International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 1 ISSN 2250-3153 www.ijsrp.org 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
220
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International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 1 ISSN 2250-3153
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 24 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 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
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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
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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
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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
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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
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( )( )( ) ( )
( ) ( )( )
138
( )( )(( ) ) ( )
( ) ( )( ) ( )
( ) 139
( )( ) ( ) ( )
( )
140
( )( ) (( ) ) ( )
( ) 141
Definition of ( )( ) ( )
( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
142
( )( )(( )
) ( )( )(( ) ) ( )
( ) ( ) ( ) ( )( ) 143
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
) and
( )( )( ) . (
) and ( ) are points belonging to the interval [( )( ) ( )
( )] . It is to be
noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if ( )
( )
then the function ( )( )( ) , the first augmentation coefficient attributable to the system, would
be absolutely continuous.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )( )
( )( ) ( )
( )
( )( )
144
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( ) which together with ( )( ) ( )
( ) ( )( )
( )( ) and the constants ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
Satisfy the inequalities
( )( )
( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
145
( )( )
( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
146
Where we suppose
( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
146A
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Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( ) ( )
( )
(
)( ) ( ) ( )( )
Definition of ( )
( ) ( )( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
( )( )(( )
) ( )( )(( ) ) ( )
( ) ( ) ( ) ( )( )
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
)
and( )( )( ) . (
) and ( ) are points belonging to the interval [( )( ) ( )
( )] . It is
to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if
( )( ) then the function (
)( )( ) , the first augmentation coefficient attributable to the system, would be absolutely continuous.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
Definition of ( )( ) ( )
( ) : There exists two constants ( )
( ) and ( )( ) which together with
( )( ) ( )
( ) ( )( ) ( )
( ) and the constants
( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( )
( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( )
( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 35 ISSN 2250-3153
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Theorem 1: if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
147
Theorem 2 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( )
( ) ( )( ) ( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
148
Theorem 3 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
( ) ( )( ) ( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
149
Theorem 4 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
150
Theorem 5 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
151
Theorem 6 : if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
152
Theorem 7: if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
153
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( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
Theorem 8: if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
153
A
Theorem 9: if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
153
B
Proof: Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
154
( ) ( )
( )
( ) ( )
( ) 155
( ) ( )
( ) ( )( ) 156
( ) ( )
( ) ( )( ) 157
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
158
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 37 ISSN 2250-3153
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Where ( ) is the integrand that is integrated over an interval ( )
Proof:
Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
159
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
160
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof:
Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
161
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 38 ISSN 2250-3153
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( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof: Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
162
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof: Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
163
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( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof:
Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
164
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof:
Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
( ) ( )
( )
( ) ( )
( )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 40 ISSN 2250-3153
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( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
165
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof:
Consider operator ( ) defined on the space of sextuples of continuous functions
which satisfy
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
166
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
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
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 43 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
(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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 50 ISSN 2250-3153
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If is bounded, the same property follows for and respectively.
Remark 14: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.
218
Remark 15: If is bounded from below and (( )( ) (( )( ) )) (
)( ) then
Definition of ( )( ) :
Indeed let be so that for
( )( ) (
)( )(( )( ) ) ( ) ( )( )
219
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) (( )( ) ) (
)( )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations
220
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
( )( ) ( )
( ) large to have
221
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
222
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
223
In order that the operator ( ) transforms the space of sextuples of functions satisfying
Equations into itself
224
The operator ( ) is a contraction with respect to the metric
((( )( ) ( )
( )) (( )( ) ( )
( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
Indeed if we denote
Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))
225
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It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses on Equations it follows
|( )( ) ( )
( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis the result follows
226
Remark 16: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as
not conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( )( ) and hypothesis can replaced by a usual Lipschitz condition.
227
Remark 17: There does not exist any where ( ) ( )
it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
228
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 18: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
(( )( ))
( )( )(( )
( )) (
)( )
229
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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
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 133 ISSN 2250-3153
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( )
( )
[
( )( ) (
)( )( ) ( )( )( ) (
)( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( ) ]
( )
( )
[
( )( ) (
)( )( ) ( )( )( ) (
)( )( )
( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( ) ]
Where ( )( )( ) (
)( )( ) ( )( )( ) are first augmentation coefficients for
category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are second
augmentation coefficient for category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are third
augmentation coefficient for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are fourth
augmentation coefficient for category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are fifth
augmentation coefficient for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are sixth
augmentation coefficient for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are Seventh
augmentation coefficient for category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are eighth
augmentation coefficient for 1,2,3
( )( )( ) , (
)( )( ) , ( )( )( ) are ninth
augmentation coefficient for 1,2,3
( )( )
[
( )( ) (
)( )( ) ( )( )( ) – (
)( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( )
]
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 134 ISSN 2250-3153
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( )( )
[ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
( )( )
[ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( )
– ( )( )( ) – (
)( )( ) – ( )( )( ) ]
Where ( )( )( ) (
)( )( ) ( )( )( ) are first detrition coefficients for
category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are second
detrition coefficients for category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are third detrition
coefficients for category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are fourth
detrition coefficients for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are fifth detrition
coefficients for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are sixth detrition
coefficients for category 1, 2 and 3
– ( )( )( ) , – (
)( )( ) , – ( )( )( ) are seventh detrition
coefficients for category 1, 2 and 3
– ( )( )( ) – (
)( )( ) , – ( )( )( ) are eighth detrition
coefficients for category 1, 2 and 3
– ( )( )( ) , – (
)( )( ) , – ( )( )( ) are ninth
detrition coefficients for category 1, 2 and 3
Where we suppose
( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )( ) ( )
( ) ( )( ) ( )
( )
97
( )( ) ( ) ( )
( )
( )( ) ( ) ( )
( )
98
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 135 ISSN 2250-3153
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Definition of ( )( ) ( )
( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
( )( )( ) (
)( )( ) ( )( ) ( )( )
99
With the Lipschitz condition, we place a restriction on the behavior of functions
( )( )(
) and( )( )( ) (
) and ( ) are points belonging to the interval
[( )( ) ( )
( )] . It is to be noted that ( )( )( ) is uniformly continuous. In the eventuality of
the fact, that if ( )( ) then the function (
)( )( ) , the first augmentation coefficient
attributable to the system, would be absolutely continuous.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
100
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( )which together With ( )( ) ( )
( ) ( )( )
and ( )( ) and the constants ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
101
Where we suppose
( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
102
( )( )( ) ( )
( ) ( )( ) ( )
( ) 103
( )( ) ( ) ( )
( ) 104
( )( ) (( ) ) ( )
( ) 105
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 136 ISSN 2250-3153
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Definition of ( )( ) ( )
( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
106
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( ) 107
( )( )(( )
) ( )( )(( ) ) ( )
( ) ( ) ( ) ( )( ) 108
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
)
and( )( )( ) . (
) and ( ) are points belonging to the interval [( )( ) ( )
( )] . It is
to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if
( )( ) then the function (
)( )( ) , the first augmentation coefficient attributable to the
system, would be absolutely continuous.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
109
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( ) which together
with ( )( ) ( )
( ) ( )( ) ( )
( ) and the constants
( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( )
satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) 110
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) 111
Where we suppose
( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )( ) ( )
( ) ( )( ) ( )
( )
112
( )( ) ( ) ( )
( ) 113
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 139 ISSN 2250-3153
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Definition of ( )( ) ( )
( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
( )( )(( )
) ( )( )(( ) ) ( )
( ) ( ) ( ) ( )( )
124
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
)
and( )( )( ) . (
) and ( ) are points belonging to the interval [( )( ) ( )
( )] . It is
to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if
( )( ) then the function (
)( )( ) , the first augmentation coefficient attributable to the
system, would be absolutely continuous.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
125
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( ) which together with
( )( ) ( )
( ) ( )( ) ( )
( ) and the constants
( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
126
Where we suppose
( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )(( ) ) ( )
( ) ( )( ) ( )
( )
127
( )( ) ( ) ( )
( )
( )( ) (( ) ) ( )
( )
128
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Definition of ( )( ) ( )
( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
( )( )(( )
) ( )( )(( ) ) ( )
( ) ( ) ( ) ( )( )
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
)
and( )( )( ) . (
) and ( ) are points belonging to the interval [( )( ) ( )
( )] . It is
to be noted that ( )( )( ) is uniformly continuous. In the eventuality of the fact, that if
( )( ) then the function (
)( )( ) , the first augmentation coefficient attributable to the
system, would be absolutely continuous.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
129
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( ) which together with
( )( ) ( )
( ) ( )( ) ( )
( ) and the constants
( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( )
satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
130
Where we suppose
(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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 145 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 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
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 147 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
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
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( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof: Consider operator ( ) defined on the space of sextuples of continuous functions which satisfy
166A
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )
By
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
167
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 1
168
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Analogous inequalities hold also for
The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious
that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
169
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
170
Analogous inequalities hold also for
The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious
that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
171
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
172
Analogous inequalities hold also for
The operator ( ) maps the space of functions satisfying into itself .Indeed it is obvious that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
173
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 4
174
The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious
that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
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From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 5
175
The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious
that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
176
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 6
Analogous inequalities hold also for
177
(b) The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is
obvious that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
178
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 7
The operator ( ) maps the space of functions satisfying Equations into itself .Indeed it is obvious that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
180
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 8
181
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Analogous inequalities hold also for
The operator ( ) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious
that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 9
Analogous inequalities hold also for
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
( )( ) ( )
( ) large to have
182
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
183
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
184
In order that the operator ( ) transforms the space of sextuples of functions satisfying
Equations into itself
The operator ( ) is a contraction with respect to the metric
(( ( ) ( )) ( ( ) ( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
185
Indeed if we denote
Definition of : ( ) ( )( )
It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
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∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses it follows
| ( ) ( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) (( ( ) ( ) ( ) ( )))
And analogous inequalities for . Taking into account the hypothesis the result follows
186
Remark 1: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( ) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any where ( ) ( )
From 19 to 24 it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 3: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
(( )( ))
( )( )(( )
( )) (
)( )
If is bounded, the same property follows for and respectively.
187
Remark 4: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.
188
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Remark 5: If is bounded from below and (( )( ) ( ( ) )) (
)( ) then
Definition of ( )( ) :
Indeed let be so that for
( )( ) (
)( )( ( ) ) ( ) ( )( )
189
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) ( ( ) ) (
)( )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
( )( ) ( )
( ) large to have
190
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
191
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
192
In order that the operator ( ) transforms the space of sextuples of functions satisfying
Equations into itself
193
The operator ( ) is a contraction with respect to the metric
((( )( ) ( )
( )) (( )( ) ( )
( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
194
Indeed if we denote
Definition of : ( ) ( )( )
195
It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
196
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( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses it follows
197
|( )( ) ( )
( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis the result follows 198
Remark 6: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( )( ) and hypothesis can replaced by a usual Lipschitz condition.
199
Remark 7: There does not exist any where ( ) ( )
it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
200
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 8: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
(( )( ))
( )( )(( )
( )) (
)( )
If is bounded, the same property follows for and respectively.
201
Remark 9: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.
202
Remark 10: If is bounded from below and (( )( ) (( )( ) )) (
)( ) then 203
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Definition of ( )( ) :
Indeed let be so that for
( )( ) (
)( )(( )( ) ) ( ) ( )( )
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
204
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) (( )( ) ) (
)( )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations
205
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
( )( ) ( )
( ) large to have
207
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
208
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
209
In order that the operator ( ) transforms the space of sextuples of functions satisfying
Equations into itself
210
The operator ( ) is a contraction with respect to the metric
((( )( ) ( )
( )) (( )( ) ( )
( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
211
Indeed if we denote
Definition of :( ( ) ( ) ) ( )(( ) ( ))
212
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It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses it follows
213
| ( )
( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis the result follows
214
Remark 11: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as
not conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( )( ) and hypothesis can replaced by a usual Lipschitz condition.
215
Remark 12: There does not exist any where ( ) ( )
it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
216
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 13: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
(( )( ))
( )( )(( )
( )) (
)( )
217
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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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 162 ISSN 2250-3153
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Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))
It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses on it follows
|( )( ) ( )
( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis the result follows
237
Remark 21: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as
not conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( )( ) and hypothesis can replaced by a usual Lipschitz condition.
238
Remark 22: There does not exist any where ( ) ( )
it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
239
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 23: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
240
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 163 ISSN 2250-3153
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(( )( ))
( )( )(( )
( )) (
)( )
If is bounded, the same property follows for and respectively.
Remark 24: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.
241
Remark 25: If is bounded from below and (( )( ) (( )( ) )) (
)( ) then
Definition of ( )( ) :
Indeed let be so that for
( )( ) (
)( )(( )( ) ) ( ) ( )( )
242
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) (( )( ) ) (
)( )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations
Analogous inequalities hold also for
243
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
( )( ) ( )
( ) large to have
244
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
245
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
246
In order that the operator ( ) transforms the space of sextuples of functions satisfying
Equations into itself
The operator ( ) is a contraction with respect to the metric
((( )( ) ( )
( )) (( )( ) ( )
( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
247
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 164 ISSN 2250-3153
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Indeed if we denote
Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))
It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses it follows
|( )( ) ( )
( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis the result follows
248
Remark 26: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as
not conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( )( ) and hypothesis can replaced by a usual Lipschitz condition.
249
Remark 27: There does not exist any where ( ) ( )
it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
250
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 28: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
251
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
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 167 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 34: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.
263
Remark 35: If is bounded from below and (( )( ) (( )( ) )) (
)( ) then
Definition of ( )( ) :
Indeed let be so that for
( )( ) (
)( )(( )( ) ) ( ) ( )( )
264
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) (( )( ) ) (
)( )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations
265
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
( )( ) ( )
( ) large to have
266
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
267
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
268
In order that the operator ( ) transforms the space of sextuples of functions satisfying
Equations into itself
The operator ( ) is a contraction with respect to the metric
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 168 ISSN 2250-3153
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((( )( ) ( )
( )) (( )( ) ( )
( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
269
Indeed if we denote
Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( ))
270
It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
271
Where ( ) represents integrand that is integrated over the interval
From the hypotheses it follows
272
|( )( ) ( )
( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis the result follows
273
Remark 36: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as
not conformal with the reality, however we have put this hypothesis ,in order that we can postulate
condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of
If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on
( )( ) and hypothesis can replaced by a usual Lipschitz condition.
274
Remark 37 There does not exist any where ( ) ( )
it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
275
Definition of (( )( ))
(( )
( )) (( )
( ))
:
276
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Remark 38: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
(( )( ))
( )( )(( )
( )) (
)( )
If is bounded, the same property follows for and respectively.
Remark 39: If bounded, from below, the same property holds for The proof is
analogous with the preceding one. An analogous property is true if is bounded from below.
277
Remark 40: If is bounded from below and (( )( ) (( )( ) )) (
)( ) then
Definition of ( )( ) :
Indeed let be so that for
( )( ) (
)( )(( )( ) ) ( ) ( )( )
278
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) (( )( ) ( ) ) (
)( )
279
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose ( )( ) ( )
( ) large to have
279A
( )( )
( )( ) [( )( ) (( )
( ) )
(( )( )
)
] ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )] ( )
( )
In order that the operator ( ) transforms the space of sextuples of functions satisfying 39,35,36 into itself
The operator ( ) is a contraction with respect to the metric
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 170 ISSN 2250-3153
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((( )( ) ( )
( )) (( )( ) ( )
( )))
| ( )( )
( )( )| ( )( )
| ( )( )
( )( )| ( )( )
Indeed if we denote
Definition of ( ) ( ) : ( ( ) ( ) ) ( )(( ) ( )) It results
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
Where ( ) represents integrand that is integrated over the interval
From the hypotheses on 45,46,47,28 and 29 it follows
|( )( ) ( )| ( )( )
( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
And analogous inequalities for . Taking into account the hypothesis (39,35,36) the result follows
Remark 41: The fact that we supposed ( )( ) (
)( ) depending also on can be considered as not conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition necessary to prove the uniqueness of the solution bounded by
( )( ) ( )( ) ( )
( ) ( )( ) respectively of If instead of proving the existence of the solution on , we have to prove it only on a compact then it
suffices to consider that ( )( ) (
)( ) depend only on and respectively on ( )( ) and hypothesis can replaced by a usual Lipschitz condition.
Remark 42: There does not exist any where ( ) ( )
From 99 to 44 it results
( ) [ ∫ {(
)( ) ( )( )( ( ( )) ( ))} ( )
]
( ) ( (
)( ) ) for
Definition of (( )( ))
(( )
( )) (( )
( ))
:
Remark 43: if is bounded, the same property have also . indeed if
( )( ) it follows
(( )
( )) (
)( ) and by integrating
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 171 ISSN 2250-3153
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(( )( ))
( )( )(( )
( )) (
)( )
In the same way , one can obtain
(( )( ))
( )( )(( )
( )) (
)( )
If is bounded, the same property follows for and respectively. Remark 44: If bounded, from below, the same property holds for The proof is analogous with the preceding one. An analogous property is true if is bounded from below.
Remark 45: If is bounded from below and (( )( ) (( )( ) )) (
)( ) then Definition of ( )( ) : Indeed let be so that for
( )( ) (
)( )(( )( ) ) ( ) ( )( )
Then
( )
( )( )( ) which leads to
(( )( )( )( )
) ( )
If we take such that
it results
(( )( )( )( )
)
By taking now sufficiently small one sees that is unbounded.
The same property holds for if ( )( ) (( )( ) ) (
)( )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 92
Behavior of the solutions of equation
Theorem If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
( )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
280
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations
( )( )( ( ))
( )
( ) ( ) ( )( ) and ( )
( )( ( )) ( )
( ) ( ) ( )( )
281
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations
( )( )( ( ))
( )
( ) ( ) ( )( ) and ( )
( )( ( )) ( )
( ) ( ) ( )( )
282
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) :- 283
International Journal of Scientific and Research Publications, Volume 3, Issue 9, September 2013 172 ISSN 2250-3153
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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