(̂ )( ) ( )In order that the operator transforms the space of sextuples of functions satisfying IN to itself ( )The 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 )( ) ( )( ) | (̂ )( )|( (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )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 notconformal with the reality, however we have put this hypothesis ,in order that we can postulate condition ( ) ( )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. 203
  • 32. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Remark 2: There does not exist any where ( ) ( )From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 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 isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is unbounded.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 ( )( ) ( )( ) It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have 204
  • 33. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 (̂ )( ) ( )( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )(̂ )( ) (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( )The 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 )( ) ( )( ) | (̂ )( )|( (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis (35,35,36) the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as notconformal with the reality, however we have put this hypothesis ,in order that we can postulate condition ( ) ( )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 ( ) ( ) 205
  • 34. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012From GLOBAL EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 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 isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is unbounded.The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutionsAnalogous inequalities hold also for ( )( ) ( )( )It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have 206
  • 35. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 (̂ )( ) ( )( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )(̂ )( ) (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( )In order that the operator transforms the space of sextuples of functions into itself ( )The 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 )( ) ( )( ) | (̂ )( )|( (( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))(̂ )( )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 notconformal with the reality, however we have put this hypothesis ,in order that we can postulate condition ( ) ( )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 ( ) ( ) 207
  • 36. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012From 69 to 32 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) :Remark 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 isanalogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) thenDefinition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is unbounded. ( ) ( )The same property holds for if ( ) (( )( ) ( ) ) ( )We now state a more precise theorem about the behaviors at infinity of the solutionsBehavior of the solutionsIf we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying 208
  • 37. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) :(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( )Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) and ( ) ( ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) 209
  • 38. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutionsIf we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(d) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots(e) of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) :By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) theroots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( )and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 210
  • 39. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012and analogously( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution satisfies the inequalities (( )( ) ( )( ) ) ( ) ( )( )( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) ) Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutionsIf we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) (( ) ) ( )( ) 211
  • 40. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Definition of ( )( ) ( )( ) ( )( ) ( )( ) :(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :-(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( ) ( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) ) 212
  • 41. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutionsIf we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(d) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) :(e) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) 213
  • 42. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined by 59 and 64 respectivelyThen the solution satisfies the inequalities (( )( ) ( )( ) ) ( ) ( )( ) where ( )( ) is defined (( )( ) ( )( ) ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) ( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutionsIf we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(g) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) :(h) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 214
  • 43. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(i) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( ) ( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 215
  • 44. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutionsIf we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(j) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) :(k) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(l) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectively 216
  • 45. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Then the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( )( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( ) ( )( ) (( )( ) ( )( ) ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Proof : From GLOBAL EQUATIONS we obtain ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(a) For ( )( ) ( )( ) ( ̅ )( ) 217
  • 46. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) (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 ( ) 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 ( )( ) 218
  • 47. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 we obtain ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( )(( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( )(( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( ) ((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) (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 ( ) :- 219
  • 48. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) . 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 ( ̅ )( ) From GLOBAL EQUATIONS we obtain ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) Definition of ( ̅ )( ) :- From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) 220
  • 49. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(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 ( ) 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 ( ̅ )( ) : From GLOBAL EQUATIONS we obtain ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) 221
  • 50. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( , ( )( ) [ ( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) ( ̅ )( ) [ ( ( )( ) (̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) (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 ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) 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 ( )( ) From GLOBAL EQUATIONS we obtain 222
  • 51. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 2012 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(g) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(h) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) (i) 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 ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) 223
  • 52. Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.2, No.4, 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 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 ( )( ) ( ) ( ) we obtain ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(j) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(k) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) 224
  • 53. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ̅ )( ) ( ̅ )( )(l) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we haveDefinition of ( ) ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )In a completely analogous way, we obtainDefinition of ( ) ( ) :- ( )( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( )Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in thetheorem.Particular case :If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( ) ( ) ( ) ( )( ) then ( ) ( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )( ) forthe special case .Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( ) ( ) ( ) ( ) ( ̅ ) if in addition ( ) ( )( ) then ( ) ( )( ) ( ) This is an importantconsequence of the relation between ( ) and ( ̅ ) and definition of ( )( ) ( ) ( )We can prove the followingTheorem 3: If ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined, then the systemIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 225
  • 54. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the system( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )] 226
  • 55. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )has a unique positive solution , which is an equilibrium solution for the system( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )has a unique positive solution , which is an equilibrium solution for( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )has a unique positive solution , which is an equilibrium solution( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) ( )( ) ( )( ) (( ))( )( ) ( )( ) ( )( ) (( ))( )( ) ( )( ) ( )( ) (( ))has a unique positive solution , which is an equilibrium solution for the system( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )] 227
  • 56. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012( )( ) [( )( ) ( )( ) ( )]( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )has a unique positive solution , which is an equilibrium solution for the system( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) [( )( ) ( )( ) ( )]( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( )has a unique positive solution , which is an equilibrium solution for the system(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( ) 228
  • 57. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012(a) Indeed the first two equations have a nontrivial solution if ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )(a) Indeed the first two equations have a nontrivial solution if ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows that thereexists a unique for which ( ) . With this value , we obtain from the three first equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows that thereexists a unique for which ( ) . With this value , we obtain from the three first equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows that thereexists a unique for which ( ) . With this value , we obtain from the three first equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :-After 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 :- 229
  • 58. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012After 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 :-After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )](e) By the same argument, the equations 92,93 admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) Where in ( ) must be replaced by their values from 96. It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows that thereexists a unique such that ( )(f) By the same argument, the equations 92,93 admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows that thereexists a unique such that (( ) )(g) By the same argument, the concatenated equations admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( ) must be replaced by their values from 96. It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows that thereexists a unique such that (( ) )(h) By the same argument, the equations of modules admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that isa decreasing function in taking into account the hypothesis ( ) ( ) it follows that thereexists a unique such that (( ) ) 230
  • 59. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012(i) By the same argument, the equations (modules) admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that isa decreasing function in taking into account the hypothesis ( ) ( ) it follows that thereexists a unique such that (( ) )(j) By the same argument, the equations (modules) admit solutions if ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows that thereexists a unique such that ( )Finally we obtain the unique solution of 89 to 94 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( )( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )] Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Obviously, these values represent an equilibrium solution 231
  • 60. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012Finally we obtain the unique solution ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionASYMPTOTIC STABILITY ANALYSISTheorem 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 :- , ( )( ) ( )( ) ( ) ( )( ) , ( )Then taking into account equations (global) and neglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 232
  • 61. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )taking into account equations (global)and neglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 233
  • 62. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 234
  • 63. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations(global) and neglecting the terms of power 2, we obtain (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) )The characteristic equation of this system is(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) 235
  • 64. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) 236
  • 65. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) 237
  • 66. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, andthis proves the theorem.Acknowledgments:The introduction is a collection of information from various articles, Books, News Paper reports, HomePages Of authors, Journal Reviews, Nature ‘s L:etters,Article Abstracts, Research papers, Abstracts OfResearch Papers, Stanford Encyclopedia, Web Pages, Ask a Physicist Column, Deliberations withProfessors, the internet including 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 of the authors, we regretwith great deal of compunction, contrition, regret, trepidiation and remorse. As Newton said, it is onlybecause erudite and eminent people allowed one to piggy ride on their backs; probably an attempt has beenmade to look slightly further. Once again, it is stated that the references are only illustrative and notcomprehensive REFERENCES 238
  • 67. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 20121. Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi - MEASUREMENT DISTURBSEXPLANATION OF QUANTUM MECHANICAL STATES-A HIDDEN VARIABLE THEORY -published at: "International Journal of Scientific and Research Publications, Volume 2, Issue 5, May 2012Edition".2. DR K N PRASANNA KUMAR, PROF B S KIRANAGI and PROF C S BAGEWADI -CLASSIC 2FLAVOUR COLOR SUPERCONDUCTIVITY AND ORDINARY NUCLEAR MATTER-A NEWPARADIGM STATEMENT - published at: "International Journal of Scientific and 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 80 th 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 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 201013. Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Annalen der Physik 18: 639 Bibcode 1905AnP...323..639E,DOI:10.1002/andp.19053231314. See also the English translation.14. Paul Allen Tipler, Ralph A. Llewellyn (2003-01), Modern Physics, W. H. Freeman and Company, pp. 87–88, ISBN 0-7167-4345-0 b15. 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 2005.16. In F. Fernflores. The Equivalence of Mass and Energy. Stanford Encyclopedia of Philosophy17. Note that the relativistic mass, in contrast to the rest mass m0, is not a relativistic invariant, and that the velocity is not a Minkowski four-vector, in contrast to the quantity , where is the differential of 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 distinction between dτ and dt.18. Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0 239
  • 68. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 201219. Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-820. Hans, H. S.; Puri, S. P. (2003). Mechanics (2 ed.). Tata McGraw-Hill. p. 433. ISBN 0-07-047360- 9., Chapter 12 page 43321. 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 nuclear bombs, until heat is allowed to escape.22. Mould, Richard A. (2002). Basic relativity (2 ed.). Springer. p. 126. ISBN 0-387-95210-1., Chapter 5 page 12623. Chow, Tail L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones & Bartlett Learning. p. 392. ISBN 0-7637-3827-1., Chapter 10 page 39224. Cockcroft-Walton experiment25. 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 ≡ 1000 calories used.26. Assuming the dam is generating at its peak capacity of 6,809 MW.27. Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominated average 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 1.5 prototype, which are ± micrograms. 228. Article on Earth rotation energy. Divided by c^2.29. Earths gravitational self-energy is 4.6 × 10-10 that of Earths total mass, or 2.7 trillion metric tons. 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.).30. There is usually more than one possible way to define a field energy, because any field can be made to couple to gravity in many different ways. By general scaling arguments, the correct answer at everyday distances, which are long compared to the quantum gravity scale, should be minimal coupling, 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.31. G. t Hooft, "Computation of the quantum effects due to a four-dimensional pseudoparticle", Physical Review D14:3432–3450 (1976).32. A. Belavin, A. M. Polyakov, A. Schwarz, Yu. Tyupkin, "Pseudoparticle Solutions to Yang Mills Equations", Physics Letters 59B:85 (1975).33. F. Klinkhammer, N. Manton, "A Saddle Point Solution in the Weinberg Salam Theory", Physical Review D 30:2212.34. Rubakov V. A. "Monopole Catalysis of Proton Decay", Reports on Progress in Physics 51:189–241 (1988).35. S.W. Hawking "Black Holes Explosions?" Nature 248:30 (1974). 240
  • 69. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 201236. 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.37. 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.pdf38. Max Jammer (1999), Concepts of mass in contemporary physics and philosophy, Princeton University Press, p. 51, ISBN 0-691-01017-X39. Eriksen, Erik; Vøyenli, Kjell (1976), "The classical and relativistic concepts of mass",Foundations of Physics (Springer) 6: 115–124, Bibcode 1976FoPh....6..115E,DOI:10.1007/BF0070867040. 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): 65–13441. 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: Nelson42. Miller, Arthur I. (1981), Albert Einsteins special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-243. Darrigol, O. (2005), "The Genesis of the theory of relativity." (PDF), Séminaire Poincaré1: 1–2244. Philip Ball (Aug 23, 2011). "Did Einstein discover E = mc2?” Physics World.45. Ives, Herbert E. (1952), "Derivation of the mass-energy relation", Journal of the Optical Society of America 42 (8): 540–543, DOI:10.1364/JOSA.42.00054046. Jammer, Max (1961/1997). Concepts of Mass in Classical and Modern Physics. New York: Dover. ISBN 0-486-29998-8.47. Stachel, John; Torretti, Roberto (1982), "Einsteins first derivation of mass-energy equivalence", American Journal of Physics 50 (8): 760– 763, Bibcode1982AmJPh..50..760S, DOI:10.1119/1.1276448. Ohanian, Hans (2008), "Did Einstein prove E=mc2?", Studies In History and Philosophy of Science Part B 40 (2): 167–173, arXiv:0805.1400,DOI:10.1016/j.shpsb.2009.03.00249. Hecht, Eugene (2011), "How Einstein confirmed E0=mc2", American Journal of Physics 79 (6): 591– 600, Bibcode 2011AmJPh..79..591H, DOI:10.1119/1.354922350. Rohrlich, Fritz (1990), "An elementary derivation of E=mc2", American Journal of Physics 58 (4): 348–349, Bibcode 1990AmJPh..58..348R, DOI:10.1119/1.1616851. 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- 241
  • 70. Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.2, No.4, 2012First Author: 1Mr. K. N.Prasanna Kumar has three doctorates one each in Mathematics, Economics,Political Science. 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 guidedover 25 students and he has received many encomiums and laurels for his contribution to Co homology Groupsand Mathematical Sciences. Known for his prolific writing, and one of the senior most Professors of thecountry, he has over 150 publications to his credit. A prolific writer and a prodigious thinker, he has to his creditseveral books 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 242
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