Issuu on Google+

Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

The General Theory Of Einstein Field Equations, Heisenberg’s uncertainty Principle, Uncertainty Principle In Time And Energy, Schrodinger’s Equation And Planck’s Equation- A “Gesellschaft- Gemeinschaft 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: A Theory is Universal and it holds good for various systems. Systems have all characteristics based on parameters. There is nothing in this measurement world that is not classified based on parameters, regardless of the generalization of a Theory. Here we give a consummate model for the well knows models in theoretical physics. That all the theories hold good means that they are interlinked with each other. Based on this premise and under the consideration that the theories are also violated and this acts as a detritions on the part of the classificatory theory, we consolidate the Model. Kant and Husserl both vouchsafe for this order and mind-boggling, misnomerliness and antinomy, in nature and systems of corporeal actions and passions. Note that some of the theories have been applied to Quantum dots and Kondo resonances. Systemic properties are analyzed in detail. Key words Einstein field equations Introduction: Following Theories are taken in to account to form a consummated theory: 1.

Einstein’s Field Equations

The Einstein field equations (EFE) may be written in the form:

where is the Ricci curvature tensor, the scalar curvature, the metric tensor, is the cosmological constant, is Newton's gravitational constant, the speed of light in vacuum, and the stress–energy tensor. 2.

Heisenberg’s Uncertainty Principle

A more formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Kennard later that year (and independently by Weyl in 1928),This essentially implies that the first term namely the momentum is subtracted from the term on RHS with the second term

237


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

on LHS in the denominator

3.

Uncertainty of time and energy

Time is to energy as position is to momentum, so it's natural to hope for a similar uncertainty relation between time and energy. This implies that the first term is dissipated by the inverse of the second term what with the Planck’s constant h is involved (ΔT) (ΔE) ≥ ℏ/2

4.

Schrödinger’s equation:

Time-dependent equation The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time: Time-dependent Schrödinger equation (general)

where Ψ is the wave function of the quantum system, i is the imaginary unit, ħ is the reduced Planck constant, and is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.LHS is subtrahend by the RHS .Model finds the prediction value for the term on the LHS with imaginary factor and Planck’s constant (5)Planck’s Equations: Planck's law describes the amount of electromagnetic energy with a certain wavelength radiated by a black body in thermal equilibrium (i.e. the spectral radiance of a black body). The law is named after Max Planck, who originally proposed it in 1900. The law was the first to accurately describe black body radiation, and resolved the ultraviolet catastrophe. It is a pioneer result of modern physics and quantum theory. In terms of frequency ( ) or wavelength (λ), Planck's law is written:

Where B is the spectral radiance, T is the absolute temperature of the black body, kB is the Boltzmann constant, h is the Planck, and c is the speed of light. However these are not the only ways to express the law; expressing it in terms of wave number rather than frequency or wavelength is also common, as are expressions in terms of the number of photons emitted at a certain wavelength, rather than energy emitted. In the limit of low frequencies (i.e. long wavelengths), Planck's law becomes the Rayleigh–Jeans law, while in the limit of high frequencies (i.e. small wavelengths) it tends to the Wien. Again there are constants involved and finding and or predicting the value of RHS and LHS is of great practical importance. We set to out do that in unmistakable terms.

238


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

Einstein Field Equations: Module Numbered One NOTATION : : Category One Of The First Term : Category Two Of The First Term : Category Three Of The First Term : Category One Of The Second Term : Category Two Of The Second Term :Category Three Of The Seond Term Einstein Field Equations(Third And Fourth Terms):Module Numbered Two : Category One Of The Third Term In Efe : Category Two Of The Third Term In Efe : Category Three Of The Third Term In Efe :Category One Of The Fourth Term In Efe : Category Two Of The Fourth Term In Efe : Category Three Of The Fourth Term On Rhs Of Efe Heisenberg’s Uncertainty Principle: Module Numbered Three : Category One Of lhs In The Hup(Note The Position Factor Is Inversely Proportional To The Momentum Factor) :Category Two Of Lhs In Hup : Category Three Of Lhs In Hup : Category One Of Rhs(Note The Momentum Term Is In The Denominator And Hence Rhs Dissipates Lhs With The Amount Equal To Plack Constant In The Numerator And Twice Of The Momentum Factor) : Category Two Of Rhs (We Are Talking Of The Different Systems To Which The Model Is Applied And Systems Therefore Are Categories. To Give A Bank Example Or That Of A Closed Economy The Total Shall Remain Constant While The Transactions Take Place In The Subsystems) : Category Three Ofrhs Of Hup(Same Bank Example Assets Equal To Liabilities But The Transactions Between Accounts Or Systems Take Place And These Are Classified Notwithstanding The Universalistic Law) Uncertainty Of Time And Energy(Explanation Given In hup And Bank’s Example Holds Good Here

239


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

Also): Module Numbered Four: : Category One Of Lhs Of Upte : Category Two Of Lhs In Upte : Category Three Of Lhs In Upte :Category One Of Rhs In Upte :Category Two Of Rhs In Upte : Category Three Of Rhs In Upte Schrodinger’s Equations(Lhs And Rhs) Same Explanations And Expatiations And Enucleation Hereinbefore Mentioned Hold Good: Module Numbered Five: : Category One Of Lhs Of Se : Category Two Of lhs Of Se :Category Three Of Lhs Of Se :Category One Of Rhs Of Se :Category Two Of Rhs Of Se :Category Three Of Rhs In Se Planck’s Equation: Module Numbered Six: : Category One Of Lhs Of Planck’s Equation : Category Two Of Lhs Of Planck’s Equation : Category Three Oflhs Of Planck’s Equation : Category One Of Rhs Of Planck’s Equation : Category Two Of Rhs Of Planck’s Equation : Category Three Of Rhs Of Planck’s Equation (

)(

)

(

)(

)

(

)(

)

(

)(

(

)(

)

(

)(

)

(

)( ) : (

)(

(

)(

)

(

)(

)

(

)(

)

(

)(

,(

)(

)

(

)(

)

(

)(

)

(

)(

)

(

)(

)

(

)(

)

(

)(

)

(

) ) )

)(

)( ) (

( (

)(

(

)(

)

(

)(

)

(

)

)(

)

)

)(

)

)

(

)(

( (

)(

(

)(

)

)

( )(

(

)(

(

)(

)

)

(

)(

)

)

)(

( )(

(

)

)

)(

)

are Accentuation coefficients )(

( (

)

)(

(

)(

(

)(

( )

) )

(

)( )(

)

( )

)(

(

)(

) ) )

)(

( (

(

)(

)

(

)(

(

)(

)

(

)( ) , (

(

)

)( )(

)

)(

)(

( )

(

)

)( )(

( (

)

)

)(

)(

( )

(

)

)( )(

( (

)

)(

( )

)

(

)(

)

(

)( )(

(

240

)

)(

( )

)

(

)(

)

(

)( )(

(

)

)(

( )

)

(

)(

)

(

)( )(

(

) )

)

)(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

are Dissipation coefficients Einstein Field Equations: Module Numbered One The differential system of this model is now (First Two terms in EFE) Governing Equations (

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)( ) (

(

)( ) (

) )

First augmentation factor First detritions factor

Einstein Field Equations(Third And Fourth Terms):Module Numbered Two Governing Equations The differential system of this model is now (

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)( ) (

(

)( ) ((

)

First augmentation factor

) )

First detritions factor

Heisenberg’s Uncertainty Principle: Module Numbered Three Governing Equations The differential system of this model is now (

)(

)

[(

)(

)

(

)( ) (

)]

241


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)( ) (

)

First augmentation factor

(

)( ) (

)

First detritions factor

Uncertainty Of Time And Energy(Explanation Given In hup And Bank’s Example Holds Good Here Also): Module Numbered Four Governing Equations:: The differential system of this model is now (

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)( ) (

)

(

)( ) ((

) )

First augmentation factor First detritions factor

Governing Equations: Schrodinger’s Equations(Lhs And Rhs) Same Explanations And Expatiations And Enucleation Hereinbefore Mentioned Hold Good: Module Numbered Five The differential system of this model is now (

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

242


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)( ) (

)

(

)( ) ((

) )

First augmentation factor First detritions factor

Planck’s Equation: Module Numbered Six Governing Equations:: The differential system of this model is now (

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)(

)

[(

)(

)

(

)( ) ((

) )]

(

)( ) (

)

(

)( ) ((

) )

First augmentation factor First detritions factor

Holistic Concatenated Sytemal Equations Henceforth Referred To As “Global Equations” (1) (2) (3) (4) (5) (6)

Einstein Field Equations(First Term and Second Term) Einstein Field Equations(Third and Fourth Terms) Heisenberg’s Principle Of Uncertainty Uncertainty of Time and Energy Schrodinger’s Equations Planck’s Equation.

(

)(

)

(

)(

)

[

[

)(

(

)( (

)

)(

)( ) (

(

)(

( (

)

)

(

)

)( ) (

( )

(

)

)

(

(

)(

)

(

(

)(

)(

) )

)(

243

( )

)

(

(

) )

(

(

)

( (

)

)(

( )

)(

)( )(

( )

)

) (

( )

) )

(

)

]

]


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)(

(

Where ( 1, 2 and 3

)

[

)( ) (

)(

(

)

)

)(

(

)( ) (

)

( )( ) ( ) , category 1, 2 and 3

(

)(

)

( )( ) ( ) category 1, 2 and 3

(

)(

)

) ( )( ( ) , for category 1, 2 and 3

)( ) (

(

(

(

(

)

(

)(

)

)

(

)(

)

(

(

)

( )

)

( )

(

Where and 3

[

[

)(

)

)(

(

)

(

(

)

)

)

(

)(

)

( )( ) ( 1, 2 and 3

(

)(

)

(

)(

)

) (

)

]

(

) are second augmentation coefficient for

(

) are third augmentation coefficient for

)(

(

)

(

) are fourth augmentation coefficient

(

) are fifth augmentation coefficient

) ,

(

)(

)

) are sixth augmentation coefficient

)

)(

( (

)

)

(

)(

)(

)

(

) –(

(

) –(

)(

(

)

(

)(

) –(

)(

)

(

(

)

(

)

(

(

)

)

(

)(

)

)(

)(

(

(

)

( (

) )

( (

)

)( ) (

(

(

)

)

(

)

)

)(

)( ) (

(

(

)(

(

(

)

(

)

)

)( ) (

(

( )( ) ( ) category 1, 2 and 3 )

(

) ,

)( ) ( )

)(

(

(

(

)(

(

[

)( ) (

)

(

( (

)

(

(

) are first augmentation coefficients for category

)

) ( )( ( ), for category 1, 2 and 3

)

)( ) (

(

) )

)(

)

(

)(

(

)(

)(

(

) ,

(

(

(

(

) ( )( ( ) for category 1, 2 and 3

( )

)(

) )

)

)(

(

www.iiste.org

)

)(

) )

)

(

(

) )

)

)

(

(

) )

)

(

)

]

]

]

)

(

)( ) (

)

(

)(

)

(

) are second detritions coefficients for

)

(

)(

)

) are third detritions coefficients for category

(

) are first detrition coefficients for category 1, 2

(

) ( )( ( ) for category 1, 2 and 3

(

)(

)

(

)

(

)(

)

) ( )( ( category 1, 2 and 3

) ,

(

)(

)

(

) ,

(

)(

)

(

) are fifth detritions coefficients for

) ,

(

)(

)

(

) ,

(

)(

)

(

) are sixth detritions coefficients for

(

)(

)

(

244

(

) are fourth detritions coefficients


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

category 1, 2 and 3

(

)(

)

(

)(

)

(

)(

)

[

)(

(

[

[

)(

)(

(

)(

)

)

)

(

( )( ) ( ) , category 1, 2 and 3

(

)(

(

(

)

(

)(

) ,

)

(

)(

)(

)

(

(

)(

)(

)

)

(

)(

)(

)

(

)(

) ( )( ( ) , ( )( coefficient for category 1, 2 and 3

)

(

) ,

(

) ( )( ( ) , ( )( coefficient for category 1, 2 and 3

)

) ,

(

(

)(

)

(

)(

)

[

[

[

where ( )( ) ( category 1, 2 and 3 ( )( ) ( 1,2 and 3

)

(

)(

(

)(

(

)(

(

)(

(

)(

(

)(

t) ,

(

)(

)

)( ) (

( )

)

) )( ) (

( )

)

(

( )( ) (

( )

(

(

)( ) (

)

) ,

(

)

)

)

(

(

)(

)

(

(

)(

)

(

(

)(

t) ,

(

)(

)

(

)

(

)(

(

)

(

)(

(

)

(

)(

)

(

)

(

)(

(

)

(

)(

)

(

) )

)

(

(

) )

)

)

(

)

(

) )

(

)

]

]

]

) are second augmentation coefficient for

)

)

)(

) are first augmentation coefficients for

(

(

)(

(

)

)

(

)

) )

) ( )( ( ) ( )( coefficient for category 1, 2 and 3

(

(

)

)( ) (

(

(

)

)(

(

) )

(

( (

)

)( ) (

)

(

)

)( ) (

(

)(

) )

)( ) (

(

)(

(

(

)

)

(

Where ( )( ) ( category 1, 2 and 3

) ( )( ( ) category 1, 2 and 3

)( ) (

(

)(

( (

)

(

) are third augmentation coefficient for

)

(

) are fourth augmentation

)(

)

) are fifth augmentation

)(

)

)(

) )

)(

(

(

)( )

(

( )

(

(

) are sixth augmentation

) –(

(

)(

(

)(

) –(

)(

)

)(

)

) )

(

(

) –( )

(

)( )(

)

(

) )

)

(

) )

)

)

(

(

(

) )

)

(

)

]

]

]

(

)( ) (

(

) are second detrition coefficients for category

245

t) are first detrition coefficients for


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

) ( )( ( ) category 1,2 and 3

)(

(

)

(

) ( )( ( ) ( )( coefficients for category 1,2 and 3 ) ( )( ( ) , for category 1,2 and 3

)

(

)

(

)(

)

(

)(

)

[

[

[

(

)(

(

)(

(

)(

(

)(

(

)(

(

(

( )( ) ( ), ( category 1, 2 and 3

)

)

)

)

)( ) (

)(

) ( )( ( ) ( for category 1, 2 and 3

)(

)(

(

) ,

(

)(

(

(

)(

(

(

(

)(

)

(

)

(

)

)( ) (

(

(

) ,

(

)(

)

)

(

)(

)

(

) ( )( ( ) , ( )( coefficients for category 1, 2 and 3 ) ( )( ( ) ( )( coefficients for category 1, 2 and 3 ) ( )( ( ) ( )( coefficients for category 1, 2 and 3

)

)

(

(

)

(

)

) are fifth detritions coefficients

(

) are sixth detritions

)

(

)

(

)(

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

(

(

)

)(

)

)

),

(

)

(

(

) are fourth detritions

)

)(

)

(

)(

) )

)( ) (

(

)

) ( )( ( ) ( for category 1, 2 and 3

(

(

) are third detrition coefficients for

)

) ,

)( ) (

(

(

)(

(

)( ) ( )

)

)

)

(

(

)

)(

(

(

)

) ( )( ( ) ( )( coefficients for category 1,2 and 3

( )

)

)(

(

www.iiste.org

(

)

)

)

(

) )

)

(

) )

)

)

(

(

(

) )

)

(

)

]

]

]

) are first augmentation coefficients for

)

(

(

) are second augmentation coefficients

)

) are third augmentation coefficients

(

)(

)

)

(

)(

)

)

(

)(

)

246

(

(

(

) are fourth augmentation

) are fifth augmentation

) are sixth augmentation


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

)

(

)(

)

(

)(

)

[

)

)

)

)

)( ) (

(

)(

)(

(

)( ) (

(

)(

) –( )

(

)(

)

(

) –(

)(

(

)

(

)(

) –(

)(

)(

)

)(

(

(

(

)

) –( )

)

)

) –( )

)

)(

(

)( ) (

(

)(

(

)

(

)(

) –(

)( ) ( )

)(

(

(

(

)(

(

(

)( ) (

( )

)(

(

[

)

)(

(

[

)( ) (

(

)(

(

www.iiste.org

)

(

(

)

)

) )

)

(

(

)

) )

)

)(

(

(

(

(

) )

)

(

)

]

]

]

) 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

(

)(

)

(

)(

)

(

)(

)

)(

(

(

(

)

)

)

)

)(

)( ) (

)

)

(

)

)( ) (

(

(

( )( ) (

(

)(

(

[

)

)(

(

)

)(

(

[

)( ) (

(

)(

(

[

)

(

)

)( ) (

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)(

(

)

(

)(

)

(

)

(

) )

)(

( )

)

)(

(

) )

(

( )

)

(

)

( )

)(

(

)(

( )

)(

( (

)

)(

(

)

)(

(

)(

( )

)

(

( )

)

) ) )

(

)

(

)

)

(

)

]

]

]

)

(

)

)(

)(

(

)

(

) )

(

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

247


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

)

(

),

(

)(

)

(

)(

)

(

),

(

)(

)

)

( )

(

)

( )

(

)(

(

)( ) (

(

[

[

)

[

)(

(

(

)(

)

),

(

)(

)

)( ) ( )

)(

)

)

)( ) (

)

)

)( ) (

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

)

(

(

)(

)

)

(

)(

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

) ,

(

)(

(

)(

)

(

),

(

)(

)

–(

)(

)

(

) –(

)(

)

)(

(

)

)(

(

)

[

(

)

(

)( ) (

(

)(

(

)(

)

(

)(

)

(

( )

)

(

)

) –(

)(

)

)

)

)(

)( )

)(

)

(

(

) )

(

(

(

)

(

)(

)

(

(

)(

)

(

)

(

)(

)

)

(

)( ) (

)

(

)(

( )

)( (

)

)

)(

(

)(

) –(

(

)(

) –( )

(

)(

) –(

(

)(

(

)

(

(

)

)(

) –( )

)

)

)( ) ( )

)( ) (

)

)

(

)(

(

)

(

)( ) ( )

(

)(

)(

) –(

)(

) –(

(

)

)

(

)

)

(

)

)

(

)

) )

(

)

(

)

]

]

]

) )

(

(

)

(

)( ) (

(

)(

(

)( (

(

)(

( (

)(

(

(

)

(

)(

[

)

),

(

( ( )

(

)(

(

[

)

)

(

)

)(

(

)

)(

(

)

)(

(

)

(

(

(

)

(

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

) )

(

(

)( ) (

( )(

(

(

)( ) (

( )(

(

(

)(

(

)

(

)

(

)(

( (

)

(

)

www.iiste.org

(

) )

)(

(

)

(

)(

(

)

(

)(

)

(

)

(

)(

(

)

(

)(

)

(

)

(

)(

(

)

(

)(

)

)(

)

)

(

) )

)

(

(

) )

)

]

)

(

(

)

]

) )

(

)

]

) ) )

(

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

3

248


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

)

(

)

(

)(

)

(

)(

)

(

)

(

)(

)

(

)

( )

[

( )

(

)

(

)(

)( ) (

–( )

)(

(

)

)(

(

)(

)

(

(

)(

)

(

)

(

)(

)

–(

)(

)

(

) , –(

)(

)

(

)

(

)(

)

(

)(

)(

)

)( ) (

(

) )

(

)

( (

)

)

)( ) (

(

) )

(

)( ) ( )

)( ) ( )

(

(

(

)

)(

)

)(

( (

(

)

)

)

)(

)

)(

(

(

(

(

)(

(

)

)

)(

(

[

(

)

)(

(

[

)

)(

(

(

www.iiste.org

(

)

)

(

(

)(

)

)

(

(

(

)(

)

(

(

)(

)

(

(

)(

)

( )

( )

(

) –(

)(

)

(

)(

(

)(

(

)(

(

)(

(

)(

)(

( )

)

) –( (

) )

) –( (

) –(

( )

)(

)

)

)(

(

) )

)

)(

)

(

)(

) –(

(

)

)(

) –(

(

)(

)(

) –(

(

)

( )

) (

)

)( ) (

(

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

)

(

(

)(

)

(

(

)(

)

(

(

)(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

)

)

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)

(

)(

(

)(

)

)

(

)(

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

) - are fourth augmentation coefficients

(

)(

)

(

)

(

)(

)

(

)

(

)(

)

(

) - fifth augmentation coefficients

(

)(

)

(

),

(

)(

)

)

(

)(

)

) sixth augmentation coefficients

(

[

[ )( ) (

(

) )

( )

(

)

)

)

(

)

)( ) (

(

(

)

)( ) (

( )

(

)

)( ) (

( ) )

(

)

)(

(

)(

)(

)

)

(

( )

(

( )

(

)

(

249

) )

) (

) )

)

)

(

) )(

(

]

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

)(

)

]

(

(

[

]

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

)(

(

)

(

)

(

)

)

)

)

)(

)(

)

(

(

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

)

)(

(

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

(

)( ) (

)(

(

(

)

]

]

]


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

)

(

(

)

(

)(

(

(

)

(

)(

(

(

)(

)

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)

(

)(

(

)(

)

)

(

)(

(

)(

)

(

(

)(

)

(

),

(

)(

)

–(

)(

)

(

) , –(

)(

)

[

[

[

(

)

)

)

)

( )

(

)

)( ) (

)

)

(

)(

)

(

)(

)

(

(

( )

)

(

) )

(

(

(

) –(

)( )

) –(

)(

)

)

) –(

)( (

)

( )(

)

( )(

(B)

The functions (

Definition of ( )( (

)( ) (

(

)( ) (

)

li

(

Definition of ( ̂

)(

)

( )( ( )(

)( ) ( )( ) ( )(

Where ( ̂

)

)

(

)

(

)(

)

(

) –(

)(

)

)

( )(

)

)

)(

( )(

) ( ̂

)

( )(

)

( ̂

)(

(

)

)

( ̂

)(

( )(

)

)

)

)

)( ) : )(

)

( )(

)

are positive constants and

They satisfy Lipschitz condition: (

)( ) (

)

(

)( ) (

)

)(

]

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

( ̂ )

)

(

)

( )(

(

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

)( ) are positive continuous increasing and bounded.

)

) )

]

(

(

)(

(

)

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

(

( )(

(

]

)

( )( ) :

)

(

)(

(

)

(

(C)

)

) )

)

)

(

(

(

) –(

) )

Where we suppose (A)

(

(

) –(

) –(

)

)

)(

(

)(

(

(

) )

)

) –(

(

)

)(

(

(

) )

)

) –(

)

(

(

(

(

( )

)

)( ) (

(

) )

(

)(

) –(

)( ) (

(

)

)(

(

)

)

)( ) (

(

)( ) (

(

(

www.iiste.org

( ̂

)

250

)( )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)( ) (

)

)( ) (

(

)

www.iiste.org

)(

)( )

( ̂

)

With the Lipschitz condition, we place a restriction on the behavior of functions ) and( )( ) ( ) ( ) and ( ) are points belonging to the interval ( )( ) ( ( ) ( ) ( ) ̂ ̂ ) is uniformly continuous. In the eventuality of the [( ) ( ) ] . It is to be noted that ( ) ( ) , the first augmentation coefficient WOULD be fact, that if ( ̂ )( ) then the function ( )( ) ( absolutely continuous. Definition of ( ̂ ( ̂

(D)

)(

)( )

( )( ) ( ̂ )( )

)

)(

)( ) : )

are positive constants

( )( ) ( ̂ )( )

Definition of ( ̂ )(

( ̂

)

)( ) :

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

(E)

)

)( ) which together )( ) and the constants

satisfy the inequalities ( ̂

)( )

( ̂

)( )

( )(

)

( )(

)

( ̂

)(

)

( ̂ )( ) ( ̂

( )(

)

( )(

)

)(

)

( ̂

)(

)(

)

)

)(

)

Where we suppose (F)

( )(

)

( )(

(G)

The functions (

Definition of ( )(

(H)

)

)

)(

(

)(

)

)

( )(

)( ) (

)

(

)( ) (

)

( )(

( ̂

)

( )(

)

)( ) (

li

(

)( ) ((

) )

( ̂

)( ) :

)

)(

)

( ̂

)(

( )(

)

)

( )(

)

( )(

(

)(

)

)(

(

)

)( ) are positive continuous increasing and bounded.

(

li

Where ( ̂

( )(

(r )( ) :

(

Definition of ( ̂

)

)

( )

( ̂

)

)(

)

)

( )(

)

( )( ) are positive constants and

They satisfy Lipschitz condition: (

)( ) (

)

(

)( ) ((

)

( )

)( ) ( (

)( ) ((

)

(̂ ) )

)( (̂

)

)(

)

(

)

( ̂

)( )

(

)

( ̂

)( )

) With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ( ) ( ) ̂ ̂ ) .( ) and ( ) are points belonging to the interval [( and( ) ( ) ( )( ) ] . It is to ( ) ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) be noted that ( ) (

251


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)( ) (

then the function ( Definition of ( ̂ ( ̂

(I)

)(

( )( ) ( ̂ )( )

)( (̂

)

) , the SECOND augmentation coefficient would be absolutely continuous.

)

www.iiste.org

)( ) :

)(

)

are positive constants

( )( ) ( ̂ )( )

Definition of ( ̂ )(

( ̂

)

)( ) :

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

)( )

( ̂

)( )

( )(

)

( )(

)

( ̂

)(

)

( ̂ )( ) ( ̂

( )(

)

( )(

)

)(

)

( ̂

)(

)

)(

)

)(

)

Where we suppose ( )(

(J)

)

( )( )(

The functions (

Definition of ( )(

)

)

(

)( ) (

)

)( ) ( )( ) (

Definition of ( ̂ Where ( ̂

)(

)

)(

)

)

( )(

)

( )(

)

(

)(

)

are positive continuous increasing and bounded.

(r )( ) :

)( ) (

(

)(

(

(

(

( li

)

)

( )(

( ̂

)

( )(

)

)

( )(

)

)

( )(

)

)(

)

( ̂ )(

)

)(

)

( ̂

( )(

)

( )(

)

)(

)(

)

)( ) : ( )(

)

are positive constants and

They satisfy Lipschitz condition: (

)( ) (

(

)( ) (

)

)( ) (

(

)

(

)

)( ) (

)

( ̂

)

)(

)( ) ( ̂

)

)( )

) With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) .( ) And ( ) are points belonging to the interval [( ̂ )( ) ( ̂ )( ) ] . It is to and( )( ) ( ) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) be noted that ( )( ) ( ( ) ) , the THIRD augmentation coefficient, would be absolutely continuous. then the function ( ) ( Definition of ( ̂ (K)

( ̂

)(

)( )

)

(̂ )(

)( ) : )

are positive constants

252


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )( ) ( ̂ )( )

www.iiste.org

( )( ) ( ̂ )( )

There exists two constants There exists two constants ( ̂ )( ) and ( ̂ ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) ( ̂ )( ) and the constants ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) satisfy the inequalities ( ̂

)( )

( ̂

)( )

( )(

)

( )(

)

( ̂

)(

)

( ̂

)( ) ( ̂

( )(

)

( )(

)

)(

)

( ̂

)(

)

)(

)( ) which together with

)

)(

)

Where we suppose ( )(

(L)

)

( )(

)

)(

(

)

)(

Definition of ( )(

)

)( ) (

)

(

)( ) ((

) )

Definition of ( ̂ Where ( ̂

)(

)(

)

)

)

)(

(

)

(

)( ) are positive continuous increasing and bounded.

( )(

)

( ̂

( )(

( )( ) ( ( )( ) ((

li

( )(

( )( ) :

(

(N)

)

)

The functions (

(M)

( )(

( )(

( )( ) ) ) ( )(

)

( ̂

)

)(

)

)

( ̂ )(

)

)(

)

)

)( ) : ( )(

)

( )(

)

)(

are positive constants and

They satisfy Lipschitz condition: (

)( ) (

)

(

)( ) ((

)

)( ) (

( )

)

)( ) ((

(

) )

( ̂

)

)(

)

(

)

(

)( )

)

( ̂

)( )

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ and( )( ) (

) )( ) ] . It is to

) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) be noted that ( )( ) ( ( ) ) , the FOURTH augmentation coefficient WOULD be absolutely then the function ( ) ( continuous. Definition of ( ̂ ( ̂

)(

)

( )( ) ( ̂ )( )

)(

)( )

)

)( ) :

are positive constants

( )( ) ( ̂ )( )

253


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

Definition of ( ̂

( ̂

)

)( ) :

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

(O)

( ̂

)(

www.iiste.org

( )(

)( )

( )(

( )(

)( )

( ̂

)

)

( ̂

)

( )(

)( (̂

)

)( ) ( ̂

( ̂

)

)(

( ̂

)

)(

)(

)

)

)(

)

Where we suppose (P)

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

)( ) (

)

(

)( ) ((

) )

(

(R) (

li

Definition of ( ̂ Where ( ̂

)(

( )(

)(

)

)

)

)( ( )(

)

( ̂

)

)(

)

( )( ) ( )( )

)

( ̂

)

)

( )(

)( ) ( )( ) ( )(

( ̂

)

)( ) : ( )(

)

( )(

)

are positive constants and

They satisfy Lipschitz condition: ( (

)( ) (

)

)( ) ((

)

)( ) (

( )

)( ) ((

(

)

) )

)(

)

)(

)

(

)

( ̂

)( )

(

)

( ̂

)( )

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ and( )( ) (

) )( ) ] . It is to

) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) be noted that ( )( ) ( ( ) ) , theFIFTH augmentation coefficient attributable would be absolutely then the function ( ) ( continuous.

(S)

Definition of ( ̂

)(

)(

( ̂

)( ( ( ̂

)

)( )

)( )

Definition of ( ̂ (T)

( ( ̂

)(

) )

)( )

)( ) :

are positive constants

)( ) )

( ̂

)( ) :

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

)( ) and ( ̂ )( ) which together with ( ̂ )( ) and the constants

254


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )( ( ̂

)( )

( ̂

)( )

)

( )(

)

( )(

)

( )(

)

( )(

www.iiste.org

)

( )(

)

satisfy the inequalities

( )(

)

( )(

)

( ̂

)(

)

( ̂

)( ) ( ̂

( )(

)

( )(

)

)(

)

( ̂

)(

)

)(

)

)(

)

Where we suppose ( )( (U)

)

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

)( ) (

)

(

)( ) ((

) )

Definition of ( ̂

)(

Where ( ̂

)(

)

)(

)

( )(

)

( ̂

)(

)

)

( )(

) )

( ̂

)

)

( )(

)

)( ) ((

(

( ̂

)

( )(

)( ) (

( li

( )(

)

)( ) :

)(

)

( )(

)

( )(

)

are positive constants and

They satisfy Lipschitz condition: (

)( ) (

)

(

)( ) ((

)

)( ) (

( )

)( ) ((

(

)

)( (̂

) )

( ̂

)

)(

)

(

)

(

)( )

)

( ̂

)( )

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( ) ( ) .( ) and ( ) are points belonging to the interval [( ̂ )( ) ( ̂ and( )( ) (

) )( ) ] . It is to

) is uniformly continuous. In the eventuality of the fact, that if ( ̂ )( ) be noted that ( )( ) ( ( ) ) , the SIXTH augmentation coefficient would be absolutely continuous. then the function ( ) ( Definition of ( ̂ ( ̂

)(

)

)(

( )( ) ( ̂ )( )

Definition of ( ̂

)( )

)

)( ) :

are positive constants

( )( ) ( ̂ )( )

)(

)

( ̂

)( ) :

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

)( )

( ̂

)( )

( )(

)

( )(

)

( ̂

)(

)

( ̂

)( ) ( ̂

( )(

)

( )(

)

)(

)

( ̂

)(

)

255

)(

)

)(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

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

Definition of ( ̂ )

( )

( ̂

( )

( ) ( ̂

) ( ̂

)(

Definition of ( ̂ )(

( ) ( )

( )

) ( ̂

)( )

) ( ̂

)( )

( ̂

)(

) ( ̂

)( )

( )

( ):

( ) ( ̂

)(

Definition of ( ̂

( )

( ̂

)

)(

Definition of ( ̂

( )

( ̂

)

)(

)( )

( )

( ): )( )

( )

,

( ) ( )

( )

,

)( )

( )

( ):

( )

,

)( )

( )

,

)( )

( )

, ( )

Proof: Consider operator which satisfy ( )

,

,

) ( ̂

) ( ̂

( )

,

)( )

( ) ( ̂

( )

,

) ( ̂

( ) ( ̂

( )

,

)( )

( ̂

( )

( )

) ( ̂

)

( )

,

)(

( ̂

( )

)( )

( ̂

( )

( )

,

)(

Definition of

( )

)( )

( ̂

( ) ( )

( ):

( ̂ )(

( )

:

defined on the space of sextuples of continuous functions ( ̂

)

( )

( ̂ )(

) ( ̂

)( )

( )

( ̂

)(

) ( ̂

)( )

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

)(

)

By

(

256

)( ) (

(

)( ) (

(

(

(

)) ))

(

(

) )) ) ))

( (

(

(

) )] ) )]

(

(

)

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

̅ ( )

∫ [(

̅ ( )

∫ [(

)(

)

(

(

̅ ( )

∫ [(

)(

)

(

∫ [(

)(

)

(

̅ (t) T

T

Where

(

)

)(

)

www.iiste.org

((

)(

))

((

)(

(

))

((

(

))

((

(

))

(

)

(

)( ) (

(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

is the integrand that is integrated over an interval (

))

(

) ))

(

(

) )]

(

(

)

)

Proof: ( )

Consider operator satisfy ( )

:

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

( )

( ̂

)

( )

( ̂ )(

) ( ̂

)( )

( )

( ̂

)(

) ( ̂

)( )

̅ ( )

∫ [(

)(

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

̅ ( )

∫ [(

)(

)

(

(

))

((

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

which

)

By

Where

(

)

)

(

))

(

((

)(

)

)( ) (

(

(

))

(

) ))

(

(

) )]

(

)

(

)( ) (

(

(

))

(

) ))

(

(

) )]

(

)

(

)( ) (

(

(

))

(

) ))

(

(

) )]

(

)

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

is the integrand that is integrated over an interval (

)

Proof: ( )

Consider operator satisfy ( )

:

defined on the space of sextuples of continuous functions ( ̂

( ) ( )

( ̂

)(

) ( ̂

)( )

( )

( ̂

)(

) ( ̂

)( )

∫ [(

)(

)(

( ̂

)

)(

which

)

By ̅ ( )

)

(

(

))

((

)(

)

)( ) (

257

(

(

))

(

) ))

(

(

) )]

(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

̅ ( )

∫ [(

)(

̅ ( )

∫ [(

)(

̅ ( )

∫ [(

)(

)

(

(

̅ ( )

∫ [(

)(

)

(

∫ [(

)(

)

(

̅ (t) T

T

Where

(

)

((

)(

((

)(

))

((

)(

(

))

((

(

))

((

(

)

(

))

(

))

(

)

(

)( ) (

(

(

)( ) (

(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)

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

Consider operator satisfy ( )

)

www.iiste.org

))

(

))

(

(

) ))

(

) ))

(

(

( ̂

)(

( ̂

)

( )

( ̂

)(

) ( ̂

)( )

( )

( ̂

)(

) ( ̂

)( )

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

((

)(

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

̅ ( )

∫ [(

)(

)

(

(

))

((

∫ [(

)(

)

(

(

))

((

)(

(

) )]

(

(

)

)

) :

defined on the space of sextuples of continuous functions

( )

) )]

(

which

)

By

̅ (t) T

T

Where

(

)

(

))

(

(

(

)( ) (

(

(

)( ) (

(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)

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

Consider operator satisfy ( )

)

)( ) (

))

(

))

(

(

) ))

(

))

(

(

) ))

( ̂

( )

( ̂

)(

) ( ̂

)( )

( )

( ̂

)(

) ( ̂

)( )

)(

)

( ̂

)(

By

258

)

(

) ))

(

) )]

(

(

(

) )]

(

) )]

(

(

)

)

)

)

defined on the space of sextuples of continuous functions

( )

(

:

which


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

̅ ( )

∫ [(

)(

)

(

(

))

̅ ( )

∫ [(

)(

)

(

(

))

̅ ( )

∫ [(

)(

)

̅ ( )

∫ [(

)(

)

(

(

̅ ( )

∫ [(

)(

)

(

∫ [(

)(

)

(

̅ (t) T

T

Where

(

)

((

)(

((

)(

)

((

)(

)

))

((

)(

(

))

((

(

))

((

(

))

(

)

)( ) (

(

(

))

(

) ))

(

(

) )]

(

)

(

)( ) (

(

(

))

(

) ))

(

(

) )]

(

)

(

)( ) (

(

(

))

(

) ))

(

(

) )]

(

)

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

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

Consider operator satisfy ( )

www.iiste.org

) :

defined on the space of sextuples of continuous functions ( ̂

( )

)(

( ̂

)

( )

( ̂

)(

) ( ̂

)( )

( )

( ̂

)(

) ( ̂

)( )

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

)

̅ ( )

∫ [(

)(

((

)(

̅ ( )

∫ [(

)(

)

(

(

))

((

)(

̅ ( )

∫ [(

)(

)

(

(

))

((

∫ [(

)(

)

(

(

))

((

)(

which

)

By

̅ (t) T Where

T (

)

( )

(

))

(

)( ) (

(

(

)( ) (

(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)(

)

(

)( ) ( (

(

))

(

) ))

(

(

) )]

(

)

)

(

)(

))

(

))

(

(

) ))

(

))

(

(

(

) ))

(

) ))

(

(

(

(

) )]

(

) )]

(

) )]

(

)

)

)

)

maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it ( ̂ )(

)( ) (

∫ [( (

(

(

is the integrand that is integrated over an interval (

(a) The operator is obvious that ( )

)

)( ) (

)

)

(

) ( ̂

)( ) (

)( ) ( ̂ )( ) ( (̂ ( ̂ )( )

)

)]

)( )

From which it follows that

259

(

)

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( (

( )

)( )

( ̂

)

)( )

(

̂ )

[(( )( )

( ̂

www.iiste.org

( )

(

)

)( )

)

( ̂ )( ) ]

) is as defined in the statement of theorem 1

Analogous inequalities hold also for (b) The operator is obvious that ( ) (

(

( )

maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it

)(

)

)( ) ( ̂ )( ) ( ̂ )( )

(

)

( ̂ )(

)( ) (

∫ [(

(

( ̂

) ( ̂ )( )

)( ) (

)

)]

(

)

)

From which it follows that (

( )

)( )

( ̂

)

(

)( )

( ̂

)(

̂ )( ) [((

)

(

)

)( )

)

( ̂ )( ) ]

Analogous inequalities hold also for (a) The operator is obvious that ( )

( )

maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it

(

(

)(

( ̂

)( ) (

∫ [( )

(

)

)(

) ( ̂

)( ) (

)( ) ( ̂ )( ) ( (̂ ( ̂ )( )

)

)]

)( )

(

)

)

From which it follows that (

( )

( ̂

)

)( )

(

)( )

( ̂

)(

̂ ) [((

)(

)

)

(

)( )

)

( ̂

)( ) ]

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

(

)(

)

(

)

)( ) ( ̂ )( ) ( (̂ ( ̂ )( )

)( )

)

From which it follows that (

( ) (

)

( ̂

)( )

(

)( )

( ̂

)(

̂ ) [((

( )

)

)

(

)( )

)

( ̂

)( ) ]

) is as defined in the statement of theorem 1

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

260


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

(

)

)( ) ( ̂ )( ) ( (̂ ( ̂ )( )

(

)

www.iiste.org

)( )

)

From which it follows that (

( )

(

( ̂

)

)( )

(

)( )

[(( )( )

( ̂

̂

( )

)

(

)

)( )

)

)( ) ]

) is as defined in the statement of theorem 1

(d) The operator is obvious that

( )

( )

maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it

(

)(

(

( ̂

)( ) (

∫ [( )

) ( ̂

)( ) (

)( ) ( ̂ )( ) ( (̂ ( ̂ )( )

(

)

)(

)

)]

(

)( )

)

)

From which it follows that (

( ) (

( ̂

)

)( )

(

)( )

( ̂

)(

̂ ) [((

)(

)

(

)

)( )

)

( ̂

)( ) ]

) is as defined in the statement of theorem 6

Analogous inequalities hold also for It is now sufficient to take ̂ )( ) an ( ̂ (P

( )( ) ( ̂ )( )

( )( ) ( ̂ )( )

)( ) large to have (̂

( )( ) [( ̂ ( ̂ )( )

( (̂

)( ) )(

)(

̂ ) [((

(( ̂ )(

)

)(

)

( )

((

( )

( )

|

)(

( )

)

(

)

) (̂

( )

In order that the operator EQUATIONS into itself The operator

and to choose

)( )

(

)

)( )

( ̂

)

]

( ̂ )(

)

)( ) ]

( ̂

)(

)

transforms the space of sextuples of functions

is a contraction with respect to the metric ( )

( )

( )

( )

))

( )|

)( )

|

( )

( )

Indeed if we denote Definition of ̃ ̃ : ( ̃ ̃ )

( )

(

)

261

( )

( )|

)( )

satisfying GLOBAL


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

It results ( )

̃ ( )|

( )

)( ) (

(

( )

)( ) |

∫ (

( )

(

)

( )

) )|

( )

)( ) (

(

Where

(

( )

)( ) |

∫(

( )

(

)( ) (

|

))

( )

( )

|

)( ) (

( )

)( ) (

(

)( ) (

)

)( ) (

|

(

)

(

)

)

)

)( ) (

)

)( ) (

)( ) (

))

)

)

)( ) (

)

(

)

t

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 ( )( ) an ( )( ) depending also on t 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 ( )( ) an ( )( ) depend only on T 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 t

Definition of (( ̂ )( ) ) (( ̂ )( ) ) Remark 3: if

(

(( ̂ )( ) ) :

is bounded, the same property have also

( ̂ )( ) it follows (( ̂ )( ) )

(( ̂ )( ) ) (

(

)(

. indeed if

)

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 T

is bounded from below and li

((

262

)( ) ( ( ) ))

(

)( ) then


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

Definition of ( )( ) an Indeed let )(

(

www.iiste.org

:

be so that for

)

)( ) ( ( ) )

(

)( ) ( )(

(

Then (

(

)( ) ( )( )

(

(

)( ) ( )( )

( )(

( ) )

)

which leads to

)(

)

If we take t such that

)

sufficiently small one sees that T

By taking now if li

The same property holds for

( )

(

)

it results

( ( ) )

is unbounded.

( )

(

)

We now state a more precise theorem about the behaviors at infinity of the solutions It is now sufficient to take ( ̂ )(

( ̂

)

( )( ) ( ̂ )( )

( )( ) ( ̂ )( )

)( ) large to have )( )

( )( ) [( ̂ ( ̂ )( )

)(

( )( ) [(( ( ̂ )( )

̂

(( ̂ )(

)

( )

)

( )

)(

(((

)

|

( )

(

) )( )

( )

)

)

( ̂

]

( ̂ )(

)

)( ) ]

( ̂

)(

)

transforms the space of sextuples of functions

satisfying

is a contraction with respect to the metric

)( ) ) ((

(

)

(

)

In order that the operator The operator

and to choose

( )

( )

)(

)

( )|

)( ) ))

(

)( )

|

( )

( )

( )

( )|

)( )

Indeed if we denote Definition of ̃ ̃ : ( ̃ ̃ )

( )

(

)

It results |̃

( )

̃ ( )|

)( ) (

( ( )

( )

)( ) |

∫ (

)( ) (

(

Where

( )

(

)

(

( )

) )|

( )

( )

)( ) |

∫(

(

|

))

)( ) (

( )

( )

( )

(

|

)

)( ) (

)( ) (

|

)( ) (

(̂ )( ) ( ( )

(

)

))

)( ) (

)

)

(

)

)

)( ) ( (̂

)

)( ) (

)

represents integrand that is integrated over the interval

From the hypotheses it follows

263

)( ) (

)

(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)(

|( (̂

)

)( ) |e

( )(

(( )( )

)

)(

(

www.iiste.org

)( )

( ̂ )(

)

)

( ̂P )( ) ( ̂ )( ) ) (((

)(

)

(

)(

)

(

)(

)

(

)( ) ))

an T . Taking into account the hypothesis the result follows

And analogous inequalities for

Remark 1: The fact that we supposed ( )( ) an ( )( ) depending also on t 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 ( ̂P )( ) e( an ( ̂ )( ) e( 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 ( )( ) an ( )( ) depend only on T and respectively on ( )(an not on t) and hypothesis can replaced by a usual Lipschitz condition. (t)

Remark 2: There does not exist any t where

an T (t)

From 19 to 24 it results (t)

e[

T (t)

T e(

)( ) (

∫ {( (

)( ) )

)( ) (

( (

)) (

) )}

(

)]

for t

Definition of (( ̂ )( ) ) (( ̂ )( ) ) an (( ̂ )( ) ) : Remark 3: if

an

is bounded, the same property have also

( ̂ )( ) it follows

(( ̂ )( ) )

(( ̂ )( ) )

(

)(

(

)

. indeed if

and by integrating

)( ) (( ̂ )( ) ) (

)(

)

)( ) (( ̂ )( ) ) (

)(

)

In the same way , one can obtain (( ̂ )( ) ) or

If

(

is bounded, the same property follows for

and

respectively.

Remark 4: If is bounded, from below, the same property holds for an The proof is analogous with the preceding one. An analogous property is true if is bounded from below. Remark 5: If T

is bounded from below and li

Definition of ( )( ) an

)(

)

(

)( ) (( (

Then (

)( ) ( )( )

(

)( ) ( )( )

T

(

T

(

)(

)

same property holds for T

)(t) t))

(

)( ) then T

t

)(t) t)

)( ) ( ) (

)( ) ((

:

Indeed let t be so that for t (

((

)

log if li

)

T which leads to )

e

( )(

T (t)

T e

If we take t such that e

By taking now (

)

( )

((

it results

sufficiently small one sees that T is unbounded. The )(t) t)

264

(

)(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

We now state a more precise theorem about the behaviors at infinity of the solutions It is now sufficient to take ̂ )( ) an ( ̂ (P

( )( ) ( ̂ )( )

( )( ) ( ̂ )( )

and to choose

)( ) large to have )( )

( )( ) [( ̂ ( ̂ )( )

)(

)( )

( (̂

̂

[(( )( )

(( ̂

)

)(

)

)(

(

)

( )

)(

(((

)

|

( )

)( )

)

)

( ̂

( ̂

]

)(

( ̂

)( ) ]

)

)(

)

transforms the space of sextuples of functions

into itself

is a contraction with respect to the metric

)( ) ) ((

(

)

( )

In order that the operator The operator

)

(

( )

( )

)(

)

( )|

)( ) ))

(

)( )

( )

|

( )

( )

)( )

( )|

Indeed if we denote Definition of ̃ ̃ :( (̃) (̃) )

( )

)(

((

))

It results |̃

( )

∫(

( )

( )

)( ) |

∫ (

( )

)( ) (

(

̃ ( )|

( )

)( ) (

(

Where

(

(

)

) )|

( )

))

( )

)( ) (

|

( )

(

( )

)( ) |

( )

(

|

)( ) (

)( ) (

)

)( ) ( ( )

|

(

)

))

)( ) (

)( ) (

)

)

(

)

)

)( ) ( (̂

)

)( ) (

)

represents integrand that is integrated over the interval

)( ) (

)

(

)

t

From the hypotheses it follows | (̂

( )

( )

(( )( )

| )(

)( )

)

(

)(

)

And analogous inequalities for

)(

)

( ̂ )( ) ( ̂ )( ) ) (((

)(

)

(

)(

)

(

)(

)

(

)( ) ))

. Taking into account the hypothesis the result follows

Remark 1: The fact that we supposed ( )( ) an ( )( ) depending also on t 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 suffices to consider that ( )( ) an ( )( )

265

, we have to prove it only on a compact then it depend only on T and respectively on


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

www.iiste.org

) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any

( )

where

( )

From 19 to 24 it results ( )

[ ∫ {(

( )

( (

)( ) (

)( ) (

)( ) )

( (

)) (

) )}

for t

Definition of (( ̂ )( ) ) (( ̂ )( ) ) Remark 3: if

)]

(

(( ̂ )( ) ) :

is bounded, the same property have also

( ̂ )( ) it follows

(( ̂ )( ) )

(( ̂ )( ) )

(

)(

(

. indeed if

)

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 T

is bounded from below and li

Definition of ( )( ) an Indeed let (

)(

)

((

)( ) ((

)( ) ))

(

)( ) then

:

be so that for )( ) ((

(

(

Then

)( ) )

)( ) ( ) (

(

(

)( ) ( )( )

(

(

)( ) ( )( )

)

)(

( )(

( )

which leads to )

)

If we take t such that

if li

(

)

( )

it results

sufficiently small one sees that T

By taking now

The same property holds for

)

((

)( ) )

(

)

( )

We now state a more precise theorem about the behaviors at infinity of the solutions It is now sufficient to take

( )( ) ( ̂ )( )

( )( ) ( ̂ )( )

(̂ P )( ) an ( ̂

)( ) large to have

( )( ) [( ̂ ( ̂ )( )

(( ̂

( )

)

( )

)

)

(

and to choose

)( )

)

]

( ̂

266

)(

)

is unbounded.


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )( ) [(( ( ̂ )( )

̂

)(

)

(

)

( )

In order that the operator ( )

The operator )(

(((

)

|

( )

)

( ̂

( ̂

)( ) ]

)(

)

transforms the space of sextuples of functions

satisfying IN to itself

is a contraction with respect to the metric

)( ) ) ((

(

)( )

www.iiste.org

)(

( )

( )

( )|

)

)( ) ))

(

)( )

( )

|

( )

( )

( )|

)( )

Indeed if we denote Definition of (̃) (̃) : ( (̃) (̃) )

( )

)(

((

))

It results |̃

( )

̃ ( )|

( )

)( ) (

(

( )

Where

( )

)( ) |

∫ (

(

)

(

( )

) )|

)( ) (

(

( )

)( ) |

∫(

|

)( ) (

( )

( )

( )

( )

))

(

)( ) (

)

)( ) (

(

)( ) (

|

)( ) (

|

)

)( ) (

)

(

)

)

)( ) (

( )

)

(

))

)

represents integrand that is integrated over the interval

)( ) (

)

)( ) (

)

(

)

t

From the hypotheses it follows |( (̂

)(

)

(( )( )

)( ) |

( )(

)

(

)(

)( ) )

And analogous inequalities for

)(

( ̂ )( ) ( ̂ )( ) ) (((

)

)(

)

(

)(

)

(

)(

)

(

)( ) ))

. Taking into account the hypothesis the result follows

Remark 1: The fact that we supposed ( )( ) an ( )( ) depending also on t 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 ( ) an ( ) depend only on T 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 ( )

[ ∫ {(

)( ) (

)( ) (

( (

)) (

) )}

(

)]

267


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)( ) )

( (

( )

www.iiste.org

for t

Definition of (( ̂ )( ) ) (( ̂ )( ) ) Remark 3: if

(( ̂ )( ) ) :

is bounded, the same property have also

( ̂ )( ) it follows

(( ̂ )( ) )

(( ̂ )( ) )

(

)(

(

. indeed if

)

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 T

is bounded from below and li

Definition of ( )( ) an Indeed let (

)(

)

)( ) ((

((

)( ) ))

(

)( ) then

:

be so that for (

)( ) ((

)( ) ( ) (

(

Then

)( ) )

(

(

)( ) ( )( )

(

(

)( ) ( )( )

( )(

( )

)

)

which leads to

)(

)

)

If we take t such that

sufficiently small one sees that T

By taking now if li

The same property holds for

(

)

( )

it results

)( ) )

((

(

)

is unbounded.

( )

We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS inequalities hold also for It is now sufficient to take ̂ )( ) an ( ̂ (P

( )( ) ( ̂ )( )

( )( ) ( ̂ )( )

)( ) large to have (̂

( )( ) [( ̂ ( ̂ )( )

( )( ) [(( ( ̂ )( )

)(

̂

(( ̂

)

)(

and to choose

)

)(

)

In order that the operator

)

(

)

( )

)( )

(

)

)( )

( ̂

)

( ̂

]

( ̂

)( ) ]

)(

)

)(

)

transforms the space of sextuples of functions

268

into itself


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )

The operator )(

(((

)

( )

|

is a contraction with respect to the metric

)( ) ) ((

(

www.iiste.org

( )

( )

)(

)

)( ) ))

(

)( )

( )|

( )

|

( )

( )

)( )

( )|

Indeed if we denote Definition of (̃) (̃) : ( (̃) (̃) )

( )

)(

((

))

It results ( )

̃ ( )|

( )

)( ) (

( ( )

( )

)( ) |

∫ (

)( ) (

(

Where

(

)

( )

) )|

(

( )

( )

)( ) |

∫(

(

)( ) (

|

( )

))

( )

( )

(

)( ) ( ( )

)( ) (

)( ) (

)

|

)( ) (

|

(

)( ) (

)

(

)

)

)

)

)( ) ( (̂

))

)

)( ) (

)

represents integrand that is integrated over the interval

)( ) (

)

(

)

t

From the hypotheses it follows )(

|( (̂

)

(( )( )

)( ) |

( )(

)

(

)(

)( ) )

)(

( ̂ )( ) ( ̂ )( ) ) (((

)

And analogous inequalities for

)(

)

(

)(

)

(

)(

)

(

)( ) ))

. Taking into account the hypothesis the result follows

Remark 1: The fact that we supposed ( )( ) an ( )( ) depending also on t 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 ( ) an ( ) depend only on T and respectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition. Remark 2: There does not exist any

( )

where

( )

From GLOBAL EQUATIONS it results ( )

[ ∫ {(

( )

( (

)( ) (

)( ) )

)( ) (

( (

)) (

(

)]

for t

Definition of (( ̂ )( ) ) (( ̂ )( ) ) Remark 3: if

) )}

(( ̂ )( ) ) :

is bounded, the same property have also

269

. indeed if


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( ̂ )( ) it follows

www.iiste.org

(( ̂ )( ) )

(( ̂ )( ) )

(

)(

(

)

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 T

is bounded from below and li

Definition of ( )( ) an Indeed let )(

(

)

)( ) ((

((

)( ) ))

(

)( ) then

:

be so that for )( ) ((

(

)( ) ( )(

(

Then

)( ) )

(

(

)( ) ( )( )

(

(

)( ) ( )( )

( )

)

( )(

)

which leads to

)(

)

If we take t such that

)

sufficiently small one sees that T

By taking now if li

The same property holds for

(

)

( )

it results

)( ) )

((

(

)

is unbounded.

( )

We now state a more precise theorem about the behaviors at infinity of the solutions Analogous inequalities hold also for It is now sufficient to take ̂ )( ) an ( ̂ (P

( )( ) ( ̂ )( )

( )( ) ( ̂ )( )

)( ) large to have (̂

( (̂

)( ) )(

̂ )( ) [(

( )( ) [(( ( ̂ )( )

̂

(( ̂

)

)(

)

)(

)

In order that the operator ( )

The operator (((

)(

)

(

and to choose

)

(

)

( )

)( )

(

)

)( )

( ̂

)

( ̂

]

( ̂

)( ) ]

)(

)

)(

)

transforms the space of sextuples of functions

is a contraction with respect to the metric

)( ) ) ((

)(

)

(

)( ) ))

270

into itself


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )

|

( )

( )

)( )

( )|

www.iiste.org

( )

|

( )

( )

)( )

( )|

Indeed if we denote Definition of (̃) (̃) : ( (̃) (̃) )

( )

)(

((

))

It results ( )

̃ ( )|

( )

)( ) (

( ( )

( )

)( ) |

∫ (

)( ) (

(

Where

(

)

( )

) )|

(

( )

( )

)( ) |

∫(

( )

(

)( ) (

|

))

( )

( )

)( ) ( ( )

)( ) (

(

)( ) (

)

|

)( ) (

|

(

)

(

)

)

)

)( ) (

)

)( ) ( (̂

))

)

)( ) (

)

represents integrand that is integrated over the interval

)( ) (

)

(

)

t

From the hypotheses it follows )(

|( (̂

)

(( )( )

)( ) |

( )(

)

(

)(

)( )

)

)(

( ̂ )( ) ( ̂ )( ) ) (((

)

And analogous inequalities for

)(

)

(

)(

)

(

)(

)

(

)( ) ))

. Taking into account the hypothesis the result follows

Remark 1: The fact that we supposed ( )( ) an ( )( ) depending also on t 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 ( )( ) an ( )( ) depend only on T and respectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition. Remark 2: There does not exist any

( )

where

( )

From 69 to 32 it results ( )

[ ∫ {(

( )

( (

)( ) (

)( ) )

)( ) (

( (

)) (

) )}

(( ̂ )( ) ) :

is bounded, the same property have also

( ̂ )( ) it follows (( ̂ )( ) )

)]

for t

Definition of (( ̂ )( ) ) (( ̂ )( ) ) Remark 3: if

(

(( ̂ )( ) ) (

(

)( ) (( ̂ )( ) ) (

)(

. indeed if

)

)(

271

and by integrating )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

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 T

is bounded from below and li

Definition of ( )( ) an Indeed let )(

(

)( ) ((

((

)( ) ))

)( ) then

(

:

be so that for

)

)( ) ((

(

)( ) ( ) (

(

Then

)( ) )

(

(

)( ) ( )( )

(

(

)( ) ( )( )

( )(

( )

)

)

which leads to

)(

)

If we take t such that

)

sufficiently small one sees that T

By taking now if li

The same property holds for

)( ) ((

(

it results

)( ) ( ) )

)(

(

is unbounded.

)

We now state a more precise theorem about the behaviors at infinity of the solutions Behavior of the solutions If we denote and define Definition of ( )( )(

(a)

)

( )(

)

)

( )(

( )(

)

)

( )(

( )(

)

)

( )( ) :

four constants satisfying

( )(

)

(

)(

)

(

)(

)

(

)( ) (

( )(

)

(

)(

)

(

)(

)

(

)( ) (

)

)(

( )

Definition of ( )( (b) By ( )( (

)

( )

(

) ( )

) ( )

)

( )(

)

)

( ̅ )(

( ̅ )( ) ( )( )

)

)(

(

)

)(

( )(

)

(

)(

)

)

(

)

)

( )

(

( ̅ )(

)

)

)( ) (

( )

)( ( )

)

( )(

)(

)

( )(

)

)

)

:

)(

)

and (

)(

( )

( )

(

( )

)

)

the roots of ( )(

) ( )

(

the equations )(

)

( ̅ )( ) :

)

( )(

( )(

( )(

)

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

( )

)(

(

)( ) (

( (

and respectively (

( )

(c) If we define ( )(

)

)

Definition of (

(

( )(

( ) ( )

Definition of ( ̅ )( By ( ̅ )( ( )( ) (

)

)

)

( )( )

)

( )(

( )( ) :)

( )(

by )

( )(

272

)

)

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


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)(

(

)

( )(

and ( )( )(

(

)

)

)(

(

)

( ̅ )(

www.iiste.org

)

( )(

)

( )(

)

( ̅ )(

)

)

( )(

)

(

)(

( )(

)

)

( )(

)

( ̅ )(

)

( )(

)

and analogously ( )( )(

(

)

)

)(

(

)(

( )(

and (

)(

(

)

)

( )(

)(

(

)

( ̅ )(

)

)(

(

)

)

)(

(

)

)(

)

(

)(

)

( ̅ )(

( ̅ )(

)

(

)(

)

where (

(

)( )

(

)

)

)

( )(

)

( )(

)

)(

(

)

)(

)

( ̅ )(

)

are defined respectively Then the solution satisfies the inequalities )( ) (

((

)( ) )

)( )

(

( )

where ( )( ) is defined (

(

)( ) (( (

(

)( ) (( (

)( ) ( )( )

)( )

)( )

(

)( ) ((

(

)( ) ((

(

)( )

)( )

( ) (

[ )( ) )

)( ) (

)( ) )( ) ( )( ) (

(

Definition of ( )( Where ( )(

)

)

( )( (

)(

(

)(

[ )( ) )

( )(

)

)

((

(

)(

)

(

)( ) (

(

)(

)

)( ) (

)( )

(

)( ) )

)(

)( ) :-

( )

( )(

( )( (

(

]

)( )

)

)

)(

)( ) )

]

)( ) (

)(

)( )

(

((

)( )

)

)( )

(

)( ) )

(

)( ) (

( )

)( ) (

)( )

( )

(

)( ) )

)( )

(

((

( )

)( )

(

)( ) (

((

)( )

(

)( ) )

( )

[ )( ) )

)( ) (

)( ) (

(

(

)( ) )

)( )

( (

)( ) (

((

)( )

(

)(

)( )

)( )

]

)

)

(

)(

)

)

Behavior of the solutions

273

( ) (

)( )

( )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

If we denote and define Definition of (

)(

)

)(

)

( )(

(d)

)

)(

(

(

)(

)

( )(

)

( )(

)

)

( )( ) :

four constants satisfying

)(

)

(

)(

)

(

)(

)

(

)( ) (T

( )(

)

(

)(

)

(

)(

)

(

)( ) ((

Definition of ( )(

)

( )(

(

By ( )(

)

( )(

)

)

)( ) (

and (

( )

Definition of ( ̅ )( By ( ̅ )(

)

)

)

( )

( )(

) ( )

)

)( ) (

( )

)(

Definition of ( (f) If we define (

(

)(

(

)(

)

( ̅ )(

)( ) (

( )(

)

)(

)

)

(

)(

)

( )

)( (

)(

( )(

(

)

( )(

)(

)

)

(

)

)

) ( )

)

)(

)(

)

)(

(

)

)

the

)

( )( ) :-

( )(

)

( )(

)

by

( )(

)

(

)(

)

( )(

)

( )(

)

( )(

)

(

)(

)

( )(

)

(

)(

)

( ̅ )(

)

( )(

)

( )(

)

( ̅ )(

)

( ̅ )(

)

)

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

( )(

)

and analogously (

)(

( )(

) )

)(

and ( (

)(

)

(

)(

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

( )(

)

( )(

)

(

)(

(

)(

)

(

)(

)

(

)(

)

)

(

)(

)

(

)(

)

)

( ̅ )(

)

(

)(

)

)

Then the solution satisfies the inequalities e((

)( ) (

)( ) )

( )

)(

e(

)( )

( )( ) is defined

274

)

( )(

the roots

)

)

)(

(

( ̅ )(

)(

(

) )

and

(

and ( )(

(

( ̅ )( ) :

)

) ( )

)( ) ((

(

) ( )

and respectively ( ̅ )(

roots of the equations ( and (

)

)

)( ) :

(

)( ) (

( ̅ )(

( ̅ )(

)

)( ) (T

(

) )

and respectively (

the equations (

(e) of

)(

(

)

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

( )( ) ((

(

)( ) ( ( ) ) )( ) (

(

T )( )

(

)( ) ((

(

)( ) ((

)( )

e(

( )

)( )

(

[e )( ) )

)( ) (

)( ) (

Definition of ( )( Where ( )(

)

( )(

)

( )(

)(

(

(R )(

)

)

)(

)

)( ) (

]

e

(

)( )

( )

)

)( )

(

)( )

(

e

( )

]

T e

(

)( )

(R )( ) :)

)(

(

)( ) ( )(

(

)(

)(

(

(

)

)( ) )

T e

)( ) )

(R )(

)(

(

)

)

)( ) (

( )

)( )

)( ) (

[e(( )( ) )

)( ) (

]

)( )

(

e

e((

(

e

)( )

(

e

)( ) )

T )( )

(

)( )

(

)( )

(

)( ) (

T e((

( )

)( ) )

)( )

(

e

)( )

e(

)( )

(

)( ) (

((

)( )

e(

)( ) )

( )

[e )( ) )

)( ) (

)( )

T e(

)( ) )

)( )

( )( ) ((

(

)( ) (

e((

)( )

(

www.iiste.org

)

)

)(

(

)(

(

)

)

)

Behavior of the solutions If we denote and define Definition of ( )( )(

(a)

)

( )(

)

)

( )(

( )(

)

)

( )(

( )(

)

)

( )( ) :

four constants satisfying

( )(

)

(

)(

)

(

)(

)

(

)( ) (

( )(

)

(

)(

)

(

)(

)

(

)( ) (

Definition of ( )( (b) By ( )( (

)

( )

) ( )

(

)

( )(

)

( )(

)

)( ) (

By ( ̅ )(

)

( ) ( )

( )(

)

( ̅ )(

)

)( ) (

Definition of ( (c) If we define (

( )

)

)(

)

(

)(

)

( )( )( (

)( ) ((

(

)

) ( )

( )

)

( )(

)(

)

( )(

)

) )

( )(

)

)(

)(

)

)(

(

( )(

)(

)

( ̅ )(

)

(

the roots of

and

( )(

)

)

)

( )

(

) ( )

)

)( ) :

(

(

)( ) (

)( ) (

(

and respectively ( ̅ )(

roots of the equations ( and (

)

)

and respectively (

( ) ( )

)

and (

)(

(

)

) ( )

)

(

)

( )( ) :)

( )(

)

by

275

)(

)

the

the equations


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

)

(

)(

( )(

)

( )(

and ( )( )(

(

)

)

(

)(

(

)(

)

)

( )(

)

www.iiste.org

)

( ̅ )(

)

( )(

)

( )(

)

( )(

)

( )(

)

( ̅ )(

)

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

( )(

)

and analogously ( )(

)

(

)(

)

( )(

)

(

)(

)

(

)(

)

( )(

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

(

)(

)

(

)(

)

)(

)

(

)(

)

( )(

)

(

)(

)

( ̅ )(

)

(

)(

)

(

( ̅ )(

)

and (

)(

)

Then the solution satisfies the inequalities ((

)( ) (

)( ) )

)( )

(

( )

( )( ) is defined (

( (

((

)( )

)( ) (

)( ) )

)( ) )( ) ( )( ) (

( (

)( ) (( ( )( ) ((

)( )

(

)( )

(

)( )

(

( )( ) ((

(

)( ) ((

)( ) (

(

)( )

)( ) )( ) (

( )

)( ) )( ) ( )( ) (

(

Definition of ( )( Where ( )(

)

( )( ( (

)

[ )( ) ) ( )(

)(

)(

( )

)(

( )

(

)( ) (

( )

)

((

)( ) :-

)

)(

)(

(

)

)

(

)(

( )(

(

)( ) (

(

(

)(

(

)( )

(

] )

)( )

)( )

]

)

)

)(

( )(

)

)

Behavior of the solutions If we denote and define Definition of ( )( (d) ( )(

)

( )(

)

)

( )(

( )(

)

)

( )(

( )(

)

)

)( )

)( ) )

]

)( ) )

)(

)

)( ) (

)( ) (

)

)( )

(

)( ) )

)( )

(

)( )

(

((

)( )

(

)( )

(

[ )( ) )

)( ) (

((

)( )

)( ) )

)( )

(

( )

(

)( )

(

)( ) (

((

[ )( ) )

(

)( ) )

( )

( )( ) :

four constants satisfying

276

( ) (

)( )

( )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

( )(

)

(

)(

)

(

)(

)

(

)( ) (

)

( )(

)

(

)(

)

(

)(

)

(

)( ) ((

) )

Definition of ( )( (e) By ( )( ( )( ) (

)

)

( )

)

( ̅ )(

( ̅ )(

)

)

)(

(

) ( )

( ̅ )(

)

( )

( )

)

( )

)(

(

)

(

( )

)

( )

( ) ( )

)(

(

)(

)

(

)(

( )(

)

( )(

and ( )( )(

(

)

) )

)

)(

(

(

)(

(

)(

)

)

)

)

)(

)

( ̅ )(

)

(

the roots of

and

)

( ) ( )

the

( )

(

)

( )

( )(

( )(

)

( )(

) )

:

)(

)

)

( ̅ )( ) :

and ( ) ( ) ( ) ( ) ( ) ( ) ( ) Definition of ( ) ( ) ( ) ( )( (f) If we define (

( )

and respectively ( ̅ )(

roots of the equations ( ( )

)

( )(

)

)( ) ((

(

and respectively ( ( )( )

( )

( )(

)

)

)(

(

( )( ) ( )( )

Definition of ( ̅ )( By ( ̅ )(

( )(

) ( )

)( ) (

and (

)

)( ) (

(

)

( )( ) :-

( )(

)

( ̅ )(

)

)

)

by

( )(

)

( )(

)

( )(

)

( )(

)

( ̅ )(

)

( ̅ )(

)

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

( )(

)

and analogously ( )(

)

(

)(

)

( )(

)

(

)(

)

(

)(

)

(

)(

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

(

)(

)

(

)(

)

( ̅ )(

)

)(

and (

)

( )( ) ( )( ) ( )( ) ( )( ) are defined by 59 and 64 respectively

)(

(

)

where ( )(

)

( ̅ )(

)

Then the solution satisfies the inequalities ((

)( ) (

)( ) )

)( )

(

( )

where ( )( ) is defined (

(

(

(

((

)( )

)( ) (

)( ) )

)( ) )( ) ( )( ) (

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

)( ) )( ) ( )( )

[ )( ) ) ( )

( )

[ )( ) ) (

)( )

((

)( ) ( (

((

(

)( ) (

)( ) )

)( )

]

(

)( )

)( )

)( )

( (

)( )

)( ) )

277

(

] )

)( )

( )

the equations


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)( )

(

(

)( )

(

( )( ) ((

(

)( ) ((

)( ) )( ) (

( )

)( )

(

)( ) (

Where ( )(

)

( )(

)

)

(

)(

)( ) :-

)(

)

( )(

)

(

)(

(

)(

)

(

)( ) (

)(

(

)(

(

)(

)

(

( )(

)

( )(

)

)

(

)( )

)( )

(

)(

)

)(

)

(

(

)( )

(

)( ) )

)

)( ) )

]

)( ) (

((

)( ) (

(

)( )

(

[ )( ) )

)( ) (

Definition of ( )(

)( )

(

[ )( ) )

)( ) (

((

)( )

(

www.iiste.org

( ) )( )

(

]

)

( )(

)

Behavior of the solutions If we denote and define Definition of ( )( (g) ( )(

)

( )(

)

)

( )(

)

( )(

)

)

( )( ) :

four constants satisfying

( )(

)

(

)(

)

(

)(

)

(

)( ) (

)

( )(

)

(

)(

)

(

)(

)

(

)( ) ((

) )

Definition of ( )( (h) By ( )( (

)

( )

) ( )

(

)

( )(

)

( )

Definition of ( ̅ )( By ( ̅ )(

( )(

)

( )(

( ̅ )(

( ̅ )(

)

)(

(

)

( )

)

)

(

)

)(

)

( )

(

(

)(

)

(

)(

( )(

)

( )(

and ( )( (

)(

)

) )

)

)(

(

(

)(

(

)(

) )

)

)

)

(

)(

)

( )

( ) ( )

)

( )

( )(

( )(

)

( ̅ )(

( )(

)

( ̅ )(

)

the roots of

and

)

)

)

(

)

)

the

( )

) )

( )( ) :-

( )(

)

by

( )(

)

( )(

)

( )(

)

( )(

)

)

( )(

)(

(

( ̅ )( ) :

and ( )( ) ( ( ) ) ( )( ) ( ) ( )( Definition of ( )( ) ( )( ) ( )( ) ( )( )(

)(

and respectively ( ̅ )(

roots of the equations (

(i) If we define (

)

( ̅ )(

)

)

:

)

) ( )

)

( )(

) )

( )

(

( ̅ )(

( )(

)

)( ) ((

(

( )

and respectively (

( )

)

)

)(

(

( ) ( )

)

)( ) (

and (

)

)( ) (

(

( )(

and analogously

278

)

( ̅ )(

)

the equations


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

( )(

)

(

)(

)

( )(

)

(

)(

)

(

)(

)

(

)(

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

(

)(

)

(

)(

)

( ̅ )(

)

)(

and (

( ̅ )(

)

)

( )( ) ( )( ) ( )( are defined respectively

)

)(

(

)

)(

(

)

where ( )(

)

( ̅ )(

)

Then the solution satisfies the inequalities )( ) (

((

)( ) )

)( )

(

( )

where ( )( ) is defined (

(

(

(

)( ) )( ) ( )( ) (

( )( ) ((

)( )

(

)( ) ((

)( )

)( )

(

)( ) ((

(

[ )( ) )

)( ) ( )( )

)( )

( )

)( ) )( ) ( )( ) (

(

Definition of ( )( Where ( )(

)(

[ )( ) )

)

)

)(

( )

)( )

)( ) :-

)(

)

( )

)

(

( )(

)

( )(

( ) (

]

)( )

)

)

)

)(

)( )

(

)(

( )(

(

( )

)

)( )

(

)(

)

)( )

)( ) )

]

)( ) )

)(

(

)( ) (

)( ) (

) ( ) ( )(

(

)

(

(

]

)( )

(

]

((

(

((

)( )

(

)( ) )

)( )

)( ) (

(

)(

(

)( ) (

(

( )(

)

( )(

(

)

(

)( )

(

[ )( ) )

)( ) (

)( ) )

)( )

)( )

(

)( )

(

)( ) (

((

((

( )

)( )

(

( )

[ )( ) )

(

(

)( ) )

( )( ) ((

(

)( ) (

((

)( )

)(

(

)

)

Behavior of the solutions If we denote and define Definition of ( )( (j) ( )(

)

( )(

)

)

( )(

)

( )(

)

)

( )( ) :

four constants satisfying

( )(

)

(

)(

)

(

)(

)

(

)( ) (

)

( )(

)

(

)(

)

(

)(

)

(

)( ) ((

) )

Definition of ( )( (k) By ( )(

)

)

( )(

)

( )(

)

(

)(

)

(

)(

)

( )

and respectively (

)( ) (

( (

( )

( )(

)

)( ) ((

) )

)

( )(

)

:

)(

)

279

(

)(

)

the roots of

the equations


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)( ) (

(

( )

)( ) (

and (

( )(

) ( )

Definition of ( ̅ )( By ( ̅ )(

)

( )(

)

)

) ( )

( ̅ )(

( ̅ )(

)

)

) ( )

)

)(

(

)

( )

)

( )

( )

(

)

( )

( ) ( )

)(

(

)(

)

(

)(

( )(

)

( )(

and ( )( )(

(

)

) )

)

and

( ̅ )( ) :

)(

(

(

)(

(

)(

)

)

( )(

( )(

)

)

( ) ( )

( ̅ )(

)

the

( )

(

)

( )

and ( ) ( ) ( ) ( ) Definition of ( )( ) ( )( ) ( )( ) ( )( (l) If we define (

)

and respectively ( ̅ )(

roots of the equations ( ( )

)(

(

( ̅ )(

www.iiste.org

)

( )( ) :-

( )(

)

( ̅ )(

)

)

by

( )(

)

( )(

)

( )(

)

( )(

)

)

( ̅ )(

)

( ̅ )(

)

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

( )(

)

and analogously ( )(

)

(

)(

)

( )(

)

(

)(

)

(

)(

)

(

)(

)

( )(

)

(

)(

)

( )(

)

( ̅ )(

)

(

)(

)

(

)(

)

( ̅ )(

)

)(

and (

)

( )( ) ( )( ) ( )( are defined respectively

)

)(

(

)

)(

(

)

where ( )(

)

( ̅ )(

)

Then the solution satisfies the inequalities )( ) (

((

)( ) )

(

( )

)( )

where ( )( ) is defined (

(

(

(

((

)( )

)( ) (

)( ) )

)( ) )( ) ( )( ) (

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

)( ) )( ) ( )( )

(

(

)( )

(

)( ) ((

(

)( ) ((

(

[ )( ) )

)( ) )( ) (

)( ) )( ) ( )( ) (

(

(

)( )

(

(

)( )

[ )( ) )

)( ) (

)( ) (

((

)( ) (

)( ) )

)( )

)( )

(

)( )

(

)( ) ( (

((

( ) [ )( ) )

((

[ )( ) )

( ) )( )

( )

)( )

(

)( )

(

]

(

]

)( )

)

)( ) )

)( ) (

((

)( )

)( ) )

(

]

)( ) )

(

)( )

)( )

]

280

( ) (

)( )

( )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

Definition of ( )( Where ( )(

)

)

( )(

)

)(

(

)( ) (

(

( )(

)

(

)(

)(

)

(

)( ) (

(

)

)(

)

)( ) :-

( )

( )(

( )(

www.iiste.org

)

)(

)

)(

)

)

(

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

)(

(

) ( )

Definition of

)(

((

)

)(

(

)

)( ) (

(

))

)( ) (

(

)

( )

(

)(

( )(

) ( )

) ( )

( )

:-

It follows )( ) (

((

( )

( )(

)

) ( )

( )

)( ) )

(

)( ) (

((

( )

)

From which one obtains Definition of ( ̅ )(

( )(

(a) For ( )

( )( ) :-

)

( )(

)

( ̅ )( )( ) ((

[ ( )( )

)( ) ( )( ) (

(

( )

)

)( ) ((

[ ( ( )( )

)

)( ) (

)( ) (

)( ) ) ]

,

)( ) ) ]

it follows ( )(

)

( )

( )( ( )

)

(

)( ) (

)( )

(

)( ) (

)( )

( )(

)

In the same manner , we get ( )

[ ( ( ̅ )( )

( )(

( )(

)

(

)

( )(

[ ( )( )

[ ( ( )( )

(c) If

( )(

)

( )

( ̅ )(

)

)( ) ((

)( ) (

)( ) ((

)( ) (

)( )((̅ )( ) (̅ )( ) ) ]

( ̅ )(

)

( ̅ )(

( )

( )(

)

)

(̅ )( ) ( (

)

)( ) ) ]

( )

)( ) ) ]

( ̅ )(

)

, we obtain

281

( )

)( )

)( ) (̅ )( )

we find like in the previous case,

)( )((̅ )( ) (̅ )( ) ) ]

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

)

)

)( ) ( )( ) (

, ( ̅ )(

)( )((̅ )( ) (̅ )( ) ) ]

From which we deduce ( )(

(b) If

)( )((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

(

)( ) )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )(

)

( )

www.iiste.org

)( ) ((̅ )( ) (̅ )( )) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

( )(

)( ) ((̅ )( ) (̅ )( )) ]

[ ( ( ̅ )( )

)

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

Definition of )(

(

)

( ) :-

( )

( )

)( ) ,

(

( )

( )

( )

( )

In a completely analogous way, we obtain ( )

Definition of (

)(

)

( )

( ) :( )( ) ,

( )

( )

( )

( )

( )

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

)

( )( ) ( )( ) and in this case ( )( ) ( )( ) and as a consequence ( ) ( )( )

Analogously if (

)(

)

)(

(

)

( )(

)

( ̅ )( ) if in addition ( )( ) ( ) this also defines ( )( ) for the

( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then consequence of the relation between ( )( ) and ( ̅ )(

)

( ) ( )( ) ( ) This is an important and definition of ( )( )

we obtain ( )

(

)(

Definition of

)

)(

((

( )

)

)(

(

)

)( ) (T

(

t))

(

)( ) (T

t)

( )

(

)(

) ( )

( )

:-

It follows )( ) (

((

( )

)

(

)(

) ( )

( )

)( ) )

(

((

)( ) (

( )

)

( )(

From which one obtains Definition of ( ̅ )( (d) For ( )

( )

( )( (

)

( )( ) :-

)

( )(

)( ) ( )( ) (

it follows ( )(

[ ( )( )

[ ( ( )( )

)

( )

)

( )

( ̅ )( )( )((

)( )((

( )(

)( ) (

)

)( ) (

)( ) ) ]

)( ) ) ]

,

)

In the same manner , we get

282

(C)(

)

(

)( ) (

)( )

(

)( ) (

)( )

) ( )

(

)( ) )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )

( )(

(e) If (

)

)

( )(

( )

( ̅ )(

( )

)( ) ((

)( ) ((

)

)( ) (

)

( ̅ )(

)

( )(

)( ) ) ]

( )

( ̅ )(

)

(

( )

)

, we obtain

)( )((̅ )( ) (̅ )( ) ) ]

( )(

)( )((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

)( )

)( ) (̅ )( )

)

)( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

(̅ )( ) (

)

we find like in the previous case,

)( ) (

)( )((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

( )

( ̅ )(

( )

)( )((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )(

( )

)

[ ( ( )( )

(f) If

)

[ ( )( )

)( ) ( )( ) (

̅ )( , (C

)( ) ((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

From which we deduce ( )(

( )(

)( ) ((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

www.iiste.org

)

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

Definition of (

)(

)

( )

( ) :-

( )

)( ) ,

(

( )

( )

( )

( )

In a completely analogous way, we obtain ( )

Definition of (

)(

)

( )

( ) :( )( ) ,

( )

( )

( )

( )

( )

Particular case : If ( )( ) ( )( ( )( ) then ( ) ( )

)

Analogously if (

)(

( )( ) ( )( ) and in this case ( )( ( )( ) and as a consequence ( ) ( )( )

)(

(

)

( )(

)

)

( ̅ )( ) if in addition ( )( ( )

)

)

( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then consequence of the relation between ( )( ) and ( ̅ )(

( )

)

(

)(

)

( ) This is an important

From GLOBAL EQUATIONS we obtain ( )

(

)(

Definition of

)

((

( )

:-

)(

)

(

)(

)

(

)( ) (

))

(

)( ) (

)

( )

(

)(

( )(

) ( )

) ( )

( )

It follows ((

)( ) (

( )

)

( )(

) ( )

(

)( ) )

( )

((

283

)( ) (

( )

)

(

)( ) )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

From which one obtains ( )(

(a) For

( )

( )

(

)

( )(

)( ) ((

[ ( ( )( )

it follows ( )(

)

( )

( ̅ )(

)( ) ((

[ ( )( )

)( ) ( )( ) (

)

)( ) (

)( ) (

( )(

( )

)

)( ) ) ]

( )(

,

)( ) ) ]

)

(

)( ) (

)( )

(

)( ) (

)( )

)

In the same manner , we get ( )

( )

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

)( )((̅ )( ) (̅ )( ) ) ]

)( )((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

, ( ̅ )(

(̅ )( ) (

)

(

)( ) )( ) (̅ )( )

Definition of ( ̅ )( ) :From which we deduce ( )( ( )(

(b) If

( )

( )

)

( )(

)

( )

( ̅ )(

)( ) ((

)( ) ((

)

( ̅ )(

)

)( ) (

( )(

)

)( ) (

)

we find like in the previous case,

)( ) ) ]

( )

)( ) ) ]

)( ) ((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

( )(

( ̅ )(

( )

)( ) ((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )(

)

[ ( ( )( )

(c) If

( )

[ ( )( )

)( ) ( )( ) (

(

)

)

( ̅ )(

( )

)

, we obtain

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

( )

( )(

)

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

Definition of (

)(

)

( )

( ) :-

( )

)( ) ,

(

( )

( )

( ) ( )

In a completely analogous way, we obtain ( )

Definition of (

)(

)

( )

( ) :-

( )

( )( ) ,

( )

( )

( ) ( )

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

)

( )( ) ( )( ) and in this case ( )( ( ) and as a consequence ( ) ( )( ( )

284

) )

( ̅ )( ) if in addition ( )( ( )

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)(

Analogously if (

)

)(

(

)

www.iiste.org

( )(

)

( )( ) and then

( )( ) ( ̅ )( ) if in addition ( )( ) ( )( ) then consequence of the relation between ( )( ) and ( ̅ )(

( )

)

)(

(

)

( ) This is an important

From GLOBAL EQUATIONS we obtain ( )

(

)(

)

( )

Definition of

)(

((

)

)(

(

)

)( ) (

(

))

)( ) (

(

( )

)

(

)(

) ( )

( )

:-

It follows (( )( ) ( ( ) ) ( )( From which one obtains Definition of ( ̅ )( ( )(

(d) For

( )

( )

(

)

) ( )

((

)( ) (

( )(

)

( )

( )(

)

( )( ) :-

)

( )(

)( ) ((

[ ( ( )( )

)

( )

)

)

)( ) (

)( ) (

( )(

( )

( ̅ )(

)( ) ((

[ ( )( )

)( ) ( )( ) (

it follows ( )(

( )

)( ) )

(

)( ) ) ]

,

)( ) ) ]

(

)( ) (

)( )

(

)( ) (

)( )

)

In the same manner , we get ( )

)( )((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

[ ( ( ̅ )( )

From which we deduce ( )( ( )(

(e) If

(

( )

( )

)

( )(

( )(

)

)

( )

( ̅ )(

( )

( ̅ )(

)

)( ) ((

)( ) (

)( ) ((

[ ( ( )( )

( )(

( )

[ ( )( )

)( ) (

( )

)

( )(

)

)( ) ) ]

( ̅ )(

(

( )

( )

)

)( ) ((̅ )( ) (̅ )( )) ]

)( ) ((̅ )( ) (̅ )( )) ]

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

)(

)

( )

( )

(

)( ) ,

( )

( )

)( )

)( ) (̅ )( )

, we obtain

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

(̅ )( ) (

)

)( ) ) ]

)( )((̅ )( ) (̅ )( ) ) ]

( ̅ )(

)

we find like in the previous case,

)( )((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

(f) If

)

)

)( ) ( )( ) (

[ ( ( ̅ )( )

, ( ̅ )(

)( )((̅ )( ) (̅ )( ) ) ]

( ) ( )

In a completely analogous way, we obtain

285

( )(

)

) ( )

(

)( ) )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( )

Definition of (

)(

)

( )

www.iiste.org

( ) :( )( ) ,

( )

( )

( )

( )

( )

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

)

( )( ) ( )( ) and in this case ( )( ( ) and as a consequence ( ) ( )( ( )

)

( ̅ )( ) if in addition ( )( ) ( ) this also defines ( )( ) for

)

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

)(

)

)(

(

)

)( ) (

(

)( ) (

( )

( )(

)

)( ) (

(

)

( )

(

) ( )

( )

)( ) )

(

)( ) (

((

( )

( )(

)

) ( )

From which one obtains Definition of ( ̅ )( ( )(

(g) For

( )

( )( ) :-

)

( )(

[ ( ( )( )

it follows ( )(

)

( )

)

[ ( )( )

)( ) ( )( ) (

(

( )

)

)( ) ((

)( ) ((

( )(

( )

( ̅ )(

)

)( ) (

)( ) (

)( ) ) ]

,

)( ) ) ]

( )(

)

(

)( ) (

)( )

(

)( ) (

)( )

)

In the same manner , we get ( )

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

( )(

( )(

)

(

)

( )(

)( ) ( )( ) (

)

)

[ ( )( )

[ ( ( )( )

)( )((̅ )( ) (̅ )( ) ) ]

)( )((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

From which we deduce ( )( (h) If

)(

) ( )

( )

:-

It follows ((

))

)( ) ((

( )

( ̅ )(

( )

( ̅ )(

)

)( ) ((

)( ) (

)( ) (

, ( ̅ )(

)

(̅ )( ) ( (

)

we find like in the previous case, )( ) ) ]

)( ) ) ]

286

( )

( )

)( )

)( ) (̅ )( )

(

)( ) )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

)( )((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )(

( )(

)

)

( )

( ̅ )(

)( )((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

(i) If

www.iiste.org

( ̅ )(

)

( )(

)

, we obtain )( ) ((̅ )( ) (̅ )( )) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

)

( )(

)( ) ((̅ )( ) (̅ )( )) ]

[ ( ( ̅ )( )

)

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

)(

)

( )

( )

(

)( ) ,

( )

( )

( )

( )

In a completely analogous way, we obtain Definition of ( ) ( ) :(

)(

)

( )

( )( ) ,

( )

( )

( )

( )

( )

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

)

( )( ) ( )( ) and in this case ( )( ) ( ) and as a consequence ( ) ( )( )

( ̅ )( ) if in addition ( )( ) ( ) this also defines ( )( ) for

( )

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

(

)(

)

((

( )

Definition of

)(

)

)(

(

)

)( ) (

(

))

)( ) (

(

)

( )

(

)(

( )(

) ( )

) ( )

( )

:-

It follows ((

)( ) (

( )

( )(

)

) ( )

(

( )

)( ) )

((

)( ) (

( )

)

From which one obtains Definition of ( ̅ )( ( )(

(j) For

( )

( )

(

)

( )( ) :-

)

( )(

)( ) ( )( ) (

)

[ ( )( )

[ ( ( )( )

)( ) ((

( ̅ )( )( ) (( )( ) (

)

)( ) (

)( ) ) ]

,

)( ) ) ]

287

( )(

)

(

)( ) (

)( )

(

)( ) (

)( )

(

)( ) )


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

it follows ( )(

)

( )

( )(

( )

www.iiste.org

)

In the same manner , we get ( )

( )(

( )(

)

)

( )(

( )

( )

( )

)

( ̅ )(

)

)( ) ((

)( ) (

)( ) ((

)( ) (

)( ) ((̅ )( ) (̅ )( )) ]

( ̅ )(

)

( )(

(̅ )( ) (

)

(

)( ) )( ) (̅ )( )

)

we find like in the previous case, )( ) ) ]

( )

)( ) ) ]

( ̅ )(

)

( )

)

, we obtain

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

( ̅ )(

( )

)( ) ((̅ )( ) (̅ )( )) ]

[ ( ( ̅ )( )

( )(

( )

[ ( )( )

[ ( ( )( )

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

(l) If

)

)

)( ) ( )( ) (

(

, ( ̅ )(

)( )((̅ )( ) (̅ )( ) ) ]

[ ( ( ̅ )( )

From which we deduce ( )( (k) If

)( )((̅ )( ) (̅ )( ) ) ]

[ ( (̅ )( ) ( ̅ )( ) (̅ )( )

( )

)( ) ((̅ )( ) (̅ )( )) ]

)( ) ((̅ )( ) (̅ )( )) ]

[ ( ( ̅ )( )

( )(

)

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

)(

)

( )

( )

(

)( ) ,

( )

( )

( ) ( )

In a completely analogous way, we obtain Definition of ( ) ( ) :(

)(

)

( )

( )( ) ,

( )

( )

( )

( )

( )

Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case : If ( )( ) ( )( ) ( )( ) ( )( ) and in this case ( )( ) ( ̅ )( ) if in addition ( )( ) ( ) ( ) ( ) ( ) ( ) and as a consequence ( ) 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 ( )( ) We can prove the following Theorem 3: If (

)(

)

(

)( ) are independent on , and the conditions

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

)( ) (

(

)(

)

(

,

288

)( ) (

)(

)

(

)( ) (

)(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)( ) (

If (

)(

)

(

)

(

)( ) as defined, then the system

(

)( ) are independent on t , and the conditions

)(

( )(

)

)( ) (

)(

)

www.iiste.org

)( ) (

(

)(

)

)( ) (

(

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

)

(

)( ) as defined are satisfied , then the system

(

)( ) are independent on , and the conditions

)(

( If (

)(

)

)( ) (

(

)(

)

(

)( ) (

)( ) (

(

)(

)

)( ) (

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

)

(

)( ) as defined are satisfied , then the system

(

)( ) are independent on , and the conditions

)(

If (

)(

)

)( ) (

(

)(

)

(

)( ) (

(

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

)

(

)( ) as defined are satisfied , then the system

If (

)(

)

(

)( ) (

(

)(

)

(

)( ) (

(

)(

)

(

)( ) (

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

)

(

)( ) as defined satisfied , then the system

(

)( ) are independent on , and the conditions

(

)(

)

)( ) (

)(

)

(

)( ) (

)

)(

)

)

)(

)( ) (

(

)

(

)(

)( ) (

)

)(

)

)(

)

)(

)( ) (

(

)

(

)(

)( ) (

)

)(

)

)(

)

)( ) (

(

)(

)

)( ) are independent on , and the conditions

)(

If (

)

)(

,

)( ) (

)(

)(

)( ) (

(

(

)( ) (

(

,

)( ) (

)(

)(

)( ) (

(

(

)

,

(

(

)(

(

)( ) (

)(

)( ) (

(

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

, (

)( ) (

)(

)

(

)

289

)( ) (

)(

)

(

)( ) (

)(

)

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

)

)

(

)( ) as defined are satisfied , then the system

)(

(

)( ) (

(

)(

)

(

)( ) (

)(

)

(

)( ) (

)(

, )( ) (

(

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

(

)(

)

(

)( ) ( )

(

)(

)

(

)(

)

(

)( ) ( )

(

)(

)

(

)(

)

(

)( ) ( )

)(

)

(

)( ) (

)(

)

(

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

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)(

)

(

)( ) (

)

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

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)(

)

(

)( ) (

)

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

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

290

)( ) (

)(

)

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

(

)(

)

(

)(

)

(

)( ) ((

))

(

)(

)

(

)(

)

(

)( ) ((

))

(

)(

)

(

)(

)

(

)( ) ((

))

www.iiste.org

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

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)

( )

)]

( )

(

)

(

)(

( (

( )

(

[(

)

)

(

)(

)

(

)( ) (

)

)(

)

(

)(

)

(

)( ) (

)

)(

)

(

)(

)

(

)( ) (

)

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

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

[(

)(

)

(

)( ) (

)]

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)(

)

(

)( ) (

)

(

)(

)

(

)(

)

(

)( ) (

)

has a unique positive solution , which is an equilibrium solution for the system (a) Indeed the first two equations have a nontrivial solution )( ) ( )( )( )( ) (

( ) ( ( )( ) (

)

(

)( ) (

)(

)

)( ) (

(

if )( ) (

)

(

) )

( )

( (

)( ) (

)

)

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

)( ) (

(

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

)( ) (

)(

)

(

)( ) (

291

if )( ) (

)

(

)( ) (

)( ) (

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

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

) )

(

( )

(

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

)( ) (

)(

)

)( ) (

(

if )( ) (

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

) )

(

( )

(

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

)( ) (

)(

)

)( ) (

(

(

) )

(

( )

(

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

)( ) (

(

)(

)

)( ) (

(

)( ) (

(

) )

(

( )

(

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

)( ) (

Definition and uniqueness of T After hypothesis exists a unique ( [(

)( ) (

)]

)( ) (

(

)

)

(

)( ) (

)( ) (

)

)( ) (

)

(

)( ) (

)( ) (

)

(

)( ) (

)( ) (

)

if )( ) (

)

and the functions ( )( ) ( ) being increasing, it follows that there ) . With this value , we obtain from the three first equations (

,

Definition and uniqueness of T

)

)( ) (

:-

( ) ( ) for which (

)( )

)( ) (

)(

)( ) (

if

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

(

if

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

)

[(

)( )

)( ) (

)( ) (

)]

:-

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

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

)]

,

Definition and uniqueness of T After hypothesis exists a unique ( [(

)( ) (

)]

After hypothesis exists a unique ( [(

)( ) (

[(

)]

( )

( )

)( )

)( ) (

)( ) (

)]

:and the functions ( )( ) ( ) being increasing, it follows that there ) . With this value , we obtain from the three first equations (

,

Definition and uniqueness of T After hypothesis

(

( ) ( ) for which (

)( )

)( ) (

)]

and the functions ( )( ) ( ) being increasing, it follows that there ) . With this value , we obtain from the three first equations

,

Definition and uniqueness of T

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

:-

( ) ( ) for which (

)( )

)( ) (

[(

[(

)( )

)( ) (

)( ) (

)]

:and the functions (

292

)( ) (

) being increasing, it follows that there


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

exists a unique (

)( )

)( ) (

[(

(

for which )( ) (

)]

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

[(

)( ) (

)]

:and the functions ( )( ) ( ) being increasing, it follows that there ) . With this value , we obtain from the three first equations

,

)]

)( )

(

( ) ( ) for which (

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

[(

)

,

Definition and uniqueness of T After hypothesis exists a unique

www.iiste.org

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

[(

)]

(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

)

)( ) (

)] (

)( ) (

)(

)( ) (

)

) 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 (( ) ) (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 ( ) decreasing function in taking into account the hypothesis ( ) it follows that there exists a unique such that (( ) ) (h) By the same argument, the equations of modules admit solutions ( [(

) )( ) (

(

)( ) ( )( ) (

)( )

)

(

)( ) (

(

)( ) (

)(

)( ) (

if

)

)] (

)( ) (

)(

)( ) (

)

) Where in ( )( must be replaced by their values from 96. It is easy to see that is a ( ) decreasing function in taking into account the hypothesis ( ) it follows that there exists a unique such that (( ) ) (i) By the same argument, the equations (modules) admit solutions

293

if


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

( [(

)

)( ) (

(

)( ) (

)( ) (

)( )

)

(

)( ) (

(

)( ) (

www.iiste.org

)(

)

)( ) (

)( ) (

)] (

)(

)( ) (

)

) 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 (( ) ) (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 ( ) Finally we obtain the unique solution of 89 to 94 gi en y (

)

gi en y (

,

[(

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

[(

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

)]

, ,

)]

)

and

[(

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

[(

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

)]

)]

Obviously, these values represent an equilibrium solution Finally we obtain the unique solution gi en y ((

T

))

[(

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

[(

)( ) (

(

, T gi en y (T ) )]

,

)( ) ((

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

[(

)( )

[(

)]

)( )

(

, T

) )]

and

)( ) (

)( ) ((

) )]

Obviously, these values represent an equilibrium solution Finally we obtain the unique solution gi en y ((

))

[(

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

[(

)( ) (

(

)]

)( ) )( ) (

gi en y (

,

)]

,

and

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

[(

,

)

( [(

)( ) (

)]

)( ) )( ) (

Obviously, these values represent an equilibrium solution Finally we obtain the unique solution

294

)]


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

gi en y (

)

[(

[(

)( ) (

gi en y (

,

)( )

(

)( ) (

)( ) (

)]

)( ) ((

)

)( ) (

[(

)( ) (

)( ) (

[(

)]

)( )

(

,

) )]

and

)( )

(

,

)( )

(

www.iiste.org

)( ) ((

) )]

Obviously, these values represent an equilibrium solution Finally we obtain the unique solution gi en y ((

))

)( )

( [(

)( ) (

)( ) (

[(

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

gi en y (

, )]

) )( )

(

,

)( ) (

[(

,

) )]

and )( ) (

)]

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

[(

) )]

Obviously, these values represent an equilibrium solution Finally we obtain the unique solution gi en y ((

))

)( )

( [(

)( ) (

)( ) (

[(

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

gi en y (

, )]

)( ) (

[(

,

) )]

and

)( )

(

,

) )( ) (

)]

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

[(

) )]

Obviously, these values represent an equilibrium solution 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 ((

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

∑

(

(

(

)(

)

295

)( )

)

)(

)

(

)(

)


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

((

)(

)

(

)( ) )

(

)(

)

(

(

)( )

)

((

)(

)

(

)( ) )

(

)(

)

(

(

)( )

)

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

)(

)

Belong to

Denote Definition of

:,T

(

)( )

(T )

)(

(

T )

,

)( )

(

((

) )

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

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

(

)( ) T

)

((

)(

)

(

)( ) )

(

)(

)

(

(

)( ) T

)

((

)(

)

(

)( ) )

(

)(

)

(

(

)( ) T

)

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

)(

)

(

Denote Definition of

:, )( )

(

(

)

)(

(

)

,

(

)( )

((

) )

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

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

296

)(

)

Belong to


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

((

)(

)

(

)( ) )

(

)(

)

(

(

)( )

)

((

)(

)

(

)( ) )

(

)(

)

(

(

)( )

)

((

)(

)

(

)( ) )

(

)(

)

(

(

)( )

)

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

)(

)

)(

(

)

Belong to

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

)(

)

(

Denote Definition of

:,

(

)( )

(

)

(

)(

)

,

(

)( )

((

) )

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

)(

)

(

)( ) )

(

)(

)

(

)(

)

((

)(

)

(

)( ) )

(

)(

)

(

)(

)

297

)(

)

Belong to


Advances in Physics Theories and Applications ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 7, 2012

www.iiste.org

((

)(

)