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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
www.ijsrp.org
( )( )( ) , (
)( )( ) , ( )( )( ) 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
www.ijsrp.org
– ( )( )( ) , – (
)( )( ) , – ( )( )( ) 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
www.ijsrp.org
( )
( )
[ (
)( ) ( )( )( ) – (
)( )( ) – ( )( )( )
( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
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
www.ijsrp.org
– ( )( )( ) , – (
)( )( ) , – ( )( )( )
– ( )( )( ) , – (
)( )( ) – ( )( )( )
– ( )( )( ) , – (
)( )( ) , – ( )( )( ) 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
www.ijsrp.org
( )
( )
[
(
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
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
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 31 ISSN 2250-3153
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 34 ISSN 2250-3153
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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
( ) ( )
( )
( ) ( )
( )
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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
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 41 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
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
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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
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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
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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
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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
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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
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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
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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
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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 ( )
( ) ( )( )( )
( ) ( )( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
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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
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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,
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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
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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
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
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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 (( )
)
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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
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 124 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 125 ISSN 2250-3153
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( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) , (
)( )( ) ( )( )( )
( )( )( ) , (
)( )( ) , ( )( )( )
( )( )( ) , (
)( )( ) , ( )( )( )
( )( )( ) , (
)( )( ) , ( )( )( )
( )( )( ) , (
)( )( ) , ( )( )( ) 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 126 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 127 ISSN 2250-3153
www.ijsrp.org
( )
( )
[
(
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
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
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 132 ISSN 2250-3153
www.ijsrp.org
( )( )
[
( )( ) (
)( )( ) ( )( )( ) – (
)( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( )
]
( )( )
[ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
( )( )
[ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
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
www.ijsrp.org
( )
( )
[
( )( ) (
)( )( ) ( )( )( ) (
)( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( ) ]
( )
( )
[
( )( ) (
)( )( ) ( )( )( ) (
)( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( ) ]
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
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 152 ISSN 2250-3153
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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
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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
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 165 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 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
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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
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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
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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
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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
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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
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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
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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
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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 ( )
( ) ( )( )( )
( ) ( )( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
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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
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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,
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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
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 206 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
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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 212 ISSN 2250-3153
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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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((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
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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((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
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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((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( )) ( )
( )
(( )( ) ( )( ) ( )
( )) (( )( )( )
( ) ( )
( )( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
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