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Hierarchial Problem-Part Three: Forschungsgemeinschaft: Aggregate

Statement

Dr. K.N.P.Kumar

Post-Doctoral Scholar, Department of Mathematics, Kuvempu university, Karnataka, India

Abstract: Warped extra dimensions pioneered by the aforementioned Lisa Randall along with Raman

Sundrum — hold that gravity is just as strong as the other forces, but not in our three-spatial-dimension

Universe. It lives in a different three-spatial-dimension Universe that’s offset by some tiny amount —

like 10-31 meters — from our own Universe in the fourth spatial dimension. (Or, as the diagram above

indicates, in the fifth dimension, once time is included.) This is interesting, because it would be stable, and it

could provide a possible explanation as to why our Universe began expanding so rapidly at the beginning

(warped spacetime can do that), so it’s got some compelling perks. A concatenated model representing

various aspectionalities, attributions, predicational anteriories, and ontological consonance is presented.

Keywords: extra dimensions hierarchy, hierarchy problem large extra dimensions particle

physics, supersymmetry SUSY, Technicolor warped extra dimensions, AdS/CFT correspondence, Graviton

probability function, Randall–Sundrum models

INTRODUCTION—VARIABLES USED

Warped dimensions (Source: Wikipedia)

Randall–Sundrum models (also called 5-dimensional warped geometry theory) imagines that the real world

is a higher-dimensional universe described by warped geometry. More concretely, our universe is a five-

dimensional anti-de Sitter space and the elementary particles except for the graviton are localized on a (3 +

1)-dimensional brane or branes.

The models were proposed in 1999 by Lisa Randall and Raman Sundrum because they were dissatisfied

with the universal extra-dimensional models then in vogue. Such models require two fine tunings; one for

the value of the bulk cosmological constant and the other for the brane tensions. Later, while studying RS

models in the context of the AdS/CFT correspondence, they showed how it can be dual to Technicolor

models.

There are two popular models. The first, called RS1 has a finite size for the extra dimension with two branes,

one at each end. The second, RS2, is similar to the first, but one brane has been placed infinitely far away, so

that there is only one brane left in the model.

Overview

The model is a braneworld theory developed while trying to solve the hierarchy problem of the Standard

Model. It involves a finite five-dimensional bulk that is extremely warped and contains two branes: the

Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane") and the Tevbrane (our

home with the Standard Model particles; also called "Weakbrane"). In this model, the two branes are

separated in the not-necessarily large fifth dimension by approximately 16 units (the units based on the brane

and bulk energies). The Planckbrane has positive brane energy, and the Tevbrane has negative brane energy.

These energies are the cause of the extremely warped spacetime.

Graviton probability function

In this warped spacetime that is only warped along the fifth dimension, the graviton's probability function is

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extremely high at the Planckbrane, but it drops exponentially as it moves closer towards the Tevbrane. In

this, gravity would be much weaker on the Tevbrane than on the Planckbrane.

Growing, shrinking, and changing weight: solving the hierarchy problem

Also in this model, any objects moving from the Planckbrane to the Tevbrane in the bulk would be growing,

becoming lighter, and moving more slowly through time. Distance and time expand near the Tevbrane, and

mass and energy shrink near it. This creates an alternate explanation of gravity's weakness in the Tevbrane:

everything is lighter. An interesting part of this is that the hierarchy problem is automatically solved. The

main scale on the Planckbrane would be the Planck scale. However, the change by 16 units causes the scales

to change by 16 orders of magnitude. On the Planckbrane, strings would be 10−33 cm in size, but on the

Tevbrane they'd be 10−17 cm. In fact, this makes the mass scale for the Weakbrane based on about a TeV.

Therefore, you no longer have the strange range of masses and energies. If this is the case, we should see

evidence when the Large Hadron Collider begins working.

National Geographic Society: The Greatest Unsolved Problem in Theoretical Physics

Posted by Ethan on September 19, 2012

There are two other possibilities, one which is much more promising than the other, and both involving extra

dimensions.

Image credit: Cetin BAL, as far as I can tell.

(1) Warped Extra Dimensions. This theory — pioneered by the aforementioned Lisa Randall along

with Raman Sundrum — hold that gravity is just (=) as strong as the other forces, but not in

our three-spatial-dimension Universe.

(2) It lives in a different three-spatial-dimension Universe that’s offset (=, e, eb) by some tiny

amount — like 10-31 meters — from our own Universe in the fourth spatial dimension.

(3) (Or, as the diagram above indicates, in the fifth dimension, once time is included.) This is

interesting, because it would be stable, and it could (eb) provide a possible explanation as to

why our Universe began expanding so rapidly at the beginning (warped spacetime can do that),

so it’s got some compelling perks.

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(4) What it should also (e) include are an extra set of particles; not supersymmetric particles, but

Kaluza-Klein particles, which are a direct consequence (e) of there being extra dimensions.

(5) For what it’s worth, there has been a hint from one experiment in space that there might (eb)

be a Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the Higgs.

Image credit: J. Chang et al. (2008), Nature, from the Advanced Thin Ionization Calorimeter (ATIC).

(6) This is by no means a certainty, as it’s(=) just an excess of observed electrons over(e) the

expected background, but it’s worth keeping in mind as the LHC eventually ramps up to full

energy; a new particle that’s below 1,000 GeV in mass should be within range of this machine.

And finally…

Image credit: Universe-review.ca.

(7) Large Extra Dimensions. Instead of being warped, the extra dimensions could be “large”,

where large is only large relative (e) to the warped ones, which were 10-31 meters in scale.

(8) The “large” extra dimensions would be around millimeter-sized, which meant that new

particles would start showing (eb) up right around the scale that the LHC is capable of

probing.

(9) Again, there would be new Kaluza-Klein particles, and this could be a possible solution (e) to

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the hierarchy problem.

NOTATION

Module One

Warped Extra Dimensions. This theory — pioneered by the aforementioned Lisa Randall along with Raman

Sundrum — hold that gravity is just as strong as the other forces, but not in our three-spatial-dimension

Universe

: Category one of strong as the other forces, but not in our three-spatial-dimension Universe

: Category two of strong as the other forces, but not in our three-spatial-dimension Universe

: Category three of strong as the other forces, but not in our three-spatial-dimension Universe

: Category one of gravity. We are talking of the systemic differentiation. This does not mean that there

are three gravities. Systems are legion that hold the rule of gravity. Characterstics of those systems form the

bastion of the characterization portfolio. perspicuous characterstics, sophisticated seasoning, constituent

structure, concept formulation, beat cadence, interiority of perfect causes, incompatibility of conjunctions’,

resonantial resumptions, hypothetical propositions, logical compatibilities, on causal correspondences,

predicational inherence, emphasis related phenomenological methodologies, Partcipational observation,

background actuality, addressed aestheticism, polished perception, refined proficiency, existential

worldliness, existential strife, predicational anteriorities, apophthegmatic axiom, brocard- gnome,

implantation and inculcation, pronouncement and proposition, declaration dogma, character consonance,

shift and stratagem, ontological consonance, primordial exactitude, phenomenological correlates, accolytish

representation, atrophied asseveration, suppositional surmise, theorization anamensial alienisms, anchorite

aperitif, Arcadian Atticism all delineated in the conditionalities and functionalities and orinetatilities of the

system which incorporate the rules and regulations, axiomatic predications and postulation alcovishness of

the foregoing state. Characterstics of such systems are taken in to consideration in the complete

consolidation and consummation of the systems

: Category two of gravity

: Category three of gravity

Module Two

Gravity lives in a different three-spatial-dimension Universe that’s offset by some tiny amount — like 10-

31 meters — from our own Universe in the fourth spatial dimension

: Category one of some tiny amount — like 10-31 meters — from our own Universe in the fourth spatial

dimension

: Category two of some tiny amount — like 10-31 meters — from our own Universe in the fourth

spatial dimension

: Category three of some tiny amount — like 10-31 meters — from our own Universe in the fourth

spatial dimension

: Category one of Gravity lives in a different three-spatial-dimension Universe

: Category two of Gravity lives in a different three-spatial-dimension Universe

: Category three of Gravity lives in a different three-spatial-dimension Universe

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Module three

(Or, as the diagram above indicates, in the fifth dimension, once time is included.) This is interesting,

because it would be stable, and it could (eb) provide a possible explanation as to why our Universe began

expanding so rapidly at the beginning (warped spacetime can do that), so it’s got some compelling perks

: Category one of it would be stable

: Category two of it would be stable

: Category three of it would be stable

: Category one of possible explanation as to why our Universe began expanding so rapidly at the

beginning (warped spacetime can do that), so it’s got some compelling perks

: Category two of possible explanation as to why our Universe began expanding so rapidly at the

beginning (warped spacetime can do that), so it’s got some compelling perks

: Category three of possible explanation as to why our Universe began expanding so rapidly at the

beginning (warped spacetime can do that), so it’s got some compelling perks

Module four

What it should also (e) include are an extra set of particles; not supersymmetric particles, but Kaluza-Klein

particles, which are a direct consequence of there being extra dimensions

: Category one of extra set of particles; not supersymmetric particles, but Kaluza-Klein particles, which

are a direct consequence of there being extra dimensions

: Category two of extra set of particles; not supersymmetric particles, but Kaluza-Klein particles, which

are a direct consequence of there being extra dimensions

: Category three of extra set of particles; not supersymmetric particles, but Kaluza-Klein particles,

which are a direct consequence of there being extra dimensions

: Category one of possible explanation as to why our Universe began expanding so rapidly at the

beginning (warped spacetime can do that)

: Category two of possible explanation as to why our Universe began expanding so rapidly at the

beginning (warped spacetime can do that)

: Category three of possible explanation as to why our Universe began expanding so rapidly at the

beginning (warped spacetime can do that)

Module five

For what it’s worth, there has been a hint from one experiment in space that there might (eb) be a Kaluza-

Klein particle at energy of about 600 GeV, or about 5 times the mass of the Higgs

: Category one of hint from one experiment in space. perspicuous characterstics, sophisticated

seasoning, constituent structure, concept formulation, beat cadence, interiority of perfect causes,

incompatibility of conjunctions’, resonantial resumptions, hypothetical propositions, logical compatibilities,

on causal correspondences, predicational inherence, emphasis related phenomenological methodologies,

Partcipational observation, background actuality, addressed aestheticism, polished perception, refined

proficiency, existential worldliness, existential strife, predicational anteriorities, apophthegmatic axiom,

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brocard- gnome, implantation and inculcation, pronouncement and proposition, declaration dogma, character

consonance, shift and stratagem, ontological consonance, primordial exactitude, phenomenological

correlates, accolytish representation, atrophied asseveration, suppositional surmise, theorization anamensial

alienisms, anchorite aperitif, Arcadian Atticism all delineated in the conditionalities and functionalities and

orinetatilities of the system which incorporate the rules and regulations, axiomatic predications and

postulation alcovishness of the foregoing state. Characterstics of such systems are taken in to

consideration in the complete consolidation and consummation of the systems

: Category two of hint from one experiment in space

: Category three of hint from one experiment in space

: Category one of Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the

Higgs

: Category two of Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the

Higgs

: Category three of Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the

Higgs

Module six

This is by no means a certainty, as it’s just an excess of observed electrons over(e) the expected

background, but it’s worth keeping in mind as the LHC eventually ramps up to full energy; a new particle

that’s below 1,000 GeV in mass should be within range of this machine

: Category one of excess of observed electrons over the expected background, but it’s worth keeping in

mind as the LHC eventually ramps up to full energy; a new particle that’s below 1,000 GeV in mass should

be within range of this machine

: Category two of excess of observed electrons over the expected background, but it’s worth keeping in

mind as the LHC eventually ramps up to full energy; a new particle that’s below 1,000 GeV in mass should

be within range of this machine

: Category three of excess of observed electrons over the expected background, but it’s worth keeping

in mind as the LHC eventually ramps up to full energy; a new particle that’s below 1,000 GeV in mass

should be within range of this machine

: Category one of Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the

Higgs

: Category two of Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the

Higgs

: Category three of Kaluza-Klein particle at energy of about 600 GeV, or about 5 times the mass of the

Higgs

Module seven

The “large” extra dimensions would be around millimeter-sized, which meant that new particles would start

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showing (eb) up right around the scale that the LHC is capable of probing

: Category one of “large” extra dimensions would be around millimeter-sized, which meant that new

particles

: Category two of “large” extra dimensions would be around millimeter-sized, which meant that new

particles

: Category three of “large” extra dimensions would be around millimeter-sized, which meant that new

particles

: Category one of right around the scale that the LHC is capable of probing

: Category two of right around the scale that the LHC is capable of probing

: Category three of right around the scale that the LHC is capable of probing

Module eight

The “large” extra dimensions would be around millimeter-sized, which meant that new particles would start

showing (eb) up right around the scale that the LHC is capable of probing.

: Category one of “large” extra dimensions would be around millimeter-sized, which meant that new

particles

: Category two of “large” extra dimensions would be around millimeter-sized, which meant that new

particles

: Category three of “large” extra dimensions would be around millimeter-sized, which meant that new

particles

: Category one of right around the scale that the LHC is capable of probing. Domain restriction.

: Category two of right around the scale that the LHC is capable of probing.

: Category three of right around the scale that the LHC is capable of probing.

Module Nine

Again, there would be new Kaluza-Klein particles, and this could be a possible solution (e) to the hierarchy

problem

: Category one of hierarchy problem. This does not mean that there are three Hierarchial problems.

Systemic differentiation. Partcipational observation, background actuality, addressed aestheticism, polished

perception, refined proficiency, existential worldliness, existential strife, predicational anteriorities,

apophthegmatic axiom, brocard- gnome, implantation and inculcation, pronouncement and proposition,

declaration dogma, character consonance, shift and stratagem, ontological consonance, primordial

exactitude, phenomenological correlates, accolytish representation, atrophied asseveration, suppositional

surmise, theorization anamensial alienisms, anchorite aperitif, Arcadian Atticism all delineated in the

conditionalities and functionalities and orinetatilities of the system which incorporate the rules and

regulations, axiomatic predications and postulation alcovishness of the foregoing state. Characterstics of

such systems are taken in to consideration in the complete consolidation and consummation of the

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systems

: Category two of hierarchy problem

: Category three of hierarchy problem

: Category one of Kaluza-Klein particles, and this could be a possible

: Category two of Kaluza-Klein particles, and this could be a possible

: Category three of Kaluza-Klein particles, and this could be a possible

The Coefficients:

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ):

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ),( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

are Accentuation coefficients

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ,

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

are Dissipation coefficients

Module Numbered One

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

( )

( ) [( )( ) (

)( )( )] 1

( )

( ) [( )( ) (

)( )( )] 2

( )

( ) [( )( ) (

)( )( )] 3

( )

( ) [( )( ) (

)( )( )] 4

( )

( ) [( )( ) (

)( )( )] 5

( )

( ) [( )( ) (

)( )( )] 6

( )( )( ) First augmentation factor

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( )( )( ) First detritions factor

Module Numbered Two

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

( )

( ) [( )( ) (

)( )( )] 7

( )

( ) [( )( ) (

)( )( )] 8

( )

( ) [( )( ) (

)( )( )] 9

( )

( ) [( )( ) (

)( )(( ) )] 10

( )

( ) [( )( ) (

)( )(( ) )] 11

( )

( ) [( )( ) (

)( )(( ) )] 12

( )( )( ) First augmentation factor

( )( )(( ) ) First detritions factor

Module Numbered Three

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

( )

( ) [( )( ) (

)( )( )] 13

( )

( ) [( )( ) (

)( )( )] 14

( )

( ) [( )( ) (

)( )( )] 15

( )

( ) [( )( ) (

)( )( )] 16

( )

( ) [( )( ) (

)( )( )] 17

( )

( ) [( )( ) (

)( )( )] 18

( )( )( ) First augmentation factor

( )( )( ) First detritions factor

Module Numbered Four

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

( )

( ) [( )( ) (

)( )( )] 19

( )

( ) [( )( ) (

)( )( )] 20

( )

( ) [( )( ) (

)( )( )] 21

( )

( ) [( )( ) (

)( )(( ) )] 22

( )

( ) [( )( ) (

)( )(( ) )] 23

( )

( ) [( )( ) (

)( )(( ) )] 24

( )( )( ) First augmentation factor

( )( )(( ) ) First detritions factor

Module Numbered Five:

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

( )

( ) [( )( ) (

)( )( )] 25

( )

( ) [( )( ) (

)( )( )] 26

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( )

( ) [( )( ) (

)( )( )] 27

( )

( ) [( )( ) (

)( )(( ) )] 28

( )

( ) [( )( ) (

)( )(( ) )] 29

( )

( ) [( )( ) (

)( )(( ) )] 30

( )( )( ) First augmentation factor

( )( )(( ) ) First detritions factor

Module Numbered Six

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

( )

( ) [( )( ) (

)( )( )] 31

( )

( ) [( )( ) (

)( )( )] 32

( )

( ) [( )( ) (

)( )( )] 33

( )

( ) [( )( ) (

)( )(( ) )] 34

( )

( ) [( )( ) (

)( )(( ) )] 35

( )

( ) [( )( ) (

)( )(( ) )] 36

( )( )( ) First augmentation factor

Module Numbered Seven:

The differential system of this model is now (Seventh Module)

( )

( ) [( )( ) (

)( )( )] 37

( )

( ) [( )( ) (

)( )( )] 38

( )

( ) [( )( ) (

)( )( )] 39

( )

( ) [( )( ) (

)( )(( ) )] 40

( )

( ) [( )( ) (

)( )(( ) )] 41

( )

( ) [( )( ) (

)( )(( ) )] 42

( )( )( ) First augmentation factor

Module Numbered Eight

The differential system of this model is now

( )

( ) [( )( ) (

)( )( )] 43

( )

( ) [( )( ) (

)( )( )] 44

( )

( ) [( )( ) (

)( )( )] 45

( )

( ) [( )( ) (

)( )(( ) )] 46

( )

( ) [( )( ) (

)( )(( ) )] 47

( )

( ) [( )( ) (

)( )(( ) )] 48

Module Numbered Nine

The differential system of this model is now

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( )

( ) [( )( ) (

)( )( )] 49

( )

( ) [( )( ) (

)( )( )] 50

( )

( ) [( )( ) (

)( )( )] 51

( )

( ) [( )( ) (

)( )(( ) )] 52

( )

( ) [( )( ) (

)( )(( ) )] 53

( )

( ) [( )( ) (

)( )(( ) )] 54

( )( )( ) First augmentation factor

( )( )(( ) ) First detrition factor

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

55

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

56

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

57

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second augmentation coefficient for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficient for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fourth augmentation

coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficient

for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation coefficient

for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are seventh augmentation coefficient for

1,2,3

( )( )( ) (

)( )( ) ( )( )( ) are eight augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth

augmentation coefficient for 1,2,3

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

58

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( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

59

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

60

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for category 1,

2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients

for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detrition coefficients for

category 1, 2 and 3

– ( )( )( ) – (

)( )( ) , – ( )( )( ) are seventh detrition coefficients for

category 1, 2 and 3

– ( )( )( ) – (

)( )( ) – ( )( )( ) are eight detrition coefficients for category 1,

2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth

detrition coefficients for category 1, 2 and 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

61

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

62

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

63

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 13 ISSN 2250-3153

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( )( )( ) , (

)( )( ) , ( )( )( ) are second augmentation coefficient for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficient for

category 1, 2 and 3

( )( )( ) (

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

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth augmentation

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation

coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are seventh augmentation coefficient

for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are eight augmentation coefficient for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation coefficient for

category 1, 2 and 3

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

64

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

65

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

66

( )( )( ) , (

)( )( ) , ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are second detrition coefficients for category

1,2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition coefficients for

category 1,2 and 3

( )( )( ) (

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

coefficients for category 1,2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition coefficients

for category 1,2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are sixth detrition coefficients

for category 1,2 and 3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 14 ISSN 2250-3153

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– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition coefficients for

category 1,2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eight detrition coefficients for

category 1,2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients for category

1,2 and 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

67

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

68

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

69

( )( )( ) , (

)( )( ) , ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are second augmentation coefficients

for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficients

for category 1, 2 and 3

( )( )( ) , (

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

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are seventh augmentation

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are eight augmentation coefficients

for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation coefficients for

category 1, 2 and 3

( )

( )

[

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

70

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 15 ISSN 2250-3153

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( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

71

( )

( )

[

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

72

( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for category 1,

2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are third detrition coefficients for category

1,2 and 3

( )( )( ) (

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

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are seventh detrition coefficients

for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eight detrition coefficients for

category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients for

category 1, 2 and 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

73

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

74

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

75

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 16 ISSN 2250-3153

www.ijsrp.org

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) , (

)( )( ) ( )( )( )

( )( )( ) , (

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

( )( )( ) , (

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

( )( )( ) , (

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

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth detrition coefficients for

category 1 2 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

76

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

77

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

78

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )( )( ) , (

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

– ( )( )( ) – (

)( )( ) – ( )( )( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 17 ISSN 2250-3153

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– ( )( )( ) , – (

)( )( ) , – ( )( )( )

– ( )( )( ) , – (

)( )( ) – ( )( )( )

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients for

category 1 2 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

79

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

80

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

81

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

coefficients for category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation

coefficients for category 1,2,and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation

coefficients for category 1,2, 3

( )( )( ) , (

)( )( ) , ( )( )( ) are seventh augmentation

coefficients for category 1,2, 3

( )( )( ) , (

)( )( ) , ( )( )( ) are eighth augmentation

coefficients for category 1,2, 3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation

coefficients for category 1,2, 3

( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

82

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 18 ISSN 2250-3153

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( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

83

( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

84

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for

category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition coefficients

for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition coefficients

for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients

for category 1,2, and 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

85

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

86

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

87

( )( )( ) (

)( )( ) ( )( )( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 19 ISSN 2250-3153

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( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

coefficients

( )( )( ) (

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

coefficients

( )( )( ) , (

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

coefficients

( )( )( ) , (

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

( )( )( ) , (

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

Eighth augmentation coefficients

( )( )( ) , (

)( )( ) , ( )( )( ) ninth augmentation

coefficients

( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

88

( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

89

( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

90

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients

for category 1, 2, and 3

( )( )( ) , (

)( )( ) ( )( )( ) are fifth detrition

coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 20 ISSN 2250-3153

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coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( )

are eighth detrition coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition

coefficients for category 1, 2, and 3

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

91

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

92

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

93

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are seventh

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

are eighth augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 21 ISSN 2250-3153

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coefficient for 1,2,3

( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

94

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second detrition

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition

coefficients for category 1, 2 and 3

( )( )( ) (

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

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( )

are seventh detrition coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition

coefficients for category 1, 2 and 3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 22 ISSN 2250-3153

www.ijsrp.org

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

95

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are seventh

augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are eighth

augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth

augmentation coefficient for 1,2,3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 23 ISSN 2250-3153

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( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second

detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition

coefficients for category 1, 2 and 3

( )( )( ) (

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

detrition coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , – ( )( )( ) are sixth detrition coefficients

for category 1, 2 and 3

– ( )( )( ) – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition

coefficients for category 1, 2 and 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

96

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 24 ISSN 2250-3153

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( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are Seventh

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are eighth

augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth

augmentation coefficient for 1,2,3

( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 25 ISSN 2250-3153

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( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second

detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition

coefficients for category 1, 2 and 3

( )( )( ) (

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

detrition coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1, 2 and 3

– ( )( )( ) – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth

detrition coefficients for category 1, 2 and 3

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

97

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )

98

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 26 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )( ) (

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

99

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

( )( )(

) and( )( )( ) (

) and ( ) are points belonging to the interval

[( )( ) ( )

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

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

)( )( ) , the first augmentation coefficient

attributable to the system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

100

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( )which together With ( )( ) ( )

( ) ( )( )

and ( )( ) and the constants ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

101

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

102

( )( )( ) ( )

( ) ( )( ) ( )

( ) 103

( )( ) ( ) ( )

( ) 104

( )( ) (( ) ) ( )

( ) 105

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 27 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

106

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( ) 107

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( ) 108

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

109

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together

with ( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 110

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 111

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

112

( )( ) ( ) ( )

( ) 113

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 28 ISSN 2250-3153

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( )( ) ( ) ( )

( )

Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(

) ( )( )( ) ( )

( ) ( )( )

114

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

115

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

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

116

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

117

( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )

118

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 29 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

119

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )( )

( )( ) ( )

( )

( )( )

120

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

121

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

122

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )

123

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 30 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

124

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

125

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

126

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

127

( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )

128

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

129

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

130

Where we suppose

(A) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(B) The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

131

(C) ( )( ) ( ) ( )

( )

(D)

132

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 32 ISSN 2250-3153

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( )( ) (( ) ) ( )

( )

Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

133

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

(E) ( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

134

Definition of ( )( ) ( )

( ) :

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

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

135

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

136

The functions ( )( ) (

)( ) are positive continuous increasing and bounded

Definition of ( )( ) ( )

( ):

137

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 33 ISSN 2250-3153

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( )( )( ) ( )

( ) ( )( )

138

( )( )(( ) ) ( )

( ) ( )( ) ( )

( ) 139

( )( ) ( ) ( )

( )

140

( )( ) (( ) ) ( )

( ) 141

Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

142

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( ) 143

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

) and

( )( )( ) . (

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

( )] . It is to be

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

( )

then the function ( )( )( ) , the first augmentation coefficient attributable to the system, would

be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )( )

( )( ) ( )

( )

( )( )

144

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with ( )( ) ( )

( ) ( )( )

( )( ) and the constants ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

Satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

145

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

146

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

146A

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Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( ) ( )

( )

(

)( ) ( ) ( )( )

Definition of ( )

( ) ( )( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

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

)

and( )( )( ) . (

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

( )] . It is

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

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

Definition of ( )( ) ( )

( ) : There exists two constants ( )

( ) and ( )( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 35 ISSN 2250-3153

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

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

147

Theorem 2 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( )

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

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

148

Theorem 3 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

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

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

149

Theorem 4 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

150

Theorem 5 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

151

Theorem 6 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

152

Theorem 7: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

153

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 36 ISSN 2250-3153

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( ) ( )( )

( )( ) , ( )

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

Theorem 8: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

153

A

Theorem 9: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

153

B

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

which satisfy

154

( ) ( )

( )

( ) ( )

( ) 155

( ) ( )

( ) ( )( ) 156

( ) ( )

( ) ( )( ) 157

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

158

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 37 ISSN 2250-3153

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

Proof:

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

which satisfy

159

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

160

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

Proof:

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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

161

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 38 ISSN 2250-3153

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( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

162

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

163

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( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

Proof:

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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

164

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

Proof:

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

which satisfy

( ) ( )

( )

( ) ( )

( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 40 ISSN 2250-3153

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( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

165

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

Proof:

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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

166

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

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

166A

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

167

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 1

168

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Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

169

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

170

Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

171

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

172

Analogous inequalities hold also for

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

173

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 4

174

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

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From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 5

175

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

176

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 6

Analogous inequalities hold also for

177

(a) The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is

obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

178

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 7

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

180

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 8

181

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Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 9

Analogous inequalities hold also for

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

182

( )( )

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

( ) )

(( )( )

)

] ( )( )

183

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

184

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

Equations into itself

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

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

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

185

Indeed if we denote

Definition of : ( ) ( )( )

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 45 ISSN 2250-3153

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∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

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

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

186

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

)( ) depending also on can be considered as not

conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

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

From 19 to 24 it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

Definition of (( )( ))

(( )

( )) (( )

( ))

:

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

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

187

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.

188

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

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

189

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

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

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

190

( )( )

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

( ) )

(( )( )

)

] ( )( )

191

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

192

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

Equations into itself

193

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

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

194

Indeed if we denote

Definition of : ( ) ( )( )

195

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

196

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( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

197

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows 198

Remark 6: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as not

conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

199

Remark 7: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

200

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 8: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

201

Remark 9: 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.

202

Remark 10: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then 203

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Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

204

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

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

205

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

207

( )( )

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

( ) )

(( )( )

)

] ( )( )

208

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

209

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

Equations into itself

210

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

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

211

Indeed if we denote

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

212

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It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

213

| ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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

214

Remark 11: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

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

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

215

Remark 12: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

216

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 13: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

217

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

Remark 14: 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.

218

Remark 15: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

219

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

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

220

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

221

( )( )

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

( ) )

(( )( )

)

] ( )( )

222

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

223

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

Equations into itself

224

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

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

225

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It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on Equations it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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

226

Remark 16: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

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

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

227

Remark 17: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

228

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 18: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

229

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

Remark 19: 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.

230

Remark 20: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

231

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37

to 42

Analogous inequalities hold also for

232

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

233

( )( )

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

( ) )

(( )( )

)

] ( )( )

234

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

235

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

Equations into itself

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

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

236

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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

237

Remark 21: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

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

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

238

Remark 22: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

239

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 23: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

240

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(( )( ))

( )( )(( )

( )) (

)( )

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

Remark 24: 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.

241

Remark 25: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

242

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

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

Analogous inequalities hold also for

243

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

244

( )( )

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

( ) )

(( )( )

)

] ( )( )

245

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

246

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

Equations into itself

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

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

247

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Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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

248

Remark 26: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

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

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

249

Remark 27: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

250

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 28: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

251

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In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

Remark 29: 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.

252

Remark 30: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

253

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

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

254

Analogous inequalities hold also for

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

255

( )( )

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

( ) )

(( )( )

)

] ( )( )

256

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

257

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

Equations into itself

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

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

258

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 57 ISSN 2250-3153

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Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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

259

Remark 31: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

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

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

260

Remark 32: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

261

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 33: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

262

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(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

Remark 34: 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.

263

Remark 35: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

264

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

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

265

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

266

( )( )

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

( ) )

(( )( )

)

] ( )( )

267

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

268

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

Equations into itself

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

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 59 ISSN 2250-3153

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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

269

Indeed if we denote

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

270

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

271

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

272

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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

273

Remark 36: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

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

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

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

274

Remark 37 There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

275

Definition of (( )( ))

(( )

( )) (( )

( ))

:

276

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Remark 38: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

Remark 39: 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.

277

Remark 40: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

278

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

279

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose ( )( ) ( )

( ) large to have

279A

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

In order that the operator ( ) transforms the space of sextuples of functions satisfying 39,35,36 into itself

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

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 61 ISSN 2250-3153

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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( )) It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on 45,46,47,28 and 29 it follows

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

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis (39,35,36) the result follows

Remark 41: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as not conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.

Remark 42: There does not exist any where ( ) ( )

From 99 to 44 it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 43: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 62 ISSN 2250-3153

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(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively. Remark 44: 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 45: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then Definition of ( )( ) : Indeed let be so that for

( )( ) (

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

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

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

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 92

Behavior of the solutions of equation

Theorem If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

280

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( ) ( )( )

281

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( ) ( )( )

282

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :- 283

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If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

are defined

284

Then the solution of global equations satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation

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

( )( ) ( )( )

285

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

286

( )( ) ( )

(( )( ) ( )( )) 287

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 288

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

289

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

290

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( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

Behavior of the solutions of equation

Theorem 2: If we denote and define

291

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

292

( )( ) (

)( ) ( )( ) (

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

( ) 293

( )( ) (

)( ) ( )( ) (

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

( ) 294

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : 295

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots 296

of the equations ( )( )( ( ))

( )

( ) ( ) ( )( ) 297

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and 298

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : 299

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the 300

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

( )

( ) ( ) ( )( ) 301

and ( )( )( ( ))

( )

( ) ( ) ( )( ) 302

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :- 303

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by 304

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 305

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

306

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 307

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

308

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and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 309

Then the solution of global equations satisfies the inequalities

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

( )( )

310

( )( ) is defined by equation

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

( )( ) ( )( ) 311

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

312

( )( ) ( )

(( )( ) ( )( )) 313

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 314

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

315

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):- 316

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

317

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

318

Behavior of the solutions

Theorem 3: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

319

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 66 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

320

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

321

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) and ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

Then the solution of global equations satisfies the inequalities

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

( )( )

( )( ) is defined by equation

322

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

( )( ) ( )( ) 323

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

324

( )( ) ( )

(( )( ) ( )( )) 325

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( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 326

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

327

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

328

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

( ) ( )( ) ( )

( ) ( )( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

329

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : By ( )

( ) ( )( ) and respectively ( )

( ) ( )( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

330

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 68 ISSN 2250-3153

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( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

331

Then the solution of global equations satisfies the inequalities

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

( )( ) where ( )

( ) is defined by equation

332

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

( )( ) ( )( )

333

(( )( )

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

( )( ) ( )

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) )

334

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

335

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

336

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( ) ( )

( ) ( )( ) ( )

( )

337

Behavior of the solutions of equation Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

338

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 69 ISSN 2250-3153

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( )( ) (

)( ) ( )( ) (

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

( ) ( )

( ) ( )( ) (

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

)( )(( ) ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

339

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

340

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

341

Then the solution of global equations satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation

342

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

( )( ) ( )( )

343

(( )( )

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

( )( ) ( ) 344

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 70 ISSN 2250-3153

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( )( )

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

[ ( )( ) ( )( ) ]

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

345

( )( ) ( )( ) ( )

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

346

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

347

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):- Where ( )

( ) ( )( )( )

( ) ( )( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( ) ( )

( ) ( )( ) ( )

( )

348

Behavior of the solutions of equation Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( ) ( )

( ) ( )( ) (

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

)( )(( ) ) ( )( )

349

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

350

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

351

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 71 ISSN 2250-3153

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( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

352

Then the solution of global equations satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation

353

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

( )( ) ( )( )

354

(( )( )

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

( )( ) ( )

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) )

355

( )( ) ( )

(( )( ) ( )( ))

356

( )( ) ( )( ) ( )

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

357

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

358

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

359

Behavior of the solutions of equation

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 72 ISSN 2250-3153

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Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

361

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

362

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

363

Then the solution of global equations satisfies the inequalities 364

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 73 ISSN 2250-3153

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

( )( )

where ( )( ) is defined by equation

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

( )( ) ( )( )

365

(( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

366

( )( ) ( )

(( )( ) ( )( ))

367

( )( ) ( )( ) ( )

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

368

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

369

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

370

Behavior of the solutions of equation

Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

371

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) : 372

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 74 ISSN 2250-3153

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By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

374

Then the solution of global equations satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation

375

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

( )( ) ( )( )

376

( ( )( )

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

( )( ) ( ) 377

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 75 ISSN 2250-3153

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( )( )

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

( )( ) ( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

378

( )( ) ( )( ) ( )

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

379

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

380

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

381

Behavior of the solutions of equation 37 to 92 Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

382

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) : By ( )

( ) ( )( ) and respectively ( )

( ) ( )( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )

( ) ( )( ) ( )

( ) ( )( ) by

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 76 ISSN 2250-3153

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( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined by 59 and 69 respectively

Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation 45

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

( )( ) ( )( )

(

( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):- Where ( )

( ) ( )( )( )

( ) ( )( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 77 ISSN 2250-3153

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( )( ) (

)( ) ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

383

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

384

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

385

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

386

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

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for

the special case

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( ) and definition of ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

387

Definition of ( ) :- ( )

388

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

389

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

390

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In the same manner , we get

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

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

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

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

( )( ) ( )( )

391

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

( ) 392

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

393

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

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

394

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

395

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

396

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition

( )( ) ( )

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

( ) ( )

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( )

397

Proof : From global equations we obtain 398

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( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

399

From which one obtains

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

400

In the same manner , we get

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

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

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

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

( )( ) ( )( )

Definition of ( )( ) :-

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

( )

401

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

402

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

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

Definition of ( )( ) :-

403

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 81 ISSN 2250-3153

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

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

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

404

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

405

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If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

406

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

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

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

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

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( ) and definition of ( )( )

407

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

408

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From which one obtains Definition of ( )

( ) ( )( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )

( ) ( )( ) ( )( )

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

409

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

410

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

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

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

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

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case .

411

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Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( ) and definition of ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )

( ) ( )( ) ( )( )

412

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

413

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

414

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

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

Definition of ( )( ) :-

415

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

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

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

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

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( ) and definition of ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

416

In the same manner , we get

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

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

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

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

( )( ) ( )( )

417

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

( )

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

418

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

419

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for

the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then ( )( ) ( )

( )if in addition

( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between

( )( ) and ( )

( ) and definition of ( )( )

420

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

421

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Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

422

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

423

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 ( )( ) :-

424

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for

the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( ) and definition of ( )( )

Proof : From 99,20,44,22,23,44 we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

424A

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 89 ISSN 2250-3153

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( )( )

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

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

( )( )

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

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

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

( )( )

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 99, 20,44,22,23, and 44 we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )

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

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case . Analogously if (

)( ) ( )( ) ( )

( ) ( )( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

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

( ) and definition of ( )( )

We can prove the following

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

425

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations 426

( )( )(

)( ) ( )( )( )

( ) 427

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 90 ISSN 2250-3153

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( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) 428

( )( )(

)( ) ( )( )( )

( ) , 429

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

430

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

431

We can prove the following

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

432

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

433

Theorem If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

434

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 91 ISSN 2250-3153

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( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

435

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

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

436

Theorem : If ( )( ) (

)( ) are independent on , and the conditions (with the notations

45,46,27,28)

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation 45 are satisfied , then the system

436

A

( )( ) [(

)( ) ( )( )( )] 437

( )( ) [(

)( ) ( )( )( )] 438

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 92 ISSN 2250-3153

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( )( ) [(

)( ) ( )( )( )] 439

( )( ) (

)( ) ( )( )( ) 440

( )( ) (

)( ) ( )( )( ) 441

( )( ) (

)( ) ( )( )( ) 442

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

( )( ) [(

)( ) ( )( )( )] 443

( )( ) [(

)( ) ( )( )( )] 444

( )( ) [(

)( ) ( )( )( )] 445

( )( ) (

)( ) ( )( )( ) 446

( )( ) (

)( ) ( )( )( ) 447

( )( ) (

)( ) ( )( )( ) 448

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )] 449

( )( ) [(

)( ) ( )( )( )] 450

( )( ) [(

)( ) ( )( )( )] 451

( )( ) (

)( ) ( )( )( ) 452

( )( ) (

)( ) ( )( )( ) 453

( )( ) (

)( ) ( )( )( ) 454

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

455

( )( ) [(

)( ) ( )( )( )]

456

( )( ) [(

)( ) ( )( )( )]

457

( )( ) (

)( ) ( )( )(( ))

458

( )( ) (

)( ) ( )( )(( ))

459

( )( ) (

)( ) ( )( )(( ))

460

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 93 ISSN 2250-3153

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has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

461

( )( ) [(

)( ) ( )( )( )]

462

( )( ) [(

)( ) ( )( )( )]

463

( )( ) (

)( ) ( )( )( )

464

( )( ) (

)( ) ( )( )( )

465

( )( ) (

)( ) ( )( )( )

466

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

467

( )( ) [(

)( ) ( )( )( )]

468

( )( ) [(

)( ) ( )( )( )]

469

( )( ) (

)( ) ( )( )( )

470

( )( ) (

)( ) ( )( )( )

471

( )( ) (

)( ) ( )( )( )

472

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

473

( )( ) [(

)( ) ( )( )( )]

474

( )( ) [(

)( ) ( )( )( )]

475

( )( ) (

)( ) ( )( )( )

476

( )( ) (

)( ) ( )( )( )

477

( )( ) (

)( ) ( )( )( )

478

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 94 ISSN 2250-3153

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( )( ) [(

)( ) ( )( )( )]

479

( )( ) [(

)( ) ( )( )( )]

480

( )( ) [(

)( ) ( )( )( )]

481

( )( ) (

)( ) ( )( )( )

482

( )( ) (

)( ) ( )( )( )

483

( )( ) (

)( ) ( )( )( )

484

( )( ) [(

)( ) ( )( )( )] 484

A

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

485

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

486

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

487

Proof: 488

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 95 ISSN 2250-3153

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

489

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

490

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

491

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

492

Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

492

A

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

493

Definition and uniqueness of :-

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

there exists a unique for which (

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

494

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equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

495

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

496

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

497

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

498

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

499

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

500

Definition and uniqueness of :-

501

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 97 ISSN 2250-3153

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

Definition and uniqueness of :-

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

there exists a unique for which (

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

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

501

A

By the same argument, the 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 ( )

502

By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )

503

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 (( )

)

504

By the same argument, the 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 (( )

)

505

By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

506

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[( )( )(

)( )( ) ( )( )(

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

)( )( )

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 (( )

)

By the same argument, the 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 (( )

)

507

By the same argument, the 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 (

)

508

By the same argument, the 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 (

)

509

By the same argument, the 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 (

)

510

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 (( )

)

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 99 ISSN 2250-3153

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Finally we obtain the unique solution

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )] ,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution

511

Finally we obtain the unique solution

(( )

) , (

) and 512

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

513

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )] 514

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution of global equations

515

Finally we obtain the unique solution

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

516

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

Obviously, these values represent an equilibrium solution of global equations

517

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

518

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 100 ISSN 2250-3153

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( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

Obviously, these values represent an equilibrium solution of global equations

519

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

520

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

Obviously, these values represent an equilibrium solution of global equations

521

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

522

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

523

Finally we obtain the unique solution of 89 to 99

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

523

A

ASYMPTOTIC STABILITY ANALYSIS 524

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 101 ISSN 2250-3153

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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 and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

525

((

)( ) ( )( )) ( )

( ) ( )( )

526

((

)( ) ( )( )) ( )

( ) ( )( )

527

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

528

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

529

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

530

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

531

Proof: Denote

Definition of :-

,

532

( )( )

(

) ( )( ) ,

( )( )

( ( )

) 533

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

((

)( ) ( )( )) ( )

( ) ( )( )

534

((

)( ) ( )( )) ( )

( ) ( )( )

535

((

)( ) ( )( )) ( )

( ) ( )( )

536

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

537

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 102 ISSN 2250-3153

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((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

538

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

539

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 :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

540

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

((

)( ) ( )( )) ( )

( ) ( )( )

541

((

)( ) ( )( )) ( )

( ) ( )( )

542

((

)( ) ( )( )) ( )

( ) ( )( )

543

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

544

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

545

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

546

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

547

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

(( )

)

548

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

((

)( ) ( )( )) ( )

( ) ( )( )

549

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((

)( ) ( )( )) ( )

( ) ( )( )

550

((

)( ) ( )( )) ( )

( ) ( )( )

551

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

552

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

553

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

554

ASYMPTOTIC STABILITY ANALYSIS

Theorem 5: 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

555

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

556

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

((

)( ) ( )( )) ( )

( ) ( )( )

557

((

)( ) ( )( )) ( )

( ) ( )( )

558

((

)( ) ( )( )) ( )

( ) ( )( )

559

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

560

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

561

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

562

ASYMPTOTIC STABILITY ANALYSIS

Theorem 6: 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

563

Definition of :-

,

564

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(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

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

((

)( ) ( )( )) ( )

( ) ( )( )

565

((

)( ) ( )( )) ( )

( ) ( )( )

566

((

)( ) ( )( )) ( )

( ) ( )( )

567

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

568

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

569

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

570

ASYMPTOTIC STABILITY ANALYSIS

Theorem 7: 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

571

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

572

Then taking into account equations and neglecting the terms of power 2, we obtain from

((

)( ) ( )( )) ( )

( ) ( )( )

573

((

)( ) ( )( )) ( )

( ) ( )( )

574

((

)( ) ( )( )) ( )

( ) ( )( )

575

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

576

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

578

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

579

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Obviously, these values represent an equilibrium solution

ASYMPTOTIC STABILITY ANALYSIS

Theorem 8: 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 :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

580

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

((

)( ) ( )( )) ( )

( ) ( )( )

581

((

)( ) ( )( )) ( )

( ) ( )( )

582

((

)( ) ( )( )) ( )

( ) ( )( )

583

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

584

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

585

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586

ASYMPTOTIC STABILITY ANALYSIS Theorem 9: 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

586A

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

Then taking into account equations 89 to 99 and neglecting the terms of power 2, we obtain from 99 to 44

((

)( ) ( )( )) ( )

( ) ( )( )

586B

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((

)( ) ( )( )) ( )

( ) ( )( )

586 C

((

)( ) ( )( )) ( )

( ) ( )( )

586 D

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586 E

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586 F

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586

G The characteristic equation of this system is 587

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

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((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

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((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

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((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

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((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and

this proves the theorem.

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Hierarchial Problem-Part Four: Fingerspitzengefuhl: An Intuitive

Exposition

INTRODUCTION—VARIABLES USED

Source: Wikipedia: What is Hierarchial problem Matt Strassler: emphasis mine

An important feature of nature that puzzles scientists like is known as the hierarchy, meaning the vast

discrepancy between aspects of the weak nuclear force and gravity. There are several different ways to

describe this hierarchy, each emphasizing a different feature of it. Here is one:

The mass of the smallest possible black hole defines what is known as the Planck Mass. [Planck was the

scientist who took the first step towards quantum mechanics.] (A more precise way to define this is as a

combination of Newton’s gravitational constant G, Planck’s quantum constant h-bar, and the speed of

light c: the Planck mass is the square root of h-bar times c divided by G.) The masses of the W and Z

particles, the force carriers of the weak nuclear force, are about 10,000,000,000,000,000 times smaller than

the Planck Mass. Thus there is a huge hierarchy in the mass scales of weak nuclear forces and gravity.

When faced with such a large number as 10,000,000,000,000,000, ten quadrillion, the question that

physicists are naturally led to ask is: where did that number come from? It might have some sort of

interesting explanation. But while trying to figure out a possible explanation, physicists in the 1970s realized

there was actually a serious problem, even a paradox, behind this number. The issue, now called the

hierarchy problem, has to do with the size of the non-zero Higgs field, which in turn determines the mass of

the W and Z particles. The non-zero Higgs field has a size of about 250 GeV, and that gives us the W and Z

particles with masses of about 100 GeV. But it turns out that quantum mechanics would lead us to

expect that this size of a Higgs field is unstable, something like (warning: imperfect analogy ahead) a vase

balanced precariously on the edge of a table. With the physics we know about so far, the tendency of

quantum mechanics to jostle — those quantum fluctuations I’ve mentioned elsewhere — would seem to

imply that there are two natural values for the Higgs field — in analogy to the two natural places for the

vase, firmly placed on the table or smashed on the floor. Naively, the Higgs field should either be zero, or it

should be as big as the Planck Energy, 10,000,000,000,000,000 times larger than it is observed to be. Why is

it at a value that is non-zero and tiny, a value that seems, at least naively, so unnatural?

Many theoretical physicists have devoted significant fractions of their careers to trying to solve this problem.

Some have argued that new particles and new forces are needed (and their theories go by names such as

supersymmetry, Technicolor, little Higgs, etc.) Some have argued that our understanding of gravity is

mistaken and that there are new unknown dimensions (“extra dimensions”) of space that will become

apparent to our experiments at the Large Hadron collider in the near future. Others have argued that there is

nothing to explain, because of a selection effect: the universe is far larger and far more diverse than the part

that we can see, and we live in an apparently unnatural part of the universe mainly because the rest of it is

uninhabitable — much the way that although rocky planets are rare in the universe, we live on one because

it’s the only place we could have evolved and survived. There may be other solutions to this problem that

have not yet been invented.

Many of these solutions — certainly all the ones with new particles and forces or with new dimensions —

predict that new phenomena should be visible at the Large Hadron Collider. Even as I write this, the Large

Hadron Collider is slowly but surely excluding many of these possibilities. So far it has not seen any

unexpected new phenomena. But these are still early days. By the way, you will often read the

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hierarchy problem stated as a problem with the Higgs particle mass. This is incorrect. The problem

is with how big the non-zero Higgs field is. (For experts — quantum mechanics corrects not the Higgs

particle mass but the Higgs mass-squared parameter, changing the Higgs field potential energy and thus the

field’s value, making it zero or immense. That’s a disaster because the W and Z masses are known. The

Higgs mass is unknown, and therefore it could be very large — if the W and Z masses were very large too.

So it is the W and Z masses — and the size of the non-zero Higgs field — that are the problem, both

logically and scientifically.)

National Geographic Society: The Greatest Unsolved Problem in Theoretical Physics

Posted by Ethan on September 19, 2012

(1) New Kaluza-Klein particles could be a possible solution to the hierarchy problem with

one extra (eb) consequence of this model would be that gravity would radically depart from

Newton’s law at distances below a millimeter, something that’s been incredibly difficult to test.

Modern experimentalists, however, are more than up to the challenge.

Images credit: Cryogenic Helium Turbulence and Hydrodynamics activity at cnrs.fr.

(2) Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release electrical

energies when their shape is changed / when they are torqued) can be (eb) created

with spacings of mere microns between them, as shown above.

(3) This new technique allows us to place constraints that if there are “large” extra dimensions,

they’re smaller than around 5-10 microns.

(4) This means that if there are large extra dimensions, they’re at energies that are both not (e)

accessible to the LHC and that do not solve the hierarchy problem.

Of course, there either could be a completely different solution to the hierarchy problem or there may be no

solution at all; this could just be the way nature is, and there may be no explanation for it. But science will

never progress unless we try, and that’s what these ideas and searches are: our attempt to move our

knowledge of the Universe forward. And as always, I can’t wait to see what — beyond the already-

discovered Higgs boson — the LHC turns up!

Bjoern September 20, 2012 (Post for Ethan’s Blog)

(5) @crd2: Possibly all the supersymmetric partner particles are unstable, and all the ones which

originally existed have long decayed, so (eb) that now only virtual ones exist (as Wow said).

(6) But quite a lot of physicists think that at least one of these particles still exists, the so-called

“neutralino”. These particles could (e) make up the Dark Matter in the universe. (so, in a

sence, they are really “bouncing around with all the other stuff” – but in another sense, they

don’t, because they probably interact with other matter only very weakly or even not at all; and

no, they are definitely not components of other, “normal” particles)

I don’t understand what your question “are they supposed to be in this dimension alone or in another one, or

in both at the same time” is supposed to mean. Particles don’t exist “in a dimension”, that makes little

sense…

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(7) Regardless if these particles really exist in the universe or only virtually – if their existence is

possible at all, the LHC should be able to create them.

(8) “Possibly all the supersymmetric partner particles are unstable” It is more like there are so

many things that these particles CAN decay in to and these decay products release (eb) energy.

Since this is energetically beneficial, it does so. Because so much energy is released, it does so

with alacrity.

September 20, 2012

(9) But why is gravity considered as a force? All the other 3 fundamental forces have (e) particles,

while we know that gravity is a curvature of space time. Which means it is (=) a property of

our universe and not a force. Then why consider it to be a fundamental force?

NOTATION

Module One

New Kaluza-Klein particles could be a possible solution to the hierarchy problem with one extra (eb)

consequence of this model would be that gravity would radically depart from Newton’s law at distances

below a millimeter, something that’s been incredibly difficult to test. Modern experimentalists, however,

are more than up to the challenge

: Category one of Kaluza-Klein particles could be a possible solution to the hierarchy problem

: Category two of Kaluza-Klein particles could be a possible solution to the hierarchy problem

: Category three of Kaluza-Klein particles could be a possible solution to the hierarchy problem

: Category one of gravity would radically depart from Newton’s law at distances below a millimeter,

something that’s been incredibly difficult to test. Modern experimentalists, however, are more than up to the

challenge

: Category two of gravity would radically depart from Newton’s law at distances below a millimeter,

something that’s been incredibly difficult to test. Modern experimentalists, however, are more than up to the

challenge

: Category three of gravity would radically depart from Newton’s law at distances below a millimeter,

something that’s been incredibly difficult to test. Modern experimentalists, however, are more than up to the

challenge

Module Two

Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release electrical energies

when their shape is changed / when they are torqued) can be (eb) created with spacings of mere microns

between them, as shown above

: Category one of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued)

: Category two of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued)

: Category three of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued)

: Category one of spacings of mere microns between them, as shown above

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: Category two of spacings of mere microns between them, as shown above

: Category three of spacings of mere microns between them, as shown above

Module three

This new technique allows us to place constraints that if there are “large” extra dimensions, they’re smaller

than around 5-10 microns

: Category one of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued) can be created with spacings of

mere microns between them

: Category two of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued) can be created with spacings of

mere microns between them

: Category three of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that

release electrical energies when their shape is changed / when they are torqued) can be created

with spacings of mere microns between them

: Category one of constraints that if there are “large” extra dimensions, they’re smaller than around 5-10

microns

: Category two of constraints that if there are “large” extra dimensions, they’re smaller than around 5-

10 microns

: Category three of constraints that if there are “large” extra dimensions, they’re smaller than around 5-

10 microns

Module four

This means that if there are large extra dimensions, they’re at energies that are both not (e) accessible to the

LHC and that do not solve the hierarchy problem

: Category one of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued) can be created with spacings of

mere microns between them allows us to place constraints that if there are “large” extra dimensions, they’re

smaller than around 5-10 microns

: Category two of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that release

electrical energies when their shape is changed / when they are torqued) can be created with spacings of

mere microns between them allows us to place constraints that if there are “large” extra dimensions, they’re

smaller than around 5-10 microns

: Category three of Tiny, supercooled cantilevers, loaded with piezoelectric crystals (crystals that

release electrical energies when their shape is changed / when they are torqued) can be created

with spacings of mere microns between them allows us to place constraints that if there are “large” extra

dimensions, they’re smaller than around 5-10 microns

: Category one of there are large extra dimensions, they’re at energies that are both not accessible to the

LHC and that do not solve the hierarchy problem

: Category two of there are large extra dimensions, they’re at energies that are both not accessible to the

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LHC and that do not solve the hierarchy problem

: Category three of there are large extra dimensions, they’re at energies that are both not accessible to

the LHC and that do not solve the hierarchy problem

Module five

Possibly all the supersymmetric partner particles are unstable, and all the ones which originally existed have

long decayed, so (eb) that now only virtual ones exist

: Category one of Possibly all the supersymmetric partner particles are unstable, and all the ones which

originally existed have long decayed

: Category two of Possibly all the supersymmetric partner particles are unstable, and all the ones which

originally existed have long decayed

: Category three of Possibly all the supersymmetric partner particles are unstable, and all the ones

which originally existed have long decayed

: Category one of only virtual ones exist

: Category two of only virtual ones exist

: Category three of only virtual ones exist

Module six

But quite a lot of physicists think that at least one of these particles still exists, the so-called “neutralino”.

These particles could (e) make up the Dark Matter in the universe. (so, in a sence, they are really “bouncing

around with all the other stuff” – but in another sense, they don’t, because they probably interact with other

matter only very weakly or even not at all; and no, they are definitely not components of other, “normal”

particles)

: Category one of Dark Matter in the universe. (so, in a sence, they are really “bouncing around with all

the other stuff” – but in another sense, they don’t, because they probably interact with other matter only very

weakly or even not at all; and no, they are definitely not components of other, “normal” particles)

: Category two of Dark Matter in the universe. (so, in a sence, they are really “bouncing around with all

the other stuff” – but in another sense, they don’t, because they probably interact with other matter only very

weakly or even not at all; and no, they are definitely not components of other, “normal” particles)

: Category three of Dark Matter in the universe. (so, in a sence, they are really “bouncing around with

all the other stuff” – but in another sense, they don’t, because they probably interact with other matter only

very weakly or even not at all; and no, they are definitely not components of other, “normal” particles)

: Category one of “neutralino”. Systems that have particles neutralino are taken in to consideration and

characterstics form the basis for the classification scheme.

: Category two of “neutralino”.

: Category three of “neutralino”.

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Module seven

regardless if these particles really exist in the universe or only virtually – if their existence is possible at all,

the LHC should be able to create them

: Category one of regardless if these particles really exist in the universe or only virtually – if their

existence is possible at all, the LHC

: Category two of regardless if these particles really exist in the universe or only virtually – if their

existence is possible at all, the LHC

: Category three of regardless if these particles really exist in the universe or only virtually – if their

existence is possible at all, the LHC

: Category one of “neutralino”.

: Category two of “neutralino”.

: Category three of “neutralino”.

Module eight

“Possibly all the supersymmetric partner particles are unstable” It is more like there are so many things that

these particles CAN decay in to and these decay products release (eb) energy. Since this is energetically

beneficial, it does so. Because so much energy is released, it does so with alacrity

: Category one of “Possibly all the supersymmetric partner particles are unstable” It is more like there

are so many things that these particles CAN decay in to and these decay products

: Category two of “Possibly all the supersymmetric partner particles are unstable” It is more like there

are so many things that these particles CAN decay in to and these decay products

: Category three of “Possibly all the supersymmetric partner particles are unstable” It is more like there

are so many things that these particles CAN decay in to and these decay products

: Category one of energy. Since this is energetically beneficial, it does so. Because so much energy is

released, it does so with alacrity

: Category two of energy. Since this is energetically beneficial, it does so. Because so much energy is

released, it does so with alacrity

: Category three of energy. Since this is energetically beneficial, it does so. Because so much energy is

released, it does so with alacrity

Module Nine

But why is gravity considered as a force? All the other 3 fundamental forces have (e) particles, while we

know that gravity is a curvature of space time. Which means it is a property of our universe and not a force.

Then why consider it to be a fundamental force?

: Category one of particles, while we know that gravity is a curvature of space time; property of our

universe and not a force

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: Category two of particles, while we know that gravity is a curvature of space time; property of our

universe and not a force

: Category three of particles, while we know that gravity is a curvature of space time; property of our

universe and not a force

: Category one of 3 fundamental forces; gravity

: Category two of 3 fundamental forces; gravity

: Category three of 3 fundamental forces; gravity

The Coefficients:

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ):

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ),( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

are Accentuation coefficients

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ,

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

are Dissipation coefficients

Module Numbered One

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

( )

( ) [( )( ) (

)( )( )] 1

( )

( ) [( )( ) (

)( )( )] 2

( )

( ) [( )( ) (

)( )( )] 3

( )

( ) [( )( ) (

)( )( )] 4

( )

( ) [( )( ) (

)( )( )] 5

( )

( ) [( )( ) (

)( )( )] 6

( )( )( ) First augmentation factor

( )( )( ) First detritions factor

Module Numbered Two

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

( )

( ) [( )( ) (

)( )( )] 7

( )

( ) [( )( ) (

)( )( )] 8

( )

( ) [( )( ) (

)( )( )] 9

( )

( ) [( )( ) (

)( )(( ) )] 10

( )

( ) [( )( ) (

)( )(( ) )] 11

( )

( ) [( )( ) (

)( )(( ) )] 12

( )( )( ) First augmentation factor

( )( )(( ) ) First detritions factor

Module Numbered Three

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

( )

( ) [( )( ) (

)( )( )] 13

( )

( ) [( )( ) (

)( )( )] 14

( )

( ) [( )( ) (

)( )( )] 15

( )

( ) [( )( ) (

)( )( )] 16

( )

( ) [( )( ) (

)( )( )] 17

( )

( ) [( )( ) (

)( )( )] 18

( )( )( ) First augmentation factor

( )( )( ) First detritions factor

Module Numbered Four

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

( )

( ) [( )( ) (

)( )( )] 19

( )

( ) [( )( ) (

)( )( )] 20

( )

( ) [( )( ) (

)( )( )] 21

( )

( ) [( )( ) (

)( )(( ) )] 22

( )

( ) [( )( ) (

)( )(( ) )] 23

( )

( ) [( )( ) (

)( )(( ) )] 24

( )( )( ) First augmentation factor

( )( )(( ) ) First detritions factor

Module Numbered Five:

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

( )

( ) [( )( ) (

)( )( )] 25

( )

( ) [( )( ) (

)( )( )] 26

( )

( ) [( )( ) (

)( )( )] 27

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 119 ISSN 2250-3153

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( )

( ) [( )( ) (

)( )(( ) )] 28

( )

( ) [( )( ) (

)( )(( ) )] 29

( )

( ) [( )( ) (

)( )(( ) )] 30

( )( )( ) First augmentation factor

( )( )(( ) ) First detritions factor

Module Numbered Six

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

( )

( ) [( )( ) (

)( )( )] 31

( )

( ) [( )( ) (

)( )( )] 32

( )

( ) [( )( ) (

)( )( )] 33

( )

( ) [( )( ) (

)( )(( ) )] 34

( )

( ) [( )( ) (

)( )(( ) )] 35

( )

( ) [( )( ) (

)( )(( ) )] 36

( )( )( ) First augmentation factor

Module Numbered Seven:

The differential system of this model is now (Seventh Module)

( )

( ) [( )( ) (

)( )( )] 37

( )

( ) [( )( ) (

)( )( )] 38

( )

( ) [( )( ) (

)( )( )] 39

( )

( ) [( )( ) (

)( )(( ) )] 40

( )

( ) [( )( ) (

)( )(( ) )] 41

( )

( ) [( )( ) (

)( )(( ) )] 42

( )( )( ) First augmentation factor

Module Numbered Eight

The differential system of this model is now

( )

( ) [( )( ) (

)( )( )] 43

( )

( ) [( )( ) (

)( )( )] 44

( )

( ) [( )( ) (

)( )( )] 45

( )

( ) [( )( ) (

)( )(( ) )] 46

( )

( ) [( )( ) (

)( )(( ) )] 47

( )

( ) [( )( ) (

)( )(( ) )] 48

Module Numbered Nine

The differential system of this model is now

( )

( ) [( )( ) (

)( )( )] 49

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 120 ISSN 2250-3153

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( )

( ) [( )( ) (

)( )( )] 50

( )

( ) [( )( ) (

)( )( )] 51

( )

( ) [( )( ) (

)( )(( ) )] 52

( )

( ) [( )( ) (

)( )(( ) )] 53

( )

( ) [( )( ) (

)( )(( ) )] 54

( )( )( ) First augmentation factor

( )( )(( ) ) First detrition factor

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

55

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

56

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

57

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second augmentation coefficient for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficient for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fourth augmentation

coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficient

for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation coefficient

for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are seventh augmentation coefficient for

1,2,3

( )( )( ) (

)( )( ) ( )( )( ) are eight augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth

augmentation coefficient for 1,2,3

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

58

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( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

59

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

60

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for category 1,

2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients

for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detrition coefficients for

category 1, 2 and 3

– ( )( )( ) – (

)( )( ) , – ( )( )( ) are seventh detrition coefficients for

category 1, 2 and 3

– ( )( )( ) – (

)( )( ) – ( )( )( ) are eight detrition coefficients for category 1,

2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth

detrition coefficients for category 1, 2 and 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

61

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

62

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

63

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 122 ISSN 2250-3153

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( )( )( ) , (

)( )( ) , ( )( )( ) are second augmentation coefficient for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficient for

category 1, 2 and 3

( )( )( ) (

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

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth augmentation

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation

coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are seventh augmentation coefficient

for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are eight augmentation coefficient for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation coefficient for

category 1, 2 and 3

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

64

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

65

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

66

( )( )( ) , (

)( )( ) , ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are second detrition coefficients for category

1,2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition coefficients for

category 1,2 and 3

( )( )( ) (

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

coefficients for category 1,2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition coefficients

for category 1,2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are sixth detrition coefficients

for category 1,2 and 3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 123 ISSN 2250-3153

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– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition coefficients for

category 1,2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eight detrition coefficients for

category 1,2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients for category

1,2 and 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

67

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

68

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

69

( )( )( ) , (

)( )( ) , ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are second augmentation coefficients

for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficients

for category 1, 2 and 3

( )( )( ) , (

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

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are seventh augmentation

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are eight augmentation coefficients

for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation coefficients for

category 1, 2 and 3

( )

( )

[

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

70

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( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

71

( )

( )

[

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

72

( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for category 1,

2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are third detrition coefficients for category

1,2 and 3

( )( )( ) (

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

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are seventh detrition coefficients

for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eight detrition coefficients for

category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients for

category 1, 2 and 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

73

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

74

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

75

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( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) , (

)( )( ) ( )( )( )

( )( )( ) , (

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

( )( )( ) , (

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

( )( )( ) , (

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

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth detrition coefficients for

category 1 2 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

76

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

77

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

78

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )( )( ) , (

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

– ( )( )( ) – (

)( )( ) – ( )( )( )

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– ( )( )( ) , – (

)( )( ) , – ( )( )( )

– ( )( )( ) , – (

)( )( ) – ( )( )( )

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients for

category 1 2 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

79

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

80

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

81

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

coefficients for category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation

coefficients for category 1,2,and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation

coefficients for category 1,2, 3

( )( )( ) , (

)( )( ) , ( )( )( ) are seventh augmentation

coefficients for category 1,2, 3

( )( )( ) , (

)( )( ) , ( )( )( ) are eighth augmentation

coefficients for category 1,2, 3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation

coefficients for category 1,2, 3

( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

82

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 127 ISSN 2250-3153

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( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

83

( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

84

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for

category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition coefficients

for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition coefficients

for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition coefficients

for category 1,2, and 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

85

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

86

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

87

( )( )( ) (

)( )( ) ( )( )( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 128 ISSN 2250-3153

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( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

coefficients

( )( )( ) (

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

coefficients

( )( )( ) , (

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

coefficients

( )( )( ) , (

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

( )( )( ) , (

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

Eighth augmentation coefficients

( )( )( ) , (

)( )( ) , ( )( )( ) ninth augmentation

coefficients

( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

88

( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

89

( )

( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

90

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients

for category 1, 2, and 3

( )( )( ) , (

)( )( ) ( )( )( ) are fifth detrition

coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 129 ISSN 2250-3153

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coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( )

are eighth detrition coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition

coefficients for category 1, 2, and 3

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

91

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

92

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

93

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation

coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are seventh

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

are eighth augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth augmentation

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 130 ISSN 2250-3153

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coefficient for 1,2,3

( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

94

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second detrition

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition

coefficients for category 1, 2 and 3

( )( )( ) (

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

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( )

are seventh detrition coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition

coefficients for category 1, 2 and 3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 131 ISSN 2250-3153

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( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

95

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

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

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are seventh

augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are eighth

augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth

augmentation coefficient for 1,2,3

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 132 ISSN 2250-3153

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( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second

detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition

coefficients for category 1, 2 and 3

( )( )( ) (

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

detrition coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , – ( )( )( ) are sixth detrition coefficients

for category 1, 2 and 3

– ( )( )( ) – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth detrition

coefficients for category 1, 2 and 3

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) ]

96

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 133 ISSN 2250-3153

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( )

( )

[

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) ]

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) (

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first augmentation coefficients for

category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are second

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fourth

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth

augmentation coefficient for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are Seventh

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are eighth

augmentation coefficient for 1,2,3

( )( )( ) , (

)( )( ) , ( )( )( ) are ninth

augmentation coefficient for 1,2,3

( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 134 ISSN 2250-3153

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( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (

)( )( ) ( )( )( ) are first detrition coefficients for

category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are second

detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition

coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth

detrition coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition

coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh detrition

coefficients for category 1, 2 and 3

– ( )( )( ) – (

)( )( ) , – ( )( )( ) are eighth detrition

coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth

detrition coefficients for category 1, 2 and 3

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

97

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )

98

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 135 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )( ) (

)( )( ) ( )( ) ( )( )

99

With the Lipschitz condition, we place a restriction on the behavior of functions

( )( )(

) and( )( )( ) (

) and ( ) are points belonging to the interval

[( )( ) ( )

( )] . It is to be noted that ( )( )( ) is uniformly continuous. In the eventuality of

the fact, that if ( )( ) then the function (

)( )( ) , the first augmentation coefficient

attributable to the system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

100

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( )which together With ( )( ) ( )

( ) ( )( )

and ( )( ) and the constants ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

101

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

102

( )( )( ) ( )

( ) ( )( ) ( )

( ) 103

( )( ) ( ) ( )

( ) 104

( )( ) (( ) ) ( )

( ) 105

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 136 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

106

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( ) 107

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( ) 108

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

109

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together

with ( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 110

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 111

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

112

( )( ) ( ) ( )

( ) 113

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 137 ISSN 2250-3153

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( )( ) ( ) ( )

( )

Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(

) ( )( )( ) ( )

( ) ( )( )

114

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) And ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

115

There exists two constants There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

116

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

117

( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )

118

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 138 ISSN 2250-3153

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

119

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )( )

( )( ) ( )

( )

( )( )

120

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

121

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

122

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )

123

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

124

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

125

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

126

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

127

( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )

128

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Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

129

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

130

Where we suppose

(G) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(H) The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

131

(I) ( )( ) ( ) ( )

( )

(J)

132

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( )( ) (( ) ) ( )

( )

Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

133

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the

system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

(K) ( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

134

Definition of ( )( ) ( )

( ) :

(L) There exists two constants ( )( ) and ( )

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

135

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

136

The functions ( )( ) (

)( ) are positive continuous increasing and bounded

Definition of ( )( ) ( )

( ):

137

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( )( )( ) ( )

( ) ( )( )

138

( )( )(( ) ) ( )

( ) ( )( ) ( )

( ) 139

( )( ) ( ) ( )

( )

140

( )( ) (( ) ) ( )

( ) 141

Definition of ( )( ) ( )

( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

142

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( ) 143

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

) and

( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is to be

noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if ( )

( )

then the function ( )( )( ) , the first augmentation coefficient attributable to the system, would

be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )( )

( )( ) ( )

( )

( )( )

144

Definition of ( )( ) ( )

( ) :

There exists two constants ( )( ) and ( )

( ) which together with ( )( ) ( )

( ) ( )( )

( )( ) and the constants ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

Satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

145

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

146

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

146A

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Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( ) ( )

( )

(

)( ) ( ) ( )( )

Definition of ( )

( ) ( )( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(

)

and( )( )( ) . (

) and ( ) are points belonging to the interval [( )( ) ( )

( )] . It is

to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if

( )( ) then the function (

)( )( ) , the first augmentation coefficient attributable to the system, would be absolutely continuous.

Definition of ( )( ) ( )

( ) :

( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )

Definition of ( )( ) ( )

( ) : There exists two constants ( )

( ) and ( )( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 144 ISSN 2250-3153

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Theorem 1: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

147

Theorem 2 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( )

( ) ( )( ) ( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

148

Theorem 3 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

( ) ( )( ) ( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

149

Theorem 4 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

150

Theorem 5 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

151

Theorem 6 : if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

152

Theorem 7: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

153

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( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

Theorem 8: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

153

A

Theorem 9: if the conditions above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

( ) ( )( ) ( )( ) , ( )

153

B

Proof: Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

154

( ) ( )

( )

( ) ( )

( ) 155

( ) ( )

( ) ( )( ) 156

( ) ( )

( ) ( )( ) 157

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

158

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 146 ISSN 2250-3153

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Where ( ) is the integrand that is integrated over an interval ( )

Proof:

Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

159

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

160

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof:

Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

161

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

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( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof: Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

162

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof: Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

163

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( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof:

Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

164

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof:

Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

( ) ( )

( )

( ) ( )

( )

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( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

165

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof:

Consider operator ( ) defined on the space of sextuples of continuous functions

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

166

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

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( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

Proof: Consider operator ( ) defined on the space of sextuples of continuous functions which satisfy

166A

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

( ) ∫ [( )

( ) ( ( )) (( )( ) (

)( )( ( ( )) ( ))) ( ( ))] ( )

Where ( ) is the integrand that is integrated over an interval ( )

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

167

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 1

168

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Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

169

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

170

Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

171

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

172

Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying into itself .Indeed it is obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

173

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 4

174

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

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From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 5

175

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

176

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 6

Analogous inequalities hold also for

177

(b) The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is

obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

178

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 7

The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

180

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 8

181

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Analogous inequalities hold also for

The operator ( ) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious

that

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 9

Analogous inequalities hold also for

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

182

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

183

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

184

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

The operator ( ) is a contraction with respect to the metric

(( ( ) ( )) ( ( ) ( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

185

Indeed if we denote

Definition of : ( ) ( )( )

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

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∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) (( ( ) ( ) ( ) ( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

186

Remark 1: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as not

conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( ) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any where ( ) ( )

From 19 to 24 it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 3: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively.

187

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.

188

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Remark 5: If is bounded from below and (( )( ) ( ( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )( ( ) ) ( ) ( )( )

189

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) ( ( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

190

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

191

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

192

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

193

The operator ( ) is a contraction with respect to the metric

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

194

Indeed if we denote

Definition of : ( ) ( )( )

195

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

196

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( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

197

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows 198

Remark 6: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as not

conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

199

Remark 7: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

200

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 8: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively.

201

Remark 9: 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.

202

Remark 10: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then 203

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Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

204

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations

205

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

207

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

208

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

209

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

210

The operator ( ) is a contraction with respect to the metric

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

211

Indeed if we denote

Definition of :( ( ) ( ) ) ( )(( ) ( ))

212

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It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

213

| ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

214

Remark 11: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

215

Remark 12: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

216

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 13: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

217

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 159 ISSN 2250-3153

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If is bounded, the same property follows for and respectively.

Remark 14: 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.

218

Remark 15: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

219

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations

220

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

221

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

222

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

223

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

224

The operator ( ) is a contraction with respect to the metric

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

225

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It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on Equations it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

226

Remark 16: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

227

Remark 17: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

228

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 18: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

229

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 161 ISSN 2250-3153

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If is bounded, the same property follows for and respectively.

Remark 19: 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.

230

Remark 20: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

231

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37

to 42

Analogous inequalities hold also for

232

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

233

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

234

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

235

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

The operator ( ) is a contraction with respect to the metric

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

236

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Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

237

Remark 21: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

238

Remark 22: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

239

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 23: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

240

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 163 ISSN 2250-3153

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(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively.

Remark 24: 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.

241

Remark 25: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

242

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations

Analogous inequalities hold also for

243

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

244

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

245

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

246

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

The operator ( ) is a contraction with respect to the metric

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

247

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 164 ISSN 2250-3153

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Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

248

Remark 26: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

249

Remark 27: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

250

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 28: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

251

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In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively.

Remark 29: 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.

252

Remark 30: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

253

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations

254

Analogous inequalities hold also for

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

255

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

256

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

257

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

The operator ( ) is a contraction with respect to the metric

((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

258

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 166 ISSN 2250-3153

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Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on it follows

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

259

Remark 31: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

260

Remark 32: There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

261

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 33: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

262

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(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively.

Remark 34: 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.

263

Remark 35: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

264

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations

265

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

266

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

267

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

268

In order that the operator ( ) transforms the space of sextuples of functions satisfying

Equations into itself

The operator ( ) is a contraction with respect to the metric

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 168 ISSN 2250-3153

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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

269

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))

270

It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

271

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

272

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis the result follows

273

Remark 36: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as

not conformal with the reality, however we have put this hypothesis ,in order that we can postulate

condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on

( )( ) and hypothesis can replaced by a usual Lipschitz condition.

274

Remark 37 There does not exist any where ( ) ( )

it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

275

Definition of (( )( ))

(( )

( )) (( )

( ))

:

276

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Remark 38: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively.

Remark 39: 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.

277

Remark 40: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

278

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ( ) ) (

)( )

279

It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose ( )( ) ( )

( ) large to have

279A

( )( )

( )( ) [( )( ) (( )

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

In order that the operator ( ) transforms the space of sextuples of functions satisfying 39,35,36 into itself

The operator ( ) is a contraction with respect to the metric

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 170 ISSN 2250-3153

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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( )) It results

| ( )

( )

| ∫ ( )( )

|

( )

( )| ( )( ) ( ) ( )( ) ( ) ( )

∫ ( )( )|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )( )(

( ) ( ))|

( )

( )| ( )( ) ( ) ( )( ) ( )

( )

( )( )(

( ) ( )) (

)( )( ( )

( )) ( )( ) ( ) ( )( ) ( ) ( )

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on 45,46,47,28 and 29 it follows

|( )( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

And analogous inequalities for . Taking into account the hypothesis (39,35,36) the result follows

Remark 41: The fact that we supposed ( )( ) (

)( ) depending also on can be considered as not conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )

( ) ( )( ) respectively of If instead of proving the existence of the solution on , we have to prove it only on a compact then it

suffices to consider that ( )( ) (

)( ) depend only on and respectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.

Remark 42: There does not exist any where ( ) ( )

From 99 to 44 it results

( ) [ ∫ {(

)( ) ( )( )( ( ( )) ( ))} ( )

]

( ) ( (

)( ) ) for

Definition of (( )( ))

(( )

( )) (( )

( ))

:

Remark 43: if is bounded, the same property have also . indeed if

( )( ) it follows

(( )

( )) (

)( ) and by integrating

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 171 ISSN 2250-3153

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(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively. Remark 44: 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 45: If is bounded from below and (( )( ) (( )( ) )) (

)( ) then Definition of ( )( ) : Indeed let be so that for

( )( ) (

)( )(( )( ) ) ( ) ( )( )

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded.

The same property holds for if ( )( ) (( )( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 92

Behavior of the solutions of equation

Theorem If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

280

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( ) ( )( )

281

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( ) ( )( )

282

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :- 283

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If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

are defined

284

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined by equation

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

285

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

286

( )( ) ( )

(( )( ) ( )( )) 287

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 288

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

289

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

290

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( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

Behavior of the solutions of equation

Theorem 2: If we denote and define

291

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

292

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( ) 293

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( ) 294

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : 295

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots 296

of the equations ( )( )( ( ))

( )

( ) ( ) ( )( ) 297

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and 298

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : 299

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the 300

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( ) 301

and ( )( )( ( ))

( )

( ) ( ) ( )( ) 302

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :- 303

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by 304

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 305

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

306

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 307

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

308

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and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 309

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

310

( )( ) is defined by equation

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( ) 311

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

312

( )( ) ( )

(( )( ) ( )( )) 313

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 314

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

315

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):- 316

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

317

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

318

Behavior of the solutions

Theorem 3: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )(( ) ) ( )

( )

319

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Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

320

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

321

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) and ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

( )( ) is defined by equation

322

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( ) 323

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

324

( )( ) ( )

(( )( ) ( )( )) 325

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( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( )) 326

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

327

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

328

Behavior of the solutions of equation Theorem: If we denote and define Definition of ( )

( ) ( )( ) ( )

( ) ( )( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

329

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : By ( )

( ) ( )( ) and respectively ( )

( ) ( )( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

330

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( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

331

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( ) where ( )

( ) is defined by equation

332

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

333

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) )

334

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

335

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

336

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( ) ( )

( ) ( )( ) ( )

( )

337

Behavior of the solutions of equation Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

338

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( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( ) ( )

( ) ( )( ) (

)( ) ( )( )(( ) ) (

)( )(( ) ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

339

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

340

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

341

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined by equation

342

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

343

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( ) 344

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( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

345

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

346

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

347

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):- Where ( )

( ) ( )( )( )

( ) ( )( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( ) ( )

( ) ( )( ) ( )

( )

348

Behavior of the solutions of equation Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( ) ( )

( ) ( )( ) (

)( ) ( )( )(( ) ) (

)( )(( ) ) ( )( )

349

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

350

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

351

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 180 ISSN 2250-3153

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( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

352

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined by equation

353

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

354

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) )

355

( )( ) ( )

(( )( ) ( )( ))

356

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

357

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

358

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

359

Behavior of the solutions of equation

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Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

361

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

362

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

363

Then the solution of global equations satisfies the inequalities 364

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 182 ISSN 2250-3153

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(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined by equation

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

365

(( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

366

( )( ) ( )

(( )( ) ( )( ))

367

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

368

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

369

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

370

Behavior of the solutions of equation

Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

( )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )

371

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) : 372

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 183 ISSN 2250-3153

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By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

374

Then the solution of global equations satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined by equation

375

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

376

( ( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( ) 377

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 184 ISSN 2250-3153

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( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

378

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

379

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

380

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

381

Behavior of the solutions of equation 37 to 92 Theorem 2: If we denote and define

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) : ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

)( )( ) ( )( )( ) ( )

( )

( )( ) (

)( ) ( )( ) (

)( )(( ) ) ( )( )(( ) ) ( )

( )

382

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) : By ( )

( ) ( )( ) and respectively ( )

( ) ( )( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) :

By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the

roots of the equations ( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )

( ) ( )( ) ( )

( ) ( )( ) by

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 185 ISSN 2250-3153

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( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined by 59 and 69 respectively

Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities

(( )( ) ( )( )) ( )

( )( )

where ( )( ) is defined by equation 45

( )( ) (( )( ) ( )( )) ( )

( )( ) ( )( )

(

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ))

( )( ) ( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

( )( ) (( )( ) ( )( ))

( )( )

( )( )(( )( ) ( )( ))

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]

( )( )

Definition of ( )( ) ( )

( ) ( )( ) ( )

( ):- Where ( )

( ) ( )( )( )

( ) ( )( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 186 ISSN 2250-3153

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( )( ) (

)( ) ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

383

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

384

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

385

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

386

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And so with the notation of the first part of condition (c) , we have

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for

the special case

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

387

Definition of ( ) :- ( )

388

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

389

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

390

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In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

391

From which we deduce ( )( ) ( )( ) ( )

( ) 392

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

393

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

And so with the notation of the first part of condition (c) , we have

394

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

395

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

396

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition

( )( ) ( )

( ) then ( )( ) ( )( ) and as a consequence ( ) ( )

( ) ( )

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( )

397

Proof : From global equations we obtain 398

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( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

399

From which one obtains

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

400

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

Definition of ( )( ) :-

From which we deduce ( )( ) ( )( ) ( )

( )

401

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

402

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

And so with the notation of the first part of condition (c) , we have

Definition of ( )( ) :-

403

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 190 ISSN 2250-3153

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( )

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

404

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

405

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If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

406

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

And so with the notation of the first part of condition (c) , we have

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

407

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

408

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From which one obtains Definition of ( )

( ) ( )( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )

( ) ( )( ) ( )( )

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

409

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

410

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

And so with the notation of the first part of condition (c) , we have

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case .

411

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Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )

( ) ( )( ) ( )( )

412

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

413

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

414

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

And so with the notation of the first part of condition (c) , we have

Definition of ( )( ) :-

415

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 194 ISSN 2250-3153

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

416

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

417

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 195 ISSN 2250-3153

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From which we deduce ( )( ) ( )( ) ( )

( )

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

418

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

419

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for

the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then ( )( ) ( )

( )if in addition

( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important consequence of the relation between

( )( ) and ( )

( ) and definition of ( )( )

420

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

421

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 196 ISSN 2250-3153

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Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

422

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

423

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 ( )( ) :-

424

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 197 ISSN 2250-3153

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the

theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for

the special case .

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

Proof : From 99,20,44,22,23,44 we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains

Definition of ( )( ) ( )

( ) :-

For ( )( )

( )

( ) ( )( )

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

424A

In the same manner , we get

( )( ) ( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

, ( )( ) ( )( ) ( )( )

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 198 ISSN 2250-3153

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( )( )

( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

( )( ) ( )( )( )( )

[ ( )( )(( )( ) ( )( )) ]

( )( ) [ ( )( )(( )( ) ( )( )) ]

( )( )

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 99, 20,44,22,23, and 44 we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition ( )( )

( )( ) then ( )( ) ( )

( ) and as a consequence ( ) ( )( ) ( ) this also defines ( )

( ) for the special case . Analogously if (

)( ) ( )( ) ( )

( ) ( )( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

We can prove the following

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

425

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations 426

( )( )(

)( ) ( )( )( )

( ) 427

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 199 ISSN 2250-3153

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( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) 428

( )( )(

)( ) ( )( )( )

( ) , 429

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

430

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

431

We can prove the following

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

432

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

433

Theorem If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

434

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( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

435

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system

436

Theorem : If ( )( ) (

)( ) are independent on , and the conditions (with the notations

45,46,27,28)

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation 45 are satisfied , then the system

436

A

( )( ) [(

)( ) ( )( )( )] 437

( )( ) [(

)( ) ( )( )( )] 438

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( )( ) [(

)( ) ( )( )( )] 439

( )( ) (

)( ) ( )( )( ) 440

( )( ) (

)( ) ( )( )( ) 441

( )( ) (

)( ) ( )( )( ) 442

has a unique positive solution , which is an equilibrium solution for the system

( )( ) [(

)( ) ( )( )( )] 443

( )( ) [(

)( ) ( )( )( )] 444

( )( ) [(

)( ) ( )( )( )] 445

( )( ) (

)( ) ( )( )( ) 446

( )( ) (

)( ) ( )( )( ) 447

( )( ) (

)( ) ( )( )( ) 448

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )] 449

( )( ) [(

)( ) ( )( )( )] 450

( )( ) [(

)( ) ( )( )( )] 451

( )( ) (

)( ) ( )( )( ) 452

( )( ) (

)( ) ( )( )( ) 453

( )( ) (

)( ) ( )( )( ) 454

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

455

( )( ) [(

)( ) ( )( )( )]

456

( )( ) [(

)( ) ( )( )( )]

457

( )( ) (

)( ) ( )( )(( ))

458

( )( ) (

)( ) ( )( )(( ))

459

( )( ) (

)( ) ( )( )(( ))

460

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has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

461

( )( ) [(

)( ) ( )( )( )]

462

( )( ) [(

)( ) ( )( )( )]

463

( )( ) (

)( ) ( )( )( )

464

( )( ) (

)( ) ( )( )( )

465

( )( ) (

)( ) ( )( )( )

466

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

467

( )( ) [(

)( ) ( )( )( )]

468

( )( ) [(

)( ) ( )( )( )]

469

( )( ) (

)( ) ( )( )( )

470

( )( ) (

)( ) ( )( )( )

471

( )( ) (

)( ) ( )( )( )

472

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )]

473

( )( ) [(

)( ) ( )( )( )]

474

( )( ) [(

)( ) ( )( )( )]

475

( )( ) (

)( ) ( )( )( )

476

( )( ) (

)( ) ( )( )( )

477

( )( ) (

)( ) ( )( )( )

478

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( )( ) [(

)( ) ( )( )( )]

479

( )( ) [(

)( ) ( )( )( )]

480

( )( ) [(

)( ) ( )( )( )]

481

( )( ) (

)( ) ( )( )( )

482

( )( ) (

)( ) ( )( )( )

483

( )( ) (

)( ) ( )( )( )

484

( )( ) [(

)( ) ( )( )( )] 484

A

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

485

Proof:

(b) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

486

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

487

Proof: 488

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(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

489

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

490

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

491

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

492

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

492

A

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

493

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

494

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 205 ISSN 2250-3153

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equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

495

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

496

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

497

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

498

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

499

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

500

Definition and uniqueness of :-

501

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After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows that

there exists a unique for which (

) . With this value , we obtain from the three first

equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

501

A

By the same argument, the 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 ( )

502

By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

503

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 (( )

)

504

By the same argument, the 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 (( )

)

505

By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

506

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 207 ISSN 2250-3153

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[( )( )(

)( )( ) ( )( )(

)( )( )] ( )( )( )(

)( )( )

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 (( )

)

By the same argument, the 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 (( )

)

507

By the same argument, the 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 (

)

508

By the same argument, the 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 (

)

509

By the same argument, the 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 (

)

510

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 (( )

)

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 208 ISSN 2250-3153

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Finally we obtain the unique solution

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )] ,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution

511

Finally we obtain the unique solution

(( )

) , (

) and 512

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

513

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )] 514

Obviously, these values represent an equilibrium solution

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution of global equations

515

Finally we obtain the unique solution

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

516

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

Obviously, these values represent an equilibrium solution of global equations

517

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

518

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( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

Obviously, these values represent an equilibrium solution of global equations

519

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

520

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

Obviously, these values represent an equilibrium solution of global equations

521

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

522

Finally we obtain the unique solution

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

523

Finally we obtain the unique solution of 89 to 99

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

523

A

ASYMPTOTIC STABILITY ANALYSIS 524

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 210 ISSN 2250-3153

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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 and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

525

((

)( ) ( )( )) ( )

( ) ( )( )

526

((

)( ) ( )( )) ( )

( ) ( )( )

527

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

528

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

529

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

530

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

531

Proof: Denote

Definition of :-

,

532

( )( )

(

) ( )( ) ,

( )( )

( ( )

) 533

taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

534

((

)( ) ( )( )) ( )

( ) ( )( )

535

((

)( ) ( )( )) ( )

( ) ( )( )

536

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

537

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 211 ISSN 2250-3153

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((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

538

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

539

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 :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

540

Then taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

541

((

)( ) ( )( )) ( )

( ) ( )( )

542

((

)( ) ( )( )) ( )

( ) ( )( )

543

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

544

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

545

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

546

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

547

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

(( )

)

548

Then taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

549

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((

)( ) ( )( )) ( )

( ) ( )( )

550

((

)( ) ( )( )) ( )

( ) ( )( )

551

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

552

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

553

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

554

ASYMPTOTIC STABILITY ANALYSIS

Theorem 5: 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

555

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

556

Then taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

557

((

)( ) ( )( )) ( )

( ) ( )( )

558

((

)( ) ( )( )) ( )

( ) ( )( )

559

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

560

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

561

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

562

ASYMPTOTIC STABILITY ANALYSIS

Theorem 6: 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

563

Definition of :-

,

564

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 213 ISSN 2250-3153

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(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

Then taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

565

((

)( ) ( )( )) ( )

( ) ( )( )

566

((

)( ) ( )( )) ( )

( ) ( )( )

567

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

568

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

569

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

570

ASYMPTOTIC STABILITY ANALYSIS

Theorem 7: 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

571

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

572

Then taking into account equations and neglecting the terms of power 2, we obtain from

((

)( ) ( )( )) ( )

( ) ( )( )

573

((

)( ) ( )( )) ( )

( ) ( )( )

574

((

)( ) ( )( )) ( )

( ) ( )( )

575

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

576

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

578

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

579

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Obviously, these values represent an equilibrium solution

ASYMPTOTIC STABILITY ANALYSIS

Theorem 8: 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 :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

580

Then taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

581

((

)( ) ( )( )) ( )

( ) ( )( )

582

((

)( ) ( )( )) ( )

( ) ( )( )

583

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

584

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

585

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586

ASYMPTOTIC STABILITY ANALYSIS Theorem 9: 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

586A

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

Then taking into account equations 89 to 99 and neglecting the terms of power 2, we obtain from 99 to 44

((

)( ) ( )( )) ( )

( ) ( )( )

586B

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 215 ISSN 2250-3153

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((

)( ) ( )( )) ( )

( ) ( )( )

586 C

((

)( ) ( )( )) ( )

( ) ( )( )

586 D

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586 E

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586 F

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586

G The characteristic equation of this system is 587

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 216 ISSN 2250-3153

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((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 217 ISSN 2250-3153

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((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

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)( ) ( )( ) ( )

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((( )( )) ( (

)( ) ( )( ) ( )

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( )

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( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

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( ) )]

((( )( ) ( )( ) ( )

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)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 218 ISSN 2250-3153

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((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

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( )

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+

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( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

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( )( )( )

( ) )]

((( )( ) ( )( ) ( )

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((( )( ) ( )( ) ( )

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( )( )( )

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((( )( ) ( )( ) ( )

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((( )( )) ( (

)( ) ( )( ) ( )

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((( )( )) ( (

)( ) ( )( ) ( )

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((( )( )) ( (

)( ) ( )( ) ( )

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( )

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( )) (( )( )( )

( ) ( )

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+

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( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

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( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 219 ISSN 2250-3153

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((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( ))

((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

( )) (( )( ) ( )( ) ( )

( ))

[((( )( ) ( )( ) ( )

( ))( )( )

( )( )( )

( ) )]

((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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( ))( )( )

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( )) ( ) ( ) ( )

( ) ( ) ( ) )

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)( ) ( )( ) ( )

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((( )( )) ( (

)( ) ( )( ) ( )

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((( )( )) ( (

)( ) ( )( ) ( )

( ) ( )( )) ( )( )) ( )

( )

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( ) ( )

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((( )( ) ( )( ) ( )

( )) ( ) ( ) ( )

( ) ( ) ( ) )

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and

this proves the theorem.

References

(1) ^ http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-hierarchy-problem/

(2) ^ R. Barbieri, G. F. Giudice (1988). "Upper Bounds on Supersymmetric Particle Masses". Nucl. Phys. B

306: 63. doi:10.1016/0550-3213(88)90171-X.

International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 220 ISSN 2250-3153

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(3) ^ Stephen P. Martin, A Supersymmetry Primer

(4) ^ K. Meissner, H. Nicolai (2006). "Conformal Symmetry and the Standard Model". Physics Letters B648:

312–317. arXiv:hep-th/0612165. Bibcode:2007PhLB..648..312M. doi:10.1016/j.physletb.2007.03.023.

(5) ^ Zee, A. (2003). Quantum field theory in a nutshell. Princeton University Press.

Bibcode:2003qftn.book.....Z.

(6) ^ N. Arkani-Hamed, S. Dimopoulos, G. Dvali (1998). "The Hierarchy problem and new dimensions at a

millimeter". Physics Letters B429: 263–272. arXiv:hep-ph/9803315. Bibcode:1998PhLB..429..263A.

doi:10.1016/S0370-2693(98)00466-3.

(7) ^ N. Arkani-Hamed, S. Dimopoulos, G. Dvali (1999). "Phenomenology, astrophysics and cosmology of

theories with submillimeter dimensions and TeV scale quantum gravity". Physical Review D59: 086004.

arXiv:hep-ph/9807344. Bibcode:1999PhRvD..59h6004A. doi:10.1103/PhysRevD.59.086004.

(8) ^ For a pedagogical introduction, see M. Shifman (2009). "Large Extra Dimensions: Becoming acquainted

with an alternative paradigm". Crossing the boundaries: Gauge dynamics at strong coupling. Singapore:

World Scientific. arXiv:0907.3074.

(9) ^ M. Gogberashvili, Hierarchy problem in the shell universe model, Arxiv:hep-ph/9812296.

(10) M. Gogberashvili, Our world as an expanding shell, Arxiv:hep-ph/9812365.

(11) M. Gogberashvili, Four dimensionality in noncompact Kaluza-Klein model, Arxiv:hep-ph/9904383.

(12) CMS Collaoration, "Search for Microscopic Black Hole Signatures at the Large Hadron Collider,"

http://arxiv.org/abs/1012.3375