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

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

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  • 1. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 SOME CONTRIBUTIONS TO YANG MILLS THEORY FORTIFICATION –DISSIPATION MODELS 1 DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3 PROF C S BAGEWADIABSTRACT. We provide a series of Models for the problems that arise in Yang Mills Theory. No claimis made that the problem is solved. We do factorize the Yang Mills Theory and give a Model for thevalues of LHS and RHS of the yang Mills theory. We hope these forms the stepping stone for furtherfactorizations and solutions to the subatomic denominations at Planck’s scale. Work also throws light onsome important factors like mass acquisition by symmetry breaking, relation between strong interactionand weak interaction, Lagrangian Invariance despite transformations, Gauge field, Noncommutativesymmetry group of Gauge Theory and Yang Mills Theory itself. Key Words: Acquisition of mass, Symmetry Breaking, Strong interaction ,Unified Electroweak interaction, Continuous group of local transformations, Lagrangian Variance, Group generator in Gauge Theory, Vector field or Gauge field, commutative symmetry group in Gauge Theory, Yang Mills TheoryThe outlay of the paper is as follows: I. INTRODUCTION II. FORMULATION OF THE PROBLEM III. STATEMENT OF GOVERNING EQUATIONS IV. THE SOLUTION-BODY FABRIC OF THE THESIS V. ACKNOWLEDGEMENTS VI. REFRENCES I. INTRODUCTION:We take in to consideration the following parameters, processes and concepts: (1) Acquisition of mass (2) Symmetry Breaking (3) Strong interaction (4) Unified Electroweak interaction (5) Continuous group of local transformations (6) Lagrangian Variance (7) Group generator in Gauge Theory (8) Vector field or Gauge field (9) Non commutative symmetry group in Gauge Theory (10) Yang Mills Theory (We repeat the same Bank’s example. Individual debits and Credits are conservative so also the holistic one. Generalized theories are applied to various systems which are parameterized. And we live in ‘measurement world’. Classification is done on the parameters of various systems to which the Theory is applied. ). (11) First Term of the Lagrangian of the Yang Mills Theory(LHS) 304
  • 2. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (12) RHS of the Yang Mills Theory II. FORMULATION OF THE PROBLEM SYMMETRY BREAKING AND ACQUISITION OF MASS: MODULE NUMBERED ONENOTATION : : CATEGORY ONE OF SYMMETRY BREAKING : CATEGORY TWO OF SYMMETRY BREAKING : CATEGORY THREE OF SYMMETRY BREAKING : CATEGORY ONE OF ACQUISITION OF MASS : CATEGORY TWO OF ACQUISITION OF MASS :CATEGORY THREE OF ACQUISITION OF MASS UNIFIED ELECTROWEAK INTERACTION AND STRONG INTERACTION: MODULE NUMBERED TWO:========================================================================== === : CATEGORY ONE OF UNIFIED ELECTROWEAK INTERACTION : CATEGORY TWO OFUNIFIED ELECTROWEAK INTERACTION : CATEGORY THREE OFUNIFIED ELECTROWEAK IONTERACTION :CATEGORY ONE OF STRONG INTERACTION : CATEGORY TWO OF STRONG INTERACTION : CATEGORY THREE OF STRONG INTERACTION LAGRANGIAN INVARIANCE AND CONTINOUS GROUP OF LOCAL TRANSFORMATIONS: 305
  • 3. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 MODULE NUMBERED THREE:============================================================================= : CATEGORY ONE OF CONTINUOUS GROUP OF LOCAL TRANSFORMATIONS :CATEGORY TWO OFCONTINUOUS GROUP OF LOCAL TRANSFORMATIONS : CATEGORY THREE OF CONTINUOUS GROUP OF LOCAL TRANSFORMATION : CATEGORY ONE OF LAGRANGIAN INVARIANCE :CATEGORY TWO OF LAGRANGIAN INVARIANCE : CATEGORY THREE OF LAGRANGIAN INVARIANCE GROUP GENERATOR OF GAUGE THEORY AND VECTOR FIELD(GAUGE FIELD): : MODULE NUMBERED FOUR:============================================================================ : CATEGORY ONE OF GROUP GENERATOR OF GAUGE THEORY : CATEGORY TWO OF GROUP GENERATOR OF GAUGE THEORY : CATEGORY THREE OF GROUP GENERATOR OF GAUGE THEORY :CATEGORY ONE OF VECTOR FIELD NAMELY GAUGE FIELD :CATEGORY TWO OF GAUGE FIELD : CATEGORY THREE OFGAUGE FIELD YANG MILLS THEORYAND NON COMMUTATIVE SYMMETRY GROUP IN GAUGE THEORY: MODULE NUMBERED FIVE:============================================================================= : CATEGORY ONE OF NON COMMUTATIVE SYMMETRY GROUP OF GAUGETHEORY : CATEGORY TWO OF NON COMMUTATIVE SYMMETRY GROUP OPF GAUGETHEORY :CATEGORY THREE OFNON COMMUTATIVE SYMMETRY GROUP OF GAUGE 306
  • 4. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012THEORY : CATEGORY ONE OFYANG MILLS THEORY (Theory is applied to various subatomicparticle systems and the classification is done based on the parametricization of these systems.There is not a single system known which is not characterized by some properties) :CATEGORY TWO OF YANG MILLS THEORY :CATEGORY THREE OF YANG MILLS THEORY LHS OF THE YANG MILLS THEORY AND RHS OF THE YANG MILLS THEORY.TAKENTO THE OTHER SIDE THE LHS WOULD DISSIPATE THE RHS WITH OR WITHOUT TIME LAG : MODULE NUMBERED SIX:============================================================================= : CATEGORY ONE OF LHS OF YANG MILLS THEORY : CATEGORY TWO OF LHS OF YANG MILLS THEORY : CATEGORY THREE OF LHS OF YANG MILLS THEORY : CATEGORY ONE OF RHS OF YANG MILLS THEORY : CATEGORY TWO OF RHS OF YANG MILLS THEORY : CATEGORY THREE OF RHS OF YANG MILLS THEORY (Theory applied to variouscharacterized systems and the systemic characterizations form the basis for the formulation of theclassification).===============================================================================( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) : ( )( ) ( )( ) ( ) ( ) ( ) (( ) ( ) ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ( ) ( ) ) ( ) ( ) ( ( ) ( ) ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )are Accentuation coefficients( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 307
  • 5. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012are Dissipation coefficients III. STATEMENT OF GOVERNING EQUATIONS: SYMMETRY BREAKING AND ACQUISITION OF MASS: 1 MODULE NUMBERED ONEThe differential system of this model is now (Module Numbered one) ( )( ) [( )( ) ( )( ) ( )] 2 ( )( ) [( )( ) ( )( ) ( )] 3 ( )( ) [( )( ) ( )( ) ( )] 4 ( )( ) [( )( ) ( )( ) ( )] 5 ( )( ) [( )( ) ( )( ) ( )] 6 ( )( ) [( )( ) ( )( ) ( )] 7 ( )( ) ( ) First augmentation factor 8 ( )( ) ( ) First detritions factor UNIFIED ELECTROWEAK INTERACTION AND STRONG INTERACTION: 9 MODULE NUMBERED TWOThe differential system of this model is now ( Module numbered two) ( )( ) [( )( ) ( )( ) ( )] 10 ( )( ) [( )( ) ( )( ) ( )] 11 ( )( ) [( )( ) ( )( ) ( )] 12 ( )( ) [( )( ) ( )( ) (( ) )] 13 ( )( ) [( )( ) ( )( ) (( ) )] 14 ( )( ) [( )( ) ( )( ) (( ) )] 15 ( )( ) ( ) First augmentation factor 16 ( )( ) (( ) ) First detritions factor 17 308
  • 6. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 LAGRANGIAN INVARIANCE AND CONTINOUS GROUP OF LOCAL 18 TRANSFORMATIONS: MODULE NUMBERED THREEThe differential system of this model is now (Module numbered three) ( )( ) [( )( ) ( )( ) ( )] 19 ( )( ) [( )( ) ( )( ) ( )] 20 ( )( ) [( )( ) ( )( ) ( )] 21 ( )( ) [( )( ) ( )( ) ( )] 22 ( )( ) [( )( ) ( )( ) ( )] 23 ( )( ) [( )( ) ( )( ) ( )] 24 ( )( ) ( ) First augmentation factor ( )( ) ( ) First detritions factor 25 26 GROUP GENERATOR OF GAUGE THEORY AND VECTOR FIELD(GAUGE FIELD): : MODULE NUMBERED FOUR:============================================================================The differential system of this model is now (Module numbered Four) ( )( ) [( )( ) ( )( ) ( )] 27 ( )( ) [( )( ) ( )( ) ( )] 28 ( )( ) [( )( ) ( )( ) ( )] 29 ( )( ) [( )( ) ( )( ) (( ) )] 30 ( )( ) [( )( ) ( )( ) (( ) )] 31 ( )( ) [( )( ) ( )( ) (( ) )] 32 ( )( ) ( ) First augmentation factor 33 ( )( ) (( ) ) First detritions factor 34 309
  • 7. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 YANG MILLS THEORYAND NON COMMUTATIVE SYMMETRY GROUP IN GAUGE 35 THEORY: MODULE NUMBERED FIVEThe differential system of this model is now (Module number five) ( )( ) [( )( ) ( )( ) ( )] 36 ( )( ) [( )( ) ( )( ) ( )] 37 ( )( ) [( )( ) ( )( ) ( )] 38 ( )( ) [( )( ) ( )( ) (( ) )] 39 ( )( ) [( )( ) ( )( ) (( ) )] 40 ( )( ) [( )( ) ( )( ) (( ) )] 41 ( )( ) ( ) First augmentation factor 42 ( )( ) (( ) ) First detritions factor 43 LHS OF THE YANG MILLS THEORY AND RHS OF THE YANG MILLS THEORY.TAKEN 44TO THE OTHER SIDE THE LHS WOULD DISSIPATE THE RHS WITH OR WITHOUT TIME 45 LAG : MODULE NUMBERED SIX :The differential system of this model is now (Module numbered Six) ( )( ) [( )( ) ( )( ) ( )] 46 ( )( ) [( )( ) ( )( ) ( )] 47 ( )( ) [( )( ) ( )( ) ( )] 48 ( )( ) [( )( ) ( )( ) (( ) )] 49 ( )( ) [( )( ) ( )( ) (( ) )] 50 310
  • 8. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) [( )( ) ( )( ) (( ) )] 51 ( )( ) ( ) First augmentation factor 52 ( )( ) (( ) ) First detritions factor 53HOLISTIC CONCATENATE SYTEMAL EQUATIONS HENCEFORTH REFERRED TO AS “GLOBAL 54EQUATIONS”We take in to consideration the following parameters, processes and concepts: (1) Acquisition of mass (2) Symmetry Breaking (3) Strong interaction (4) Unified Electroweak interaction (5) Continuous group of local transformations (6) Lagrangian Variance (7) Group generator in Gauge Theory (8) Vector field or Gauge field (9) Non commutative symmetry group in Gauge Theory (10) Yang Mills Theory (We repeat the same Bank’s example. Individual debits and Credits are conservative so also the holistic one. Generalized theories are applied to various systems which are parameterized. And we live in ‘measurement world’. Classification is done on the parameters of various systems to which the Theory is applied. ). (11) First Term of the Lagrangian of the Yang Mills Theory(LHS) (12) RHS of the Yang Mills Theory ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 55 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 56 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 57 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 311
  • 9. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 58 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficient for category 1, 2 and 3 59 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fourth augmentation coefficient for category 1, 2 and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficient for category 1, 2 and 3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentation coefficient for category 1, 2 and 3 60 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 61 ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 62 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 63 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 64 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1, 2 and 3 65 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 66 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 67 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 68 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( )Where ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 69 312
  • 10. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficient for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficient for category 1, 2 and3 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are fifth augmentation coefficient for category 1, 2 and3 70 ( )( ) ( ), ( )( ) ( ) , ( )( ) ( ) are sixth augmentation coefficient for category 1, 2 and3 71 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 72 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 73 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 74 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are first detrition coefficients for category 1, 2 and 3 75 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1,2 and 3 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are fifth detritions coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are sixth detritions coefficients for category 1,2 and 3 76 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 77 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 78 313
  • 11. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 79 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are first augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are second augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are third augmentation coefficients for category 1, 2 and 3 ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1,2 and 3 80 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1, 2and 3 81 82 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 83 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( )( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 84 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( )( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are first detritions coefficients for category 1, 2 and 3 85 ( )( ) ( ) , ( )( ) ( ) , ( )( ) ( ) are second detritions coefficients for category 1, 2 and 3 ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) are third detrition coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detritions coefficients for category 1,2 and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detritions coefficients for category 1, 2and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth detritions coefficients for category 1, 2and 3 86 314
  • 12. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 87 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 88 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 89 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 90 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 91 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1, 2,and3 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1, 2,and 3 ( )( ) ( ), ( )( ) ( ), ( )( ) ( ) are sixth augmentation coefficients for category 1, 2,and 3 92 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 93 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 94 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 95 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 96 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) , ( )( ) ( ) ( )( ) ( ), ( )( ) ( ), ( )( ) ( )–( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 315
  • 13. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 97 98 99 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 100 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 101 ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 102 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth augmentation coefficients for category 1,2,and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth augmentation coefficients for category 1,2,and3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are sixth augmentation coefficients for category 1,2, 3 103 104 ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 105 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) 106 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 107 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1,2, and 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1,2, and 3 316
  • 14. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detrition coefficients for category 1,2, and 3 108 109 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 110 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 111 ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 112 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - are fourth augmentation coefficients ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) - fifth augmentation coefficients ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) sixth augmentation coefficients 113 114 ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) ( ) ( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 115 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( )( ) ( ) –( )( ) ( ) –( )( ) ( ) 116 ( )( ) [ ] ( )( ) ( ) ( )( ) ( ) –( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) 117 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) are fourth detrition coefficients for category 1, 2, and 3 317
  • 15. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( ), ( )( ) ( ) ( )( ) ( ) are fifth detrition coefficients for category 1, 2, and3–( )( ) ( ) , –( )( ) ( ) –( )( ) ( ) are sixth detrition coefficients for category 1, 2, and3 118Where we suppose 119(A) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 120(B) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) 121 ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( )(C) ( )( ) ( ) ( )( ) 122 ( )( ) ( ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 123 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 124 )( ) 125( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions 126( )( ) ( ) and( )( ) ( ) ( ) and ( ) are points belonging to the interval[( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality ofthe fact, that if ( ̂ )( ) then the function ( ) ( ( ) ) , the first augmentation coefficientWOULD be absolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 127(D) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 128(E) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together 129 with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and ( ̂ )( ) and the constants 130 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 131 satisfy the inequalities 318
  • 16. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) 132 ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( )Where we suppose 134(F) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 135(G) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 136Definition of ( )( ) ( )( ) : 137 ( ) ( )( ) ( ) ( )( ) ( ̂ ) 138 ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) 139(H) ( )( ) ( ) ( )( ) 140 ( )( ) (( ) ) ( )( ) 141Definition of ( ̂ )( ) ( ̂ )( ) : 142Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 143 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 144 )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ 145With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 146and( ) ( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It isto be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) ( ) then the function ( ) ( ) , the SECOND augmentation coefficient would be absolutelycontinuous.Definition of ( ̂ )( ) (̂ )( ) : 147(I) ( ̂ )( ) (̂ )( ) are positive constants 148 ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 149There exists two constants ( ̂ )( ) and ( ̂ )( ) which togetherwith ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) ( ̂ )( ) 150(̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) 151( ̂ )( ) 319
  • 17. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 Where we suppose 152 (J) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 153 The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) ( )( ) ( ̂ )( ) ( )( ) ( ) ( )( ) 154 ( )( ) ( ) ( )( ) 155 Definition of ( ̂ )( ) ( ̂ )( ) : 156 Where ( ̂ )( ) (̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 157 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ 158 )( ) 159 ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 160 and( )( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) ( ) then the function ( ) ( ) , the THIRD augmentation coefficient, would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) : 161 (K) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with 162 ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 163 satisfy the inequalities 164 ( ) ( ) ( ) ( ) ( ̂ ) ( ) ( ̂ ) ( ) (̂ ( ) ) 165 ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) 166 ( ̂ )( ) 167 Where we suppose 168(L) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 169 (M) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. 320
  • 18. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) 170 (N) ( )( ) ( ( )( ) ) ( )( ) (( ) ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 171 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 172 and( )( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ ) ( ) then the function ( )( ) ( ) , the FOURTH augmentation coefficient WOULD be absolutely continuous. 173 Defi174nition of ( ̂ )( ) (̂ )( ) : 174(O) ( ̂ ) ( ) (̂ )( ) are positive constants(P) ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 175 (Q) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose 176(R) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 177 (S) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : 321
  • 19. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) ( ̂ )( ) 178 (T) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants and They satisfy Lipschitz condition: 179 )( ) ( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂ With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 180 and( ) (( ) ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) then the function ( )( ) ( ) , theFIFTH augmentation coefficient attributable would be absolutely continuous. Definition of ( ̂ )( ) (̂ )( ) : 181(U) ( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) Definition of ( ̂ )( ) ( ̂ )( ) : 182(V) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( ) ( ̂ )( ) Where we suppose 183 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 184 (W) The functions ( )( ) ( )( ) are positive continuous increasing and bounded. Definition of ( )( ) ( )( ) : ( )( ) ( ) ( )( ) ( ̂ )( ) ( )( ) (( ) ) ( )( ) ( )( ) (̂ )( ) 322
  • 20. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 185(X) ( )( ) ( ) ( )( ) ( ) ( ) (( ) ) ( )( )Definition of ( ̂ )( ) ( ̂ )( ) : Where ( ̂ )( ) ( ̂ )( ) ( )( ) ( )( ) are positive constants andThey satisfy Lipschitz condition: 186 )( )( )( ) ( ) ( )( ) ( ) (̂ )( ) ( ̂ )( )( )( ) (( ) ) ( )( ) (( ) ) (̂ )( ) ( ) ( ) ( ̂With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) 187and( ) (( ) ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It isto be noted that ( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if( ̂ ) ( ) then the function ( )( ) ( ) , the SIXTH augmentation coefficient would beabsolutely continuous.Definition of ( ̂ )( ) (̂ )( ) : 188( ̂ )( ) (̂ )( ) are positive constants ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( )Definition of ( ̂ )( ) ( ̂ )( ) : 189There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )satisfy the inequalities ( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( )( ) ( )( ) (̂ )( ) ( ̂ )( ) (̂ )( )( ̂ )( ) 190Theorem 1: if the conditions IN THE FOREGOING above are fulfilled, there exists a solution 191satisfying the conditionsDefinition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 192 193 323
  • 21. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Definition of ( ) ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 194 195 ) ( ̂ )( ) ( ) ( ̂ )( , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Definition of ( ) ( ): 196 ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 197Definition of ( ) ( ): ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( ) 198Definition of ( ) ( ): 199 ( ) ( ̂ )( ) ( ) ( ̂ ) , ( ) ) ( ̂ )( ) ( ) ( ̂ )( , ( )Proof: Consider operator ( ) defined on the space of sextuples of continuous functions 200 which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 201 ) ( ̂ )( ) ( ) ( ̂ )( 202 ) ( ̂ )( ) ( ) ( ̂ )( 203By 204 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 205 324
  • 22. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 206̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 207̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 208̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 209Where ( ) is the integrand that is integrated over an interval ( ) 210Proof: 211 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 212 ) ( ̂ )( ) ( ) ( ̂ )( 213 ) ( ̂ )( ) ( ) ( ̂ )( 214By 215 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 216 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 217̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 218̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 219̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 220Where ( ) is the integrand that is integrated over an interval ( )Proof: 221 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 222 ) ( ̂ )( ) ( ) ( ̂ )( 223 ) ( ̂ )( ) ( ) ( ̂ )( 224By 225 325
  • 23. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 226 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 227̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 228̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 229̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 230Where ( ) is the integrand that is integrated over an interval ( ) ( ) 231Consider operator defined on the space of sextuples of continuous functions which satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 232 ) ( ̂ )( ) ( ) ( ̂ )( 233 ) ( ̂ )( ) ( ) ( ̂ )( 234By 235 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 236 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 237̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 238̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 239̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 240Where ( ) is the integrand that is integrated over an interval ( ) ( ) 241Consider operator defined on the space of sextuples of continuous functionswhich satisfy 242 ( ) ( ) ( ̂ )( ) ( ̂ )( ) 243 ) ( ̂ )( ) ( ) ( ̂ )( 244 ) ( ̂ )( ) ( ) ( ̂ )( 245 326
  • 24. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012By 246 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 247 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 248̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 249̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 250̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 251Where ( ) is the integrand that is integrated over an interval ( ) 252 ( )Consider operator defined on the space of sextuples of continuous functionswhich satisfy ( ) ( ) ( ̂ )( ) ( ̂ )( ) 253 ) ( ̂ )( ) ( ) ( ̂ )( 254 ) ( ̂ )( ) ( ) ( ̂ )( 255By 256 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 257 ̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 258̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 259̅ ( ) ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 260̅ () ∫ [( )( ) ( ( )) (( )( ) ( )( ) ( ( ( )) ( ) )) ( ( ) )] ( ) 261Where ( ) is the integrand that is integrated over an interval ( ) 262(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 263 .Indeed it is obvious that 327
  • 25. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 264 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )( ( ) ) [(( ( ) ) ( ̂ )( ) ] ( ̂ )( )( ) is as defined in the statement of theorem 1Analogous inequalities hold also for 265(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 266 .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( ( )( ) ) 267( )( ) ( ̂ )( ) )( ) ( (̂ ) ( ̂ )( )From which it follows that 268 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂ )(( ( ) ) ) [(( ) ) ( ̂ )( ) ] ( ̂ )(Analogous inequalities hold also for 269(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 270 .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 271 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )(Analogous inequalities hold also for 272(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 273 .Indeed it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 274 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( 328
  • 26. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) is as defined in the statement of theorem 1(c) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 275 .Indeed it is obvious that ̂ )( ) ( ) ( ) ∫ [( )( ) ( ( ̂ )( ) ( )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 276 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) (̂ )( ) ] ( ̂ )( )( ) is as defined in the statement of theorem 1(d) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself 277 .Indeed it is obvious that ) ( ̂ )( ) ( ( ) ∫ [( )( ) ( ( ̂ )( ) )] ( ) ( )( ) ( ̂ )( ) )( ) ( ( )( ) ) ( (̂ ) ( ̂ )( )From which it follows that 278 (̂ )( ) ( )( ) ( ) ( ̂ )( ) ̂( ( ) ) [(( )( ) ) ( ̂ )( ) ] ( ̂ )( ) ( ) is as defined in the statement of theorem 6Analogous inequalities hold also for 279 280 ( )( ) ( )( ) 281It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have 282 (̂ )( ) 283 ( ) ( )( ) [( ̂ ) ( ) (( ̂ ) ( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 284 ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 285In order that the operator transforms the space of sextuples of functions satisfying 329
  • 27. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012GLOBAL EQUATIONS into itself ( ) 286The operator is a contraction with respect to the metric ( ) ( ) ( ) ( ) (( )( )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 287Definition of ̃ ̃ : ( ̃ ̃) ( ) ( )It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( ) 288| | (( )( ) ( )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ) (( ( ) ( ) ( ) ( ) ))(̂ )( )And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 289not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by ( ) ( )( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact then ( ) ( )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 ( ) ( ) 290From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] 291 ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) : 292Remark 3: if is bounded, the same property have also . indeed if 330
  • 28. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 293analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) ( ( ) )) ( )( ) then 294Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) ( ( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 295 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that is ( ) ( )unbounded. The same property holds for if ( ) ( ( ) ) ( )We now state a more precise theorem about the behaviors at infinity of the solutions 296 ( )( ) ( )( ) 297It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) ( ̂ )( ) large to have (̂ )( ) 298 ( ) ( )( ) [( ̂ )( ) (( ̂ ) ( ) ) ] ( ̂ )( )( ̂ )( ) 299 (̂ )( ) ( ) ( )( ) [(( ̂ )( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 300In order that the operator transforms the space of sextuples of functions satisfying ( ) 301The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | | 331
  • 29. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Indeed if we denote 302Definition of ̃ ̃ : ( ̃ ̃ ) ( ) ( )It results 303 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the interval 304From the hypotheses it follows )( ) ( )( ) | (̂ )( ) 305|( (( )( ) ( )( ) ( ̂ )( )(̂ )( )(̂ ) ( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result follows 306Remark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered as 307not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by ̂ )( ) ̂ )( )( ̂ )( ) ( ( ̂ )( ) ( respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where () () 308From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] () ( () ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 309Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( ) 310If is bounded, the same property follows for and respectively. 332
  • 30. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Remark 4: If bounded, from below, the same property holds for The proof is 311analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 312Definition of ( )( ) :Indeed let be so that for( )( ) ( )( ) (( )( ) ) () ( )( )Then ( )( ) ( ) ( ) which leads to 313 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) 314 ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions 315 ( )( ) ( )( ) 316It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 317 ( ) ( )( ) [( ̂ )( ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 318 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ ( ) ) ] ( ̂ ( ) )( ̂ )( ) ( ) 319In order that the operator transforms the space of sextuples of functions into itself ( ) 320The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denote 321Definition of ̃ ̃ :( (̃) ( ) ) ̃ ( ) (( )( ))It results 322 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) 333
  • 31. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) 323 ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows ( ) ( ) (̂ )( ) 324| | (( )( ) ( )( ) (̂ )( )(̂ )( ) ( ) ̂(̂ ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))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 325not conformal with the reality, however we have put this hypothesis ,in order that we can postulatecondition necessary to prove the uniqueness of the solution bounded by ( ) ( )( ̂ )( ) ( ̂ ) ( ̂ )( ) ( ̂ ) respectively ofIf instead of proving the existence of the solution on , we have to prove it only on a compact thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 326From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 327Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 328analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 329Definition of ( )( ) : 330Indeed let be so that for 334
  • 32. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to 331 ( )( ) ( )( ) ( )( ) 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 332 ( )( ) ( )( ) 333It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 334 ( ) ( )( ) [( ̂ )( ) (( ̂ ) ( ) ) ] ( ̂ ) ( )( ̂ )( ) (̂ )( ) 335 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 336In order that the operator transforms the space of sextuples of functions satisfying IN toitself ( ) 337The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) ( |̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the interval 335
  • 33. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012From the hypotheses it follows 338 )( ) ( )( ) | (̂ )( ) 339|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 340as not conformal with the reality, however we have put this hypothesis ,in order that we canpostulate 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 ( ) ( ) 341From 19 to 24 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 342Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 343analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 344Definition of ( )( ) :Indeed let be so that for 336
  • 34. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( )( ) which leads to 345 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUSinequalities hold also for 346 ( )( ) ( )( ) 347 It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 348 ( ) ( )( ) [( ̂ ( ) ) (( ̂ )( ) ) ] ( ̂ )( )( ̂ )( ) (̂ )( ) 349 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 350In order that the operator transforms the space of sextuples of functions into itself ( ) 351The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) 337
  • 35. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( )Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows 352 )( ) ( )( ) | (̂ )( ) 353|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis (35,35,36) the resultfollowsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 354as not conformal with the reality, however we have put this hypothesis ,in order that we canpostulate 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 ( ) ( ) 355From GLOBAL EQUATIONS it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 356Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively.Remark 4: If bounded, from below, the same property holds for The proof is 357analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 358 338
  • 36. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Definition of ( )( ) :Indeed let be so that for 359 ( )( ) ( )( ) (( )( ) ) ( ) ( )( )Then ( )( ) ( ) ( ) which leads to 360 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutionsAnalogous inequalities hold also for 361 ( )( ) ( )( ) 362It is now sufficient to take and to choose ( ̂ )( ) ( ̂ )( )( ̂ )( ) (̂ )( ) large to have (̂ )( ) 363 ( ) ( )( ) [( ̂ ) ( ) (( ̂ )( ) ) ] ( ̂ ) ( )(̂ )( ) (̂ )( ) 364 ( ) ( )( ) [(( ̂ ) ( ) ) ( ̂ )( ) ] ( ̂ )( )( ̂ )( ) ( ) 365In order that the operator transforms the space of sextuples of functions into itself ( ) 366The operator is a contraction with respect to the metric ((( )( ) ( )( ) ) (( )( ) ( )( ) )) ( ) ( ) ( ) ( )| (̂ )( ) ( ) ( ) ( ) ( )| (̂ )( ) | |Indeed if we denoteDefinition of (̃) ( ) : ( (̃) ( ) ) ̃ ̃ ( ) (( )( ))It results 339
  • 37. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (|̃ ∫( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (∫ ( )( ) | | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) (( )( ) ( ( ) )| | ) ) ( ) ( ) ( ) (̂ )( ) ( (̂ )( ) ( ( )( ) ( ( )) ( )( ) ( ( )) ) ) ( ) 367Where ( ) represents integrand that is integrated over the intervalFrom the hypotheses it follows )( ) ( )( ) | (̂ )( ) 368|( (( )( ) ( )( ) (̂ )( )(̂ )( )( ̂ )( ) ( ̂ )( ) ) ((( )( ) ( )( ) ( )( ) ( )( ) ))And analogous inequalities for . Taking into account the hypothesis the result followsRemark 1: The fact that we supposed ( )( ) ( )( ) depending also on can be considered 369as not conformal with the reality, however we have put this hypothesis ,in order that we canpostulate 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 thenit suffices to consider that ( )( ) ( )( ) depend only on and respectively on( )( ) and hypothesis can replaced by a usual Lipschitz condition.Remark 2: There does not exist any where ( ) ( ) 370From 69 to 32 it results [ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )] ( ) ( ( ) ( ( )( ) ) forDefinition of (( ̂ )( ) ) (( ̂ )( ) ) (( ̂ )( ) ) : 371Remark 3: if is bounded, the same property have also . indeed if ( ̂ )( ) it follows (( ̂ )( ) ) ( )( ) and by integrating (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )In the same way , one can obtain (( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )If is bounded, the same property follows for and respectively. 340
  • 38. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Remark 4: If bounded, from below, the same property holds for The proof is 372analogous with the preceding one. An analogous property is true if is bounded from below.Remark 5: If is bounded from below and (( )( ) (( )( ) )) ( )( ) then 373Definition of ( )( ) :Indeed let be so that for ( )( ) ( )( ) (( )( ) ) ( ) ( )( ) 374Then ( )( ) ( )( ) which leads to 375 ( )( ) ( )( ) ( )( ) If we take such that it results ( )( ) ( )( ) ( ) By taking now sufficiently small one sees that isunbounded. The same property holds for if ( )( ) (( )( ) ( ) ) ( )( )We now state a more precise theorem about the behaviors at infinity of the solutions 376Behavior of the solutions 377If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 378(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( )Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 379 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) and ( ) ( ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :- 380(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 341
  • 39. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 381 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( ) 382are defined respectivelyThen the solution satisfies the inequalities 383 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 384( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 385 ( ) ( )( ) (( )( ) ( )( ) ) 386 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 387 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 388 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 342
  • 40. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Behavior of the solutions 389If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) : 390(d) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 391 ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( ) 392Definition of ( )( ) ( )( ) ( )( ) ( )( ) : 393By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots 394(e) of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 395 and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and 396Definition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 397By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the 398roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 399and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 400Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 401(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 402( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 403( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) 404and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 405and analogously 406( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( )and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) 407Then the solution satisfies the inequalities 408 (( )( ) ( )( ) ) ( ) ( )( )( )( ) is defined 409 343
  • 41. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( )( ) 410 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 411( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( ) ( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 412 ( ) ( )( ) (( )( ) ( )( ) ) 413 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 414 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) ) Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 415 Where ( )( ) ( )( ) ( )( ) ( )( ) 416 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 417 ( )( ) ( )( ) ( )( ) 418Behavior of the solutions 419If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(a) )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) : 420(b) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the roots of the equations ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 421 344
  • 42. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(c) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 422 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )Then the solution satisfies the inequalities (( )( ) ( )( ) ) ( )( ) ( )( )( ) is defined 423 (( )( ) ( )( ) ) ( )( ) 424 ( ) ( )( ) ( )( ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 425( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 426 ( ) ( )( ) (( )( ) ( )( ) ) 427 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 428 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 429 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 430 431Behavior of the solutions 432 345
  • 43. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(d) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 433(e) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 434 435 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( ) 436Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(f) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 437 438 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined by 59 and 64 respectivelyThen the solution satisfies the inequalities 439 440 (( )( ) ( )( ) ) ( ) ( )( ) 441 442 where ( )( ) is defined 443 444 445 346
  • 44. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( ) ( )( ) 446 ( )( ) ( )( ) 447 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 448( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) ( ) (( )( ) ( )( ) ) 449 ( )( ) (( )( ) ( )( ) ) 450 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 451 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 452 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 453Behavior of the solutions 454If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(g) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 455(h) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 456 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(i) If we define ( )( ) ( )( ) ( )( ) ( )( ) by 347
  • 45. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 457 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 458 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) 459 ( ) ( )( ) ( )( ) 460 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 461( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 462 ( ) ( )( ) (( )( ) ( )( ) ) 463 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 464 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 465 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )Behavior of the solutions 466 348
  • 46. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012If we denote and defineDefinition of ( )( ) ( )( ) ( )( ) ( )( ) :(j) ( )( ) ( )( ) ( )( ) ( )( ) four constants satisfying ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (( ) ) ( )( ) (( ) ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) : 467(k) By ( )( ) ( )( ) and respectively ( )( ) ( )( ) the roots of the ( ) ( ) ( ) ( ) ( ) equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) andDefinition of ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) ( ̅ )( ) : 468 By ( ̅ )( ) ( ̅ )( ) and respectively ( ̅ )( ) ( ̅ )( ) the ( ) ( ) ( ) ( ) ( ) roots of the equations ( ) ( ) ( ) ( ) and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )Definition of ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) :-(l) If we define ( )( ) ( )( ) ( )( ) ( )( ) by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 470 ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( )and analogously 471 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) ( )( ) ( ̅ )( ) and ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( ) ( )( ) where ( )( ) ( ̅ )( )are defined respectivelyThen the solution satisfies the inequalities 472 (( )( ) ( )( ) ) ( )( ) ( )where ( )( ) is defined (( )( ) ( )( ) ) ( )( ) 473 ( ) ( )( ) ( )( ) 349
  • 47. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) 474( [ ] ( ) ( )( ) (( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) 475 ( ) ( )( ) (( )( ) ( )( ) ) 476 ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 477 [ ] ( )( )( ) (( )( ) ( )( ) ) ( )( ) (( )( ) ( )( ) ) ( )( ) ( )( ) [ ]( )( ) (( )( ) ( )( ) ( )( ) )Definition of ( )( ) ( )( ) ( )( ) ( )( ) :- 478 Where ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 479Proof : From GLOBAL EQUATIONS we obtain 480 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 481Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtainsDefinition of ( ̅ )( ) ( )( ) :-(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 482 350
  • 48. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 483 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 484 (c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL E486QUATIONS we get easily the result stated in the theorem. Particular case : 485 If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) ( ) ( ) then ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also ( ) defines ( ) for the special case Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( )( ) and ( ̅ )( ) and definition of ( )( ) 486 we obtain 487 351
  • 49. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 488 Definition of :- It follows 489 ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains 490 Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( )(( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( )(( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 491 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( ) ((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( ) 492(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 493 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) (f) If ( )( ) ( ̅ )( ) ( )( ) , we obtain 494 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have ( ) Definition of ( ) :- 495 ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain 496 ( ) Definition of ( ) :- 352
  • 50. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) . 497 Particular case : 498 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 ( ̅ )( ) 499 From GLOBAL EQUATIONS we obtain 500 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) Definition of ( ) :- ( ) 501 It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) 502 From which one obtains(a) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 503 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) Definition of ( ̅ )( ) :- From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(b) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 504 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) 353
  • 51. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( )(c) If ( )( ) ( ̅ )( ) ( )( ) , we obtain 505 [ ( )( ) ((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain ( ) Definition of ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( )( ) and ( ̅ )( ) 506 : From GLOBAL EQUATIONS we obtain 507 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- 508 It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(d) For ( )( ) ( )( ) ( ̅ )( ) 354
  • 52. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( , ( )( ) [ ( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 509 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) ( ̅ )( ) [ ( ( )( ) (̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(e) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 510 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 511 ( ) ( ) ( ) 512 (f) If ( ) ( ̅ ) ( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) And so with the notation of the first part of condition (c) , we have Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) 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 513 ( )( ) ( )( ) 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 ( )( ) 514 355
  • 53. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 From GLOBAL EQUATIONS we obtain 515 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(g) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 516 [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(h) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 517 [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) )( ) (( ( ) [ ( )( ) ( )( ) ) ] ( )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( )((̅ )( ) (̅ )( ) ) ] ( ̅ )( ) ( ̅ )( ) 518 (i) If ( )( ) ( ̅ )( ) ( )( ) , we obtain [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) ( )( ) ( ) ( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( )( ) ( ̅ )( ) 519 And so with the notation of the first part of condition (c) , we have Definition of ( ) ( ) :- ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) In a completely analogous way, we obtain Definition of ( ) ( ) :- 356
  • 54. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012 ( ) ( )( ) ( ) ( ) ( )( ) , ( ) ( ) ( ) Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( ) ( ) ( ) ( ) ( ) then ( ) ( )( ) and as a consequence ( ) ( )( ) ( ) this also ( ) defines ( ) for the special case . Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( ) ( ) ( ) ( ) ( ̅ ) if in addition ( ) ( )( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between ( ) and ( ̅ ) and definition of ( )( ) ( ) ( ) 520 we obtain 521 ( ) ( )( ) (( )( ) ( )( ) ( )( ) ( )) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) Definition of :- It follows ( ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) (( )( ) ( ( ) ) ( )( ) ( ) ( )( ) ) From which one obtains Definition of ( ̅ )( ) ( )( ) :-(j) For ( )( ) ( )( ) ( ̅ )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )( ) (( )( ) ( )( ) ) ] , ( )( ) [ ( ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) In the same manner , we get 522 [ ( )( )((̅ )( ) (̅ )( ) ) ] 523 (̅ )( ) ( ̅ )( ) (̅ )( ) (̅ )( ) ( )( ) ( ) ( ) )( )((̅ )( ) (̅ )( ) ) ] , ( ̅ )( ) [ ( ( )( ) (̅ )( ) ( ̅ )( ) From which we deduce ( )( ) ( ) ( ) ( ̅ )( )(k) If ( )( ) ( )( ) ( ̅ )( ) we find like in the previous case, 524 357
  • 55. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 [ ( )( ) (( )( ) ( )( ) ) ] ( )( ) ( )( ) ( )( ) ( )( ) [ ( )( ) (( )( ) ( )( ) ) ] ( ) ( ) ( )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] (̅ )( ) ( ̅ )( ) (̅ )( ) [ ( )( ) ((̅ )( ) (̅ )( )) ] ( ̅ )( ) ( ̅ )( ) 525(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 ( ) for the special case .Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then ( ) ( ) ( ) ( ) ( ̅ ) if in addition ( ) ( )( ) then ( ) ( )( ) ( ) This is an importantconsequence of the relation between ( ) and ( ̅ ) and definition of ( )( ) ( ) ( ) 526 527 527We can prove the following 528Theorem 3: If ( )( ) ( )( ) are independent on , and the conditions( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined, then the system 529If ( )( ) ( )( ) are independent on , and the conditions 530. 358
  • 56. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012( )( ) ( )( ) ( )( ) ( )( ) 531( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 532( )( ) ( )( ) ( )( ) ( )( ) , 533( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 534 ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 535( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 536( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined are satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 537( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ,( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) as defined satisfied , then the systemIf ( )( ) ( )( ) are independent on , and the conditions 538( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) , 539 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 359
  • 57. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) as defined are satisfied , then the system( )( ) [( )( ) ( )( ) ( )] 540( )( ) [( )( ) ( )( ) ( )] 541( )( ) [( )( ) ( )( ) ( )] 542( )( ) ( )( ) ( )( ) ( ) 543( )( ) ( )( ) ( )( ) ( ) 544( )( ) ( )( ) ( )( ) ( ) 545has a unique positive solution , which is an equilibrium solution for the system 546( )( ) [( )( ) ( )( ) ( )] 547( )( ) [( )( ) ( )( ) ( )] 548( )( ) [( )( ) ( )( ) ( )] 549( )( ) ( )( ) ( )( ) ( ) 550( )( ) ( )( ) ( )( ) ( ) 551( )( ) ( )( ) ( )( ) ( ) 552has a unique positive solution , which is an equilibrium solution for 553( )( ) [( )( ) ( )( ) ( )] 554( )( ) [( )( ) ( )( ) ( )] 555( )( ) [( )( ) ( )( ) ( )] 556( )( ) ( )( ) ( )( ) ( ) 557( )( ) ( )( ) ( )( ) ( ) 558( )( ) ( )( ) ( )( ) ( ) 559has a unique positive solution , which is an equilibrium solution 560( )( ) [( )( ) ( )( ) ( )] 561( )( ) [( )( ) ( )( ) ( )] 563( )( ) [( )( ) ( )( ) ( )] 564( )( ) ( )( ) ( )( ) (( )) 565( )( ) ( )( ) ( )( ) (( )) 566( )( ) ( )( ) ( )( ) (( )) 567 360
  • 58. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012has a unique positive solution , which is an equilibrium solution for the system 568( )( ) [( )( ) ( )( ) ( )] 569( )( ) [( )( ) ( )( ) ( )] 570( )( ) [( )( ) ( )( ) ( )] 571( )( ) ( )( ) ( )( ) ( ) 572( )( ) ( )( ) ( )( ) ( ) 573( )( ) ( )( ) ( )( ) ( ) 574has a unique positive solution , which is an equilibrium solution for the system 575( )( ) [( )( ) ( )( ) ( )] 576( )( ) [( )( ) ( )( ) ( )] 577( )( ) [( )( ) ( )( ) ( )] 578( )( ) ( )( ) ( )( ) ( ) 579( )( ) ( )( ) ( )( ) ( ) 580( )( ) ( )( ) ( )( ) ( ) 584has a unique positive solution , which is an equilibrium solution for the system 582 583 584(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( ) 585(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( ) 586 361
  • 59. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 587(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( ) 588(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )( )( ) ( )( ) 589(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( )( ) 560(a) Indeed the first two equations have a nontrivial solution if ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )( )( ) ( )( )Definition and uniqueness of :- 561After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 562After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) 563 , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 564After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows thatthere exists a unique for which ( ) . With this value , we obtain from the three firstequations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] 362
  • 60. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 565Definition and uniqueness of :- 566After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 567After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Definition and uniqueness of :- 568After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it followsthat there exists a unique for which ( ) . With this value , we obtain from the threefirst equations ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )](e) By the same argument, the equations 92,93 admit solutions if 569 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) Where in ( ) must be replaced by their values from 96. It is easy to see thatis a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that ( )(f) By the same argument, the equations 92,93 admit solutions if 570 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to see that 571 is a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ) )(g) By the same argument, the concatenated equations admit solutions if 572 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( ) 363
  • 61. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Where in ( ) must be replaced by their values from 96. It is easy to see thatis a decreasing function in taking into account the hypothesis ( ) ( ) it followsthat there exists a unique such that (( ) ) 573(h) By the same argument, the equations of modules admit solutions if 574 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to seethat is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that (( ) )(i) By the same argument, the equations (modules) admit solutions if 575 ( ) ( )( ) ( )( ) ( )( ) ( )( )[( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )] ( )( ) ( )( )( ) ( )Where in ( )( ) must be replaced by their values from 96. It is easy to seethat is a decreasing function in taking into account the hypothesis ( ) ( ) itfollows that there exists a unique such that (( ) )(j) By the same argument, the equations (modules) admit solutions if 578 579 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 580 ( ) ( ) ( ) ( ) ( ) ( )[( ) ( ) ( ) ( ) ( ) ( )] ( ) ( )( ) ( ) 581Where in ( )( ) must be replaced by their values It is easy to see that is adecreasing function in taking into account the hypothesis ( ) ( ) it follows thatthere exists a unique such that ( )Finally we obtain the unique solution of 89 to 94 582 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( )( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 583 (( )) , ( ) and 584 364
  • 62. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ( )( ) ( )( ) 585 , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 586 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )] Obviously, these values represent an equilibrium solution 587Finally we obtain the unique solution 588 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 589 ( ) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 590 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 591 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 592 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionFinally we obtain the unique solution 593 (( )) , ( ) and ( )( ) ( )( ) , [( )( ) ( )( ) ( )] [( )( ) ( )( ) ( )] ( )( ) ( )( ) 594 , [( )( ) ( )( ) (( ) )] [( )( ) ( )( ) (( ) )]Obviously, these values represent an equilibrium solutionASYMPTOTIC STABILITY ANALYSIS 595 365
  • 63. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012Theorem 4: If the conditions of the previous theorem are satisfied and if the functions( )( ) ( )( ) Belong to ( ) ( ) then the above equilibrium point is asymptotically stable.Proof: DenoteDefinition of :- , 596 ( )( ) ( ) ( )( ) ( ) ( ) , ( )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 597 (( )( ) ( )( ) ) ( )( ) ( )( ) 598 (( )( ) ( )( ) ) ( )( ) ( )( ) 599 (( )( ) ( )( ) ) ( )( ) ( )( ) 600 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 601 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 602 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 603If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 604Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenote 605Definition of :- , 606 ( )( ) ( )( ) 607 ( ) ( )( ) , (( ) )taking into account equations (global)and neglecting the terms of power 2, we obtain 608 (( )( ) ( )( ) ) ( )( ) ( )( ) 609 (( )( ) ( )( ) ) ( )( ) ( )( ) 610 (( )( ) ( )( ) ) ( )( ) ( )( ) 611 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 612 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 613 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 614 366
  • 64. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 615Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- , ( )( ) ( )( ) ( ) ( )( ) , (( ) ) 616Then taking into account equations (global) and neglecting the terms of power 2, we obtain 617 (( )( ) ( )( ) ) ( )( ) ( )( ) 618 (( )( ) ( )( ) ) ( )( ) ( )( ) 619 (( )( ) ( )( ) ) ( )( ) ( )( ) 6120 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 621 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 622 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 623If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 624Belong to ( ) ( ) then the above equilibrium point is asymptotically stablDenoteDefinition of :- 625 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 626 (( )( ) ( )( ) ) ( )( ) ( )( ) 627 (( )( ) ( )( ) ) ( )( ) ( )( ) 628 (( )( ) ( )( ) ) ( )( ) ( )( ) 629 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 630 367
  • 65. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 631 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 632 633 If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( )Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- 634 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations (global) and neglecting the terms of power 2, we obtain 635 (( )( ) ( )( ) ) ( )( ) ( )( ) 636 (( )( ) ( )( ) ) ( )( ) ( )( ) 637 (( )( ) ( )( ) ) ( )( ) ( )( ) 638 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 639 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 640 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 641If the conditions of the previous theorem are satisfied and if the functions ( )( ) ( )( ) 642Belong to ( ) ( ) then the above equilibrium point is asymptotically stableDenoteDefinition of :- 643 , ( )( ) ( )( ) ( ) ( )( ) , (( ) )Then taking into account equations(global) and neglecting the terms of power 2, we obtain 644 (( )( ) ( )( ) ) ( )( ) ( )( ) 645 (( )( ) ( )( ) ) ( )( ) ( )( ) 646 368
  • 66. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 (( )( ) ( )( ) ) ( )( ) ( )( ) 647 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 648 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 649 (( )( ) ( )( ) ) ( )( ) ∑ ( ( )( ) ) 650 651The characteristic equation of this system is 652(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )] 653((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) 369
  • 67. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) 370
  • 68. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )+(( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) )[((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) )]((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) )( )( ) ( )( ) ( )( ) ) ((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) 371
  • 69. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012 ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( ) ( ) ) ((( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ) ( )( ) ) ( )( ) (( )( ) ( )( ) ( )( ) ) (( )( ) ( )( ) ( )( ) ( )( ) ( )( ) )((( )( ) ( )( ) ( )( ) ) ( )( ) ( )( ) ( )( ) )And as one sees, all the coefficients are positive. It follows that all the roots have negative real part,and this proves the theorem. IV. Acknowledgments:=======================================================================The introduction is a collection of information from various articles, Books, NewsPaper reports, Home Pages Of authors, Journal Reviews, Nature ‘s L:etters,ArticleAbstracts, Research papers, Abstracts Of Research Papers, StanfordEncyclopedia, Web Pages, Ask a Physicist Column, Deliberations with Professors,the internet including Wikipedia. We acknowledge all authors who have contributedto the same. In the eventuality of the fact that there has been any act of omission onthe part of the authors, we regret with great deal of compunction, contrition, regret,trepidiation and remorse. As Newton said, it is only because erudite and eminentpeople allowed one to piggy ride on their backs; probably an attempt has been madeto look slightly further. Once again, it is stated that the references are onlyillustrative and not comprehensive V. REFERENCES ================================================================ =========1. Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi - MEASUREMENTDISTURBS EXPLANATION OF QUANTUM MECHANICAL STATES-A HIDDEN VARIABLETHEORY - published at: "International Journal of Scientific and Research Publications, Volume 2,Issue 5, May 2012 Edition".2. DR K N PRASANNA KUMAR, PROF B S KIRANAGI and PROF C S BAGEWADI -CLASSIC 2 FLAVOUR COLOR SUPERCONDUCTIVITY AND ORDINARY NUCLEARMATTER-A NEW PARADIGM STATEMENT - published at: "International Journal of Scientificand Research Publications, Volume 2, Issue 5, May 2012 Edition".3. A HAIMOVICI: “On the growth of a two species ecological system divided on age groups”. Tensor, Vol 37 (1982),Commemoration volume dedicated to Professor Akitsugu Kawaguchi on his 80th birthday4. FRTJOF CAPRA: “The web of life” Flamingo, Harper Collins See "Dissipative structures” pages 172-188 372
  • 70. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 20125. 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 2010 373
  • 71. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(9)^ a b c Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhaltabhängig?", Annalen der Physik 18:639 Bibcode 1905AnP...323..639E,DOI:10.1002/andp.19053231314. See also the English translation.(10)^ a b Paul Allen Tipler, Ralph A. Llewellyn (2003-01), Modern Physics, W. H. Freeman andCompany, pp. 87–88, ISBN 0-7167-4345-0(11)^ a b Rainville, S. et al. World Year of Physics: A direct test of E=mc2. Nature 438, 1096-1097(22 December 2005) | doi: 10.1038/4381096a; Published online 21 December 2005.(12)^ In F. Fernflores. The Equivalence of Mass and Energy. Stanford Encyclopedia of Philosophy(13)^ Note that the relativistic mass, in contrast to the rest mass m0, is not a relativistic invariant, andthat the velocity is not a Minkowski four-vector, in contrast to the quantity , where is the differentialof the proper time. However, the energy-momentum four-vector is a genuine Minkowski four-vector,and the intrinsic origin of the square-root in the definition of the relativistic mass is the distinctionbetween dτ and dt.(14)^ Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0(15)^ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8(16)^ Hans, H. S.; Puri, S. P. (2003). Mechanics (2 ed.). Tata McGraw-Hill. p. 433. ISBN 0-07-047360-9., Chapter 12 page 433(17)^ E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. Freeman and Co., NY. 1992.ISBN0-7167-2327-1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclearbombs, until heat is allowed to escape.(18)^ Mould, Richard A. (2002). Basic relativity (2 ed.). Springer. p. 126. ISBN 0-387-95210-1., Chapter 5 page 126(19)^ Chow, Tail L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones &Bartlett Learning. p. 392. ISBN 0-7637-3827-1., Chapter 10 page 392(20)^ [2] Cockcroft-Walton experiment(21)^ a b c Conversions used: 1956 International (Steam) Table (IT) values where one calorie 374
  • 72. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers conversion value of one gram TNT≡ 1000 calories used.(22)^ Assuming the dam is generating at its peak capacity of 6,809 MW.(23)^ Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominatedaverage Cp of 25.8, 5.134 moles of metal, and 132 J.K-1 for the prototype. A variation of± picograms is of course, much smaller than the actual uncertainty in the mass of the international 1.5prototype, which are ± micrograms. 2(24)^ [3] Article on Earth rotation energy. Divided by c^2.(25)^ a b Earths gravitational self-energy is 4.6 × 10-10 that of Earths total mass, or 2.7 trillion metrictons. Citation: The Apache Point Observatory Lunar Laser-Ranging Operation (APOLLO), T. W.Murphy, Jr. et al. University of Washington, Dept. of Physics (132 kB PDF, here.).(26)^ There is usually more than one possible way to define a field energy, because any field can bemade to couple to gravity in many different ways. By general scaling arguments, the correct answerat everyday distances, which are long compared to the quantum gravity scale, should be minimalcoupling, which means that no powers of the curvature tensor appear. Any non-minimal couplings,along with other higher order terms, are presumably only determined by a theory of quantum gravity,and within string theory, they only start to contribute to experiments at the string scale.(27)^ G. t Hooft, "Computation of the quantum effects due to a four-dimensional pseudoparticle",Physical Review D14:3432–3450 (1976).(28)^ A. Belavin, A. M. Polyakov, A. Schwarz, Yu. Tyupkin, "Pseudoparticle Solutions to YangMills Equations", Physics Letters 59B:85 (1975).(29)^ F. Klinkhammer, N. Manton, "A Saddle Point Solution in the Weinberg Salam Theory",Physical Review D 30:2212.(30)^ Rubakov V. A. "Monopole Catalysis of Proton Decay", Reports on Progress in Physics51:189–241 (1988).(31)^ S.W. Hawking "Black Holes Explosions?" Nature 248:30 (1974).(32)^ Einstein, A. (1905), "Zur Elektrodynamik bewegter Körper." (PDF), Annalen der Physik 17:891–921, Bibcode 1905AnP...322...891E,DOI:10.1002/andp.19053221004. English translation. 375
  • 73. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(33)^ See e.g. Lev B.Okun, The concept of Mass, Physics Today 42 (6), June 1969, p. 31–36, http://www.physicstoday.org/vol-42/iss-6/vol42no6p31_36.pdf(34)^ Max Jammer (1999), Concepts of mass in contemporary physics and philosophy, PrincetonUniversity Press, p. 51, ISBN 0-691-01017-X(35)^ Eriksen, Erik; Vøyenli, Kjell (1976), "The classical and relativistic concepts ofmass",Foundations of Physics (Springer) 6: 115–124, Bibcode 1976FoPh....6..115E,DOI:10.1007/BF00708670(36)^ a b Jannsen, M., Mecklenburg, M. (2007), From classical to relativistic mechanics:Electromagnetic models of the electron., in V. F. Hendricks, et al., , Interactions: Mathematics,Physics and Philosophy (Dordrecht: Springer): 65–134(37)^ a b Whittaker, E.T. (1951–1953), 2. Edition: A History of the theories of aether and electricity,vol. 1: The classical theories / vol. 2: The modern theories 1900–1926, London: Nelson(38)^ Miller, Arthur I. (1981), Albert Einsteins special theory of relativity. Emergence (1905) andearly interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2(39)^ a b Darrigol, O. (2005), "The Genesis of the theory of relativity." (PDF), Séminaire Poincaré1:1–22(40)^ Philip Ball (Aug 23, 2011). "Did Einstein discover E = mc2?” Physics World.(41)^ Ives, Herbert E. (1952), "Derivation of the mass-energy relation", Journal of the OpticalSociety of America 42 (8): 540–543, DOI:10.1364/JOSA.42.000540(42)^ Jammer, Max (1961/1997). Concepts of Mass in Classical and Modern Physics. New York:Dover. ISBN 0-486-29998-8.(43)^ Stachel, John; Torretti, Roberto (1982), "Einsteins first derivation of mass-energyequivalence", American Journal of Physics 50 (8): 760–763, Bibcode1982AmJPh..50..760S, DOI:10.1119/1.12764(44)^ Ohanian, Hans (2008), "Did Einstein prove E=mc2?", Studies In History and Philosophy ofScience Part B 40 (2): 167–173, arXiv:0805.1400,DOI:10.1016/j.shpsb.2009.03.002 376
  • 74. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012(45)^ Hecht, Eugene (2011), "How Einstein confirmed E0=mc2", American Journal ofPhysics 79 (6): 591–600, Bibcode 2011AmJPh..79..591H, DOI:10.1119/1.3549223(46)^ Rohrlich, Fritz (1990), "An elementary derivation of E=mc2", American Journal ofPhysics 58 (4): 348–349, Bibcode 1990AmJPh..58..348R, DOI:10.1119/1.16168(47) (1996). Lise Meitner: A Life in Physics. California Studies in the History of Science. 13.Berkeley: University of California Press. pp. 236–237. ISBN 0-520-20860-(48)^ UIBK.ac.at(49)^ J. J. L. Morton; et al. (2008). "Solid-state quantum memory using the 31P nuclearspin". Nature 455 (7216): 1085–1088. Bibcode 2008Natur.455.1085M.DOI:10.1038/nature07295.(50)^ S. Weisner (1983). "Conjugate coding". Association of Computing Machinery, Special InterestGroup in Algorithms and Computation Theory 15: 78–88.(51)^ A. Zelinger, Dance of the Photons: From Einstein to Quantum Teleportation, Farrar, Straus &Giroux, New York, 2010, pp. 189, 192, ISBN 0374239665(52)^ B. Schumacher (1995). "Quantum coding". Physical Review A 51 (4): 2738–2747. Bibcode 1995PhRvA..51.2738S. DOI:10.1103/PhysRevA.51.2738.(53)^ a b Straumann, N (2000). "On Paulis invention of non-abelian Kaluza-Klein Theory in1953". ArXiv: gr-qc/0012054 [gr-qc].(54)^ See Abraham Pais account of this period as well as L. Susskinds "Superstrings, Physics Worldon the first non-abelian gauge theory" where Susskind wrote that Yang–Mills was "rediscovered"only because Pauli had chosen not to publish.(55)^ Reifler, N (2007). "Conditions for exact equivalence of Kaluza-Klein and Yang-Millstheories". ArXiv: gr-qc/0707.3790 [gr-qc].(56)^ Yang, C. N.; Mills, R. (1954). "Conservation of Isotopic Spin and Isotopic GaugeInvariance". Physical Review 96 (1): 191– 377
  • 75. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 2012195. Bibcode 1954PhRv...96...191Y.DOI:10.1103/PhysRev.96.191.(57)^ Caprini, I.; Colangelo, G.; Leutwyler, H. (2006). "Mass and width of the lowest resonance inQCD". Physical Review Letters 96 (13): 132001. ArXiv: hep-ph/0512364.Bibcode 2006PhRvL..96m2001C. DOI:10.1103/PhysRevLett.96.132001.(58)^ Yndurain, F. J.; Garcia-Martin, R.; Pelaez, J. R. (2007). "Experimental status of the ππisoscalar S wave at low energy: f0 (600) pole and scattering length". Physical Review D76 (7):074034. ArXiv:hep-ph/0701025. Bibcode 2007PhRvD..76g4034G.DOI:10.1103/PhysRevD.76.074034.(59)^ Novikov, V. A.; Shifman, M. A.; A. I. Vainshtein, A. I.; Zakharov, V. I. (1983). "Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories From Instanton Calculus”.Nuclear 229 (2): 381–393. Bibcode 1983NuPhB.229..381N.DOI:10.1016/0550-3213(83)90338-3.(60)^ Ryttov, T.; Sannino, F. (2008). "Super symmetry Inspired QCD Beta Function". PhysicalReview D 78 (6): 065001. Bibcode 2008PhRvD..78f5001R.DOI:10.1103/PhysRevD.78.065001(61)^ Bogolubsky, I. L.; Ilgenfritz, E.-M.; A. I. Müller-Preussker, M.; Sternbeck, A. (2009). "Latticegluodynamics computation of Landau-gauge Greens functions in the deep infrared". Physics LettersB 676 (1-3): 69–73. Bibcode 2009PhLB..676...69B.DOI:10.1016/j.physletb.2009.04.076. 378
  • 76. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 7, 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============================================================================== 379