Yang mills theory

Advances in Physics Theories and Applications www.iiste.org

ISSN 2224-719X (Paper) ISSN 2225-0638 (Online)

Vol 7, 2012

304

SOME CONTRIBUTIONS TO YANG MILLS THEORY

FORTIFICATION –DISSIPATION MODELS

1DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3 PROF C S BAGEWADI

ABSTRACT. We provide a series of Models for the problems that arise in Yang Mills Theory. No claim

is made that the problem is solved. We do factorize the Yang Mills Theory and give a Model for the

values of LHS and RHS of the yang Mills theory. We hope these forms the stepping stone for further

factorizations and solutions to the subatomic denominations at Planck’s scale. Work also throws light on

some important factors like mass acquisition by symmetry breaking, relation between strong interaction

and weak interaction, Lagrangian Invariance despite transformations, Gauge field, Noncommutative

symmetry 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 Theory

The 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)



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(12) RHS of the Yang Mills Theory

II. FORMULATION OF THE PROBLEM

SYMMETRY BREAKING AND ACQUISITION OF MASS:

MODULE NUMBERED ONE

NOTATION :

: 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:



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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 GAUGE

THEORY

: CATEGORY TWO OF NON COMMUTATIVE SYMMETRY GROUP OPF GAUGE

THEORY

:CATEGORY THREE OFNON COMMUTATIVE SYMMETRY GROUP OF GAUGE



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THEORY

: CATEGORY ONE OFYANG MILLS THEORY (Theory is applied to various subatomic

particle 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.TAKEN

TO 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 various

characterized systems and the systemic characterizations form the basis for the formulation of the

classification).

==========================================================================

=====

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )

( ) ( )( ): ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ),

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

are Accentuation coefficients

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) , (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )



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308

are Dissipation coefficients

III. STATEMENT OF GOVERNING EQUATIONS:

SYMMETRY BREAKING AND ACQUISITION OF MASS:

MODULE NUMBERED ONE

The differential system of this model is now (Module Numbered one)

1

( )

( ) [( )( ) (

)( )( )] 2

( )

( ) [( )( ) (

)( )( )] 3

( )

( ) [( )( ) (

)( )( )] 4

( )

( ) [( )( ) (

)( )( )] 5

( )

( ) [( )( ) (

)( )( )] 6

( )

( ) [( )( ) (

)( )( )] 7

( )( )( ) First augmentation factor 8

( )( )( ) First detritions factor

UNIFIED ELECTROWEAK INTERACTION AND STRONG INTERACTION:

MODULE NUMBERED TWO

The differential system of this model is now ( Module numbered two)

9

( )

( ) [( )( ) (

)( )( )] 10

( )

( ) [( )( ) (

)( )( )] 11

( )

( ) [( )( ) (

)( )( )] 12

( )

( ) [( )( ) (

)( )(( ) )] 13

( )

( ) [( )( ) (

)( )(( ) )] 14

( )

( ) [( )( ) (

)( )(( ) )] 15


( )( )(( ) ) First detritions factor 17



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LAGRANGIAN INVARIANCE AND CONTINOUS GROUP OF LOCAL

TRANSFORMATIONS:

MODULE NUMBERED THREE

The differential system of this model is now (Module numbered three)

18

( )

( ) [( )( ) (

)( )( )] 19

( )

( ) [( )( ) (

)( )( )] 20

( )

( ) [( )( ) (

)( )( )] 21

( )

( ) [( )( ) (

)( )( )] 22

( )

( ) [( )( ) (

)( )( )] 23

( )

( ) [( )( ) (

)( )( )] 24

( )( )( ) First augmentation factor

( )( )( ) First detritions factor 25

GROUP GENERATOR OF GAUGE THEORY AND VECTOR FIELD(GAUGE FIELD):

: MODULE NUMBERED FOUR:

==========================================================================

==

The differential system of this model is now (Module numbered Four)

26

( )

( ) [( )( ) (

)( )( )] 27

( )

( ) [( )( ) (

)( )( )] 28

( )

( ) [( )( ) (

)( )( )] 29

( )

( ) [( )( ) (

)( )(( ) )] 30

( )

( ) [( )( ) (

)( )(( ) )] 31

( )

( ) [( )( ) (

)( )(( ) )] 32





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310

YANG MILLS THEORYAND NON COMMUTATIVE SYMMETRY GROUP IN GAUGE

THEORY:

MODULE NUMBERED FIVE

The differential system of this model is now (Module number five)

35

( )

( ) [( )( ) (

)( )( )] 36

( )

( ) [( )( ) (

)( )( )] 37

( )

( ) [( )( ) (

)( )( )] 38

( )

( ) [( )( ) (

)( )(( ) )] 39

( )

( ) [( )( ) (

)( )(( ) )] 40

( )

( ) [( )( ) (

)( )(( ) )] 41



LHS OF THE YANG MILLS THEORY AND RHS OF THE YANG MILLS THEORY.TAKEN

TO THE OTHER SIDE THE LHS WOULD DISSIPATE THE RHS WITH OR WITHOUT TIME

LAG :

MODULE NUMBERED SIX

:

The differential system of this model is now (Module numbered Six)

44

45

( )

( ) [( )( ) (

)( )( )] 46

( )

( ) [( )( ) (

)( )( )] 47

( )

( ) [( )( ) (

)( )( )] 48

( )

( ) [( )( ) (

)( )(( ) )] 49

( )

( ) [( )( ) (

)( )(( ) )] 50



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( )

( ) [( )( ) (

)( )(( ) )] 51



HOLISTIC CONCATENATE SYTEMAL EQUATIONS HENCEFORTH REFERRED TO AS “GLOBAL EQUATIONS”

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

54

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

55

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

56

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

57



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Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fourth augmentation coefficient for category 1, 2 and

3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation coefficient for category 1, 2 and 3

58

59

60

( )

( ) [

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

61

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

62

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

63

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second detritions coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detritions coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth detritions coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detritions coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detritions coefficients for category 1, 2 and 3

64

65

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

66

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

67

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

68

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for category 1, 2 and 3 69



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313

( )( )( ) , (

)( )( ) , ( )( )( ) are second augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth augmentation coefficient for category 1, 2 and

3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth augmentation coefficient for category 1, 2 and

3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation coefficient for category 1, 2 and

3

70

71

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

72

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

73

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

74

( )( )( ) , (

)( )( ) , ( )( )( ) are first detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are second detrition coefficients for category 1,2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition coefficients for category 1,2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth 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

75

( )( ) [

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

76

( )( ) [

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

77

78



Vol 7, 2012

314

( )( ) [

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )( )( ) , (

)( )( ) , ( )( )( ) 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

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficients for category 1, 2

and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation coefficients for category 1, 2

and 3

79

80

81

( )( ) [

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

82

( )( ) [

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

83

( )( ) [

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

84

( )( )( ) (

)( )( ) ( )( )( ) are first detritions coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) 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, 2

and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth detritions coefficients for category 1, 2

and 3

85

86



Vol 7, 2012

315

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

87

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

88

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

89

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth augmentation coefficients for category 1, 2,and

3

( )( )( ) , (

)( )( ) ( )( )( ) are fifth augmentation coefficients for category 1, 2,and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation coefficients for category 1, 2,and 3

90

91

92

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

93

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

94

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

95

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) , ( )( )( )

( )( )( ) , (

)( )( ) , ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

96



Vol 7, 2012

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97

98

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

99

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

100

( )

( ) [

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

101

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth augmentation coefficients for category 1,2,

and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficients for category 1,2,and

3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation coefficients for category 1,2, 3

102

103

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

104

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

105

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

106

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition coefficients for category 1,2, and 3

107



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– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition coefficients for category 1,2, and 3

108

( )

( )

[(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

109

( )

( )

[(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

110

( )

( )

[(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

111

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) - are fourth augmentation coefficients

( )( )( ) (

)( )( ) ( )( )( ) - fifth augmentation coefficients

( )( )( ) , (

)( )( ) ( )( )( ) sixth augmentation coefficients

112

113

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

114

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

115

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

116

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for category 1, 2, and 3

117



Vol 7, 2012

318

( )( )( ) , (

)( )( ) ( )( )( ) are fifth detrition coefficients for category 1, 2, and

3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition coefficients for category 1, 2, and

3

118

Where we suppose 119

(A) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(B) The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

120

121

(C) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

122

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )( ) (

)( )( ) ( )( ) ( )( )

123

124

125

With the Lipschitz condition, we place a restriction on the behavior of functions

( )( )(

) and( )( )( ) (

) and ( ) are points belonging to the interval

[( )( ) ( )

( )] . It is to be noted that ( )( )( ) is uniformly continuous. In the eventuality of

the fact, that if ( )( ) then the function (

)( )( ) , the first augmentation coefficient

WOULD be absolutely continuous.

126


( ) :

(D) ( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

127


( ) :

(E) There exists two constants ( )( ) and ( )

( ) which together

with ( )( ) ( )

( ) ( )( ) and ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

128

129

130

131



Vol 7, 2012

319

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

132


(F) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( ) 135

(G) The functions ( )( ) (

)( ) are positive continuous increasing and bounded. 136


( ): 137

( )( )( ) ( )

( ) ( )( )

138

( )( )( ) ( )

( ) ( )( ) ( )

( ) 139

(H) ( )( ) ( ) ( )

( ) 140

( )( ) (( ) ) ( )

( ) 141


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


142

They satisfy Lipschitz condition: 143

( )( )(

) ( )( )( ) ( )

( ) ( )( ) 144

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( ) 145

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) And ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if ( )

( )

then the function ( )( )( ) , the SECOND augmentation coefficient would be absolutely

continuous.

146


( ) : 147

(I) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

148


( ) :

There exists two constants ( )( ) and ( )

( ) which together

with ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

149

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 150

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 151



Vol 7, 2012

320


(J) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

153

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


154

155

156


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(

) ( )( )( ) ( )

( ) ( )( )

157

158

159


)

and( )( )( ) . (


( )] . 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.

160


( ) :

(K) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

161

There exists two constants There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

162

163

164

165

166

167


(L) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(M) The functions ( )( ) (


169



Vol 7, 2012

321


( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

(N) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


170


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

171


)

and( )( )( ) . (


( )] . 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.

172

173

Defi174nition of ( )( ) ( )

( ) :

(O) ( ) ( ) ( )( ) are positive constants

(P) ( )

( )

( )( ) ( )

( )

( )( )

174


( ) :

(Q) There exists two constants ( )( ) and ( )


( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

175


(R) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(S) The functions ( )( ) (



( ):

177



Vol 7, 2012

322

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

(T) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


178


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

179


)

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


)( )( ) , theFIFTH augmentation coefficient attributable

would be absolutely continuous.

180


( ) :

(U) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

181


( ) :

(V) There exists two constants ( )( ) and ( )


( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

182


( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(W) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

184



Vol 7, 2012

323

(X) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


185


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

186


)

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


)( )( ) , the SIXTH augmentation coefficient would be

absolutely continuous.

187


( ) :

( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

188


( ) :

There exists two constants ( )( ) and ( )


( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

189

190

Theorem 1: if the conditions IN THE FOREGOING above are fulfilled, there exists a solution

satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

191

192

193



Vol 7, 2012

324

Definition of ( ) ( )

( ) ( )( ) ( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

194

( ) ( )( ) ( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

195


( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

196


( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

197

198


( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

199

Proof: Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

200

( ) ( )

( )

( ) ( )

( ) 201

( ) ( )

( ) ( )( ) 202

( ) ( )

( ) ( )( ) 203

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

204

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

205



Vol 7, 2012

325

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

206

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

207

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

208

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

209

210

Proof:

Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

211

( ) ( )

( )

( ) ( )

( ) 212

( ) ( )

( ) ( )( ) 213

( ) ( )

( ) ( )( ) 214

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

215

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

216

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

217

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

218

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

219

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )


220

Proof:


which satisfy

221

( ) ( )

( )

( ) ( )

( ) 222

( ) ( )

( ) ( )( ) 223

( ) ( )

( ) ( )( ) 224

By 225



Vol 7, 2012

326

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

226

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

227

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

228

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

229

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )


230


which satisfy

231

( ) ( )

( )

( ) ( )

( ) 232

( ) ( )

( ) ( )( ) 233

( ) ( )

( ) ( )( ) 234

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

235

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

236

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

237

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

238

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

239

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )


240


which satisfy

241

242

( ) ( )

( )

( ) ( )

( ) 243

( ) ( )

( ) ( )( ) 244

( ) ( )

( ) ( )( ) 245



Vol 7, 2012

327

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

246

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

247

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

248

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

249

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

250

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )


251


which satisfy

252

( ) ( )

( )

( ) ( )

( ) 253

( ) ( )

( ) ( )( ) 254

( ) ( )

( ) ( )( ) 255

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

256

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

257

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

258

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

259

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

260

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )


261

262

(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself

.Indeed it is obvious that

263



Vol 7, 2012

328

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 1

264

Analogous inequalities hold also for 265

(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself


266

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( ) ( ( )

( ) )

( )( )( )( )

( )( ) ( ( )( ) )

267


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

268


(a) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself


( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

270


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

271


(b) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

273


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

274



Vol 7, 2012

329


(c) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

275


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


276

(d) The operator ( ) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

277


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


Analogous inequalities hold also for

278

279

280

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

281

282

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

283

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

284

In order that the operator ( ) transforms the space of sextuples of functions satisfying 285



Vol 7, 2012

330

GLOBAL EQUATIONS into itself

The operator ( ) is a contraction with respect to the metric

(( ( ) ( )) ( ( ) ( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

286

Indeed if we denote

Definition of :

( ) ( )( )

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

287

| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) (( ( ) ( ) ( ) ( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

288

Remark 1: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If 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.

289

Remark 2: There does not exist any where ( ) ( )

From 19 to 24 it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

290

291

Definition of (( )( ))

(( )

( )) :

Remark 3: if is bounded, the same property have also . indeed if

292



Vol 7, 2012

331

( )( ) 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

analogous with the preceding one. An analogous property is true if is bounded from below.

293

Remark 5: If is bounded from below and (( )( ) ( ( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )( ( ) ) ( ) ( )( )

294

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is

unbounded. The same property holds for if ( )( ) ( ( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions

295

296


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

297

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

298

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

299

In order that the operator ( ) transforms the space of sextuples of functions satisfying 300


((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

301



Vol 7, 2012

332

Indeed if we denote

Definition of : ( ) ( )( )

302

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

303



304

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( )

( )( )( )

( )) ((( )( ) ( )

( ) ( )( ) ( )

( )))

305

And analogous inequalities for . Taking into account the hypothesis the result follows 306





( )( ) ( )( ) ( )





( )( ) and hypothesis can replaced by a usual Lipschitz condition.

307



( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

308


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


309

310



Vol 7, 2012

333



311

Remark 5: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then



( )( ) (

)( )(( )( ) ) ( ) ( )( )

312

Then

( )


(( )( )( )( )

) ( )


it results

313

(( )( )( )( )

)


unbounded. The same property holds for if ( )( ) (( )( ) ) (

)( )


314

315


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

316

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

317

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

318

In order that the operator ( ) transforms the space of sextuples of functions into itself 319


((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

320

Indeed if we denote

Definition of :( ( ) ( ) ) ( )(( ) ( ))

321

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

322



Vol 7, 2012

334

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )



323

| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( )

( )( )( )

( )) ((( )( ) ( )

( ) ( )( ) ( )

( )))


324





( )( ) ( )( ) ( )






325



( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

326


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


327



328


)( ) then



329

330



Vol 7, 2012

335

( )( ) (

)( )(( )( ) ) ( ) ( )( )

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )


331

332


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

333

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

334

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

335

In order that the operator ( ) transforms the space of sextuples of functions satisfying IN to

itself

336


((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )


337



Vol 7, 2012

336


338

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( )

( )( )( )

( )) ((( )( ) ( )

( ) ( )( ) ( )

( )))


339


)( ) depending also on can be considered

as not conformal with the reality, however we have put this hypothesis ,in order that we can

postulate condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )






340



( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

341


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


342



343


)( ) then



344



Vol 7, 2012

337

( )( ) (

)( )(( )( ) ) ( ) ( )( )

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )

We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS

inequalities hold also for

345

346


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

347

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

348

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

349



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

351



Vol 7, 2012

338

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )



352

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( )

( )( )( )

( )) ((( )( ) ( )

( ) ( )( ) ( )

( )))

And analogous inequalities for . Taking into account the hypothesis (35,35,36) the result

follows

353





( )( ) ( )( ) ( )






354


From GLOBAL EQUATIONS it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

355


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


356



357


)( ) then 358



Vol 7, 2012

339



( )( ) (

)( )(( )( ) ) ( ) ( )( )

359

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )


Analogous inequalities hold also for

360

361


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

362

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

363

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

364



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

It results

366



Vol 7, 2012

340

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )



367

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( )

( )( )( )

( )) ((( )( ) ( )

( ) ( )( ) ( )

( )))


368





( )( ) ( )( ) ( )






369



( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

370


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


371



Vol 7, 2012

341



372


)( ) then



( )( ) (

)( )(( )( ) ) ( ) ( )( )

373

374

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


unbounded. The same property holds for if ( )( ) (( )( ) ( ) ) (

)( )


375

376

Behavior of the solutions

If we denote and define


( ) ( )( ) ( )

( ) :

(a) )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

377


( ) ( )( ) ( )

( ) ( ) ( ) :

(b) By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the

equations ( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( )

( )( )

378


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( ) ( )( )

379


( ) ( )( ) ( )

( ) ( )( ) :-

(c) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

380



Vol 7, 2012

342

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

are defined respectively

381

382

Then the solution satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

383

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

384

( )( ) ( )

(( )( ) ( )( )) 385

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 386

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

387


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

388



Vol 7, 2012

343



389


( ) ( )( ) ( )

( ) :

(d) )( ) ( )

( ) ( )( ) ( )


390

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( ) 391

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( ) 392


( ) ( )( ) ( )

( ) : 393

By ( )( ) ( )


( ) the roots 394

(e) of the equations ( )( )( ( ))

( )

( ) ( ) ( )( ) 395

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and 396


( ) ( )( ) ( )

( ) : 397

By ( )( ) ( )


( ) the 398

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( ) 399

and ( )( )( ( ))

( )

( ) ( ) ( )( ) 400


( ) ( )( ) ( )

( ) :- 401

(f) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by 402

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 403

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

404

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 405

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

406

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 407


(( )( ) ( )( )) ( )

( )( )

408

( )( ) is defined 409



Vol 7, 2012

344

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( ) 410

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

411

( )( ) ( )

(( )( ) ( )( )) 412

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 413

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

414


( ) ( )( ) ( )

( ):- 415

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

416

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

417

418




( ) ( )( ) ( )

( ) :

(a) )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )(( ) ) ( )

( )

419


( ) ( )( ) ( )

( ) :

(b) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

420


( ) ( )( ) ( )

( ) :- 421



Vol 7, 2012

345

(c) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) and ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )


(( )( ) ( )( )) ( )

( )( )

( )( ) is defined

422

423

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( ) 424

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

425

( )( ) ( )

(( )( ) ( )( )) 426

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 427

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

428


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

429

430

431

Behavior of the solutions 432



Vol 7, 2012

346



( ) ( )( ) ( )

( ) :

(d) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(e) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

433


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(f) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

434 435 436

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined by 59 and 64 respectively

437 438


(( )( ) ( )( )) ( )

( )( )


439 440 441 442 443 444 445



Vol 7, 2012

347

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

446 447

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) )

448

( )( ) ( )

(( )( ) ( )( ))

449

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

450

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

451


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

452 453

Behavior of the solutions If we denote and define


( ) ( )( ) ( )

( ) :

(g) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )

454


( ) ( )( ) ( )

( ) ( ) ( ) :

(h) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

455


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(i) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

456



Vol 7, 2012

348

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined respectively

457


(( )( ) ( )( )) ( )

( )( )


458

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

459 460

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) )

461

( )( ) ( )

(( )( ) ( )( ))

462

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

463

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

464


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

465

Behavior of the solutions 466



Vol 7, 2012

349



( ) ( )( ) ( )

( ) :

(j) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(k) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

467


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(l) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

468 470

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined respectively

471


(( )( ) ( )( )) ( )

( )( )


472

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

473



Vol 7, 2012

350

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) )

474

( )( ) ( )

(( )( ) ( )( ))

475

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

476

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

477


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

478

479

Proof : From GLOBAL EQUATIONS we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains


( ) :-

(a) For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

480

481

In the same manner , we get 482



Vol 7, 2012

351

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

(b) If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

483

(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


( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in GLOBAL E486QUATIONS 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 ( )( )

484

485

486

we obtain 487



Vol 7, 2012

352

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


488

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( )

( )( ))

489



( ) :-

(d) For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

490

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

491


( ) 492

(e) If ( )( ) ( )

( )

( )


( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

493

(f) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )


494


( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

495



496



Vol 7, 2012

353

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

. 497

Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( )


)( ) ( )( ) ( )

( ) and then

( )( ) ( )




( )

498

499

From GLOBAL EQUATIONS we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

500


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( )

( )( ))

501


(a) For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

502


( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )



( )

503

(b) If ( )( ) ( )

( )

( )


( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

504



Vol 7, 2012

354

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

(c) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in

the theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( )


)( ) ( )( ) ( )

( ) and then

( )( ) ( )




( )

505

506

: From GLOBAL EQUATIONS we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( )

( )( ))



( ) :-

(d) For ( )( )

( )

( ) ( )( )

507 508



Vol 7, 2012

355

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )


( )

509

(e) If ( )( ) ( )

( )

( )


( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

510

511

(f) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( ) this also

defines ( )( ) for the special case .


)( ) ( )( ) ( )

( ) and then

( )( ) ( )





512 513

514



Vol 7, 2012

356

From GLOBAL EQUATIONS we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(g) For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

515


( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )


( )

516

(h) If ( )( ) ( )

( )

( )


( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

517

(i) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



518 519



Vol 7, 2012

357

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( ) this also



)( ) ( )( ) ( )

( ) and then

( )( ) ( )





520

we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( )

( )( ))



( ) :-

(j) For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

521


( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )


( )

522 523

(k) If ( )( ) ( )

( )

( )

( ) we find like in the previous case, 524



Vol 7, 2012

358

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

(l) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( ) this also



)( ) ( )( ) ( )

( ) and then

( )( ) ( )





525

526 527 527

We can prove the following

Theorem 3: If ( )( ) (

)( ) are independent on , and the conditions

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined, then the system

528

529

If ( )( ) (

)( ) are independent on , and the conditions 530.



Vol 7, 2012

359

( )( )(

)( ) ( )( )( )

( ) 531

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) 532

( )( )(

)( ) ( )( )( )

( ) , 533

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined are satisfied , then the system

534

If ( )( ) (


( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


535

If ( )( ) (


( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


536

If ( )( ) (


( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined satisfied , then the system

537

If ( )( ) (


( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

538

539



Vol 7, 2012

360

( )( ) ( )


( )( ) [(

)( ) ( )( )( )] 540

( )( ) [(

)( ) ( )( )( )] 541

( )( ) [(

)( ) ( )( )( )] 542

( )( ) (

)( ) ( )( )( ) 543

( )( ) (

)( ) ( )( )( ) 544

( )( ) (

)( ) ( )( )( ) 545

has a unique positive solution , which is an equilibrium solution for the system 546

( )( ) [(

)( ) ( )( )( )] 547

( )( ) [(

)( ) ( )( )( )] 548

( )( ) [(

)( ) ( )( )( )] 549

( )( ) (

)( ) ( )( )( ) 550

( )( ) (

)( ) ( )( )( ) 551

( )( ) (

)( ) ( )( )( ) 552

has a unique positive solution , which is an equilibrium solution for 553

( )( ) [(

)( ) ( )( )( )] 554

( )( ) [(

)( ) ( )( )( )] 555

( )( ) [(

)( ) ( )( )( )] 556

( )( ) (

)( ) ( )( )( ) 557

( )( ) (

)( ) ( )( )( ) 558

( )( ) (

)( ) ( )( )( ) 559

has a unique positive solution , which is an equilibrium solution 560

( )( ) [(

)( ) ( )( )( )]

561

( )( ) [(

)( ) ( )( )( )] 563

( )( ) [(

)( ) ( )( )( )]

564

( )( ) (

)( ) ( )( )(( ))

565

( )( ) (

)( ) ( )( )(( ))

566

( )( ) (

)( ) ( )( )(( )) 567



Vol 7, 2012

361


( )( ) [(

)( ) ( )( )( )]

569

( )( ) [(

)( ) ( )( )( )]

570

( )( ) [(

)( ) ( )( )( )]

571

( )( ) (

)( ) ( )( )( )

572

( )( ) (

)( ) ( )( )( )

573

( )( ) (

)( ) ( )( )( )

574


( )( ) [(

)( ) ( )( )( )]

576

( )( ) [(

)( ) ( )( )( )]

577

( )( ) [(

)( ) ( )( )( )]

578

( )( ) (

)( ) ( )( )( )

579

( )( ) (

)( ) ( )( )( )

580

( )( ) (

)( ) ( )( )( )

584


583

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

584


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

585

586



Vol 7, 2012

362


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

587


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

588


( )

( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

589


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

560

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows

that there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

561





equations

562

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

563


After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (


equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

564



Vol 7, 2012

363

565




) . With this value , we obtain from the three

first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

566





first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

567





first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

568

(e) By the same argument, the equations 92,93 admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

Where in ( ) must be replaced by their values from 96. It is easy to see that

is a decreasing function in taking into account the hypothesis ( ) ( ) it follows

that there exists a unique such that ( )

569

(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


that there exists a unique such that (( )

)

571

(g) By the same argument, the concatenated equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

572



Vol 7, 2012

364

Where in ( ) must be replaced by their values from 96. It is easy to see that


that there exists a unique such that (( )

)

573

(h) By the same argument, the equations of modules admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

Where in ( )( ) must be replaced by their values from 96. It is easy to see

that is a decreasing function in taking into account the hypothesis ( ) ( ) it

follows that there exists a unique such that (( )

)

574

(i) By the same argument, the equations (modules) admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

Where in ( )( ) must be replaced by their values from 96. It is easy to see

that is a decreasing function in taking into account the hypothesis ( ) ( ) it

follows that there exists a unique such that (( )

)

575

(j) By the same argument, the equations (modules) admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

Where in ( )( ) must be replaced by their values It is easy to see that is a

decreasing function in taking into account the hypothesis ( ) ( ) it follows that

there exists a unique such that ( )

578

579

580

581

Finally we obtain the unique solution of 89 to 94

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )] ,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution

582

Finally we obtain the unique solution 583

(( )

) , (

) and 584



Vol 7, 2012

365

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

585

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

586

Obviously, these values represent an equilibrium solution 587

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]


588


( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

589

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]


590


(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

591

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]


592


(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

593

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]


594

ASYMPTOTIC STABILITY ANALYSIS 595



Vol 7, 2012

366

Theorem 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: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( )

596

Then taking into account equations (global) and neglecting the terms of power 2, we obtain 597

((

)( ) ( )( )) ( )

( ) ( )( )

598

((

)( ) ( )( )) ( )

( ) ( )( )

599

((

)( ) ( )( )) ( )

( ) ( )( )

600

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

601

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

602

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

603

If the conditions of the previous theorem are satisfied and if the functions ( )( ) (

)( )

Belong to ( )( ) then the above equilibrium point is asymptotically stable

604

Denote

Definition of :-

605

,

606

( )( )

(

) ( )( ) ,

( )( )

( ( )

) 607

taking into account equations (global)and neglecting the terms of power 2, we obtain 608

((

)( ) ( )( )) ( )

( ) ( )( )

609

((

)( ) ( )( )) ( )

( ) ( )( )

610

((

)( ) ( )( )) ( )

( ) ( )( )

611

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

612

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

613

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

614



Vol 7, 2012

367


)( )

Belong to ( )( ) then the above equilibrium point is asymptotically stabl

Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

615

616


((

)( ) ( )( )) ( )

( ) ( )( )

618

((

)( ) ( )( )) ( )

( ) ( )( )

619

((

)( ) ( )( )) ( )

( ) ( )( )

6120

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

621

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

622

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

623


)( )

Belong to ( )( ) then the above equilibrium point is asymptotically stabl

Denote

624

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

(( )

)

625


((

)( ) ( )( )) ( )

( ) ( )( )

627

((

)( ) ( )( )) ( )

( ) ( )( )

628

((

)( ) ( )( )) ( )

( ) ( )( )

629

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

630



Vol 7, 2012

368

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

631

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

632


)( )


Denote

633

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

634


((

)( ) ( )( )) ( )

( ) ( )( )

636

((

)( ) ( )( )) ( )

( ) ( )( )

637

((

)( ) ( )( )) ( )

( ) ( )( )

638

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

639

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

640

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

641


)( )


Denote

642

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

643

Then taking into account equations(global) and neglecting the terms of power 2, we obtain 644

((

)( ) ( )( )) ( )

( ) ( )( )

645

((

)( ) ( )( )) ( )

( ) ( )( )

646



Vol 7, 2012

369

((

)( ) ( )( )) ( )

( ) ( )( )

647

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

648

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

649

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

650

651

The characteristic equation of this system is

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

652

653



Vol 7, 2012

370

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))



Vol 7, 2012

371

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))



Vol 7, 2012

372

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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, News

Paper reports, Home Pages Of authors, Journal Reviews, Nature ‘s L:etters,Article

Abstracts, Research papers, Abstracts Of Research Papers, Stanford

Encyclopedia, Web Pages, Ask a Physicist Column, Deliberations with Professors,

the internet including Wikipedia. We acknowledge all authors who have contributed

to the same. In the eventuality of the fact that there has been any act of omission on

the part of the authors, we regret with great deal of compunction, contrition, regret,

trepidiation and remorse. As Newton said, it is only because erudite and eminent

people allowed one to piggy ride on their backs; probably an attempt has been made

to look slightly further. Once again, it is stated that the references are only

illustrative and not comprehensive

V. REFERENCES

================================================================

=========

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2. DR K N PRASANNA KUMAR, PROF B S KIRANAGI and PROF C S BAGEWADI -

CLASSIC 2 FLAVOUR COLOR SUPERCONDUCTIVITY AND ORDINARY NUCLEAR

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373

5. HEYLIGHEN F. (2001): "The Science of Self-organization and Adaptivity", in L. D. Kiel, (ed) .

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http://en.wikipedia.org/wiki/Conservation_of_mass-energy#cite_ref-3








Vol 7, 2012

375

≡ 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.

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average Cp of 25.8, 5.134 moles of metal, and 132 J.K-1

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±1.5 picograms is of course, much smaller than the actual uncertainty in the mass of the international

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Vol 7, 2012

379

First 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:[email protected]

Second Author: 2Prof. B.S Kiranagi is the Former Chairman of the Department of Studies in Mathematics,

Manasa Gangotri and present Professor Emeritus of UGC in the Department. Professor Kiranagi has guided

over 25 students and he has received many encomiums and laurels for his contribution to Co homology Groups

and Mathematical Sciences. Known for his prolific writing, and one of the senior most Professors of the

country, he has over 150 publications to his credit. A prolific writer and a prodigious thinker, he has to his credit

several books on Lie Groups, Co Homology Groups, and other mathematical application topics, and excellent

publication history.-- UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri,

University of Mysore, Karnataka, India

Third Author: 3Prof. C.S. Bagewadi is the present Chairman of Department of Mathematics and Department

of Studies in Computer Science and has guided over 25 students. He has published articles in both national and

international journals. Professor Bagewadi specializes in Differential Geometry and its wide-ranging

ramifications. 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 Computer

science, Jnanasahyadri Kuvempu University, Shankarghatta, Shimoga district, Karnataka, India

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