Quantum mechanical behaviour, quantum tunneling, higgs boson

Journal of Natural Sciences Research www.iiste.org
ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)
Vol.2, No.4, 2012

Quantum Mechanical Behaviour, Quantum Tunneling, Higgs
Boson , Distorted Space And Time, Schrödinger’s Wave
Function, Neuron DNA, Particles (Hypothetical signature Less
Particles) And Consciousness
A “Syncopated Syncretism And Atrophied Asseveration”
Model
*1
Dr K N Prasanna Kumar, 2Prof B S Kiranagi And 3Prof C S Bagewadi

*1
Dr K N Prasanna Kumar, Post doctoral researcher, Dr KNP Kumar has three PhD’s, one each in Mathematics,
Economics and Political science and a D.Litt. in Political Science, Department of studies in Mathematics, Kuvempu
University, Shimoga, Karnataka, India Correspondence Mail id : drknpkumar@gmail.com

2
Prof B S Kiranagi, UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri, University
of Mysore, Karnataka, India

3
Prof C S Bagewadi, Chairman , Department of studies in Mathematics and Computer science, Jnanasahyadri
Kuvempu university, Shankarghatta, Shimoga district, Karnataka, India

Abstract:

We give a holistic model for the systems mentioned in the foregoing. Most important implication is that
Higgs Boson is the one, which warps space and time. Concept of Neuron DNA and signature less particles
are introduced.

Key Words:

Quantum Information, Space warp, Quantum Tunneling, Environmental decoherence, Schrödinger’s wave
function, Gravitational lensing, Black holes, Higgs Boson, Consciousness

Introduction:

We take in to consideration the following to build the 36 story model which consummates and consolidates
the parameters and processes involved:

1. Quantum Information
2. Quantum Mechanical behaviour
3. Quantum Tunneling
4. Non adiabatic multi photon process in the strong vibronic coupling limit
5. Environmental Decoherence(Green House Effects for example)
6. Schrodinger’s wave function
7. Gravitational lensing
8. Black holes
9. Faster than Light Particles (Neuron DNA- Mind, a signature less particles. How do you classify that?
Total energy =Existing matter-Energy attributable to signature less particles. Einstein did not take in
consideration psychic energy which is taken to be holistically conservational ,but individually and
collectively non conservative)
10. Consciousness( Total awareness- use ASDCII and Information field capacity to find the total storage-
Please refer Gesellshaft-Gememshaft paper on the subject matter)
11. Higgs Boson

83

Vol.2, No.4, 2012

12. Distorted Space and Time

Notation :

Quantum Mechanical Behaviour And Quantum Information
Module Numbered One

: Category One Of Quantum Mechanical Behaviour

: Category Two Of Quantum Mechanical Behaviour

: Category Three Of Quantum Mechanical Behaviour

: Category One Of Quantum Information

: Category Two Of Quantum Information

:Category Three Of Quantum Information

Non Adiabatic Multi Phonon Process In The Strong Vibronic Coupling And Quantum Tunneling

Module Numbered Two:

: Category One Of Non Adiabatic Multi Phonon Process

: Category Two Of Non Adiabatic Multi phonon Process

: Category Three Of Non Adiabatic Multi Phonon Process

:Category One Of Quantum Tunneling(There Are Lot Of Tunnels)

: Category Two Of Quantum Tunneling

: Category Three Of Quantum Tunneling

Environmental Decoherence (For Example Green House Effects) And Collapse of Schrodinger’s Wave
Function:

Module Numbered Three:

: Category One Of Collapse Of Schrodinger’s Wave Function(There Are Lot Of Potentialities)

:Category Two Of Collapse Of Schrodinger’s Wave Function

: Category Three Of Collapse Of Schrodinger’s Wave Function

: Category One Of Environmental Decoherence

:Category Two Of Environmental Decoherence

: Category Three Of Environmental Decoherence

Gravitational Lensing And Black holes

Module Numbered Four:

: Category One Of Black holes

: Category Two Of Black holes

84

Vol.2, No.4, 2012

: Category Three Ofblack Holes

:Category One Of gravitational Lensing

:Category Two Of Gravitational Lensing

: Category Three Of Gravitational Lensing

Faster Than Light Particles(Hypothetical Particles Of Neuron DNA-Mind) And Consciousness(Total
Awareness With Visual Images: Calculated Based On Ascii And Information Field Capacity)

Module Numbered Five:

: Category One Of Faster Than Light Particles(Signatureless Neuron Dna)

: Category Two Of Faster Than Light(Signatureless neuron Dna)

:Category Three Of Faster Thank Light Neuron Dna Particles Without Signature

:Category One Of Consciousness(Just Total Knowledge That Is Stored Like In Computer-See
Gratification Deprivation Model For Details)

:Category Two Of Consciousness

:Category Three Of Consciousness

Distorted Space And Time (St Warp) And Higgs Boson

Module Numbered Six:

: Category One Of Higgs Boson

: Category Two Of Higgs Boson

: Category Three Of Higgs Boson

: Category One Of Distorted Space And Time

: Category Two Of Distorted Space And Time

: Category Three Of Distorted Space And Time

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

( )( )
( )( )
( )( ) : ( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

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

are Accentuation coefficients
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

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

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

85

Vol.2, No.4, 2012

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

are Dissipation coefficients
Quantum Mechanical Behaviour And Quantum Information

Module Numbered One

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

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

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

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

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

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

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

( )( ) ( ) First augmentation factor

( )( ) ( ) First detritions factor

Non Adiabatic Multi Phonon Process In The Strong Vibronic Coupling And Quantum Tunneling

Module Numbered Two:

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

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

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

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

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

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

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


( )( ) (( ) ) First detritions factor

Environmental Decoherence (For Example Green House Effects) And Collapse of Schrodinger’s Wave
Function:

Module Numbered Three

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

86

Vol.2, No.4, 2012

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

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

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

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

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

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


( )( ) ( ) First detritions factor

Gravitational Lensing And Black holes

Module Numbered Four

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

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

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

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

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

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

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



Faster Than Light Particles (Hypothetical Particles Of Neuron Dna-Mind) And Consciousness(Total
Awareness With Visual Images: Calculated Based On Ascii And Information Field Capacity)

Module Numbered Five

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

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

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

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

87

Vol.2, No.4, 2012

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

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

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



Distorted Space And Time(St Warp) And Higgs Boson:

Module Numbered Six

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

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

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

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

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

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

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



Holistic Concatenated Equations Henceforth Referred To As “Global Equations”

1. Quantum Information
2. Quantum Mechanical behavior
3. Quantum Tunneling
4. Non adiabatic multi photon process in the strong vibronic coupling limit
5. Environmental Decoherence(Green House Effects for example)
6. Schrodinger’s wave function
7. Gravitational Lensing
8. Black holes
9. Faster than Light Particles (Neuron DNA- Mind, a signature less particles How do you classify that?
Total energy =Existing matter-Energy attributable to signature less particles. Einstein did not take in
consideration psychic energy which is taken to be holistically conservational ,but individually and
collectively non conservative)
10. Consciousness( Total awareness- use ASCII and Information field capacity to find the total storage-
Please refer Gesellschaft- Gemeinschaft paper on the subject matter)
11. Higgs Boson
12. Distorted Space and Time

88

Vol.2, No.4, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

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

89

Vol.2, No.4, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

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

90

Vol.2, No.4, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

91

Vol.2, No.4, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

–( )( )
( ) –( )( )
( ) –( )( )
( )

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

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

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

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

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

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

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

92

Vol.2, No.4, 2012

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

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

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

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

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

–( )( ) ( ) ( )( ) ( ) ( )( ) ( )

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

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

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

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

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

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

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

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

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

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

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

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

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

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

93

Vol.2, No.4, 2012

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

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

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

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

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

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

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

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

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

Where we suppose

(A) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

(B) The functions ( )( )
( )( ) are positive continuous increasing and bounded.

Definition of ( )( )
( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) ( ) ( )( )
( )( )
( ̂ )( )

(C) ( )( ) ( ) ( )( )

( )( ) ( ) ( )( )

Definition of ( ̂ )( )
( ̂ )( ) :

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( )
are positive constants and

They satisfy Lipschitz condition:

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

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

94

Vol.2, No.4, 2012

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.

(̂ )( ) :

(D) ( ̂ )( )
(̂ )( )
are positive constants

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) :

(E) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together
with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and ( ̂ )( ) and the constants
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )

satisfy the inequalities

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

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

Where we suppose

(F) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

(G) The functions ( )( )

( )( ) :
( )
( )( ) ( ) ( )( )
( ̂ )

( )( ) ( ) ( )( )
( )( )
( ̂ )( )

(H) ( )( ) ( ) ( )( )

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

( ̂ )( ) :

Where ( ̂ )( )
( ̂ )( )
( )( )
( )( ) are positive constants and


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

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

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.

95

Vol.2, No.4, 2012

(̂ )( ) :

(I) ( ̂ )( )
(̂ )( )

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) :

There exists two constants ( ̂ )( ) and ( ̂ )( ) which together
with ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )


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

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

Where we suppose

(J) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

The functions ( )( )

( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) ( ) ( )( )
( )( )
( ̂ )( )

( )( ) ( ) ( )( )

( )( ) ( ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
(̂ )( )
( )( )
( )( )


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

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

and( ) ( ( )
) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to
then the function ( )( ) ( ) , the THIRD augmentation coefficient, would be absolutely continuous.

(̂ )( ) :

(K) ( ̂ )( )
(̂ )( )

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

96

Vol.2, No.4, 2012

There exists two constants There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with
( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants
( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )


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

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

Where we suppose

(L) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( )

(M) The functions ( )( )

( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) (( ) ) ( )( )
( )( )
( ̂ )( )

(N) ( )( ) ( )( )( )
( )( ) (( ) ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
(̂ )( )
( )( )
( )( )


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

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

then the function ( )( ) ( ) , the FOURTH augmentation coefficient WOULD be absolutely
continuous.

(̂ )( ) :

(O) ( ̂ )( )
(̂ )( )
(P)
( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) :

(Q) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with

97

Vol.2, No.4, 2012

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


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

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

Where we suppose

(R) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
(S) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) (( ) ) ( )( )
( )( )
( ̂ )( )

(T) ( )( ) ( ) ( )( )
( )
( ) ( ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
(̂ )( )
( )( )
( )( )


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

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

then the function ( )( ) ( ) , theFIFTH augmentation coefficient attributable would be absolutely
continuous.

(̂ )( ) :

(U) ( ̂ )( )
(̂ )( )
( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( ) :

(V) There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with
( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )

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

98

Vol.2, No.4, 2012

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

Where we suppose

( )( )
( )( ) ( )( ) ( )( ) ( )( ) ( )( )
(W) The functions ( )( ) ( )( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )( ) :

( )( ) ( ) ( )( )
( ̂ )( )

( )( ) (( ) ) ( )( )
( )( )
( ̂ )( )

(X) ( )( ) ( ) ( )( )
( )
( ) (( ) ) ( )( )

( ̂ )( ) :

Where ( ̂ )( )
(̂ )( )
( )( )
( )( )


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

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

then the function ( )( ) ( ) , the SIXTH augmentation coefficient would be absolutely continuous.

(̂ )( ) :

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

( ̂ )( ) :

There exists two constants ( ̂ )( ) and ( ̂ )( ) which together with
( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )


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

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

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

99

Vol.2, No.4, 2012

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

) ( ̂ )( )
( ) ( ̂ )( , ( )

Definition of ( ) ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )


( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )


( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )


( ) ( ̂ )( )
( ) ( ̂ ) , ( )

) ( ̂ )( )
( ) ( ̂ )( , ( )

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

( ) ( ) ( ̂ )( )
( ̂ )( )

) ( ̂ )( )
( ) ( ̂ )(

) ( ̂ )( )
( ) ( ̂ )(

By

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

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

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

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Vol.2, No.4, 2012

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

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

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

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

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

( ) ( ) ( ̂ )( )
( ̂ )( )

) ( ̂ )( )
( ) ( ̂ )(

) ( ̂ )( )
( ) ( ̂ )(

By

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

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

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

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

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

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


Proof:
( )
satisfy

( ) ( ) ( ̂ )( )
( ̂ )( )

) ( ̂ )( )
( ) ( ̂ )(

) ( ̂ )( )
( ) ( ̂ )(

By

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

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

101

Vol.2, No.4, 2012

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

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

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

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


( )
satisfy

( ) ( ) ( ̂ )( )
( ̂ )( )

) ( ̂ )( )
( ) ( ̂ )(

) ( ̂ )( )
( ) ( ̂ )(

By

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

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

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

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

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

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


( )
satisfy

( ) ( ) ( ̂ )( )
( ̂ )( )

) ( ̂ )( )
( ) ( ̂ )(

) ( ̂ )( )
( ) ( ̂ )(

By

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

102

Vol.2, No.4, 2012

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

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

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

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

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


( )
satisfy

( ) ( ) ( ̂ )( )
( ̂ )( )

) ( ̂ )( )
( ) ( ̂ )(

) ( ̂ )( )
( ) ( ̂ )(

By

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

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

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

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

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

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


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

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

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

From which it follows that

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

103

Vol.2, No.4, 2012

( ) is as defined in the statement of theorem 1

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

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


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

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

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

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


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


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

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


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


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

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

104

Vol.2, No.4, 2012


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


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

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


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

( ) is as defined in the statement of theorem


( )( ) ( )( )
It is now sufficient to take and to choose
( ̂ )( ) ( ̂ )( )

( ̂ )( )
(̂ )( )
large to have

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

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

( )
In order that the operator transforms the space of sextuples of functions satisfying GLOBAL
EQUATIONS into itself
( )
The operator is a contraction with respect to the metric

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

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

Indeed if we denote

Definition of ̃ ̃ :

( ̃ ̃) ( )
( )

It results

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Vol.2, No.4, 2012

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

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

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

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

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

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.

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

From 19 to 24 it results

[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

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

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

( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )

In the same way , one can obtain

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )

If is bounded, the same property follows for and respectively.

Remark 4: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.

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

:

106

Vol.2, No.4, 2012

Indeed let be so that for

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

Then ( )( ) ( )( )
which leads to

( )( ) ( )( )
( )( ) If we take such that it results

( )( ) ( )( )
( ) By taking now sufficiently small one sees that is unbounded.
( ) ( )
The same property holds for if ( ) ( ( ) ) ( )

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

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( )
( ̂ )( )
large to have

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

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

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

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

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

Indeed if we denote

Definition of ̃ ̃ : ( ̃ ̃ ) ( )
( )

It results
( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

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

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



)( )
( )( ) | (̂ )( )
|(

107

Vol.2, No.4, 2012

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


̂ )( ) ̂ )( )
necessary to prove the uniqueness of the solution bounded by ( ̂ )( ) ( ( ̂ )( ) (
respectively of

( ) ( )
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 () ()


[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
() (

() ( ( )( ) )
for

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


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )



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

:


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

Then ( )( ) ( ) ( )
which leads to

( )( ) ( )( )

( )( ) ( )( )
( ) By taking now sufficiently small one sees that is unbounded. The
( ) ( )
same property holds for if ( ) (( )( ) ) ( )


108

Vol.2, No.4, 2012

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( )
(̂ )( ) large to have

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

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

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

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

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

Indeed if we denote

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

It results
( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

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

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



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


( ) ( )
respectively of


109

Vol.2, No.4, 2012



[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

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


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




:


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

Then ( )( ) ( ) ( )
which leads to

( )( ) ( )( )

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


( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( )

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

110

Vol.2, No.4, 2012

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

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

( )

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

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

Indeed if we denote

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

It results

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

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

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



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


( ) ( )
respectively of

( ) ( )



[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

111

Vol.2, No.4, 2012

( ) ( ( )( ) )
for

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


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




:


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

Then ( )( ) ( ) ( )
which leads to

( )( ) ( )( )

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

We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS
inequalities hold also for

( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( )

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

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

( )

112

Vol.2, No.4, 2012

( )

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

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

Indeed if we denote

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

It results

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

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

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



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

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

( ) ( )
respectively of



From GLOBAL EQUATIONS it results

[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

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


113

Vol.2, No.4, 2012

( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




:


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

Then ( )( ) ( )( )
which leads to

( )( ) ( )( )

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



( )( ) ( )( )
( ̂ )( ) ( ̂ )( )

( ̂ )( )

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

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

( )

( )

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

114

Vol.2, No.4, 2012

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

Indeed if we denote

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

It results

( ) ̃ ( )| ( ) ( ) (̂ )( ) ( (̂ )( ) (
|̃ ∫( )( ) | | ) )
( )

( ) ( ) (̂ )( ) ( (̂ )( ) (
∫ ( )( ) | | ) )

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

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



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


( ) ( )
respectively of

( ) ( )



[ ∫ {( )( ) ( )( ) ( ( ( )) ( ) )} )]
( ) (

( ) ( ( )( ) )
for

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


( ̂ )( ) it follows (( ̂ )( ) ) ( )( )
and by integrating

(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )

115

Vol.2, No.4, 2012


(( ̂ )( ) ) ( )( ) (( ̂ )( ) ) ( )( )




:


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

Then ( )( ) ( ) ( )
which leads to

( )( ) ( )( )

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


Behavior of the solutions

If we denote and define

( )( )
( )( )
( )( ) :

(a) )( )
( )( )
( )( )
( )( )
four constants satisfying

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

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

( )( )
( )( )
( )( ) ( ) ( )
:

(b) By ( )( )
( )( )
and respectively ( )( )
( )( )
the roots of the equations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( )

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

By ( ̅ )( )
( ̅ )( )
and respectively ( ̅ )( )
( ̅ )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )( )

( )( )
( )( )
( )( )
( )( ) :-

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

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

116

Vol.2, No.4, 2012

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

and ( )( )

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

and analogously

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

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

and ( )( )

( )( )
( )( )
( )( )
( )( )
( ̅ )( )
( )( )
where ( )( )
( ̅ )( )

are defined respectively

Then the solution satisfies the inequalities

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

where ( )( ) is defined

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

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

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

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

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

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

( )( )
( )( )
( )( ) :-

Where ( )( )
( )( ) ( )( )
( )( )

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

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

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



117

Vol.2, No.4, 2012

( )( )
( )( )
( )( ) :

(d) )( )
( )( )
( )( )
( )( )

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

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

( )( )
( )( )
( )( ) :

By ( )( )
( )( )
( )( )
the roots

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

and ( )( ) ( ( )
) ( )( ) ( )
( )( )
and

( ̅ )( )
( ̅ )( )
( ̅ )( ) :

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the

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

and ( )( ) ( ( )
) ( )( ) ( )
( )( )

( )( )
( )( )
( )( ) :-

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

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

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

and ( )( )

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

and analogously

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

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

and ( )( )

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


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

( )( ) is defined

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

118

Vol.2, No.4, 2012

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

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

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

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

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

( )( )
( )( )
( )( ) :-

Where ( )( )
( )( ) ( )( )
( )( )

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

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

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



( )( )
( )( )
( )( ) :

(a) )( )
( )( )
( )( )
( )( )

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

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

( )( )
( )( )
( )( ) :

(b) By ( )( )
( )( )
( )( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )

and ( )( ) ( ( )
) ( )( ) ( )
( )( )
and

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the

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

and ( )( ) ( ( )
) ( )( ) ( )
( )( )

( )( )
( )( )
( )( ) :-

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

119

Vol.2, No.4, 2012

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

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

and ( )( )

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

and analogously

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

( )( )
( )( )
( )( )
( ̅ )( )
( )( )
( )( )
( ̅ )( )
and ( )( )

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


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

( )( ) is defined

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

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

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

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

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

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

( )( )
( )( )
( )( ) :-

Where ( )( )
( )( ) ( )( )
( )( )

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

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

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


( )( )
( )( )
( )( ) :

120

Vol.2, No.4, 2012

(d) ( )( )
( )( )
( )( )
( )( )

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

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

( )( )
( )( )
( )( ) ( ) ( )
:

(e) By ( )( )
( )( )
( )( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( )( )
and

( ̅ )( )
( ̅ )( )
( ̅ )( ) :

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the
( ) ( ) ( ) ( ) ( )
roots of the equations ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( )
Definition of ( )( ) ( )( ) ( )( ) ( )( )
( )( ) :-

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

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

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

and ( )( )

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

and analogously

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

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

and ( )( )

( )( ) ( )( ) ( )( ) ( )( ) ( ̅ )( )
( )( )
where ( )( )
( ̅ )( )

are defined by 59 and 64 respectively


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

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

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

121

Vol.2, No.4, 2012

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

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

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

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

( )( )
( )( )
( )( ) :-

Where ( )( )
( )( ) ( )( )
( )( )

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

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

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


( )( )
( )( )
( )( ) :

(g) ( )( )
( )( )
( )( )
( )( )

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

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

( )( )
( )( )
( )( ) ( ) ( )
:

(h) By ( )( )
( )( )
( )( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
and ( )( ) ( ( )
) ( )( ) ( )
( )( )
and

( ̅ )( )
( ̅ )( )
( ̅ )( ) :

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the
( ) ( ) ( ) ( ) ( )
and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )

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

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

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

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

and ( )( )

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

122

Vol.2, No.4, 2012

and analogously

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

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

and ( )( )

( )( ) ( )( ) ( )( )
( )( )
( ̅ )( )
( )( )
where ( )( )
( ̅ )( )



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

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

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

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

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

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

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

( )( )
( )( )
( )( ) :-

Where ( )( )
( )( ) ( )( )
( )( )

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

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

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


( )( )
( )( )
( )( ) :

(j) ( )( )
( )( )
( )( )
( )( )

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

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

123

Vol.2, No.4, 2012

( )( )
( )( )
( )( ) ( ) ( )
:

(k) By ( )( )
( )( )
( )( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
and ( ) ( ) ( ) ( )( )
and

( ̅ )( )
( ̅ )( )
( ̅ )( ) :

By ( ̅ )( )
( ̅ )( )
( ̅ )( )
the
( ) ( ) ( ) ( ) ( )
and ( )( ) ( ( ) ) ( )( ) ( ) ( )( )

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

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

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

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

and ( )( )

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

and analogously

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

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

and ( )( )

( )( ) ( )( ) ( )( )
( )( )
( ̅ )( )
( )( )
where ( )( )
( ̅ )( )



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

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

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

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

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

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

124

Vol.2, No.4, 2012

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

( )( )
( )( )
( )( ) :-

Where ( )( )
( )( ) ( )( )
( )( )

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

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

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

Proof : From GLOBAL EQUATIONS we obtain
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )

From which one obtains

( )( ) :-

(a) For ( )( )
( )( )
( ̅ )( )

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

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

In the same manner , we get

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

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

(b) If ( )( )
( )( )
( ̅ )( )
we find like in the previous case,

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

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

125

Vol.2, No.4, 2012

(c) If ( )( )
( ̅ )( )
( )( )
, we obtain

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

And so with the notation of the first part of condition (c) , we have
( )
Definition of ( ) :-

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

In a completely analogous way, we obtain
( )

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

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

Particular case :

If ( )( ) ( )( )
( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( )
( )( ) then ( ) ( ) ( )
( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )( ) for the
special case

Analogously if ( )( )
( )( )
( )( )
( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( ) ( ) This is an important
consequence of the relation between ( )( ) and ( ̅ )( )
and definition of ( )( )

we obtain
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )


( )( ) :-

(d) For ( )( )
( )( )
( ̅ )( )

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

126

Vol.2, No.4, 2012

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


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

( ) ( ̅ )( )

(e) If ( )( )
( )( )
( ̅ )( )
[ ( )( ) (( )( ) ( )( ) ) ]
( )( ) ( )( ) ( )( )
( )( )
[ ( )( ) (( )( ) ( )( ) ) ]
( )
( )
( )( )

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

(f) If ( )( )
( ̅ )( )
( )( )
, we obtain

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

( )

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

( )

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

Particular case :

If ( )( ) ( )( )
( )( ) ( )( ) and in this case ( )( )
( ̅ )( ) if in addition ( )( )

( )( ) then ( ) ( ) ( )
( ) and as a consequence ( ) ( )( )
( )

( )( )
( )( )
( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( )
( ) This is an important

From GLOBAL EQUATIONS we obtain
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

127

Vol.2, No.4, 2012

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )


(a) For ( )( )
( )( )
( ̅ )( )

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

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


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

Definition of ( ̅ )( ) :-

( ) ( ̅ )( )

(b) If ( )( )
( )( )
( ̅ )( )

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

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

(c) If ( )( )
( ̅ )( )
( )( )
, we obtain
[ ( )( ) ((̅ )( ) (̅ )( ) ) ]
(̅ )( ) ( ̅ )( ) (̅ )( )
( )( ) ( )
( ) [ ( )( ) ((̅ )( ) (̅ )( ) ) ]
( )( )
( ̅ )( )

( )

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

( )

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

theorem.

128

Vol.2, No.4, 2012

Particular case :

If ( )( ) ( )( )
( )( ) ( )( ) and in this case ( )( )

( )( ) then ( ) ( ) ( )
( )

( )( )
( )( )
( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then ( ) ( )( )
( ) This is an important

: From GLOBAL EQUATIONS we obtain
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( ) ) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )

( )( ) :-

(d) For ( )( )
( )( )
( ̅ )( )

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

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


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

( ) ( ̅ )( )

(e) If ( )( )
( )( )
( ̅ )( )

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

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

(f) If ( )( )
( ̅ )( )
( )( )
, we obtain

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


129

Vol.2, No.4, 2012

( )

( )
( )( ) ( )
( ) ( )( ) , ( )
( )
( )
Definition of ( ) ( ) :-

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

theorem.

Particular case :

If ( )( ) ( )( )
( )( ) ( )( ) and in this case ( )( )
( )( ) then ( ) ( ) ( )
( ) this also defines ( )( ) for
the special case .

Analogously if ( )( ) ( )( ) ( )( ) ( )( ) and then
( ) ( ) ( )
( ) ( ̅ ) if in addition ( ) ( )( ) then ( ) ( )( ) ( ) This is an important
consequence of the relation between ( ) and ( ̅ ) and definition of ( )( )
( ) ( )

From GLOBAL EQUATIONS we obtain
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )


( )( ) :-

(g) For ( )( )
( )( )
( ̅ )( )

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

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


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

( ) ( ̅ )( )

130

Vol.2, No.4, 2012

(h) If ( )( )
( )( )
( ̅ )( )

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

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

(i) If ( )( )
( ̅ )( )
( )( )
, we obtain

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


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

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

theorem.

Particular case :

If ( )( ) ( )( )
( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( )
( )( ) then ( ) ( ) ( )
( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )( ) for
the special case .

( ) ( ) ( )
consequence of the relation between ( ) and ( ̅ ) and definition of ( )( )
( ) ( )

we obtain
( )
( )( )
(( )( )
( )( )
( )( ) ( )) ( )( ) ( ) ( )
( )( ) ( )

( ) ( )
Definition of :-

It follows
( )
(( )( ) ( ( )
) ( )( ) ( )
( )( ) ) (( )( ) ( ( )
) ( )( ) ( )
( )( ) )


131

Vol.2, No.4, 2012

( )( ) :-

(j) For ( )( )
( )( )
( ̅ )( )

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

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


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

( ) ( ̅ )( )

(k) If ( )( )
( )( )
( ̅ )( )

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

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

(l) If ( )( )
( ̅ )( )
( )( )
, we obtain

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


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

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

theorem.

Particular case :

If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( )
( ) ( ) ( )
( ) then ( ) ( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )( ) for
the special case .
( ) ( ) ( )
consequence of the relation between ( )( ) and ( ̅ )( ) and definition of ( )( )

132

Vol.2, No.4, 2012

We can prove the following

Theorem 3: If ( )( )
( )( ) are independent on , and the conditions

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

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

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

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

( )( )
( )( ) as defined, then the system

If ( )( )

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

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

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

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

( )( )
( )( ) as defined are satisfied , then the system

If ( )( )

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

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

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

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

( )( )
( )( )
as defined are satisfied , then the system

If ( )( )

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

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

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

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

( )( )
( )( )

If ( )( )

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

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

133

Vol.2, No.4, 2012

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

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

( )( )
( )( ) as defined satisfied , then the system

If ( )( )

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

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

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

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

( )( )
( )( )

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

134

Vol.2, No.4, 2012

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

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

has a unique positive solution , which is an equilibrium solution

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

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

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

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

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


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

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

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

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

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

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


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

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

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

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

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

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

135

Vol.2, No.4, 2012


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

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


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


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


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


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


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

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( ) ( ) being increasing, it follows that there
exists a unique for which ( ) . With this value , we obtain from the three first equations

136

Vol.2, No.4, 2012

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



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



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



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



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



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

(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 ( )

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

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

137

Vol.2, No.4, 2012

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

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

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

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

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

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

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

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

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


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

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

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


(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
exists a unique such that ( )

Finally we obtain the unique solution of 89 to 94

( ) , ( ) and

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

138

Vol.2, No.4, 2012

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

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution

(( )) , ( ) and

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

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



(( )) , ( ) and

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

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



( ) , ( ) and

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

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



(( )) , ( ) and

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

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



(( )) , ( ) and

139

Vol.2, No.4, 2012

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

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


ASYMPTOTIC STABILITY ANALYSIS

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 :-

,

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

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

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

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

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

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

If the conditions of the previous theorem are satisfied and if the functions ( )( )
( )( )
Belong to
( )
( ) then the above equilibrium point is asymptotically stable

Denote

Definition of :-

,

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

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

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

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

140

Vol.2, No.4, 2012

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

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

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

Denote

Definition of :-

,

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


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

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

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

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

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

Denote

Definition of :-

,

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


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

141

Vol.2, No.4, 2012

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

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

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

( )( )
Belong to
( )

Denote

Definition of :-

,

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


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

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

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

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

( )( )
Belong to
( )

Denote

Definition of :-

,

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

142

Vol.2, No.4, 2012

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

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

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

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

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

(( )( )
( )( ) ) ( )( )
∑ ( ( )( ) )

The characteristic equation of this system is

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

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

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

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

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

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

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

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

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

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

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

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

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

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

143

Vol.2, No.4, 2012

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

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

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

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

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

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

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

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

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

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

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

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

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

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

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

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

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

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

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

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

144

Vol.2, No.4, 2012

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

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

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

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

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

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

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

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

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

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

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

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

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

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

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

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

+
(( )( )
( )( )
( )( ) ) (( )( )
( )( )
( )( ) )

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

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

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

145

Vol.2, No.4, 2012

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

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

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

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

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

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

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and this
proves the theorem.

Acknowledgments:

The introduction is a collection of information from various articles, Books, 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, trepidation 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

REFERENCES

1. 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 birthday

2. FRTJOF CAPRA: “The web of life” Flamingo, Harper Collins See "Dissipative structures” pages 172-
188

3. 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.net

4. 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:10.1029/2005JD006097

5. 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, 980-1006

6. 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

7. Paul Allen Tipler, Ralph A. Llewellyn (2003-01), Modern Physics, W. H. Freeman and Company,

146

Vol.2, No.4, 2012

pp. 87–88, ISBN 0-7167-4345-0
8. 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.
9. F. Fernflores. The Equivalence of Mass and Energy. Stanford Encyclopedia of Philosophy
10. Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0
11. Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8
12. Hans, H. S.; Puri, S. P. (2003). Mechanics (2 ed.). Tata McGraw-Hill. p. 433. ISBN 0-07-047360-
9., Chapter 12 page 433
13. E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. Freeman and Co., NY. 1992.ISBN 0-7167-
2327-1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclear bombs, until
heat is allowed to escape.
14. Mould, Richard A. (2002). Basic relativity (2 ed.). Springer. p. 126. ISBN 0-387-95210-1., Chapter 5
page 126
15. 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
16. Cockcroft-Walton experiment
17. Earth's gravitational self-energy is 4.6 × 10-10 that of Earth's total mass, or 2.7 trillion metric tons.
Citation: The Apache Point Observatory Lunar Laser-Ranging Operation (APOLLO), T. W. Murphy, Jr. et
al. University of Washington, Dept. of Physics (132 kB PDF.).
18. There is usually more than one possible way to define a field energy, because any field can be made to
couple to gravity in many different ways. By general scaling arguments, the correct answer at everyday
distances, which are long compared to the quantum gravity scale, should be minimal coupling, which means
that no powers of the curvature tensor appear. Any non-minimal couplings, along with other higher order
terms, are presumably only determined by a theory of quantum gravity, and within string theory, they only
start to contribute to experiments at the string scale.
19. G. 't Hooft, "Computation of the quantum effects due to a four-dimensional pseudoparticle", Physical
Review D14:3432–3450 (1976).
20. A. Belavin, A. M. Polyakov, A. Schwarz, Yu. Tyupkin, "Pseudoparticle Solutions to Yang Mills
Equations", Physics Letters 59B:85 (1975).
21. F. Klinkhammer, N. Manton, "A Saddle Point Solution in the Weinberg Salam Theory", Physical
Review D 30:2212.
22. Rubakov V. A. "Monopole Catalysis of Proton Decay", Reports on Progress in Physics 51:189–241
(1988).
23. S.W. Hawking "Black Holes Explosions?" Nature 248:30 (1974).
24. Lev B.Okun, The concept of Mass, Physics Today 42 (6), June 1969, p. 31–36, http://www. Physics
today . org/vol-42/iss-6/vol42no6p31_36.pdf
25. Max Jammer (1999), Concepts of mass in contemporary physics and philosophy, Princeton University
Press, p. 51, ISBN 0-691-01017-X

147

Vol.2, No.4, 2012

26. Eriksen, Erik; Vøyenli, Kjell (1976), "The classical and relativistic concepts of mass",Foundations of
Physics (Springer) 6: 115–124, Bibcode 1976FoPh....6..115E,DOI:10.1007/BF00708670
27. 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
28. 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
29. (Miller, Arthur I. (1981), Albert Einstein's special theory of relativity. Emergence (1905) and early
interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2
30. Darrigol, O. (2005), "The Genesis of the theory of relativity." (PDF), Séminaire Poincaré1: 1–22
31. Philip Ball (Aug 23, 2011). "Did Einstein discover E = mc2?". Physics World.
32. Ives, Herbert E. (1952), "Derivation of the mass-energy relation", Journal of the Optical Society of
America 42 (8): 540–543, DOI:10.1364/JOSA.42.000540
33. Jammer, Max (1961/1997). Concepts of Mass in Classical and Modern Physics. New York:
Dover. ISBN 0-486-29998-8.
34. Stachel, John; Torretti, Roberto (1982), "Einstein's first derivation of mass-energy equivalence",
American Journal of Physics 50 (8): 760–763, Bibcode1982AmJPh..50..760S, DOI:10.1119/1.12764
35. Ohanian, Hans (2008), "Did Einstein prove E=mc2?", Studies In History and Philosophy of Science
Part B 40 (2): 167–173, arXiv:0805.1400,DOI:10.1016/j.shpsb.2009.03.002
36. Hecht, Eugene (2011), "How Einstein confirmed E0=mc2", American Journal of Physics 79 (6): 591–
600, Bibcode 2011AmJPh..79..591H, DOI:10.1119/1.3549223
37. Rohrlich, Fritz (1990), "An elementary derivation of E=mc2", American Journal of Physics 58 (4):
348–349, Bibcode 1990AmJPh..58..348R, DOI:10.1119/1.16168
38. (1996). Lise Meitner: A Life in Physics. California Studies in the History of Science. 13. Berkeley:
University of California Press. pp. 236–237.
39. "consciousness". Merriam-Webster. Retrieved June 4, 2012.
40. Robert van Gulick (2004). "Consciousness". Stanford Encyclopedia of Philosophy.
41. Farthing G (1992). The Psychology of Consciousness. Prentice Hall. ISBN 978-0-13-728668-3.
42. John Searle (2005). "Consciousness". In Honderich T. The Oxford companion to philosophy. Oxford
University Press. ISBN 978-0-19-926479-7.
43. Susan Schneider and Max Velmans (2008). "Introduction". In Max Velmans, Susan Schneider. The
Blackwell Companion to Consciousness. Wiley. ISBN 978-0-470-75145-9.
44. Güven Güzeldere (1997). Ned Block, Owen Flanagan, Güven Güzeldere. ed. The Nature of
Consciousness: Philosophical debates. Cambridge, MA: MIT Press. pp. 1–67.
45. J. J. Fins, N. D. Schiff, and K. M. Foley (2007). "Late recovery from the minimally conscious state:
ethical and policy implications". Neurology 68: 304–307.PMID 17242341.
46. Locke, John. "An Essay Concerning Human Understanding (Chapter XXVII)". Australia: University of
Adelaide. Retrieved August 20, 2010.

148

Vol.2, No.4, 2012

47. "Science & Technology: consciousness". Encyclopedia Britannica. Retrieved August 20, 2010.
48. Samuel Johnson (1756). A Dictionary of the English Language. Knapton.
49. (C. S. Lewis (1990). "Ch. 8: Conscience and conscious". Studies in words. Cambridge University
Press. ISBN 978-0-521-39831-2.
50. Thomas Hobbes (1904). Leviathan: or, the Matter, Forme & Power of a Commonwealth, Ecclesiastical
and Civill. University Press. p. 39.
51. James Ussher, Charles Richard Elrington (1613). The whole works, Volume 2. Hodges and Smith.
p. 417.
52. James Hastings and John A. Selbie (2003). Encyclopedia of Religion and Ethics Part 7. Kessinger
Publishing. p. 41. ISBN 0-7661-3677-9.
53. G. Melenaar. Mnemosyne, Fourth Series. 22. Brill. pp. 170–180.
54. Boris Hennig (2007). "Cartesian Conscientia". British Journal for the History of Philosophy 15: 455–
484.
55. Sara Heinämaa (2007). Consciousness: from perception to reflection in the history of philosophy.
Springer. pp. 205–206. ISBN 978-1-4020-6081-6.
56. Stuart Sutherland (1989). "Consciousness". Macmillan Dictionary of Psychology.
Macmillan. ISBN 978-0-333-38829-7.
57. Justin Sytsma and Edouard Machery (2010). "Two conceptions of subjective
experience". Philosophical Studies 151: 299–327. DOI: 10.1007/s11098-009-9439-x.
58. Gilbert Ryle (1949). The Concept of Mind. University of Chicago Press. pp. 156–163.ISBN 978-0-226-
73296-1.
59. Michael V. Antony (2001). "Is consciousness ambiguous?". Journal of Consciousness Studies 8: 19–44.
60. Max Velmans (2009). "How to define consciousness—and how not to define consciousness". Journal
of Consciousness Studies 16: 139–156.
61. Ned Block (1998). "On a confusion about a function of consciousness". In N. Block, O. Flanagan, G.
Guzeldere. The Nature of Consciousness: Philosophical Debates. MIT Press. pp. 375–415. ISBN 978-0-
262-52210-6.
62. Daniel Dennett (2004). Consciousness Explained. Penguin. pp. 375.

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Vol.2, No.4, 2012

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:drknpkumar@gmail.com

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

150

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