1dr k n prasanna kumar, 2prof b s kiranagi ...while we do resort to concept formulation, related...
TRANSCRIPT
International Journal of Scientific and Research Publications, Volume 2, Issue 8, August 2012 1
ISSN 2250-3153
www.ijsrp.org
A DEFACTO βDEJURE MODEL FOR POSITRONS,STELLAR NUCLOSYNTHESIS,QUANTUM COHERENCE,SIMULATION,OBJECTIVE REALITY,QUANTUM DECOHERENCE,VIRTUAL PHOTONS ,PHOTON
TUNNELLING,ENZYMES ,SPACE TIME ,INCREASE IN SPEED OF CHEMICAL REACTIONS(SVERDUPAND SEMNOVβS NUMBER)AND
QUANTUM TUNNELING
1DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3 PROF C S BAGEWADI
ABSTRACT: There exists differential relations, contiguous similarities; in compassable anti generalities, pre suppositional resemblances, dialectic transformation, portmanteau incompatibilities between different structures of stellar nucleosynthesis that occurs in stars, positrons, and Eulerian kinematic description thereof, calls for the attention of both simulation and Quantum coherence, Quantum Gravity and Objective reality. Despite the crying need for the same, and lack of comprehensive envelope of expression, there seems to be contential staticity and presciential dynamism in so far as these aspects are considered. Atrophied asseveration in stellar nucleosynthesis and environmental decoherence are probably most endearing attributions that call for both rational Leibneizism and Socratic subjectivity and of course discourse relativity. An evolutionist model is expounded with configurational entropy and morphological entity for these variables and we look at the system dispassionately without being disturbed by the state of the system. There is no clamor for participatory seriotological sermonisations or an orientation towards pedagogical pontification. We just state the facts and leave the rest to others to do the divergential affirmation, disjunctive synthesis,. While we do resort to concept formulation, related phenomenological methodologies, transformational minimal conditions we neither resort to glorification or mortification of the thesis. Warts et al are presented without any hesitation, reservation, compunction or contrition, which probably is the testimony for the fact that human knowledge is limited and all the needs to be explored stretches in front like an ocean with all its cacophonous mendacious moorings and thromboses unbenedictory singularities with splashed
contours and stigmatized boundaries.
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Parameters taken in to consideration are:
(1) Positrons
(2) Stellar Nucleosynthesis
(3) Simulation
(4) Quantum Coherence
(5) Quantum Gravity
(6) Objective reality
(7) Enzymes
(8) Space-time
(9) Virtual photons
(10) Photonic tunneling(and visibility thereof)
(11) Acceleration in Chemical reactions
(12) Quantum Tunneling
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POSITRONS AND STELLAR NUCLEOSYNTHESIS: MODULE NUMBERED ONE
NOTATION :
πΊ13 : CATEGORY ONE OF STELLAR NUCLEOSYNTHESIS
πΊ14 : CATEGORY TWO OF STELLAR NUCLEOSYNTHESIS
πΊ15 : CATEGORY THREE OF STELLAR NUCLEOSYNTHESIS
π13 : CATEGORY ONE OF POSITRONS
π14 : CATEGORY TWO OF POSITRONS
π15 :CATEGORY THREE OF POSITRONS
SIMULATIONS AND QUANTUM COHERENCEMODULE NUMBERED TWO:
Note: Every film is simulation. Every thought is simulation.
============================================================================
=
πΊ16 : CATEGORY ONE OF SIMULATIONS
πΊ17 : CATEGORY TWO OF SIMULATIONS
πΊ18 : CATEGORY THREE OF SIMULATIONS
π16 :CATEGORY ONE OF QUANTUM COHERENCE
π17 : CATEGORY TWO OF QUANTUM COHERENCE
π18 : CATEGORY THREE OF QUANTUM COHERENCE
OBJECTIVE REALITY AND QUANTUM DECOHERENCE:MODULE NUMBERED THREE:
=============================================================================
πΊ20 : CATEGORY ONE OF OBJECTIVE REALITY
πΊ21 :CATEGORY TWO OF OBJECTIVE REALITY
πΊ22 : CATEGORY THREE OF OBJECTIVE REALITY
π20 :CATEGORY ONE OF QUANTUM DECOHERENCE
π21 :CATEGORY TWO OF QUANTUM DECOHERENCE
π22 : CATEGORY THREE OFQUANTUM DECOHERENCE
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SPACE-TIME AND ENZYMES: MODULE NUMBERED FOUR:
============================================================================
πΊ24 : CATEGORY ONE OFENZYMES
πΊ25 : CATEGORY TWO OF ENZYMES
πΊ26 : CATEGORY THREE OF ENZYMES
π24 :CATEGORY ONE OF SPACE TIME
π25 :CATEGORY TWO OF SPACE TIME
π26 : CATEGORY THREE OF SPACETIME
PHOTONIC TUNNELING(VISIBILITY THEREOF) AND VIRTUAL PHOTONS:MODULE
NUMBERED FIVE:
=============================================================================
πΊ28 : CATEGORY ONE OFPHOTONIC TUNNELING
πΊ29 : CATEGORY TWO OF PHOTONIC TUNNELING
πΊ30 :CATEGORY THREE OF PHOTONIC TUNNELING
π28 :CATEGORY ONE OF VIRTUAL PHOTONS
π29 :CATEGORY TWO OFVIRTUAL PHOTONS
π30 :CATEGORY THREE OF VIRTUAL PHOTONS
QUANTUM TUNNELING AND ACCELERATED CHEMICAL REACTION:MODULE
NUMBERED SIX:
=============================================================================
πΊ32 : CATEGORY ONE OF QUANTUM TUNNELING
πΊ33 : CATEGORY TWO OF QUANTUM TUNNELING
πΊ34 : CATEGORY THREE OFQUANTUM TUNNELING
π32 : CATEGORY ONE OF ACCELERATION IN CHEMICAL REACTIONS
π33 : CATEGORY TWO OFACCELERATION IN CHEMICAL REACTIONS
π34 : CATEGORY THREE OFACCELERATION IN CHEMICAL REACTIONS
=============================================================================
==
π13 1 , π14
1 , π15 1 , π13
1 , π14 1 , π15
1 π16 2 , π17
2 , π18 2 π16
2 , π17 2 , π18
2 : π20
3 , π21 3 , π22
3 , π20 3 , π21
3 , π22 3
π24 4 , π25
4 , π26 4 , π24
4 , π25 4 , π26
4 , π28 5 , π29
5 , π30 5 , π28
5 , π29 5 , π30
5 ,
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π32 6 , π33
6 , π34 6 , π32
6 , π33 6 , π34
6
are Accentuation coefficients
π13β² 1 , π14
β² 1 , π15β² 1 , π13
β² 1 , π14β² 1 , π15
β² 1 , π16β² 2 , π17
β² 2 , π18β² 2 , π16
β² 2 , π17β² 2 , π18
β² 2
, π20β² 3 , π21
β² 3 , π22β² 3 , π20
β² 3 , π21β² 3 , π22
β² 3 π24
β² 4 , π25β² 4 , π26
β² 4 , π24β² 4 , π25
β² 4 , π26β² 4 , π28
β² 5 , π29β² 5 , π30
β² 5 π28β² 5 , π29
β² 5 , π30β² 5 ,
π32β² 6 , π33
β² 6 , π34β² 6 , π32
β² 6 , π33β² 6 , π34
β² 6
are Dissipation coefficients
POSITRONS AND STELLAR NUCLEOSYNTHESIS: MODULE NUMBERED ONE
The differential system of this model is now
ππΊ13
ππ‘= π13
1 πΊ14 β π13β² 1 + π13
β²β² 1 π14 , π‘ πΊ13
ππΊ14
ππ‘= π14
1 πΊ13 β π14β² 1 + π14
β²β² 1 π14 , π‘ πΊ14
ππΊ15
ππ‘= π15
1 πΊ14 β π15β² 1 + π15
β²β² 1 π14 , π‘ πΊ15
ππ13
ππ‘= π13
1 π14 β π13β² 1 β π13
β²β² 1 πΊ, π‘ π13
ππ14
ππ‘= π14
1 π13 β π14β² 1 β π14
β²β² 1 πΊ, π‘ π14
ππ15
ππ‘= π15
1 π14 β π15β² 1 β π15
β²β² 1 πΊ, π‘ π15
+ π13β²β² 1 π14 , π‘ = First augmentation factor
β π13β²β² 1 πΊ, π‘ = First detritions factor
SIMULATIONS AND QUANTUM COHERENCEMODULE NUMBERED TWO
The differential system of this model is now
ππΊ16
ππ‘= π16
2 πΊ17 β π16β² 2 + π16
β²β² 2 π17 , π‘ πΊ16
ππΊ17
ππ‘= π17
2 πΊ16 β π17β² 2 + π17
β²β² 2 π17 , π‘ πΊ17
ππΊ18
ππ‘= π18
2 πΊ17 β π18β² 2 + π18
β²β² 2 π17 , π‘ πΊ18
ππ16
ππ‘= π16
2 π17 β π16β² 2 β π16
β²β² 2 πΊ19 , π‘ π16
ππ17
ππ‘= π17
2 π16 β π17β² 2 β π17
β²β² 2 πΊ19 , π‘ π17
ππ18
ππ‘= π18
2 π17 β π18β² 2 β π18
β²β² 2 πΊ19 , π‘ π18
+ π16β²β² 2 π17 , π‘ = First augmentation factor
β π16β²β² 2 πΊ19 , π‘ = First detritions factor
OBJECTIVE REALITY AND QUANTUM DECOHERENCE:MODULE NUMBERED THREE
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The differential system of this model is now
ππΊ20
ππ‘= π20
3 πΊ21 β π20β² 3 + π20
β²β² 3 π21 , π‘ πΊ20
ππΊ21
ππ‘= π21
3 πΊ20 β π21β² 3 + π21
β²β² 3 π21 , π‘ πΊ21
ππΊ22
ππ‘= π22
3 πΊ21 β π22β² 3 + π22
β²β² 3 π21 , π‘ πΊ22
ππ20
ππ‘= π20
3 π21 β π20β² 3 β π20
β²β² 3 πΊ23 , π‘ π20
ππ21
ππ‘= π21
3 π20 β π21β² 3 β π21
β²β² 3 πΊ23 , π‘ π21
ππ22
ππ‘= π22
3 π21 β π22β² 3 β π22
β²β² 3 πΊ23 , π‘ π22
+ π20β²β² 3 π21 , π‘ = First augmentation factor
β π20β²β² 3 πΊ23 , π‘ = First detritions factor
SPACE-TIME AND ENZYMES: MODULE NUMBERED FOUR
:
The differential system of this model is now
ππΊ24
ππ‘= π24
4 πΊ25 β π24β² 4 + π24
β²β² 4 π25 , π‘ πΊ24
ππΊ25
ππ‘= π25
4 πΊ24 β π25β² 4 + π25
β²β² 4 π25 , π‘ πΊ25
ππΊ26
ππ‘= π26
4 πΊ25 β π26β² 4 + π26
β²β² 4 π25 , π‘ πΊ26
ππ24
ππ‘= π24
4 π25 β π24β² 4 β π24
β²β² 4 πΊ27 , π‘ π24
ππ25
ππ‘= π25
4 π24 β π25β² 4 β π25
β²β² 4 πΊ27 , π‘ π25
ππ26
ππ‘= π26
4 π25 β π26β² 4 β π26
β²β² 4 πΊ27 , π‘ π26
+ π24β²β² 4 π25 , π‘ = First augmentation factor
β π24β²β² 4 πΊ27 , π‘ = First detritions factor
PHOTONIC TUNNELING(VISIBILITY THEREOF) AND VIRTUAL PHOTONS:MODULE
NUMBERED FIVE
The differential system of this model is now
ππΊ28
ππ‘= π28
5 πΊ29 β π28β² 5 + π28
β²β² 5 π29 , π‘ πΊ28
ππΊ29
ππ‘= π29
5 πΊ28 β π29β² 5 + π29
β²β² 5 π29 , π‘ πΊ29
ππΊ30
ππ‘= π30
5 πΊ29 β π30β² 5 + π30
β²β² 5 π29 , π‘ πΊ30
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ππ28
ππ‘= π28
5 π29 β π28β² 5 β π28
β²β² 5 πΊ31 , π‘ π28
ππ29
ππ‘= π29
5 π28 β π29β² 5 β π29
β²β² 5 πΊ31 , π‘ π29
ππ30
ππ‘= π30
5 π29 β π30β² 5 β π30
β²β² 5 πΊ31 , π‘ π30
+ π28β²β² 5 π29 , π‘ = First augmentation factor
β π28β²β² 5 πΊ31 , π‘ = First detritions factor
QUANTUM TUNNELING AND ACCELERATED CHEMICAL REACTION:MODULE
NUMBERED SIX
The differential system of this model is now
ππΊ32
ππ‘= π32
6 πΊ33 β π32β² 6 + π32
β²β² 6 π33 , π‘ πΊ32
ππΊ33
ππ‘= π33
6 πΊ32 β π33β² 6 + π33
β²β² 6 π33 , π‘ πΊ33
ππΊ34
ππ‘= π34
6 πΊ33 β π34β² 6 + π34
β²β² 6 π33 , π‘ πΊ34
ππ32
ππ‘= π32
6 π33 β π32β² 6 β π32
β²β² 6 πΊ35 , π‘ π32
ππ33
ππ‘= π33
6 π32 β π33β² 6 β π33
β²β² 6 πΊ35 , π‘ π33
ππ34
ππ‘= π34
6 π33 β π34β² 6 β π34
β²β² 6 πΊ35 , π‘ π34
+ π32β²β² 6 π33 , π‘ = First augmentation factor
β π32β²β² 6 πΊ35 , π‘ = First detritions factor
HOLISTIC CONCATENATE SYTEMAL EQUATIONS HENCEFORTH REFERRED TO AS βGLOBAL EQUATIONS
POSITRONS AND STELLAR NUCLEOSYNTHESIS: MODULE NUMBERED ONE
OBJECTIVE REALITY AND QUANTUM DECOHERENCE:MODULE NUMBERED THREE
SIMULATIONS AND QUANTUM COHERENCEMODULE NUMBERED TWO
SPACE-TIME AND ENZYMES: MODULE NUMBERED FOUR
PHOTONIC TUNNELING(VISIBILITY THEREOF) AND VIRTUAL PHOTONS:MODULE
NUMBERED FIVE
QUANTUM TUNNELING AND ACCELERATED CHEMICAL REACTION:MODULE
NUMBERED SIX
ππΊ13
ππ‘= π13
1 πΊ14 β π13
β² 1 + π13β²β² 1 π14 , π‘ + π16
β²β² 2,2, π17 , π‘ + π20β²β² 3,3, π21 , π‘
+ π24β²β² 4,4,4,4, π25 , π‘ + π28
β²β² 5,5,5,5, π29 , π‘ + π32β²β² 6,6,6,6, π33 , π‘
πΊ13
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ππΊ14
ππ‘= π14
1 πΊ13 β π14
β² 1 + π14β²β² 1 π14 , π‘ + π17
β²β² 2,2, π17 , π‘ + π21β²β² 3,3, π21 , π‘
+ π25β²β² 4,4,4,4, π25 , π‘ + π29
β²β² 5,5,5,5, π29 , π‘ + π33β²β² 6,6,6,6, π33 , π‘
πΊ14
ππΊ15
ππ‘= π15
1 πΊ14 β π15
β² 1 + π15β²β² 1 π14 , π‘ + π18
β²β² 2,2, π17 , π‘ + π22β²β² 3,3, π21 , π‘
+ π26β²β² 4,4,4,4, π25 , π‘ + π30
β²β² 5,5,5,5, π29 , π‘ + π34β²β² 6,6,6,6, π33 , π‘
πΊ15
Where π13β²β² 1 π14 , π‘ , π14
β²β² 1 π14 , π‘ , π15β²β² 1 π14 , π‘ are first augmentation coefficients for category 1, 2 and 3
+ π16β²β² 2,2, π17 , π‘ , + π17
β²β² 2,2, π17 , π‘ , + π18β²β² 2,2, π17 , π‘ are second augmentation coefficient for category 1, 2 and 3
+ π20β²β² 3,3, π21 , π‘ , + π21
β²β² 3,3, π21 , π‘ , + π22β²β² 3,3, π21 , π‘ are third augmentation coefficient for category 1, 2 and 3
+ π24β²β² 4,4,4,4, π25 , π‘ , + π25
β²β² 4,4,4,4, π25 , π‘ , + π26β²β² 4,4,4,4, π25 , π‘ are fourth augmentation coefficient for category 1, 2 and 3
+ π28β²β² 5,5,5,5, π29 , π‘ , + π29
β²β² 5,5,5,5, π29, π‘ , + π30β²β² 5,5,5,5, π29, π‘ are fifth augmentation coefficient for category 1, 2 and 3
+ π32β²β² 6,6,6,6, π33 , π‘ , + π33
β²β² 6,6,6,6, π33 , π‘ , + π34β²β² 6,6,6,6, π33 , π‘ are sixth augmentation coefficient for category 1, 2 and 3
ππ13
ππ‘= π13
1 π14 β
π13β² 1 β π13
β²β² 1 πΊ, π‘ β π16β²β² 2,2, πΊ19, π‘ β π20
β²β² 3,3, πΊ23 , π‘
β π24β²β² 4,4,4,4, πΊ27 , π‘ β π28
β²β² 5,5,5,5, πΊ31 , π‘ β π32β²β² 6,6,6,6, πΊ35 , π‘
π13
ππ14
ππ‘= π14
1 π13 β π14
β² 1 β π14β²β² 1 πΊ, π‘ β π17
β²β² 2,2, πΊ19 , π‘ β π21β²β² 3,3, πΊ23 , π‘
β π25β²β² 4,4,4,4, πΊ27 , π‘ β π29
β²β² 5,5,5,5, πΊ31 , π‘ β π33β²β² 6,6,6,6, πΊ35 , π‘
π14
ππ15
ππ‘= π15
1 π14 β π15
β² 1 β π15β²β² 1 πΊ, π‘ β π18
β²β² 2,2, πΊ19 , π‘ β π22β²β² 3,3, πΊ23 , π‘
β π26β²β² 4,4,4,4, πΊ27 , π‘ β π30
β²β² 5,5,5,5, πΊ31 , π‘ β π34β²β² 6,6,6,6, πΊ35 , π‘
π15
Where β π13β²β² 1 πΊ, π‘ , β π14
β²β² 1 πΊ, π‘ , β π15β²β² 1 πΊ, π‘ are first detrition coefficients for category 1, 2 and 3
β π16β²β² 2,2, πΊ19, π‘ , β π17
β²β² 2,2, πΊ19, π‘ , β π18β²β² 2,2, πΊ19 , π‘ are second detrition coefficients for category 1, 2 and 3
β π20β²β² 3,3, πΊ23 , π‘ , β π21
β²β² 3,3, πΊ23 , π‘ , β π22β²β² 3,3, πΊ23 , π‘ are third detrition coefficients for category 1, 2 and 3
β π24β²β² 4,4,4,4, πΊ27 , π‘ , β π25
β²β² 4,4,4,4, πΊ27 , π‘ , β π26β²β² 4,4,4,4, πΊ27 , π‘ are fourth detrition coefficients for category 1, 2 and 3
β π28β²β² 5,5,5,5, πΊ31 , π‘ , β π29
β²β² 5,5,5,5, πΊ31 , π‘ , β π30β²β² 5,5,5,5, πΊ31 , π‘ are fifth detrition coefficients for category 1, 2 and 3
β π32β²β² 6,6,6,6, πΊ35 , π‘ , β π33
β²β² 6,6,6,6, πΊ35 , π‘ , β π34β²β² 6,6,6,6, πΊ35 , π‘ are sixth detrition coefficients for category 1, 2 and 3
ππΊ16
ππ‘= π16
2 πΊ17 β π16
β² 2 + π16β²β² 2 π17 , π‘ + π13
β²β² 1,1, π14 , π‘ + π20β²β² 3,3,3 π21 , π‘
+ π24β²β² 4,4,4,4,4 π25 , π‘ + π28
β²β² 5,5,5,5,5 π29 , π‘ + π32β²β² 6,6,6,6,6 π33 , π‘
πΊ16
ππΊ17
ππ‘= π17
2 πΊ16 β π17
β² 2 + π17β²β² 2 π17 , π‘ + π14
β²β² 1,1, π14 , π‘ + π21β²β² 3,3,3 π21 , π‘
+ π25β²β² 4,4,4,4,4 π25 , π‘ + π29
β²β² 5,5,5,5,5 π29 , π‘ + π33β²β² 6,6,6,6,6 π33 , π‘
πΊ17
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ππΊ18
ππ‘= π18
2 πΊ17 β π18
β² 2 + π18β²β² 2 π17 , π‘ + π15
β²β² 1,1, π14 , π‘ + π22β²β² 3,3,3 π21 , π‘
+ π26β²β² 4,4,4,4,4 π25 , π‘ + π30
β²β² 5,5,5,5,5 π29 , π‘ + π34β²β² 6,6,6,6,6 π33 , π‘
πΊ18
Where + π16β²β² 2 π17 , π‘ , + π17
β²β² 2 π17 , π‘ , + π18β²β² 2 π17 , π‘ are first augmentation coefficients for category 1, 2 and 3
+ π13β²β² 1,1, π14 , π‘ , + π14
β²β² 1,1, π14 , π‘ , + π15β²β² 1,1, π14 , π‘ are second augmentation coefficient for category 1, 2 and 3
+ π20β²β² 3,3,3 π21 , π‘ , + π21
β²β² 3,3,3 π21 , π‘ , + π22β²β² 3,3,3 π21 , π‘ are third augmentation coefficient for category 1, 2 and 3
+ π24β²β² 4,4,4,4,4 π25 , π‘ , + π25
β²β² 4,4,4,4,4 π25 , π‘ , + π26β²β² 4,4,4,4,4 π25 , π‘ are fourth augmentation coefficient for category 1, 2 and 3
+ π28β²β² 5,5,5,5,5 π29, π‘ , + π29
β²β² 5,5,5,5,5 π29 , π‘ , + π30β²β² 5,5,5,5,5 π29, π‘ are fifth augmentation coefficient for category 1, 2 and 3
+ π32β²β² 6,6,6,6,6 π33 , π‘ , + π33
β²β² 6,6,6,6,6 π33 , π‘ , + π34β²β² 6,6,6,6,6 π33 , π‘ are sixth augmentation coefficient for category 1, 2 and 3
ππ16
ππ‘= π16
2 π17 β π16
β² 2 β π16β²β² 2 πΊ19 , π‘ β π13
β²β² 1,1, πΊ, π‘ β π20β²β² 3,3,3, πΊ23 , π‘
β π24β²β² 4,4,4,4,4 πΊ27 , π‘ β π28
β²β² 5,5,5,5,5 πΊ31 , π‘ β π32β²β² 6,6,6,6,6 πΊ35 , π‘
π16
ππ17
ππ‘= π17
2 π16 β π17
β² 2 β π17β²β² 2 πΊ19 , π‘ β π14
β²β² 1,1, πΊ, π‘ β π21β²β² 3,3,3, πΊ23 , π‘
β π25β²β² 4,4,4,4,4 πΊ27 , π‘ β π29
β²β² 5,5,5,5,5 πΊ31 , π‘ β π33β²β² 6,6,6,6,6 πΊ35 , π‘
π17
ππ18
ππ‘= π18
2 π17 β π18
β² 2 β π18β²β² 2 πΊ19 , π‘ β π15
β²β² 1,1, πΊ, π‘ β π22β²β² 3,3,3, πΊ23 , π‘
β π26β²β² 4,4,4,4,4 πΊ27 , π‘ β π30
β²β² 5,5,5,5,5 πΊ31 , π‘ β π34β²β² 6,6,6,6,6 πΊ35 , π‘
π18
where β b16β²β² 2 G19, t , β b17
β²β² 2 G19, t , β b18β²β² 2 G19, t are first detrition coefficients for category 1, 2 and 3
β π13β²β² 1,1, πΊ, π‘ , β π14
β²β² 1,1, πΊ, π‘ , β π15β²β² 1,1, πΊ, π‘ are second detrition coefficients for category 1,2 and 3
β π20β²β² 3,3,3, πΊ23 , π‘ , β π21
β²β² 3,3,3, πΊ23 , π‘ , β π22β²β² 3,3,3, πΊ23 , π‘ are third detrition coefficients for category 1,2 and 3
β π24β²β² 4,4,4,4,4 πΊ27 , π‘ , β π25
β²β² 4,4,4,4,4 πΊ27 , π‘ , β π26β²β² 4,4,4,4,4 πΊ27 , π‘ are fourth detrition coefficients for category 1,2 and 3
β π28β²β² 5,5,5,5,5 πΊ31 , π‘ , β π29
β²β² 5,5,5,5,5 πΊ31 , π‘ , β π30β²β² 5,5,5,5,5 πΊ31 , π‘ are fifth detrition coefficients for category 1,2 and 3
β π32β²β² 6,6,6,6,6 πΊ35 , π‘ , β π33
β²β² 6,6,6,6,6 πΊ35 , π‘ , β π34β²β² 6,6,6,6,6 πΊ35 , π‘ are sixth detrition coefficients for category 1,2 and 3
ππΊ20
ππ‘= π20
3 πΊ21 β
π20β² 3 + π20
β²β² 3 π21 , π‘ + π16β²β² 2,2,2 π17 , π‘ + π13
β²β² 1,1,1, π14 , π‘
+ π24β²β² 4,4,4,4,4,4 π25 , π‘ + π28
β²β² 5,5,5,5,5,5 π29 , π‘ + π32β²β² 6,6,6,6,6,6 π33 , π‘
πΊ20
ππΊ21
ππ‘= π21
3 πΊ20 β π21
β² 3 + π21β²β² 3 π21 , π‘ + π17
β²β² 2,2,2 π17 , π‘ + π14β²β² 1,1,1, π14 , π‘
+ π25β²β² 4,4,4,4,4,4 π25 , π‘ + π29
β²β² 5,5,5,5,5,5 π29 , π‘ + π33β²β² 6,6,6,6,6,6 π33 , π‘
πΊ21
ππΊ22
ππ‘= π22
3 πΊ21 β π22
β² 3 + π22β²β² 3 π21 , π‘ + π18
β²β² 2,2,2 π17 , π‘ + π15β²β² 1,1,1, π14 , π‘
+ π26β²β² 4,4,4,4,4,4 π25 , π‘ + π30
β²β² 5,5,5,5,5,5 π29 , π‘ + π34β²β² 6,6,6,6,6,6 π33 , π‘
πΊ22
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+ π20β²β² 3 π21 , π‘ , + π21
β²β² 3 π21 , π‘ , + π22β²β² 3 π21 , π‘ are first augmentation coefficients for category 1, 2 and 3
+ π16β²β² 2,2,2 π17 , π‘ , + π17
β²β² 2,2,2 π17 , π‘ , + π18β²β² 2,2,2 π17 , π‘ are second augmentation coefficients for category 1, 2 and 3
+ π13β²β² 1,1,1, π14 , π‘ , + π14
β²β² 1,1,1, π14 , π‘ , + π15β²β² 1,1,1, π14 , π‘ are third augmentation coefficients for category 1, 2 and 3
+ π24β²β² 4,4,4,4,4,4 π25 , π‘ , + π25
β²β² 4,4,4,4,4,4 π25 , π‘ , + π26β²β² 4,4,4,4,4,4 π25 , π‘ are fourth augmentation coefficients for category 1, 2
and 3
+ π28β²β² 5,5,5,5,5,5 π29, π‘ , + π29
β²β² 5,5,5,5,5,5 π29, π‘ , + π30β²β² 5,5,5,5,5,5 π29 , π‘ are fifth augmentation coefficients for category 1, 2 and 3
+ π32β²β² 6,6,6,6,6,6 π33 , π‘ , + π33
β²β² 6,6,6,6,6,6 π33 , π‘ , + π34β²β² 6,6,6,6,6,6 π33 , π‘ are sixth augmentation coefficients for category 1, 2 and
3
ππ20
ππ‘= π20
3 π21 β π20
β² 3 β π20β²β² 3 πΊ23 , π‘ β π16
β²β² 2,2,2 πΊ19 , π‘ β π13β²β² 1,1,1, πΊ, π‘
β π24β²β² 4,4,4,4,4,4 πΊ27 , π‘ β π28
β²β² 5,5,5,5,5,5 πΊ31 , π‘ β π32β²β² 6,6,6,6,6,6 πΊ35 , π‘
π20
ππ21
ππ‘= π21
3 π20 β π21
β² 3 β π21β²β² 3 πΊ23 , π‘ β π17
β²β² 2,2,2 πΊ19 , π‘ β π14β²β² 1,1,1, πΊ, π‘
β π25β²β² 4,4,4,4,4,4 πΊ27 , π‘ β π29
β²β² 5,5,5,5,5,5 πΊ31 , π‘ β π33β²β² 6,6,6,6,6,6 πΊ35 , π‘
π21
ππ22
ππ‘= π22
3 π21 β π22
β² 3 β π22β²β² 3 πΊ23 , π‘ β π18
β²β² 2,2,2 πΊ19 , π‘ β π15β²β² 1,1,1, πΊ, π‘
β π26β²β² 4,4,4,4,4,4 πΊ27 , π‘ β π30
β²β² 5,5,5,5,5,5 πΊ31 , π‘ β π34β²β² 6,6,6,6,6,6 πΊ35 , π‘
π22
β π20β²β² 3 πΊ23 , π‘ , β π21
β²β² 3 πΊ23 , π‘ , β π22β²β² 3 πΊ23 , π‘ are first detrition coefficients for category 1, 2 and 3
β π16β²β² 2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2 πΊ19 , π‘ , β π18β²β² 2,2,2 πΊ19, π‘ are second detrition coefficients for category 1, 2 and 3
β π13β²β² 1,1,1, πΊ, π‘ , β π14
β²β² 1,1,1, πΊ, π‘ , β π15β²β² 1,1,1, πΊ, π‘ are third detrition coefficients for category 1,2 and 3
β π24β²β² 4,4,4,4,4,4 πΊ27 , π‘ , β π25
β²β² 4,4,4,4,4,4 πΊ27 , π‘ , β π26β²β² 4,4,4,4,4,4 πΊ27 , π‘ are fourth detrition coefficients for category 1, 2 and
3
β π28β²β² 5,5,5,5,5,5 πΊ31 , π‘ , β π29
β²β² 5,5,5,5,5,5 πΊ31 , π‘ , β π30β²β² 5,5,5,5,5,5 πΊ31 , π‘ are fifth detrition coefficients for category 1, 2 and
3
β π32β²β² 6,6,6,6,6,6 πΊ35 , π‘ , β π33
β²β² 6,6,6,6,6,6 πΊ35 , π‘ , β π34β²β² 6,6,6,6,6,6 πΊ35 , π‘ are sixth detrition coefficients for category 1, 2 and 3
ππΊ24
ππ‘= π24
4 πΊ25 β π24
β² 4 + π24β²β² 4 π25 , π‘ + π28
β²β² 5,5, π29 , π‘ + π32β²β² 6,6, π33 , π‘
+ π13β²β² 1,1,1,1 π14 , π‘ + π16
β²β² 2,2,2,2 π17 , π‘ + π20β²β² 3,3,3,3 π21 , π‘
πΊ24
ππΊ25
ππ‘= π25
4 πΊ24 β π25
β² 4 + π25β²β² 4 π25 , π‘ + π29
β²β² 5,5, π29 , π‘ + π33β²β² 6,6 π33 , π‘
+ π14β²β² 1,1,1,1 π14 , π‘ + π17
β²β² 2,2,2,2 π17 , π‘ + π21β²β² 3,3,3,3 π21 , π‘
πΊ25
ππΊ26
ππ‘= π26
4 πΊ25 β π26
β² 4 + π26β²β² 4 π25 , π‘ + π30
β²β² 5,5, π29 , π‘ + π34β²β² 6,6, π33 , π‘
+ π15β²β² 1,1,1,1 π14 , π‘ + π18
β²β² 2,2,2,2 π17 , π‘ + π22β²β² 3,3,3,3 π21 , π‘
πΊ26
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πππππ π24β²β² 4 π25 , π‘ , π25
β²β² 4 π25 , π‘ , π26β²β² 4 π25 , π‘ πππ ππππ π‘ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π28β²β² 5,5, π29 , π‘ , + π29
β²β² 5,5, π29 , π‘ , + π30β²β² 5,5, π29, π‘ πππ π πππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π32β²β² 6,6, π33 , π‘ , + π33
β²β² 6,6, π33 , π‘ , + π34β²β² 6,6, π33 , π‘ πππ π‘ππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π13β²β² 1,1,1,1 π14 , π‘ , + π14
β²β² 1,1,1,1 π14 , π‘ , + π15β²β² 1,1,1,1 π14 , π‘ are fourth augmentation coefficients for category 1, 2,and 3
+ π16β²β² 2,2,2,2 π17 , π‘ , + π17
β²β² 2,2,2,2 π17 , π‘ , + π18β²β² 2,2,2,2 π17 , π‘ are fifth augmentation coefficients for category 1, 2,and 3
+ π20β²β² 3,3,3,3 π21 , π‘ , + π21
β²β² 3,3,3,3 π21 , π‘ , + π22β²β² 3,3,3,3 π21 , π‘ are sixth augmentation coefficients for category 1, 2,and 3
ππ24
ππ‘= π24
4 π25 β π24
β² 4 β π24β²β² 4 πΊ27 , π‘ β π28
β²β² 5,5, πΊ31 , π‘ β π32β²β² 6,6, πΊ35 , π‘
β π13β²β² 1,1,1,1 πΊ, π‘ β π16
β²β² 2,2,2,2 πΊ19 , π‘ β π20β²β² 3,3,3,3 πΊ23 , π‘
π24
ππ25
ππ‘= π25
4 π24 β π25
β² 4 β π25β²β² 4 πΊ27 , π‘ β π29
β²β² 5,5, πΊ31 , π‘ β π33β²β² 6,6, πΊ35 , π‘
β π14β²β² 1,1,1,1 πΊ, π‘ β π17
β²β² 2,2,2,2 πΊ19, π‘ β π21β²β² 3,3,3,3 πΊ23 , π‘
π25
ππ26
ππ‘= π26
4 π25 β π26
β² 4 β π26β²β² 4 πΊ27 , π‘ β π30
β²β² 5,5, πΊ31 , π‘ β π34β²β² 6,6, πΊ35 , π‘
β π15β²β² 1,1,1,1 πΊ, π‘ β π18
β²β² 2,2,2,2 πΊ19, π‘ β π22β²β² 3,3,3,3 πΊ23 , π‘
π26
πππππ β π24β²β² 4 πΊ27 , π‘ , β π25
β²β² 4 πΊ27 , π‘ , β π26β²β² 4 πΊ27 , π‘ πππ ππππ π‘ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π28β²β² 5,5, πΊ31 , π‘ , β π29
β²β² 5,5, πΊ31 , π‘ , β π30β²β² 5,5, πΊ31, π‘ πππ π πππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π32β²β² 6,6, πΊ35 , π‘ , β π33
β²β² 6,6, πΊ35 , π‘ , β π34β²β² 6,6, πΊ35 , π‘ πππ π‘ππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π13β²β² 1,1,1,1 πΊ, π‘ , β π14
β²β² 1,1,1,1 πΊ, π‘ , β π15β²β² 1,1,1,1 πΊ, π‘ πππ πππ’ππ‘π πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π16β²β² 2,2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2,2 πΊ19, π‘ , β π18β²β² 2,2,2,2 πΊ19 , π‘ πππ ππππ‘π πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π20β²β² 3,3,3,3 πΊ23 , π‘ , β π21
β²β² 3,3,3,3 πΊ23 , π‘ , β π22β²β² 3,3,3,3 πΊ23 , π‘ πππ π ππ₯π‘π πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
ππΊ28
ππ‘= π28
5 πΊ29 β π28
β² 5 + π28β²β² 5 π29 , π‘ + π24
β²β² 4,4, π25 , π‘ + π32β²β² 6,6,6 π33 , π‘
+ π13β²β² 1,1,1,1,1 π14 , π‘ + π16
β²β² 2,2,2,2,2 π17 , π‘ + π20β²β² 3,3,3,3,3 π21 , π‘
πΊ28
ππΊ29
ππ‘= π29
5 πΊ28 β π29
β² 5 + π29β²β² 5 π29 , π‘ + π25
β²β² 4,4, π25 , π‘ + π33β²β² 6,6,6 π33 , π‘
+ π14β²β² 1,1,1,1,1 π14 , π‘ + π17
β²β² 2,2,2,2,2 π17 , π‘ + π21β²β² 3,3,3,3,3 π21 , π‘
πΊ29
ππΊ30
ππ‘= π30
5 πΊ29 β
π30β² 5 + π30
β²β² 5 π29 , π‘ + π26β²β² 4,4, π25 , π‘ + π34
β²β² 6,6,6 π33 , π‘
+ π15β²β² 1,1,1,1,1 π14 , π‘ + π18
β²β² 2,2,2,2,2 π17 , π‘ + π22β²β² 3,3,3,3,3 π21 , π‘
πΊ30
πππππ + π28β²β² 5 π29 , π‘ , + π29
β²β² 5 π29 , π‘ , + π30β²β² 5 π29 , π‘ πππ ππππ π‘ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
π΄ππ + π24β²β² 4,4, π25 , π‘ , + π25
β²β² 4,4, π25 , π‘ , + π26β²β² 4,4, π25 , π‘ πππ π πππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
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+ π32β²β² 6,6,6 π33 , π‘ , + π33
β²β² 6,6,6 π33 , π‘ , + π34β²β² 6,6,6 π33 , π‘ πππ π‘ππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘ πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π13β²β² 1,1,1,1,1 π14 , π‘ , + π14
β²β² 1,1,1,1,1 π14 , π‘ , + π15β²β² 1,1,1,1,1 π14 , π‘ are fourth augmentation coefficients for category 1,2, and 3
+ π16β²β² 2,2,2,2,2 π17 , π‘ , + π17
β²β² 2,2,2,2,2 π17 , π‘ , + π18β²β² 2,2,2,2,2 π17 , π‘ are fifth augmentation coefficients for category 1,2,and 3
+ π20β²β² 3,3,3,3,3 π21 , π‘ , + π21
β²β² 3,3,3,3,3 π21 , π‘ , + π22β²β² 3,3,3,3,3 π21 , π‘ are sixth augmentation coefficients for category 1,2, 3
ππ28
ππ‘= π28
5 π29 β π28
β² 5 β π28β²β² 5 πΊ31 , π‘ β π24
β²β² 4,4, πΊ27 , π‘ β π32β²β² 6,6,6 πΊ35 , π‘
β π13β²β² 1,1,1,1,1 πΊ, π‘ β π16
β²β² 2,2,2,2,2 πΊ19 , π‘ β π20β²β² 3,3,3,3,3 πΊ23 , π‘
π28
ππ29
ππ‘= π29
5 π28 β π29
β² 5 β π29β²β² 5 πΊ31 , π‘ β π25
β²β² 4,4, πΊ27 , π‘ β π33β²β² 6,6,6 πΊ35 , π‘
β π14β²β² 1,1,1,1,1 πΊ, π‘ β π17
β²β² 2,2,2,2,2 πΊ19, π‘ β π21β²β² 3,3,3,3,3 πΊ23 , π‘
π29
ππ30
ππ‘= π30
5 π29 β π30
β² 5 β π30β²β² 5 πΊ31 , π‘ β π26
β²β² 4,4, πΊ27 , π‘ β π34β²β² 6,6,6 πΊ35 , π‘
β π15β²β² 1,1,1,1,1, πΊ, π‘ β π18
β²β² 2,2,2,2,2 πΊ19 , π‘ β π22β²β² 3,3,3,3,3 πΊ23 , π‘
π30
π€ππππ β π28β²β² 5 πΊ31 , π‘ , β π29
β²β² 5 πΊ31 , π‘ , β π30β²β² 5 πΊ31 , π‘ πππ ππππ π‘ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π24β²β² 4,4, πΊ27 , π‘ , β π25
β²β² 4,4, πΊ27 , π‘ , β π26β²β² 4,4, πΊ27 , π‘ πππ π πππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1,2 πππ 3
β π32β²β² 6,6,6 πΊ35 , π‘ , β π33
β²β² 6,6,6 πΊ35 , π‘ , β π34β²β² 6,6,6 πΊ35 , π‘ πππ π‘ππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1,2 πππ 3
β π13β²β² 1,1,1,1,1 πΊ, π‘ , β π14
β²β² 1,1,1,1,1 πΊ, π‘ , β π15β²β² 1,1,1,1,1, πΊ, π‘ are fourth detrition coefficients for category 1,2, and 3
β π16β²β² 2,2,2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2,2,2 πΊ19, π‘ , β π18β²β² 2,2,2,2,2 πΊ19, π‘ are fifth detrition coefficients for category 1,2, and 3
β π20β²β² 3,3,3,3,3 πΊ23 , π‘ , β π21
β²β² 3,3,3,3,3 πΊ23 , π‘ , β π22β²β² 3,3,3,3,3 πΊ23 , π‘ are sixth detrition coefficients for category 1,2, and 3
ππΊ32
ππ‘= π32
6 πΊ33 β π32
β² 6 + π32β²β² 6 π33 , π‘ + π28
β²β² 5,5,5 π29 , π‘ + π24β²β² 4,4,4, π25 , π‘
+ π13β²β² 1,1,1,1,1,1 π14 , π‘ + π16
β²β² 2,2,2,2,2,2 π17 , π‘ + π20β²β² 3,3,3,3,3,3 π21 , π‘
πΊ32
ππΊ33
ππ‘= π33
6 πΊ32 β π33
β² 6 + π33β²β² 6 π33 , π‘ + π29
β²β² 5,5,5 π29 , π‘ + π25β²β² 4,4,4, π25 , π‘
+ π14β²β² 1,1,1,1,1,1 π14 , π‘ + π17
β²β² 2,2,2,2,2,2 π17 , π‘ + π21β²β² 3,3,3,3,3,3 π21 , π‘
πΊ33
ππΊ34
ππ‘= π34
6 πΊ33 β π34
β² 6 + π34β²β² 6 π33 , π‘ + π30
β²β² 5,5,5 π29, π‘ + π26β²β² 4,4,4, π25 , π‘
+ π15β²β² 1,1,1,1,1,1 π14 , π‘ + π18
β²β² 2,2,2,2,2,2 π17 , π‘ + π22β²β² 3,3,3,3,3,3 π21 , π‘
πΊ34
+ π32β²β² 6 π33 , π‘ , + π33
β²β² 6 π33 , π‘ , + π34β²β² 6 π33 , π‘ πππ ππππ π‘ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π28β²β² 5,5,5 π29 , π‘ , + π29
β²β² 5,5,5 π29, π‘ , + π30β²β² 5,5,5 π29, π‘ πππ π πππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π24β²β² 4,4,4, π25 , π‘ , + π25
β²β² 4,4,4, π25 , π‘ , + π26β²β² 4,4,4, π25 , π‘ πππ π‘ππππ ππ’πππππ‘ππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
+ π13β²β² 1,1,1,1,1,1 π14 , π‘ , + π14
β²β² 1,1,1,1,1,1 π14 , π‘ , + π15β²β² 1,1,1,1,1,1 π14 , π‘ - are fourth augmentation coefficients
+ π16β²β² 2,2,2,2,2,2 π17 , π‘ , + π17
β²β² 2,2,2,2,2,2 π17 , π‘ , + π18β²β² 2,2,2,2,2,2 π17 , π‘ - fifth augmentation coefficients
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+ π20β²β² 3,3,3,3,3,3 π21 , π‘ , + π21
β²β² 3,3,3,3,3,3 π21 , π‘ , + π22β²β² 3,3,3,3,3,3 π21 , π‘ sixth augmentation coefficients
ππ32
ππ‘= π32
6 π33 β π32
β² 6 β π32β²β² 6 πΊ35 , π‘ β π28
β²β² 5,5,5 πΊ31 , π‘ β π24β²β² 4,4,4, πΊ27 , π‘
β π13β²β² 1,1,1,1,1,1 πΊ, π‘ β π16
β²β² 2,2,2,2,2,2 πΊ19 , π‘ β π20β²β² 3,3,3,3,3,3 πΊ23 , π‘
π32
ππ33
ππ‘= π33
6 π32 β π33
β² 6 β π33β²β² 6 πΊ35 , π‘ β π29
β²β² 5,5,5 πΊ31 , π‘ β π25β²β² 4,4,4, πΊ27 , π‘
β π14β²β² 1,1,1,1,1,1 πΊ, π‘ β π17
β²β² 2,2,2,2,2,2 πΊ19 , π‘ β π21β²β² 3,3,3,3,3,3 πΊ23 , π‘
π33
ππ34
ππ‘= π34
6 π33 β π34
β² 6 β π34β²β² 6 πΊ35 , π‘ β π30
β²β² 5,5,5 πΊ31 , π‘ β π26β²β² 4,4,4, πΊ27 , π‘
β π15β²β² 1,1,1,1,1,1 πΊ, π‘ β π18
β²β² 2,2,2,2,2,2 πΊ19 , π‘ β π22β²β² 3,3,3,3,3,3 πΊ23 , π‘
π34
β π32β²β² 6 πΊ35 , π‘ , β π33
β²β² 6 πΊ35 , π‘ , β π34β²β² 6 πΊ35 , π‘ πππ ππππ π‘ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π28β²β² 5,5,5 πΊ31 , π‘ , β π29
β²β² 5,5,5 πΊ31 , π‘ , β π30β²β² 5,5,5 πΊ31 , π‘ πππ π πππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1, 2 πππ 3
β π24β²β² 4,4,4, πΊ27 , π‘ , β π25
β²β² 4,4,4, πΊ27 , π‘ , β π26β²β² 4,4,4, πΊ27 , π‘ πππ π‘ππππ πππ‘πππ‘πππ πππππππππππ‘π πππ πππ‘πππππ¦ 1,2 πππ 3
β π13β²β² 1,1,1,1,1,1 πΊ, π‘ , β π14
β²β² 1,1,1,1,1,1 πΊ, π‘ , β π15β²β² 1,1,1,1,1,1 πΊ, π‘ are fourth detrition coefficients for category 1, 2, and 3
β π16β²β² 2,2,2,2,2,2 πΊ19 , π‘ , β π17
β²β² 2,2,2,2,2,2 πΊ19 , π‘ , β π18β²β² 2,2,2,2,2,2 πΊ19, π‘ are fifth detrition coefficients for category 1, 2, and 3
β π20β²β² 3,3,3,3,3,3 πΊ23 , π‘ , β π21
β²β² 3,3,3,3,3,3 πΊ23 , π‘ , β π22β²β² 3,3,3,3,3,3 πΊ23 , π‘ are sixth detrition coefficients for category 1, 2, and 3
Where we suppose
(A) ππ 1 , ππ
β² 1 , ππβ²β² 1 , ππ
1 , ππβ² 1 , ππ
β²β² 1 > 0,
π, π = 13,14,15
(B) The functions ππβ²β² 1 , ππ
β²β² 1 are positive continuous increasing and bounded.
Definition of (ππ) 1 , (ππ)
1 :
ππβ²β² 1 (π14 , π‘) β€ (ππ)
1 β€ ( π΄ 13 )(1)
ππβ²β² 1 (πΊ, π‘) β€ (ππ)
1 β€ (ππβ²) 1 β€ ( π΅ 13 )(1)
(C) ππππ2ββ ππβ²β² 1 π14 , π‘ = (ππ)
1
limGββ ππβ²β² 1 πΊ, π‘ = (ππ)
1
Definition of ( π΄ 13 )(1), ( π΅ 13 )(1) :
Where ( π΄ 13 )(1), ( π΅ 13 )(1), (ππ) 1 , (ππ)
1 are positive constants and π = 13,14,15
They satisfy Lipschitz condition:
|(ππβ²β² ) 1 π14
β² , π‘ β (ππβ²β² ) 1 π14 , π‘ | β€ ( π 13 )(1)|π14 β π14
β² |πβ( π 13 )(1)π‘
|(ππβ²β² ) 1 πΊ β² , π‘ β (ππ
β²β² ) 1 πΊ, π‘ | < ( π 13 )(1)||πΊ β πΊ β² ||πβ( π 13 )(1)π‘
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With the Lipschitz condition, we place a restriction on the behavior of functions
(ππβ²β² ) 1 π14
β² , π‘ and(ππβ²β² ) 1 π14 , π‘ . π14
β² , π‘ and π14 , π‘ are points belonging to the interval
( π 13 )(1), ( π 13 )(1) . It is to be noted that (ππβ²β² ) 1 π14 , π‘ is uniformly continuous. In the eventuality of the
fact, that if ( π 13 )(1) = 1 then the function (ππβ²β² ) 1 π14 , π‘ , the first augmentation coefficient WOULD be
absolutely continuous.
Definition of ( π 13 )(1), ( π 13 )(1) :
(D) ( π 13 )(1), ( π 13 )(1), are positive constants
(ππ) 1
( π 13 )(1) ,(ππ) 1
( π 13 )(1) < 1
Definition of ( π 13 )(1), ( π 13 )(1) :
(E) There exists two constants ( π 13 )(1) and ( π 13 )(1) which together
with ( π 13 )(1), ( π 13 )(1), (π΄ 13)(1) and ( π΅ 13 )(1) and the constants
(ππ) 1 , (ππ
β²) 1 , (ππ) 1 , (ππ
β²) 1 , (ππ) 1 , (ππ)
1 , π = 13,14,15,
satisfy the inequalities
1
( π 13 )(1) [ (ππ) 1 + (ππ
β²) 1 + ( π΄ 13 )(1) + ( π 13 )(1) ( π 13 )(1)] < 1
1
( π 13 )(1) [ (ππ) 1 + (ππ
β²) 1 + ( π΅ 13 )(1) + ( π 13 )(1) ( π 13 )(1)] < 1
Where we suppose
(F) ππ 2 , ππ
β² 2 , ππβ²β² 2 , ππ
2 , ππβ² 2 , ππ
β²β² 2 > 0, π, π = 16,17,18
(G) The functions ππβ²β² 2 , ππ
β²β² 2 are positive continuous increasing and bounded.
Definition of (pi) 2 , (ri)
2 :
ππβ²β² 2 π17 , π‘ β€ (ππ)
2 β€ π΄ 16 2
ππβ²β² 2 (πΊ19 , π‘) β€ (ππ)
2 β€ (ππβ²) 2 β€ ( π΅ 16 )(2)
(H) limπ2ββ ππβ²β² 2 π17 , π‘ = (ππ)
2
limπΊββ ππβ²β² 2 πΊ19 , π‘ = (ππ)
2
Definition of ( π΄ 16 )(2), ( π΅ 16 )(2) :
Where ( π΄ 16 )(2), ( π΅ 16 )(2), (ππ) 2 , (ππ)
2 are positive constants and π = 16,17,18
They satisfy Lipschitz condition:
|(ππβ²β² ) 2 π17
β² , π‘ β (ππβ²β² ) 2 π17 , π‘ | β€ ( π 16 )(2)|π17 β π17
β² |πβ( π 16 )(2)π‘
|(ππβ²β² ) 2 πΊ19
β² , π‘ β (ππβ²β² ) 2 πΊ19 , π‘ | < ( π 16 )(2)|| πΊ19 β πΊ19
β² ||πβ( π 16 )(2)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 2 π17
β² , π‘
and(ππβ²β² ) 2 π17 , π‘ . π17
β² , π‘ and π17 , π‘ are points belonging to the interval ( π 16 )(2), ( π 16 )(2) . It is to
be noted that (ππβ²β² ) 2 π17 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 16 )(2) = 1
then the function (ππβ²β² ) 2 π17 , π‘ , the SECOND augmentation coefficient would be absolutely continuous.
Definition of ( π 16 )(2), ( π 16 )(2) :
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(I) ( π 16 )(2), ( π 16 )(2), are positive constants
(ππ) 2
( π 16 )(2) ,(ππ)
2
( π 16 )(2) < 1
Definition of ( π 13 )(2), ( π 13 )(2) :
There exists two constants ( π 16 )(2) and ( π 16 )(2) which together
with ( π 16 )(2), ( π 16 )(2), (π΄ 16)(2)πππ ( π΅ 16 )(2) and the constants
(ππ) 2 , (ππ
β²) 2 , (ππ) 2 , (ππ
β²) 2 , (ππ) 2 , (ππ)
2 , π = 16,17,18,
satisfy the inequalities
1
( M 16 )(2) [ (ai) 2 + (ai
β²) 2 + ( A 16 )(2) + ( P 16 )(2) ( k 16 )(2)] < 1
1
( π 16 )(2) [ (ππ) 2 + (ππ
β²) 2 + ( π΅ 16 )(2) + ( π 16 )(2) ( π 16 )(2)] < 1
Where we suppose
(J) ππ 3 , ππ
β² 3 , ππβ²β² 3 , ππ
3 , ππβ² 3 , ππ
β²β² 3 > 0, π, π = 20,21,22
The functions ππβ²β² 3 , ππ
β²β² 3 are positive continuous increasing and bounded.
Definition of (ππ) 3 , (ri)
3 :
ππβ²β² 3 (π21 , π‘) β€ (ππ)
3 β€ ( π΄ 20 )(3)
ππβ²β² 3 (πΊ23 , π‘) β€ (ππ)
3 β€ (ππβ²) 3 β€ ( π΅ 20 )(3)
ππππ2ββ ππβ²β² 3 π21 , π‘ = (ππ)
3
limGββ ππβ²β² 3 πΊ23 , π‘ = (ππ)
3
Definition of ( π΄ 20 )(3), ( π΅ 20 )(3) :
Where ( π΄ 20 )(3), ( π΅ 20 )(3), (ππ) 3 , (ππ)
3 are positive constants and π = 20,21,22
They satisfy Lipschitz condition:
|(ππβ²β² ) 3 π21
β² , π‘ β (ππβ²β² ) 3 π21 , π‘ | β€ ( π 20 )(3)|π21 β π21
β² |πβ( π 20 )(3)π‘
|(ππβ²β² ) 3 πΊ23
β² , π‘ β (ππβ²β² ) 3 πΊ23 , π‘ | < ( π 20 )(3)||πΊ23 β πΊ23
β² ||πβ( π 20 )(3)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 3 π21
β² , π‘
and(ππβ²β² ) 3 π21 , π‘ . π21
β² , π‘ And π21 , π‘ are points belonging to the interval ( π 20 )(3), ( π 20 )(3) . It is to
be noted that (ππβ²β² ) 3 π21 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 20 )(3) = 1
then the function (ππβ²β² ) 3 π21 , π‘ , the THIRD augmentation coefficient, would be absolutely continuous.
Definition of ( π 20 )(3), ( π 20 )(3) :
(K) ( π 20 )(3), ( π 20 )(3), are positive constants
(ππ) 3
( π 20 )(3) ,(ππ) 3
( π 20 )(3) < 1
There exists two constants There exists two constants ( π 20 )(3) and ( π 20 )(3) which together with
( π 20 )(3), ( π 20 )(3), (π΄ 20)(3)πππ ( π΅ 20 )(3) and the constants
(ππ) 3 , (ππ
β² ) 3 , (ππ) 3 , (ππ
β²) 3 , (ππ) 3 , (ππ)
3 , π = 20,21,22,
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satisfy the inequalities
1
( π 20 )(3) [ (ππ) 3 + (ππ
β²) 3 + ( π΄ 20 )(3) + ( π 20 )(3) ( π 20 )(3)] < 1
1
( π 20 )(3) [ (ππ) 3 + (ππ
β²) 3 + ( π΅ 20 )(3) + ( π 20 )(3) ( π 20 )(3)] < 1
Where we suppose
(L) ππ 4 , ππ
β² 4 , ππβ²β² 4 , ππ
4 , ππβ² 4 , ππ
β²β² 4 > 0, π, π = 24,25,26
(M) The functions ππβ²β² 4 , ππ
β²β² 4 are positive continuous increasing and bounded.
Definition of (ππ) 4 , (ππ)
4 :
ππβ²β² 4 (π25 , π‘) β€ (ππ)
4 β€ ( π΄ 24 )(4)
ππβ²β² 4 πΊ27 , π‘ β€ (ππ)
4 β€ (ππβ²) 4 β€ ( π΅ 24 )(4)
(N) ππππ2ββ ππβ²β² 4 π25 , π‘ = (ππ)
4
limGββ ππβ²β² 4 πΊ27 , π‘ = (ππ)
4
Definition of ( π΄ 24 )(4), ( π΅ 24 )(4) :
Where ( π΄ 24 )(4), ( π΅ 24 )(4), (ππ) 4 , (ππ)
4 are positive constants and π = 24,25,26
They satisfy Lipschitz condition:
|(ππβ²β² ) 4 π25
β² , π‘ β (ππβ²β² ) 4 π25 , π‘ | β€ ( π 24 )(4)|π25 β π25
β² |πβ( π 24 )(4)π‘
|(ππβ²β² ) 4 πΊ27
β² , π‘ β (ππβ²β² ) 4 πΊ27 , π‘ | < ( π 24 )(4)|| πΊ27 β πΊ27
β² ||πβ( π 24 )(4)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 4 π25
β² , π‘
and(ππβ²β² ) 4 π25 , π‘ . π25
β² , π‘ and π25 , π‘ are points belonging to the interval ( π 24 )(4), ( π 24 )(4) . It is to
be noted that (ππβ²β² ) 4 π25 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 24 )(4) = 4
then the function (ππβ²β² ) 4 π25 , π‘ , the FOURTH augmentation coefficient WOULD be absolutely
continuous.
Definition of ( π 24 )(4), ( π 24 )(4) :
(O) ( π 24 )(4), ( π 24 )(4), are positive constants (P)
(ππ) 4
( π 24 )(4) ,(ππ) 4
( π 24 )(4) < 1
Definition of ( π 24 )(4), ( π 24 )(4) :
(Q) There exists two constants ( π 24 )(4) and ( π 24 )(4) which together with
( π 24 )(4), ( π 24 )(4), (π΄ 24)(4)πππ ( π΅ 24 )(4) and the constants
(ππ) 4 , (ππ
β²) 4 , (ππ) 4 , (ππ
β²) 4 , (ππ) 4 , (ππ)
4 , π = 24,25,26, satisfy the inequalities
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1
( π 24 )(4) [ (ππ) 4 + (ππ
β²) 4 + ( π΄ 24 )(4) + ( π 24 )(4) ( π 24 )(4)] < 1
1
( π 24 )(4) [ (ππ) 4 + (ππ
β²) 4 + ( π΅ 24 )(4) + ( π 24 )(4) ( π 24 )(4)] < 1
Where we suppose
(R) ππ 5 , ππ
β² 5 , ππβ²β² 5 , ππ
5 , ππβ² 5 , ππ
β²β² 5 > 0, π, π = 28,29,30
(S) The functions ππβ²β² 5 , ππ
β²β² 5 are positive continuous increasing and bounded.
Definition of (ππ) 5 , (ππ)
5 :
ππβ²β² 5 (π29 , π‘) β€ (ππ)
5 β€ ( π΄ 28 )(5)
ππβ²β² 5 πΊ31 , π‘ β€ (ππ)
5 β€ (ππβ²) 5 β€ ( π΅ 28 )(5)
(T) ππππ2ββ ππβ²β² 5 π29 , π‘ = (ππ)
5
limGββ ππβ²β² 5 πΊ31 , π‘ = (ππ)
5
Definition of ( π΄ 28 )(5), ( π΅ 28 )(5) :
Where ( π΄ 28 )(5), ( π΅ 28 )(5), (ππ) 5 , (ππ)
5 are positive constants and π = 28,29,30
They satisfy Lipschitz condition:
|(ππβ²β² ) 5 π29
β² , π‘ β (ππβ²β² ) 5 π29 , π‘ | β€ ( π 28 )(5)|π29 β π29
β² |πβ( π 28 )(5)π‘
|(ππβ²β² ) 5 πΊ31
β² , π‘ β (ππβ²β² ) 5 πΊ31 , π‘ | < ( π 28 )(5)|| πΊ31 β πΊ31
β² ||πβ( π 28 )(5)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 5 π29
β² , π‘
and(ππβ²β² ) 5 π29 , π‘ . π29
β² , π‘ and π29 , π‘ are points belonging to the interval ( π 28 )(5), ( π 28 )(5) . It is to
be noted that (ππβ²β² ) 5 π29 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 28 )(5) = 5
then the function (ππβ²β² ) 5 π29 , π‘ , theFIFTH augmentation coefficient attributable would be absolutely
continuous.
Definition of ( π 28 )(5), ( π 28 )(5) :
(U) ( π 28 )(5), ( π 28 )(5), are positive constants
(ππ) 5
( π 28 )(5) ,(ππ)
5
( π 28 )(5) < 1
Definition of ( π 28 )(5), ( π 28 )(5) :
(V) There exists two constants ( π 28 )(5) and ( π 28 )(5) which together with
( π 28 )(5), ( π 28 )(5), (π΄ 28)(5)πππ ( π΅ 28 )(5) and the constants
(ππ) 5 , (ππ
β²) 5 , (ππ) 5 , (ππ
β²) 5 , (ππ) 5 , (ππ)
5 , π = 28,29,30, satisfy the inequalities
1
( π 28 )(5) [ (ππ) 5 + (ππ
β²) 5 + ( π΄ 28 )(5) + ( π 28 )(5) ( π 28 )(5)] < 1
1
( π 28 )(5) [ (ππ) 5 + (ππ
β²) 5 + ( π΅ 28 )(5) + ( π 28 )(5) ( π 28 )(5)] < 1
Where we suppose
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ππ 6 , ππ
β² 6 , ππβ²β² 6 , ππ
6 , ππβ² 6 , ππ
β²β² 6 > 0, π, π = 32,33,34
(W) The functions ππβ²β² 6 , ππ
β²β² 6 are positive continuous increasing and bounded.
Definition of (ππ) 6 , (ππ)
6 :
ππβ²β² 6 (π33 , π‘) β€ (ππ)
6 β€ ( π΄ 32 )(6)
ππβ²β² 6 ( πΊ35 , π‘) β€ (ππ)
6 β€ (ππβ²) 6 β€ ( π΅ 32 )(6)
(X) ππππ2ββ ππβ²β² 6 π33 , π‘ = (ππ)
6
limGββ ππβ²β² 6 πΊ35 , π‘ = (ππ)
6
Definition of ( π΄ 32 )(6), ( π΅ 32 )(6) :
Where ( π΄ 32 )(6), ( π΅ 32 )(6), (ππ) 6 , (ππ)
6 are positive constants and π = 32,33,34
They satisfy Lipschitz condition:
|(ππβ²β² ) 6 π33
β² , π‘ β (ππβ²β² ) 6 π33 , π‘ | β€ ( π 32 )(6)|π33 β π33
β² |πβ( π 32 )(6)π‘
|(ππβ²β² ) 6 πΊ35
β² , π‘ β (ππβ²β² ) 6 πΊ35 , π‘ | < ( π 32 )(6)|| πΊ35 β πΊ35
β² ||πβ( π 32 )(6)π‘
With the Lipschitz condition, we place a restriction on the behavior of functions (ππβ²β² ) 6 π33
β² , π‘
and(ππβ²β² ) 6 π33 , π‘ . π33
β² , π‘ and π33 , π‘ are points belonging to the interval ( π 32 )(6), ( π 32 )(6) . It is to
be noted that (ππβ²β² ) 6 π33 , π‘ is uniformly continuous. In the eventuality of the fact, that if ( π 32 )(6) = 6
then the function (ππβ²β² ) 6 π33 , π‘ , the SIXTH augmentation coefficient would be absolutely continuous.
Definition of ( π 32 )(6), ( π 32 )(6) :
( π 32 )(6), ( π 32 )(6), are positive constants
(ππ) 6
( π 32 )(6) ,(ππ)
6
( π 32 )(6) < 1
Definition of ( π 32 )(6), ( π 32 )(6) :
There exists two constants ( π 32 )(6) and ( π 32 )(6) which together with
( π 32 )(6), ( π 32 )(6), (π΄ 32)(6)πππ ( π΅ 32 )(6) and the constants
(ππ) 6 , (ππ
β²) 6 , (ππ) 6 , (ππ
β²) 6 , (ππ) 6 , (ππ)
6 , π = 32,33,34, satisfy the inequalities
1
( π 32 )(6) [ (ππ) 6 + (ππ
β²) 6 + ( π΄ 32 )(6) + ( π 32 )(6) ( π 32 )(6)] < 1
1
( π 32 )(6) [ (ππ) 6 + (ππ
β²) 6 + ( π΅ 32 )(6) + ( π 32 )(6) ( π 32 )(6)] < 1
Theorem 1: if the conditions IN THE FOREGOING above are fulfilled, there exists a solution satisfying
the conditions
Definition of πΊπ 0 , ππ 0 :
πΊπ π‘ β€ π 13 1
π π 13 1 π‘ , πΊπ 0 = πΊπ0 > 0
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ππ(π‘) β€ ( π 13 )(1)π( π 13 )(1)π‘ , ππ 0 = ππ0 > 0
Definition of πΊπ 0 , ππ 0
πΊπ π‘ β€ ( π 16 )(2)π( π 16 )(2)π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 16 )(2)π( π 16 )(2)π‘ , ππ 0 = ππ0 > 0
πΊπ π‘ β€ ( π 20 )(3)π( π 20 )(3)π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 20 )(3)π( π 20 )(3)π‘ , ππ 0 = ππ0 > 0
Definition of πΊπ 0 , ππ 0 :
πΊπ π‘ β€ π 24 4
π π 24 4 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 24 )(4)π( π 24 )(4)π‘ , ππ 0 = ππ0 > 0
Definition of πΊπ 0 , ππ 0 :
πΊπ π‘ β€ π 28 5
π π 28 5 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 28 )(5)π( π 28 )(5)π‘ , ππ 0 = ππ0 > 0
Definition of πΊπ 0 , ππ 0 :
πΊπ π‘ β€ π 32 6
π π 32 6 π‘ , πΊπ 0 = πΊπ0 > 0
ππ(π‘) β€ ( π 32 )(6)π( π 32 )(6)π‘ , ππ 0 = ππ0 > 0
Proof: Consider operator π(1) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+
which satisfy
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 13 )(1) , ππ
0 β€ ( π 13 )(1),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 13 )(1)π( π 13 )(1)π‘
0 β€ ππ π‘ β ππ0 β€ ( π 13 )(1)π( π 13 )(1)π‘
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By
πΊ 13 π‘ = πΊ130 + (π13) 1 πΊ14 π 13 β (π13
β² ) 1 + π13β²β² ) 1 π14 π 13 , π 13 πΊ13 π 13 ππ 13
π‘
0
πΊ 14 π‘ = πΊ140 + (π14 ) 1 πΊ13 π 13 β (π14
β² ) 1 + (π14β²β² ) 1 π14 π 13 , π 13 πΊ14 π 13 ππ 13
π‘
0
πΊ 15 π‘ = πΊ150 + (π15) 1 πΊ14 π 13 β (π15
β² ) 1 + (π15β²β² ) 1 π14 π 13 , π 13 πΊ15 π 13 ππ 13
π‘
0
π 13 π‘ = π130 + (π13 ) 1 π14 π 13 β (π13
β² ) 1 β (π13β²β² ) 1 πΊ π 13 , π 13 π13 π 13 ππ 13
π‘
0
π 14 π‘ = π140 + (π14 ) 1 π13 π 13 β (π14
β² ) 1 β (π14β²β² ) 1 πΊ π 13 , π 13 π14 π 13 ππ 13
π‘
0
T 15 t = T150 + (π15) 1 π14 π 13 β (π15
β² ) 1 β (π15β²β² ) 1 πΊ π 13 , π 13 π15 π 13 ππ 13
π‘
0
Where π 13 is the integrand that is integrated over an interval 0, π‘
Proof:
Consider operator π(2) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 16 )(2) , ππ
0 β€ ( π 16 )(2),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 16 )(2)π( π 16 )(2)π‘
0 β€ ππ π‘ β ππ0 β€ ( π 16 )(2)π( π 16 )(2)π‘
By
πΊ 16 π‘ = πΊ160 + (π16) 2 πΊ17 π 16 β (π16
β² ) 2 + π16β²β² ) 2 π17 π 16 , π 16 πΊ16 π 16 ππ 16
π‘
0
πΊ 17 π‘ = πΊ170 + (π17 ) 2 πΊ16 π 16 β (π17
β² ) 2 + (π17β²β² ) 2 π17 π 16 , π 17 πΊ17 π 16 ππ 16
π‘
0
πΊ 18 π‘ = πΊ180 + (π18 ) 2 πΊ17 π 16 β (π18
β² ) 2 + (π18β²β² ) 2 π17 π 16 , π 16 πΊ18 π 16 ππ 16
π‘
0
π 16 π‘ = π160 + (π16 ) 2 π17 π 16 β (π16
β² ) 2 β (π16β²β² ) 2 πΊ π 16 , π 16 π16 π 16 ππ 16
π‘
0
π 17 π‘ = π170 + (π17 ) 2 π16 π 16 β (π17
β² ) 2 β (π17β²β² ) 2 πΊ π 16 , π 16 π17 π 16 ππ 16
π‘
0
π 18 π‘ = π180 + (π18 ) 2 π17 π 16 β (π18
β² ) 2 β (π18β²β² ) 2 πΊ π 16 , π 16 π18 π 16 ππ 16
π‘
0
Where π 16 is the integrand that is integrated over an interval 0, π‘
Proof:
Consider operator π(3) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 20 )(3) , ππ
0 β€ ( π 20 )(3),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 20 )(3)π( π 20 )(3)π‘
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0 β€ ππ π‘ β ππ0 β€ ( π 20 )(3)π( π 20 )(3)π‘
By
πΊ 20 π‘ = πΊ200 + (π20 ) 3 πΊ21 π 20 β (π20
β² ) 3 + π20β²β² ) 3 π21 π 20 , π 20 πΊ20 π 20 ππ 20
π‘
0
πΊ 21 π‘ = πΊ210 + (π21) 3 πΊ20 π 20 β (π21
β² ) 3 + (π21β²β² ) 3 π21 π 20 , π 20 πΊ21 π 20 ππ 20
π‘
0
πΊ 22 π‘ = πΊ220 + (π22 ) 3 πΊ21 π 20 β (π22
β² ) 3 + (π22β²β² ) 3 π21 π 20 , π 20 πΊ22 π 20 ππ 20
π‘
0
π 20 π‘ = π200 + (π20) 3 π21 π 20 β (π20
β² ) 3 β (π20β²β² ) 3 πΊ π 20 , π 20 π20 π 20 ππ 20
π‘
0
π 21 π‘ = π210 + (π21) 3 π20 π 20 β (π21
β² ) 3 β (π21β²β² ) 3 πΊ π 20 , π 20 π21 π 20 ππ 20
π‘
0
T 22 t = T220 + (π22) 3 π21 π 20 β (π22
β² ) 3 β (π22β²β² ) 3 πΊ π 20 , π 20 π22 π 20 ππ 20
π‘
0
Where π 20 is the integrand that is integrated over an interval 0, π‘
Consider operator π(4) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+
which satisfy
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 24 )(4) , ππ
0 β€ ( π 24 )(4),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 24 )(4)π( π 24 )(4)π‘
0 β€ ππ π‘ β ππ0 β€ ( π 24 )(4)π( π 24 )(4)π‘
By
πΊ 24 π‘ = πΊ240 + (π24 ) 4 πΊ25 π 24 β (π24
β² ) 4 + π24β²β² ) 4 π25 π 24 , π 24 πΊ24 π 24 ππ 24
π‘
0
πΊ 25 π‘ = πΊ250 + (π25) 4 πΊ24 π 24 β (π25
β² ) 4 + (π25β²β² ) 4 π25 π 24 , π 24 πΊ25 π 24 ππ 24
π‘
0
πΊ 26 π‘ = πΊ260 + (π26) 4 πΊ25 π 24 β (π26
β² ) 4 + (π26β²β² ) 4 π25 π 24 , π 24 πΊ26 π 24 ππ 24
π‘
0
π 24 π‘ = π240 + (π24) 4 π25 π 24 β (π24
β² ) 4 β (π24β²β² ) 4 πΊ π 24 , π 24 π24 π 24 ππ 24
π‘
0
π 25 π‘ = π250 + (π25) 4 π24 π 24 β (π25
β² ) 4 β (π25β²β² ) 4 πΊ π 24 , π 24 π25 π 24 ππ 24
π‘
0
T 26 t = T260 + (π26 ) 4 π25 π 24 β (π26
β² ) 4 β (π26β²β² ) 4 πΊ π 24 , π 24 π26 π 24 ππ 24
π‘
0
Where π 24 is the integrand that is integrated over an interval 0, π‘
Consider operator π(5) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 28 )(5) , ππ
0 β€ ( π 28 )(5),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 28 )(5)π( π 28 )(5)π‘
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0 β€ ππ π‘ β ππ0 β€ ( π 28 )(5)π( π 28 )(5)π‘
By
πΊ 28 π‘ = πΊ280 + (π28 ) 5 πΊ29 π 28 β (π28
β² ) 5 + π28β²β² ) 5 π29 π 28 , π 28 πΊ28 π 28 ππ 28
π‘
0
πΊ 29 π‘ = πΊ290 + (π29) 5 πΊ28 π 28 β (π29
β² ) 5 + (π29β²β² ) 5 π29 π 28 , π 28 πΊ29 π 28 ππ 28
π‘
0
πΊ 30 π‘ = πΊ300 + (π30) 5 πΊ29 π 28 β (π30
β² ) 5 + (π30β²β² ) 5 π29 π 28 , π 28 πΊ30 π 28 ππ 28
π‘
0
π 28 π‘ = π280 + (π28) 5 π29 π 28 β (π28
β² ) 5 β (π28β²β² ) 5 πΊ π 28 , π 28 π28 π 28 ππ 28
π‘
0
π 29 π‘ = π290 + (π29) 5 π28 π 28 β (π29
β² ) 5 β (π29β²β² ) 5 πΊ π 28 , π 28 π29 π 28 ππ 28
π‘
0
T 30 t = T300 + (π30) 5 π29 π 28 β (π30
β² ) 5 β (π30β²β² ) 5 πΊ π 28 , π 28 π30 π 28 ππ 28
π‘
0
Where π 28 is the integrand that is integrated over an interval 0, π‘
Consider operator π(6) defined on the space of sextuples of continuous functions πΊπ , ππ : β+ β β+ which
satisfy
πΊπ 0 = πΊπ0 , ππ 0 = ππ
0 , πΊπ0 β€ ( π 32 )(6) , ππ
0 β€ ( π 32 )(6),
0 β€ πΊπ π‘ β πΊπ0 β€ ( π 32 )(6)π( π 32 )(6)π‘
0 β€ ππ π‘ β ππ0 β€ ( π 32 )(6)π( π 32 )(6)π‘
By
πΊ 32 π‘ = πΊ320 + (π32 ) 6 πΊ33 π 32 β (π32
β² ) 6 + π32β²β² ) 6 π33 π 32 , π 32 πΊ32 π 32 ππ 32
π‘
0
πΊ 33 π‘ = πΊ330 + (π33) 6 πΊ32 π 32 β (π33
β² ) 6 + (π33β²β² ) 6 π33 π 32 , π 32 πΊ33 π 32 ππ 32
π‘
0
πΊ 34 π‘ = πΊ340 + (π34) 6 πΊ33 π 32 β (π34
β² ) 6 + (π34β²β² ) 6 π33 π 32 , π 32 πΊ34 π 32 ππ 32
π‘
0
π 32 π‘ = π320 + (π32) 6 π33 π 32 β (π32
β² ) 6 β (π32β²β² ) 6 πΊ π 32 , π 32 π32 π 32 ππ 32
π‘
0
π 33 π‘ = π330 + (π33) 6 π32 π 32 β (π33
β² ) 6 β (π33β²β² ) 6 πΊ π 32 , π 32 π33 π 32 ππ 32
π‘
0
T 34 t = T340 + (π34) 6 π33 π 32 β (π34
β² ) 6 β (π34β²β² ) 6 πΊ π 32 , π 32 π34 π 32 ππ 32
π‘
0
Where π 32 is the integrand that is integrated over an interval 0, π‘
(a) The operator π(1) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it
is obvious that
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πΊ13 π‘ β€ πΊ130 + (π13) 1 πΊ14
0 +( π 13 )(1)π( π 13 )(1)π 13 π‘
0ππ 13 =
1 + (π13 ) 1 π‘ πΊ140 +
(π13 ) 1 ( π 13 )(1)
( π 13 )(1) π( π 13 )(1)π‘ β 1
From which it follows that
πΊ13 π‘ β πΊ130 πβ( π 13 )(1)π‘ β€
(π13 ) 1
( π 13 )(1) ( π 13 )(1) + πΊ140 π
β ( π 13 )(1)+πΊ14
0
πΊ140
+ ( π 13 )(1)
πΊπ0 is as defined in the statement of theorem 1
Analogous inequalities hold also for πΊ14 , πΊ15 , π13 , π14 , π15
(b) The operator π(2) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it
is obvious that
πΊ16 π‘ β€ πΊ160 + (π16) 2 πΊ17
0 +( π 16 )(6)π( π 16 )(2)π 16 π‘
0ππ 16 =
1 + (π16 ) 2 π‘ πΊ170 +
(π16 ) 2 ( π 16 )(2)
( π 16 )(2) π( π 16 )(2)π‘ β 1
From which it follows that
πΊ16 π‘ β πΊ160 πβ( π 16 )(2)π‘ β€
(π16 ) 2
( π 16 )(2) ( π 16 )(2) + πΊ170 π
β ( π 16 )(2)+πΊ17
0
πΊ170
+ ( π 16 )(2)
Analogous inequalities hold also for πΊ17 , πΊ18 , π16 , π17 , π18
(a) The operator π(3) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it
is obvious that
πΊ20 π‘ β€ πΊ200 + (π20) 3 πΊ21
0 +( π 20 )(3)π( π 20 )(3)π 20 π‘
0ππ 20 =
1 + (π20 ) 3 π‘ πΊ210 +
(π20 ) 3 ( π 20 )(3)
( π 20 )(3) π( π 20 )(3)π‘ β 1
From which it follows that
πΊ20 π‘ β πΊ200 πβ( π 20 )(3)π‘ β€
(π20 ) 3
( π 20 )(3) ( π 20 )(3) + πΊ210 π
β ( π 20 )(3)+πΊ21
0
πΊ210
+ ( π 20 )(3)
Analogous inequalities hold also for πΊ21 , πΊ22 , π20 , π21 , π22
(b) The operator π(4) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that
πΊ24 π‘ β€ πΊ240 + (π24) 4 πΊ25
0 +( π 24 )(4)π( π 24 )(4)π 24 π‘
0ππ 24 =
1 + (π24 ) 4 π‘ πΊ250 +
(π24 ) 4 ( π 24 )(4)
( π 24 )(4) π( π 24 )(4)π‘ β 1
From which it follows that
πΊ24 π‘ β πΊ240 πβ( π 24 )(4)π‘ β€
(π24 ) 4
( π 24 )(4) ( π 24 )(4) + πΊ250 π
β ( π 24 )(4)+πΊ25
0
πΊ250
+ ( π 24 )(4)
πΊπ0 is as defined in the statement of theorem 1
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(c) The operator π(5) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that
πΊ28 π‘ β€ πΊ280 + (π28) 5 πΊ29
0 +( π 28 )(5)π( π 28 )(5)π 28 π‘
0ππ 28 =
1 + (π28 ) 5 π‘ πΊ290 +
(π28 ) 5 ( π 28 )(5)
( π 28 )(5) π( π 28 )(5)π‘ β 1
From which it follows that
πΊ28 π‘ β πΊ280 πβ( π 28 )(5)π‘ β€
(π28 ) 5
( π 28 )(5) ( π 28 )(5) + πΊ290 π
β ( π 28 )(5)+πΊ29
0
πΊ290
+ ( π 28 )(5)
πΊπ0 is as defined in the statement of theorem 1
(d) The operator π(6) maps the space of functions satisfying GLOBAL EQUATIONS into itself .Indeed it is obvious that
πΊ32 π‘ β€ πΊ320 + (π32) 6 πΊ33
0 +( π 32 )(6)π( π 32 )(6)π 32 π‘
0ππ 32 =
1 + (π32 ) 6 π‘ πΊ330 +
(π32 ) 6 ( π 32 )(6)
( π 32 )(6) π( π 32 )(6)π‘ β 1
From which it follows that
πΊ32 π‘ β πΊ320 πβ( π 32 )(6)π‘ β€
(π32 ) 6
( π 32 )(6) ( π 32 )(6) + πΊ330 π
β ( π 32 )(6)+πΊ33
0
πΊ330
+ ( π 32 )(6)
πΊπ0 is as defined in the statement of theorem 6
Analogous inequalities hold also for πΊ25 , πΊ26 , π24 , π25 , π26
It is now sufficient to take (ππ)
1
( π 13 )(1) ,(ππ)
1
( π 13 )(1) < 1 and to choose
( P 13 )(1) and ( Q 13 )(1) large to have
(ππ) 1
(π 13 ) 1 ( π 13) 1 + ( π 13 )(1) + πΊπ0 π
β ( π 13 )(1)+πΊπ
0
πΊπ0
β€ ( π 13 )(1)
(ππ) 1
(π 13 ) 1 ( π 13 )(1) + ππ0 π
β ( π 13 )(1)+ππ
0
ππ0
+ ( π 13 )(1) β€ ( π 13 )(1)
In order that the operator π(1) transforms the space of sextuples of functions πΊπ , ππ satisfying GLOBAL
EQUATIONS into itself
The operator π(1) is a contraction with respect to the metric
π πΊ 1 , π 1 , πΊ 2 , π 2 =
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π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 13 ) 1 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 13 ) 1 π‘}
Indeed if we denote
Definition of πΊ , π :
πΊ , π = π(1)(πΊ, π)
It results
πΊ 13 1
β πΊ π 2
β€ (π13) 1 π‘
0 πΊ14
1 β πΊ14
2 πβ( π 13 ) 1 π 13 π( π 13 ) 1 π 13 ππ 13 +
{(π13β² ) 1 πΊ13
1 β πΊ13
2 πβ( π 13 ) 1 π 13 πβ( π 13 ) 1 π 13
π‘
0+
(π13β²β² ) 1 π14
1 , π 13 πΊ13
1 β πΊ13
2 πβ( π 13 ) 1 π 13 π( π 13 ) 1 π 13 +
πΊ13 2
|(π13β²β² ) 1 π14
1 , π 13 β (π13
β²β² ) 1 π14 2
, π 13 | πβ( π 13) 1 π 13 π( π 13 ) 1 π 13 }ππ 13
Where π 13 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
πΊ 1 β πΊ 2 πβ( π 13) 1 π‘ β€1
( π 13 ) 1 (π13) 1 + (π13β² ) 1 + ( π΄ 13) 1 + ( π 13) 1 ( π 13) 1 π πΊ 1 , π 1 ; πΊ 2 , π 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
Remark 1: The fact that we supposed (π13β²β² ) 1 and (π13
β²β² ) 1 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 13) 1 π( π 13 ) 1 π‘ πππ ( π 13) 1 π( π 13 ) 1 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it
suffices to consider that (ππβ²β² ) 1 and (ππ
β²β² ) 1 , π = 13,14,15 depend only on T14 and respectively on
πΊ(πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From 19 to 24 it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 1 β(ππβ²β² ) 1 π14 π 13 ,π 13 ππ 13
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 1 π‘ > 0 for t > 0
Definition of ( π 13) 1 1
, ( π 13) 1 2
πππ ( π 13) 1 3 :
Remark 3: if πΊ13 is bounded, the same property have also πΊ14 πππ πΊ15 . indeed if
πΊ13 < ( π 13) 1 it follows ππΊ14
ππ‘β€ ( π 13) 1
1β (π14
β² ) 1 πΊ14 and by integrating
πΊ14 β€ ( π 13) 1 2
= πΊ140 + 2(π14 ) 1 ( π 13) 1
1/(π14
β² ) 1
In the same way , one can obtain
πΊ15 β€ ( π 13) 1 3
= πΊ150 + 2(π15 ) 1 ( π 13) 1
2/(π15
β² ) 1
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If πΊ14 ππ πΊ15 is bounded, the same property follows for πΊ13 , πΊ15 and πΊ13 , πΊ14 respectively.
Remark 4: If πΊ13 ππ bounded, from below, the same property holds for πΊ14 πππ πΊ15 . The proof is
analogous with the preceding one. An analogous property is true if πΊ14 is bounded from below.
Remark 5: If T13 is bounded from below and limπ‘ββ((ππβ²β² ) 1 (πΊ π‘ , π‘)) = (π14
β² ) 1 then π14 β β.
Definition of π 1 and π1 :
Indeed let π‘1 be so that for π‘ > π‘1
(π14) 1 β (ππβ²β² ) 1 (πΊ π‘ , π‘) < π1, π13 (π‘) > π 1
Then ππ14
ππ‘β₯ (π14) 1 π 1 β π1π14 which leads to
π14 β₯ (π14 ) 1 π 1
π1 1 β πβπ1π‘ + π14
0 πβπ1π‘ If we take t such that πβπ1π‘ = 1
2 it results
π14 β₯ (π14 ) 1 π 1
2 , π‘ = πππ
2
π1 By taking now π1 sufficiently small one sees that T14 is unbounded.
The same property holds for π15 if limπ‘ββ(π15β²β² ) 1 πΊ π‘ , π‘ = (π15
β² ) 1
We now state a more precise theorem about the behaviors at infinity of the solutions
It is now sufficient to take (ππ)
2
( π 16 )(2) ,(ππ)
2
( π 16 )(2) < 1 and to choose
( π 16 )(2) πππ ( π 16 )(2) large to have
(ππ) 2
(π 16 ) 2 ( π 16) 2 + ( π 16 )(2) + πΊπ0 π
β ( π 16 )(2)+πΊπ
0
πΊπ0
β€ ( π 16 )(2)
(ππ) 2
(π 16 ) 2 ( π 16 )(2) + ππ0 π
β ( π 16 )(2)+ππ
0
ππ0
+ ( π 16 )(2) β€ ( π 16 )(2)
In order that the operator π(2) transforms the space of sextuples of functions πΊπ , ππ satisfying
The operator π(2) is a contraction with respect to the metric
π πΊ19 1 , π19
1 , πΊ19 2 , π19
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 16 ) 2 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 16 ) 2 π‘}
Indeed if we denote
Definition of πΊ19 , π19
: πΊ19 , π19
= π(2)(πΊ19 , π19)
It results
πΊ 16 1
β πΊ π 2
β€ (π16) 2 π‘
0 πΊ17
1 β πΊ17
2 πβ( π 16 ) 2 π 16 π( π 16 ) 2 π 16 ππ 16 +
{(π16β² ) 2 πΊ16
1 β πΊ16
2 πβ( π 16 ) 2 π 16 πβ( π 16 ) 2 π 16
π‘
0+
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(π16β²β² ) 2 π17
1 , π 16 πΊ16
1 β πΊ16
2 πβ( π 16 ) 2 π 16 π( π 16 ) 2 π 16 +
πΊ16 2
|(π16β²β² ) 2 π17
1 , π 16 β (π16
β²β² ) 2 π17 2
, π 16 | πβ( π 16) 2 π 16 π( π 16 ) 2 π 16 }ππ 16
Where π 16 represents integrand that is integrated over the interval 0, π‘
From the hypotheses it follows
πΊ19 1 β πΊ19
2 eβ( M 16 ) 2 t β€1
( M 16 ) 2 (π16 ) 2 + (π16β² ) 2 + ( A 16) 2 + ( P 16) 2 ( π 16) 2 d πΊ19
1 , π19 1 ; πΊ19
2 , π19 2
And analogous inequalities for Gπ and Tπ . Taking into account the hypothesis the result follows
Remark 1: The fact that we supposed (π16β²β² ) 2 and (π16
β²β² ) 2 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( P 16) 2 e( M 16 ) 2 t and ( Q 16) 2 e( M 16 ) 2 t
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it
suffices to consider that (ππβ²β² ) 2 and (ππ
β²β² ) 2 , π = 16,17,18 depend only on T17 and respectively on
πΊ19 (and not on t) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any t where Gπ t = 0 and Tπ t = 0
From 19 to 24 it results
Gπ t β₯ Gπ0e β (ππ
β² ) 2 β(ππβ²β² ) 2 T17 π 16 ,π 16 dπ 16
t0 β₯ 0
Tπ t β₯ Tπ0e β(ππ
β² ) 2 t > 0 for t > 0
Definition of ( M 16) 2 1
, ( M 16) 2 2
and ( M 16) 2 3 :
Remark 3: if G16 is bounded, the same property have also G17 and G18 . indeed if
G16 < ( M 16) 2 it follows dG17
dtβ€ ( M 16) 2
1β (π17
β² ) 2 G17 and by integrating
G17 β€ ( M 16) 2 2
= G170 + 2(π17) 2 ( M 16) 2
1/(π17
β² ) 2
In the same way , one can obtain
G18 β€ ( M 16) 2 3
= G180 + 2(π18) 2 ( M 16) 2
2/(π18
β² ) 2
If G17 or G18 is bounded, the same property follows for G16 , G18 and G16 , G17 respectively.
Remark 4: If G16 is bounded, from below, the same property holds for G17 and G18 . The proof is
analogous with the preceding one. An analogous property is true if G17 is bounded from below.
Remark 5: If T16 is bounded from below and limtββ((ππβ²β² ) 2 ( πΊ19 t , t)) = (π17
β² ) 2 then T17 β β.
Definition of π 2 and Ξ΅2 :
Indeed let t2 be so that for t > t2
(π17) 2 β (ππβ²β² ) 2 ( πΊ19 t , t) < Ξ΅2, T16 (t) > π 2
Then dT17
dtβ₯ (π17 ) 2 π 2 β Ξ΅2T17 which leads to
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T17 β₯ (π17 ) 2 π 2
Ξ΅2 1 β eβΞ΅2t + T17
0 eβΞ΅2t If we take t such that eβΞ΅2t = 1
2 it results
T17 β₯ (π17 ) 2 π 2
2 , π‘ = log
2
Ξ΅2 By taking now Ξ΅2 sufficiently small one sees that T17 is unbounded. The
same property holds for T18 if limπ‘ββ(π18β²β² ) 2 πΊ19 t , t = (π18
β² ) 2
We now state a more precise theorem about the behaviors at infinity of the solutions
It is now sufficient to take (ππ)
3
( π 20 )(3) ,(ππ)
3
( π 20 )(3) < 1 and to choose
( P 20 )(3) and ( Q 20 )(3) large to have
(ππ) 3
(π 20 ) 3 ( π 20) 3 + ( π 20 )(3) + πΊπ0 π
β ( π 20 )(3)+πΊπ
0
πΊπ0
β€ ( π 20 )(3)
(ππ) 3
(π 20 ) 3 ( π 20 )(3) + ππ0 π
β ( π 20 )(3)+ππ
0
ππ0
+ ( π 20 )(3) β€ ( π 20 )(3)
In order that the operator π(3) transforms the space of sextuples of functions πΊπ , ππ into itself
The operator π(3) is a contraction with respect to the metric
π πΊ23 1 , π23
1 , πΊ23 2 , π23
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 20 ) 3 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 20 ) 3 π‘}
Indeed if we denote
Definition of πΊ23 , π23
: πΊ23 , π23 = π(3) πΊ23 , π23
It results
πΊ 20 1
β πΊ π 2
β€ (π20) 3 π‘
0 πΊ21
1 β πΊ21
2 πβ( π 20 ) 3 π 20 π( π 20 ) 3 π 20 ππ 20 +
{(π20β² ) 3 πΊ20
1 β πΊ20
2 πβ( π 20 ) 3 π 20 πβ( π 20 ) 3 π 20
π‘
0+
(π20β²β² ) 3 π21
1 , π 20 πΊ20
1 β πΊ20
2 πβ( π 20 ) 3 π 20 π( π 20 ) 3 π 20 +
πΊ20 2
|(π20β²β² ) 3 π21
1 , π 20 β (π20
β²β² ) 3 π21 2
, π 20 | πβ( π 20 ) 3 π 20 π( π 20 ) 3 π 20 }ππ 20
Where π 20 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
πΊ 1 β πΊ 2 πβ( π 20) 3 π‘ β€1
( π 20 ) 3 (π20) 3 + (π20β² ) 3 + ( π΄ 20) 3 + ( π 20) 3 ( π 20) 3 π πΊ23
1 , π23 1 ; πΊ23
2 , π23 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
Remark 1: The fact that we supposed (π20β²β² ) 3 and (π20
β²β² ) 3 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
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necessary to prove the uniqueness of the solution bounded by ( π 20) 3 π( π 20 ) 3 π‘ πππ ( π 20) 3 π( π 20 ) 3 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it
suffices to consider that (ππβ²β² ) 3 and (ππ
β²β² ) 3 , π = 20,21,22 depend only on T21 and respectively on
πΊ23 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
From 19 to 24 it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 3 β(ππβ²β² ) 3 π21 π 20 ,π 20 ππ 20
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 3 π‘ > 0 for t > 0
Definition of ( π 20) 3 1
, ( π 20) 3 2
πππ ( π 20) 3 3 :
Remark 3: if πΊ20 is bounded, the same property have also πΊ21 πππ πΊ22 . indeed if
πΊ20 < ( π 20) 3 it follows ππΊ21
ππ‘β€ ( π 20) 3
1β (π21
β² ) 3 πΊ21 and by integrating
πΊ21 β€ ( π 20) 3 2
= πΊ210 + 2(π21) 3 ( π 20) 3
1/(π21
β² ) 3
In the same way , one can obtain
πΊ22 β€ ( π 20) 3 3
= πΊ220 + 2(π22) 3 ( π 20) 3
2/(π22
β² ) 3
If πΊ21 ππ πΊ22 is bounded, the same property follows for πΊ20 , πΊ22 and πΊ20 , πΊ21 respectively.
Remark 4: If πΊ20 ππ bounded, from below, the same property holds for πΊ21 πππ πΊ22 . The proof is
analogous with the preceding one. An analogous property is true if πΊ21 is bounded from below.
Remark 5: If T20 is bounded from below and limπ‘ββ((ππβ²β² ) 3 πΊ23 π‘ , π‘) = (π21
β² ) 3 then π21 β β.
Definition of π 3 and π3 :
Indeed let π‘3 be so that for π‘ > π‘3
(π21) 3 β (ππβ²β² ) 3 πΊ23 π‘ , π‘ < π3,π20 (π‘) > π 3
Then ππ21
ππ‘β₯ (π21) 3 π 3 β π3π21 which leads to
π21 β₯ (π21 ) 3 π 3
π3 1 β πβπ3π‘ + π21
0 πβπ3π‘ If we take t such that πβπ3π‘ = 1
2 it results
π21 β₯ (π21 ) 3 π 3
2 , π‘ = πππ
2
π3 By taking now π3 sufficiently small one sees that T21 is unbounded.
The same property holds for π22 if limπ‘ββ(π22β²β² ) 3 πΊ23 π‘ , π‘ = (π22
β² ) 3
We now state a more precise theorem about the behaviors at infinity of the solutions
It is now sufficient to take (ππ) 4
( π 24 )(4) ,(ππ)
4
( π 24 )(4) < 1 and to choose
( P 24 )(4) and ( Q 24 )(4) large to have
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(ππ) 4
(π 24 ) 4 ( π 24) 4 + ( π 24 )(4) + πΊπ0 π
β ( π 24 )(4)+πΊπ
0
πΊπ0
β€ ( π 24 )(4)
(ππ) 4
(π 24 ) 4 ( π 24 )(4) + ππ0 π
β ( π 24 )(4)+ππ
0
ππ0
+ ( π 24 )(4) β€ ( π 24 )(4)
In order that the operator π(4) transforms the space of sextuples of functions πΊπ , ππ satisfying IN to itself
The operator π(4) is a contraction with respect to the metric
π πΊ27 1 , π27
1 , πΊ27 2 , π27
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 24 ) 4 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 24 ) 4 π‘}
Indeed if we denote
Definition of πΊ27 , π27 : πΊ27 , π27 = π(4)( πΊ27 , π27 )
It results
πΊ 24 1
β πΊ π 2
β€ (π24) 4 π‘
0 πΊ25
1 β πΊ25
2 πβ( π 24 ) 4 π 24 π( π 24 ) 4 π 24 ππ 24 +
{(π24β² ) 4 πΊ24
1 β πΊ24
2 πβ( π 24 ) 4 π 24 πβ( π 24 ) 4 π 24
π‘
0+
(π24β²β² ) 4 π25
1 , π 24 πΊ24
1 β πΊ24
2 πβ( π 24 ) 4 π 24 π( π 24 ) 4 π 24 +
πΊ24 2
|(π24β²β² ) 4 π25
1 , π 24 β (π24
β²β² ) 4 π25 2
, π 24 | πβ( π 24 ) 4 π 24 π( π 24 ) 4 π 24 }ππ 24
Where π 24 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
πΊ27 1 β πΊ27
2 πβ( π 24 ) 4 π‘ β€1
( π 24 ) 4 (π24) 4 + (π24β² ) 4 + ( π΄ 24) 4 + ( π 24) 4 ( π 24) 4 π πΊ27
1 , π27 1 ; πΊ27
2 , π27 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
Remark 1: The fact that we supposed (π24β²β² ) 4 and (π24
β²β² ) 4 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 24) 4 π( π 24 ) 4 π‘ πππ ( π 24) 4 π( π 24) 4 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it
suffices to consider that (ππβ²β² ) 4 and (ππ
β²β² ) 4 , π = 24,25,26 depend only on T25 and respectively on
πΊ27 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
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From 19 to 24 it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 4 β(ππβ²β² ) 4 π25 π 24 ,π 24 ππ 24
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 4 π‘ > 0 for t > 0
Definition of ( π 24) 4 1
, ( π 24) 4 2
πππ ( π 24) 4 3 :
Remark 3: if πΊ24 is bounded, the same property have also πΊ25 πππ πΊ26 . indeed if
πΊ24 < ( π 24) 4 it follows ππΊ25
ππ‘β€ ( π 24) 4
1β (π25
β² ) 4 πΊ25 and by integrating
πΊ25 β€ ( π 24) 4 2
= πΊ250 + 2(π25) 4 ( π 24) 4
1/(π25
β² ) 4
In the same way , one can obtain
πΊ26 β€ ( π 24) 4 3
= πΊ260 + 2(π26) 4 ( π 24) 4
2/(π26
β² ) 4
If πΊ25 ππ πΊ26 is bounded, the same property follows for πΊ24 , πΊ26 and πΊ24 , πΊ25 respectively.
Remark 4: If πΊ24 ππ bounded, from below, the same property holds for πΊ25 πππ πΊ26 . The proof is
analogous with the preceding one. An analogous property is true if πΊ25 is bounded from below.
Remark 5: If T24 is bounded from below and limπ‘ββ((ππβ²β² ) 4 ( πΊ27 π‘ , π‘)) = (π25
β² ) 4 then π25 β β.
Definition of π 4 and π4 :
Indeed let π‘4 be so that for π‘ > π‘4
(π25) 4 β (ππβ²β² ) 4 ( πΊ27 π‘ , π‘) < π4,π24 (π‘) > π 4
Then ππ25
ππ‘β₯ (π25 ) 4 π 4 β π4π25 which leads to
π25 β₯ (π25 ) 4 π 4
π4 1 β πβπ4π‘ + π25
0 πβπ4π‘ If we take t such that πβπ4π‘ = 1
2 it results
π25 β₯ (π25 ) 4 π 4
2 , π‘ = πππ
2
π4 By taking now π4 sufficiently small one sees that T25 is unbounded.
The same property holds for π26 if limπ‘ββ(π26β²β² ) 4 πΊ27 π‘ , π‘ = (π26
β² ) 4
We now state a more precise theorem about the behaviors at infinity of the solutions ANALOGOUS
inequalities hold also for πΊ29 , πΊ30 , π28 , π29 , π30
It is now sufficient to take (ππ)
5
( π 28 )(5) ,(ππ)
5
( π 28 )(5) < 1 and to choose
( P 28 )(5) and ( Q 28 )(5) large to have
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(ππ) 5
(π 28 ) 5 ( π 28) 5 + ( π 28 )(5) + πΊπ0 π
β ( π 28 )(5)+πΊπ
0
πΊπ0
β€ ( π 28 )(5)
(ππ) 5
(π 28 ) 5 ( π 28 )(5) + ππ0 π
β ( π 28 )(5)+ππ
0
ππ0
+ ( π 28 )(5) β€ ( π 28 )(5)
In order that the operator π(5) transforms the space of sextuples of functions πΊπ , ππ into itself
The operator π(5) is a contraction with respect to the metric
π πΊ31 1 , π31
1 , πΊ31 2 , π31
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 28 ) 5 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 28 ) 5 π‘}
Indeed if we denote
Definition of πΊ31 , π31 : πΊ31 , π31 = π(5) πΊ31 , π31
It results
πΊ 28 1
β πΊ π 2
β€ (π28) 5 π‘
0 πΊ29
1 β πΊ29
2 πβ( π 28 ) 5 π 28 π( π 28 ) 5 π 28 ππ 28 +
{(π28β² ) 5 πΊ28
1 β πΊ28
2 πβ( π 28 ) 5 π 28 πβ( π 28 ) 5 π 28
π‘
0+
(π28β²β² ) 5 π29
1 , π 28 πΊ28
1 β πΊ28
2 πβ( π 28 ) 5 π 28 π( π 28 ) 5 π 28 +
πΊ28 2
|(π28β²β² ) 5 π29
1 , π 28 β (π28
β²β² ) 5 π29 2
, π 28 | πβ( π 28 ) 5 π 28 π( π 28 ) 5 π 28 }ππ 28
Where π 28 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
πΊ31 1 β πΊ31
2 πβ( π 28 ) 5 π‘ β€1
( π 28 ) 5 (π28) 5 + (π28β² ) 5 + ( π΄ 28) 5 + ( π 28) 5 ( π 28) 5 π πΊ31
1 , π31 1 ; πΊ31
2 , π31 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis (35,35,36) the result follows
Remark 1: The fact that we supposed (π28β²β² ) 5 and (π28
β²β² ) 5 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 28) 5 π( π 28 ) 5 π‘ πππ ( π 28) 5 π( π 28) 5 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it
suffices to consider that (ππβ²β² ) 5 and (ππ
β²β² ) 5 , π = 28,29,30 depend only on T29 and respectively on
πΊ31 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
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From GLOBAL EQUATIONS it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 5 β(ππβ²β² ) 5 π29 π 28 ,π 28 ππ 28
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 5 π‘ > 0 for t > 0
Definition of ( π 28) 5 1
, ( π 28) 5 2
πππ ( π 28) 5 3 :
Remark 3: if πΊ28 is bounded, the same property have also πΊ29 πππ πΊ30 . indeed if
πΊ28 < ( π 28) 5 it follows ππΊ29
ππ‘β€ ( π 28) 5
1β (π29
β² ) 5 πΊ29 and by integrating
πΊ29 β€ ( π 28) 5 2
= πΊ290 + 2(π29) 5 ( π 28) 5
1/(π29
β² ) 5
In the same way , one can obtain
πΊ30 β€ ( π 28) 5 3
= πΊ300 + 2(π30) 5 ( π 28) 5
2/(π30
β² ) 5
If πΊ29 ππ πΊ30 is bounded, the same property follows for πΊ28 , πΊ30 and πΊ28 , πΊ29 respectively.
Remark 4: If πΊ28 ππ bounded, from below, the same property holds for πΊ29 πππ πΊ30 . The proof is
analogous with the preceding one. An analogous property is true if πΊ29 is bounded from below.
Remark 5: If T28 is bounded from below and limπ‘ββ((ππβ²β² ) 5 ( πΊ31 π‘ , π‘)) = (π29
β² ) 5 then π29 β β.
Definition of π 5 and π5 :
Indeed let π‘5 be so that for π‘ > π‘5
(π29) 5 β (ππβ²β² ) 5 ( πΊ31 π‘ , π‘) < π5,π28 (π‘) > π 5
Then ππ29
ππ‘β₯ (π29) 5 π 5 β π5π29 which leads to
π29 β₯ (π29 ) 5 π 5
π5 1 β πβπ5π‘ + π29
0 πβπ5π‘ If we take t such that πβπ5π‘ = 1
2 it results
π29 β₯ (π29 ) 5 π 5
2 , π‘ = πππ
2
π5 By taking now π5 sufficiently small one sees that T29 is unbounded.
The same property holds for π30 if limπ‘ββ(π30β²β² ) 5 πΊ31 π‘ , π‘ = (π30
β² ) 5
We now state a more precise theorem about the behaviors at infinity of the solutions
Analogous inequalities hold also for πΊ33 , πΊ34 , π32 , π33 , π34
It is now sufficient to take (ππ) 6
( π 32 )(6) ,(ππ)
6
( π 32 )(6) < 1 and to choose
( P 32 )(6) and ( Q 32 )(6) large to have
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(ππ) 6
(π 32 ) 6 ( π 32) 6 + ( π 32 )(6) + πΊπ0 π
β ( π 32 )(6)+πΊπ
0
πΊπ0
β€ ( π 32 )(6)
(ππ) 6
(π 32 ) 6 ( π 32 )(6) + ππ0 π
β ( π 32 )(6)+ππ
0
ππ0
+ ( π 32 )(6) β€ ( π 32 )(6)
In order that the operator π(6) transforms the space of sextuples of functions πΊπ , ππ into itself
The operator π(6) is a contraction with respect to the metric
π πΊ35 1 , π35
1 , πΊ35 2 , π35
2 =
π π’ππ
{πππ₯π‘ββ+
πΊπ 1 π‘ β πΊπ
2 π‘ πβ(π 32 ) 6 π‘ ,πππ₯π‘ββ+
ππ 1 π‘ β ππ
2 π‘ πβ(π 32 ) 6 π‘}
Indeed if we denote
Definition of πΊ35 , π35 : πΊ35 , π35 = π(6) πΊ35 , π35
It results
πΊ 32 1
β πΊ π 2
β€ (π32) 6 π‘
0 πΊ33
1 β πΊ33
2 πβ( π 32 ) 6 π 32 π( π 32 ) 6 π 32 ππ 32 +
{(π32β² ) 6 πΊ32
1 β πΊ32
2 πβ( π 32 ) 6 π 32 πβ( π 32 ) 6 π 32
π‘
0+
(π32β²β² ) 6 π33
1 , π 32 πΊ32
1 β πΊ32
2 πβ( π 32 ) 6 π 32 π( π 32 ) 6 π 32 +
πΊ32 2
|(π32β²β² ) 6 π33
1 , π 32 β (π32
β²β² ) 6 π33 2
, π 32 | πβ( π 32 ) 6 π 32 π( π 32 ) 6 π 32 }ππ 32
Where π 32 represents integrand that is integrated over the interval 0, t
From the hypotheses it follows
πΊ35 1 β πΊ35
2 πβ( π 32 ) 6 π‘ β€1
( π 32 ) 6 (π32) 6 + (π32β² ) 6 + ( π΄ 32) 6 + ( π 32) 6 ( π 32) 6 π πΊ35
1 , π35 1 ; πΊ35
2 , π35 2
And analogous inequalities for πΊπ πππ ππ . Taking into account the hypothesis the result follows
Remark 1: The fact that we supposed (π32β²β² ) 6 and (π32
β²β² ) 6 depending also on t can be considered as not
conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition
necessary to prove the uniqueness of the solution bounded by ( π 32) 6 π( π 32 ) 6 π‘ πππ ( π 32) 6 π( π 32) 6 π‘
respectively of β+.
If instead of proving the existence of the solution on β+, we have to prove it only on a compact then it
suffices to consider that (ππβ²β² ) 6 and (ππ
β²β² ) 6 , π = 32,33,34 depend only on T33 and respectively on
πΊ35 (πππ πππ‘ ππ π‘) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any π‘ where πΊπ π‘ = 0 πππ ππ π‘ = 0
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From 69 to 32 it results
πΊπ π‘ β₯ πΊπ0π β (ππ
β² ) 6 β(ππβ²β² ) 6 π33 π 32 ,π 32 ππ 32
π‘0 β₯ 0
ππ π‘ β₯ ππ0π β(ππ
β² ) 6 π‘ > 0 for t > 0
Definition of ( π 32) 6 1
, ( π 32) 6 2
πππ ( π 32) 6 3 :
Remark 3: if πΊ32 is bounded, the same property have also πΊ33 πππ πΊ34 . indeed if
πΊ32 < ( π 32) 6 it follows ππΊ33
ππ‘β€ ( π 32) 6
1β (π33
β² ) 6 πΊ33 and by integrating
πΊ33 β€ ( π 32) 6 2
= πΊ330 + 2(π33) 6 ( π 32) 6
1/(π33
β² ) 6
In the same way , one can obtain
πΊ34 β€ ( π 32) 6 3
= πΊ340 + 2(π34) 6 ( π 32) 6
2/(π34
β² ) 6
If πΊ33 ππ πΊ34 is bounded, the same property follows for πΊ32 , πΊ34 and πΊ32 , πΊ33 respectively.
Remark 4: If πΊ32 ππ bounded, from below, the same property holds for πΊ33 πππ πΊ34 . The proof is
analogous with the preceding one. An analogous property is true if πΊ33 is bounded from below.
Remark 5: If T32 is bounded from below and limπ‘ββ((ππβ²β² ) 6 ( πΊ35 π‘ , π‘)) = (π33
β² ) 6 then π33 β β.
Definition of π 6 and π6 :
Indeed let π‘6 be so that for π‘ > π‘6
(π33 ) 6 β (ππβ²β² ) 6 πΊ35 π‘ , π‘ < π6, π32 (π‘) > π 6
Then ππ33
ππ‘β₯ (π33 ) 6 π 6 β π6π33 which leads to
π33 β₯ (π33 ) 6 π 6
π6 1 β πβπ6π‘ + π33
0 πβπ6π‘ If we take t such that πβπ6π‘ = 1
2 it results
π33 β₯ (π33 ) 6 π 6
2 , π‘ = πππ
2
π6 By taking now π6 sufficiently small one sees that T33 is unbounded.
The same property holds for π34 if limπ‘ββ(π34β²β² ) 6 πΊ35 π‘ , π‘ π‘ , π‘ = (π34
β² ) 6
We now state a more precise theorem about the behaviors at infinity of the solutions
Behavior of the solutions
If we denote and define
Definition of (π1) 1 , (π2) 1 , (π1) 1 , (π2) 1 :
(a) π1) 1 , (π2) 1 , (π1) 1 , (π2) 1 four constants satisfying
β(π2) 1 β€ β(π13β² ) 1 + (π14
β² ) 1 β (π13β²β² ) 1 π14 , π‘ + (π14
β²β² ) 1 π14 , π‘ β€ β(π1) 1
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β(π2) 1 β€ β(π13β² ) 1 + (π14
β² ) 1 β (π13β²β² ) 1 πΊ, π‘ β (π14
β²β² ) 1 πΊ, π‘ β€ β(π1) 1
Definition of (π1) 1 , (π2) 1 , (π’1) 1 , (π’2) 1 , π 1 , π’ 1 :
(b) By (π1) 1 > 0 , (π2) 1 < 0 and respectively (π’1) 1 > 0 , (π’2) 1 < 0 the roots of the equations
(π14) 1 π 1 2
+ (π1) 1 π 1 β (π13 ) 1 = 0 and (π14) 1 π’ 1 2
+ (π1) 1 π’ 1 β (π13 ) 1 = 0
Definition of (π 1) 1 , , (π 2) 1 , (π’ 1) 1 , (π’ 2) 1 :
By (π 1) 1 > 0 , (π 2) 1 < 0 and respectively (π’ 1) 1 > 0 , (π’ 2) 1 < 0 the roots of the equations
(π14) 1 π 1 2
+ (π2) 1 π 1 β (π13) 1 = 0 and (π14) 1 π’ 1 2
+ (π2) 1 π’ 1 β (π13) 1 = 0
Definition of (π1) 1 , (π2) 1 , (π1) 1 , (π2) 1 , (π0) 1 :-
(c) If we define (π1) 1 , (π2) 1 , (π1) 1 , (π2) 1 by
(π2) 1 = (π0) 1 , (π1) 1 = (π1) 1 , ππ (π0) 1 < (π1) 1
(π2) 1 = (π1) 1 , (π1) 1 = (π 1) 1 , ππ (π1) 1 < (π0) 1 < (π 1) 1 ,
and (π0) 1 =πΊ13
0
πΊ140
( π2) 1 = (π1) 1 , (π1) 1 = (π0) 1 , ππ (π 1) 1 < (π0) 1
and analogously
(π2) 1 = (π’0) 1 , (π1) 1 = (π’1) 1 , ππ (π’0) 1 < (π’1) 1
(π2) 1 = (π’1) 1 , (π1) 1 = (π’ 1) 1 , ππ (π’1) 1 < (π’0) 1 < (π’ 1) 1 ,
and (π’0) 1 =π13
0
π140
( π2) 1 = (π’1) 1 , (π1) 1 = (π’0) 1 , ππ (π’ 1) 1 < (π’0) 1 where (π’1) 1 , (π’ 1) 1
are defined respectively
Then the solution satisfies the inequalities
πΊ130 π (π1) 1 β(π13 ) 1 π‘ β€ πΊ13(π‘) β€ πΊ13
0 π(π1) 1 π‘
where (ππ) 1 is defined
1
(π1) 1 πΊ130 π (π1) 1 β(π13 ) 1 π‘ β€ πΊ14(π‘) β€
1
(π2) 1 πΊ130 π(π1) 1 π‘
( (π15 ) 1 πΊ13
0
(π1) 1 (π1) 1 β(π13 ) 1 β(π2) 1 π (π1) 1 β(π13 ) 1 π‘ β πβ(π2) 1 π‘ + πΊ15
0 πβ(π2) 1 π‘ β€ πΊ15 (π‘) β€
(π15 ) 1 πΊ130
(π2) 1 (π1) 1 β(π15β² ) 1
[π(π1) 1 π‘ β πβ(π15β² ) 1 π‘] + πΊ15
0 πβ(π15β² ) 1 π‘)
π130 π(π 1) 1 π‘ β€ π13 (π‘) β€ π13
0 π (π 1) 1 +(π13 ) 1 π‘
1
(π1) 1 π130 π(π 1) 1 π‘ β€ π13 (π‘) β€
1
(π2) 1 π130 π (π 1) 1 +(π13 ) 1 π‘
(π15 ) 1 π130
(π1) 1 (π 1) 1 β(π15β² ) 1
π(π 1) 1 π‘ β πβ(π15β² ) 1 π‘ + π15
0 πβ(π15β² ) 1 π‘ β€ π15 (π‘) β€
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(π15 ) 1 π130
(π2) 1 (π 1) 1 +(π13 ) 1 +(π 2) 1 π (π 1) 1 +(π13 ) 1 π‘ β πβ(π 2) 1 π‘ + π15
0 πβ(π 2) 1 π‘
Definition of (π1) 1 , (π2) 1 , (π 1) 1 , (π 2) 1 :-
Where (π1) 1 = (π13) 1 (π2) 1 β (π13β² ) 1
(π2) 1 = (π15) 1 β (π15 ) 1
(π 1) 1 = (π13) 1 (π2) 1 β (π13β² ) 1
(π 2) 1 = (π15β² ) 1 β (π15 ) 1
Behavior of the solutions
If we denote and define
Definition of (Ο1) 2 , (Ο2) 2 , (Ο1) 2 , (Ο2) 2 :
(d) Ο1) 2 , (Ο2) 2 , (Ο1) 2 , (Ο2) 2 four constants satisfying
β(Ο2) 2 β€ β(π16β² ) 2 + (π17
β² ) 2 β (π16β²β² ) 2 T17 , π‘ + (π17
β²β² ) 2 T17 , π‘ β€ β(Ο1) 2
β(Ο2) 2 β€ β(π16β² ) 2 + (π17
β² ) 2 β (π16β²β² ) 2 πΊ19 , π‘ β (π17
β²β² ) 2 πΊ19 , π‘ β€ β(Ο1) 2
Definition of (π1) 2 , (Ξ½2) 2 , (π’1) 2 , (π’2) 2 :
By (π1) 2 > 0 , (Ξ½2) 2 < 0 and respectively (π’1) 2 > 0 , (π’2) 2 < 0 the roots
(e) of the equations (π17 ) 2 π 2 2
+ (Ο1) 2 π 2 β (π16 ) 2 = 0
and (π14) 2 π’ 2 2
+ (Ο1) 2 π’ 2 β (π16) 2 = 0 and
Definition of (π 1) 2 , , (π 2) 2 , (π’ 1) 2 , (π’ 2) 2 :
By (π 1) 2 > 0 , (Ξ½ 2) 2 < 0 and respectively (π’ 1) 2 > 0 , (π’ 2) 2 < 0 the
roots of the equations (π17 ) 2 π 2 2
+ (Ο2) 2 π 2 β (π16 ) 2 = 0
and (π17) 2 π’ 2 2
+ (Ο2) 2 π’ 2 β (π16 ) 2 = 0
Definition of (π1) 2 , (π2) 2 , (π1) 2 , (π2) 2 :-
(f) If we define (π1) 2 , (π2) 2 , (π1) 2 , (π2) 2 by
(π2) 2 = (π0) 2 , (π1) 2 = (π1) 2 , ππ (π0) 2 < (π1) 2
(π2) 2 = (π1) 2 , (π1) 2 = (π 1) 2 , ππ (π1) 2 < (π0) 2 < (π 1) 2 ,
and (π0) 2 =G16
0
G170
( π2) 2 = (π1) 2 , (π1) 2 = (π0) 2 , ππ (π 1) 2 < (π0) 2
and analogously
(π2) 2 = (π’0) 2 , (π1) 2 = (π’1) 2 , ππ (π’0) 2 < (π’1) 2
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(π2) 2 = (π’1) 2 , (π1) 2 = (π’ 1) 2 , ππ (π’1) 2 < (π’0) 2 < (π’ 1) 2 ,
and (π’0) 2 =T16
0
T170
( π2) 2 = (π’1) 2 , (π1) 2 = (π’0) 2 , ππ (π’ 1) 2 < (π’0) 2
Then the solution satisfies the inequalities
G160 e (S1) 2 β(π16 ) 2 t β€ πΊ16 π‘ β€ G16
0 e(S1) 2 t
(ππ) 2 is defined
1
(π1) 2 G160 e (S1) 2 β(π16 ) 2 t β€ πΊ17(π‘) β€
1
(π2) 2 G160 e(S1) 2 t
( (π18 ) 2 G16
0
(π1) 2 (S1) 2 β(π16 ) 2 β(S2) 2 e (S1) 2 β(π16 ) 2 t β eβ(S2) 2 t + G18
0 eβ(S2) 2 t β€ G18(π‘) β€
(π18 ) 2 G160
(π2) 2 (S1) 2 β(π18β² ) 2
[e(S1) 2 t β eβ(π18β² ) 2 t] + G18
0 eβ(π18β² ) 2 t)
T160 e(R1) 2 π‘ β€ π16(π‘) β€ T16
0 e (R1) 2 +(π16 ) 2 π‘
1
(π1) 2 T160 e(R1) 2 π‘ β€ π16 (π‘) β€
1
(π2) 2 T160 e (R1) 2 +(π16 ) 2 π‘
(π18 ) 2 T160
(π1) 2 (R1) 2 β(π18β² ) 2
e(R1) 2 π‘ β eβ(π18β² ) 2 π‘ + T18
0 eβ(π18β² ) 2 π‘ β€ π18 (π‘) β€
(π18 ) 2 T160
(π2) 2 (R1) 2 +(π16 ) 2 +(R2) 2 e (R1) 2 +(π16 ) 2 π‘ β eβ(R2) 2 π‘ + T18
0 eβ(R2) 2 π‘
Definition of (S1) 2 , (S2) 2 , (R1) 2 , (R2) 2 :-
Where (S1) 2 = (π16) 2 (π2) 2 β (π16β² ) 2
(S2) 2 = (π18) 2 β (π18 ) 2
(π 1) 2 = (π16) 2 (π2) 1 β (π16β² ) 2
(R2) 2 = (π18β² ) 2 β (π18 ) 2
Behavior of the solutions
If we denote and define
Definition of (π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 :
(a) π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 four constants satisfying
β(π2) 3 β€ β(π20β² ) 3 + (π21
β² ) 3 β (π20β²β² ) 3 π21 , π‘ + (π21
β²β² ) 3 π21 , π‘ β€ β(π1) 3
β(π2) 3 β€ β(π20β² ) 3 + (π21
β² ) 3 β (π20β²β² ) 3 πΊ, π‘ β (π21
β²β² ) 3 πΊ23 , π‘ β€ β(π1) 3
Definition of (π1) 3 , (π2) 3 , (π’1) 3 , (π’2) 3 :
(b) By (π1) 3 > 0 , (π2) 3 < 0 and respectively (π’1) 3 > 0 , (π’2) 3 < 0 the roots of the equations
(π21) 3 π 3 2
+ (π1) 3 π 3 β (π20) 3 = 0
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and (π21) 3 π’ 3 2
+ (π1) 3 π’ 3 β (π20) 3 = 0 and
By (π 1) 3 > 0 , (π 2) 3 < 0 and respectively (π’ 1) 3 > 0 , (π’ 2) 3 < 0 the
roots of the equations (π21 ) 3 π 3 2
+ (π2) 3 π 3 β (π20 ) 3 = 0
and (π21) 3 π’ 3 2
+ (π2) 3 π’ 3 β (π20) 3 = 0
Definition of (π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 :-
(c) If we define (π1) 3 , (π2) 3 , (π1) 3 , (π2) 3 by
(π2) 3 = (π0) 3 , (π1) 3 = (π1) 3 , ππ (π0) 3 < (π1) 3
(π2) 3 = (π1) 3 , (π1) 3 = (π 1) 3 , ππ (π1) 3 < (π0) 3 < (π 1) 3 ,
and (π0) 3 =πΊ20
0
πΊ210
( π2) 3 = (π1) 3 , (π1) 3 = (π0) 3 , ππ (π 1) 3 < (π0) 3
and analogously
(π2) 3 = (π’0) 3 , (π1) 3 = (π’1) 3 , ππ (π’0) 3 < (π’1) 3
(π2) 3 = (π’1) 3 , (π1) 3 = (π’ 1) 3 , ππ (π’1) 3 < (π’0) 3 < (π’ 1) 3 , and (π’0) 3 =π20
0
π210
( π2) 3 = (π’1) 3 , (π1) 3 = (π’0) 3 , ππ (π’ 1) 3 < (π’0) 3
Then the solution satisfies the inequalities
πΊ200 π (π1) 3 β(π20 ) 3 π‘ β€ πΊ20(π‘) β€ πΊ20
0 π(π1) 3 π‘
(ππ) 3 is defined
1
(π1) 3 πΊ200 π (π1) 3 β(π20 ) 3 π‘ β€ πΊ21(π‘) β€
1
(π2) 3 πΊ200 π(π1) 3 π‘
( (π22 ) 3 πΊ20
0
(π1) 3 (π1) 3 β(π20 ) 3 β(π2) 3 π (π1) 3 β(π20 ) 3 π‘ β πβ(π2) 3 π‘ + πΊ22
0 πβ(π2) 3 π‘ β€ πΊ22(π‘) β€
(π22 ) 3 πΊ200
(π2) 3 (π1) 3 β(π22β² ) 3
[π(π1) 3 π‘ β πβ(π22β² ) 3 π‘] + πΊ22
0 πβ(π22β² ) 3 π‘)
π200 π(π 1) 3 π‘ β€ π20 (π‘) β€ π20
0 π (π 1) 3 +(π20 ) 3 π‘
1
(π1) 3 π200 π(π 1) 3 π‘ β€ π20 (π‘) β€
1
(π2) 3 π200 π (π 1) 3 +(π20 ) 3 π‘
(π22 ) 3 π200
(π1) 3 (π 1) 3 β(π22β² ) 3
π(π 1) 3 π‘ β πβ(π22β² ) 3 π‘ + π22
0 πβ(π22β² ) 3 π‘ β€ π22 (π‘) β€
(π22 ) 3 π200
(π2) 3 (π 1) 3 +(π20 ) 3 +(π 2) 3 π (π 1) 3 +(π20 ) 3 π‘ β πβ(π 2) 3 π‘ + π22
0 πβ(π 2) 3 π‘
Definition of (π1) 3 , (π2) 3 , (π 1) 3 , (π 2) 3 :-
Where (π1) 3 = (π20) 3 (π2) 3 β (π20β² ) 3
(π2) 3 = (π22) 3 β (π22 ) 3
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(π 1) 3 = (π20) 3 (π2) 3 β (π20β² ) 3
(π 2) 3 = (π22β² ) 3 β (π22) 3
Behavior of the solutions If we denote and define
Definition of (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 :
(d) (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 four constants satisfying
β(π2) 4 β€ β(π24β² ) 4 + (π25
β² ) 4 β (π24β²β² ) 4 π25 , π‘ + (π25
β²β² ) 4 π25 , π‘ β€ β(π1) 4
β(π2) 4 β€ β(π24β² ) 4 + (π25
β² ) 4 β (π24β²β² ) 4 πΊ27 , π‘ β (π25
β²β² ) 4 πΊ27 , π‘ β€ β(π1) 4
Definition of (π1) 4 , (π2) 4 , (π’1) 4 , (π’2) 4 , π 4 , π’ 4 :
(e) By (π1) 4 > 0 , (π2) 4 < 0 and respectively (π’1) 4 > 0 , (π’2) 4 < 0 the roots of the equations
(π25) 4 π 4 2
+ (π1) 4 π 4 β (π24) 4 = 0
and (π25) 4 π’ 4 2
+ (π1) 4 π’ 4 β (π24) 4 = 0 and
Definition of (π 1) 4 , , (π 2) 4 , (π’ 1) 4 , (π’ 2) 4 :
By (π 1) 4 > 0 , (π 2) 4 < 0 and respectively (π’ 1) 4 > 0 , (π’ 2) 4 < 0 the
roots of the equations (π25 ) 4 π 4 2
+ (π2) 4 π 4 β (π24 ) 4 = 0
and (π25) 4 π’ 4 2
+ (π2) 4 π’ 4 β (π24) 4 = 0
Definition of (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 , (π0) 4 :-
(f) If we define (π1) 4 , (π2) 4 , (π1) 4 , (π2) 4 by
(π2) 4 = (π0) 4 , (π1) 4 = (π1) 4 , ππ (π0) 4 < (π1) 4
(π2) 4 = (π1) 4 , (π1) 4 = (π 1) 4 , ππ (π4) 4 < (π0) 4 < (π 1) 4 ,
and (π0) 4 =πΊ24
0
πΊ250
( π2) 4 = (π4) 4 , (π1) 4 = (π0) 4 , ππ (π 4) 4 < (π0) 4
and analogously
(π2) 4 = (π’0) 4 , (π1) 4 = (π’1) 4 , ππ (π’0) 4 < (π’1) 4
(π2) 4 = (π’1) 4 , (π1) 4 = (π’ 1) 4 , ππ (π’1) 4 < (π’0) 4 < (π’ 1) 4 ,
and (π’0) 4 =π24
0
π250
( π2) 4 = (π’1) 4 , (π1) 4 = (π’0) 4 , ππ (π’ 1) 4 < (π’0) 4 where (π’1) 4 , (π’ 1) 4 are defined by 59 and 64 respectively
Then the solution satisfies the inequalities
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πΊ240 π (π1) 4 β(π24 ) 4 π‘ β€ πΊ24 π‘ β€ πΊ24
0 π(π1) 4 π‘
where (ππ) 4 is defined
1
(π1) 4 πΊ240 π (π1) 4 β(π24 ) 4 π‘ β€ πΊ25 π‘ β€
1
(π2) 4 πΊ240 π(π1) 4 π‘
(π26 ) 4 πΊ24
0
(π1) 4 (π1) 4 β(π24 ) 4 β(π2) 4 π (π1) 4 β(π24 ) 4 π‘ β πβ(π2) 4 π‘ + πΊ26
0 πβ(π2) 4 π‘ β€ πΊ26 π‘ β€
(π26 ) 4 πΊ240
(π2) 4 (π1) 4 β(π26β² ) 4
π(π1) 4 π‘ β πβ(π26β² ) 4 π‘ + πΊ26
0 πβ(π26β² ) 4 π‘
π240 π(π 1) 4 π‘ β€ π24 π‘ β€ π24
0 π (π 1) 4 +(π24 ) 4 π‘
1
(π1) 4 π240 π(π 1) 4 π‘ β€ π24 (π‘) β€
1
(π2) 4 π240 π (π 1) 4 +(π24 ) 4 π‘
(π26 ) 4 π240
(π1) 4 (π 1) 4 β(π26β² ) 4
π(π 1) 4 π‘ β πβ(π26β² ) 4 π‘ + π26
0 πβ(π26β² ) 4 π‘ β€ π26 (π‘) β€
(π26 ) 4 π24
0
(π2) 4 (π 1) 4 +(π24 ) 4 +(π 2) 4 π (π 1) 4 +(π24 ) 4 π‘ β πβ(π 2) 4 π‘ + π26
0 πβ(π 2) 4 π‘
Definition of (π1) 4 , (π2) 4 , (π 1) 4 , (π 2) 4 :-
Where (π1) 4 = (π24 ) 4 (π2) 4 β (π24β² ) 4
(π2) 4 = (π26) 4 β (π26) 4
(π 1) 4 = (π24) 4 (π2) 4 β (π24β² ) 4
(π 2) 4 = (π26β² ) 4 β (π26) 4
Behavior of the solutions If we denote and define
Definition of (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 :
(g) (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 four constants satisfying
β(π2) 5 β€ β(π28β² ) 5 + (π29
β² ) 5 β (π28β²β² ) 5 π29 , π‘ + (π29
β²β² ) 5 π29 , π‘ β€ β(π1) 5
β(π2) 5 β€ β(π28β² ) 5 + (π29
β² ) 5 β (π28β²β² ) 5 πΊ31 , π‘ β (π29
β²β² ) 5 πΊ31 , π‘ β€ β(π1) 5
Definition of (π1) 5 , (π2) 5 , (π’1) 5 , (π’2) 5 , π 5 , π’ 5 :
(h) By (π1) 5 > 0 , (π2) 5 < 0 and respectively (π’1) 5 > 0 , (π’2) 5 < 0 the roots of the equations
(π29) 5 π 5 2
+ (π1) 5 π 5 β (π28 ) 5 = 0
and (π29) 5 π’ 5 2
+ (π1) 5 π’ 5 β (π28 ) 5 = 0 and
Definition of (π 1) 5 , , (π 2) 5 , (π’ 1) 5 , (π’ 2) 5 :
By (π 1) 5 > 0 , (π 2) 5 < 0 and respectively (π’ 1) 5 > 0 , (π’ 2) 5 < 0 the
roots of the equations (π29) 5 π 5 2
+ (π2) 5 π 5 β (π28) 5 = 0
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and (π29) 5 π’ 5 2
+ (π2) 5 π’ 5 β (π28 ) 5 = 0
Definition of (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 , (π0) 5 :-
(i) If we define (π1) 5 , (π2) 5 , (π1) 5 , (π2) 5 by
(π2) 5 = (π0) 5 , (π1) 5 = (π1) 5 , ππ (π0) 5 < (π1) 5
(π2) 5 = (π1) 5 , (π1) 5 = (π 1) 5 , ππ (π1) 5 < (π0) 5 < (π 1) 5 ,
and (π0) 5 =πΊ28
0
πΊ290
( π2) 5 = (π1) 5 , (π1) 5 = (π0) 5 , ππ (π 1) 5 < (π0) 5 and analogously
(π2) 5 = (π’0) 5 , (π1) 5 = (π’1) 5 , ππ (π’0) 5 < (π’1) 5
(π2) 5 = (π’1) 5 , (π1) 5 = (π’ 1) 5 , ππ (π’1) 5 < (π’0) 5 < (π’ 1) 5 ,
and (π’0) 5 =π28
0
π290
( π2) 5 = (π’1) 5 , (π1) 5 = (π’0) 5 , ππ (π’ 1) 5 < (π’0) 5 where (π’1) 5 , (π’ 1) 5 are defined respectively
Then the solution satisfies the inequalities
πΊ280 π (π1) 5 β(π28 ) 5 π‘ β€ πΊ28(π‘) β€ πΊ28
0 π(π1) 5 π‘
where (ππ) 5 is defined
1
(π5) 5 πΊ280 π (π1) 5 β(π28 ) 5 π‘ β€ πΊ29(π‘) β€
1
(π2) 5 πΊ280 π(π1) 5 π‘
(π30 ) 5 πΊ28
0
(π1) 5 (π1) 5 β(π28 ) 5 β(π2) 5 π (π1) 5 β(π28 ) 5 π‘ β πβ(π2) 5 π‘ + πΊ30
0 πβ(π2) 5 π‘ β€ πΊ30 π‘ β€
(π30 ) 5 πΊ280
(π2) 5 (π1) 5 β(π30β² ) 5
π(π1) 5 π‘ β πβ(π30β² ) 5 π‘ + πΊ30
0 πβ(π30β² ) 5 π‘
π280 π(π 1) 5 π‘ β€ π28(π‘) β€ π28
0 π (π 1) 5 +(π28 ) 5 π‘
1
(π1) 5 π280 π(π 1) 5 π‘ β€ π28 (π‘) β€
1
(π2) 5 π280 π (π 1) 5 +(π28 ) 5 π‘
(π30 ) 5 π280
(π1) 5 (π 1) 5 β(π30β² ) 5
π(π 1) 5 π‘ β πβ(π30β² ) 5 π‘ + π30
0 πβ(π30β² ) 5 π‘ β€ π30 (π‘) β€
(π30 ) 5 π28
0
(π2) 5 (π 1) 5 +(π28 ) 5 +(π 2) 5 π (π 1) 5 +(π28 ) 5 π‘ β πβ(π 2) 5 π‘ + π30
0 πβ(π 2) 5 π‘
Definition of (π1) 5 , (π2) 5 , (π 1) 5 , (π 2) 5 :-
Where (π1) 5 = (π28 ) 5 (π2) 5 β (π28β² ) 5
(π2) 5 = (π30) 5 β (π30) 5
(π 1) 5 = (π28) 5 (π2) 5 β (π28β² ) 5
(π 2) 5 = (π30β² ) 5 β (π30) 5
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Behavior of the solutions If we denote and define
Definition of (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 :
(j) (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 four constants satisfying
β(π2) 6 β€ β(π32β² ) 6 + (π33
β² ) 6 β (π32β²β² ) 6 π33 , π‘ + (π33
β²β² ) 6 π33 , π‘ β€ β(π1) 6
β(π2) 6 β€ β(π32β² ) 6 + (π33
β² ) 6 β (π32β²β² ) 6 πΊ35 , π‘ β (π33
β²β² ) 6 πΊ35 , π‘ β€ β(π1) 6
Definition of (π1) 6 , (π2) 6 , (π’1) 6 , (π’2) 6 , π 6 , π’ 6 :
(k) By (π1) 6 > 0 , (π2) 6 < 0 and respectively (π’1) 6 > 0 , (π’2) 6 < 0 the roots of the equations
(π33) 6 π 6 2
+ (π1) 6 π 6 β (π32) 6 = 0
and (π33) 6 π’ 6 2
+ (π1) 6 π’ 6 β (π32) 6 = 0 and
Definition of (π 1) 6 , , (π 2) 6 , (π’ 1) 6 , (π’ 2) 6 :
By (π 1) 6 > 0 , (π 2) 6 < 0 and respectively (π’ 1) 6 > 0 , (π’ 2) 6 < 0 the
roots of the equations (π33 ) 6 π 6 2
+ (π2) 6 π 6 β (π32 ) 6 = 0
and (π33) 6 π’ 6 2
+ (π2) 6 π’ 6 β (π32) 6 = 0
Definition of (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 , (π0) 6 :-
(l) If we define (π1) 6 , (π2) 6 , (π1) 6 , (π2) 6 by
(π2) 6 = (π0) 6 , (π1) 6 = (π1) 6 , ππ (π0) 6 < (π1) 6
(π2) 6 = (π1) 6 , (π1) 6 = (π 6) 6 , ππ (π1) 6 < (π0) 6 < (π 1) 6 ,
and (π0) 6 =πΊ32
0
πΊ330
( π2) 6 = (π1) 6 , (π1) 6 = (π0) 6 , ππ (π 1) 6 < (π0) 6
and analogously
(π2) 6 = (π’0) 6 , (π1) 6 = (π’1) 6 , ππ (π’0) 6 < (π’1) 6
(π2) 6 = (π’1) 6 , (π1) 6 = (π’ 1) 6 , ππ (π’1) 6 < (π’0) 6 < (π’ 1) 6 ,
and (π’0) 6 =π32
0
π330
( π2) 6 = (π’1) 6 , (π1) 6 = (π’0) 6 , ππ (π’ 1) 6 < (π’0) 6 where (π’1) 6 , (π’ 1) 6 are defined respectively
Then the solution satisfies the inequalities
πΊ320 π (π1) 6 β(π32 ) 6 π‘ β€ πΊ32(π‘) β€ πΊ32
0 π(π1) 6 π‘
where (ππ) 6 is defined
1
(π1) 6 πΊ320 π (π1) 6 β(π32 ) 6 π‘ β€ πΊ33(π‘) β€
1
(π2) 6 πΊ320 π(π1) 6 π‘
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(π34 ) 6 πΊ32
0
(π1) 6 (π1) 6 β(π32 ) 6 β(π2) 6 π (π1) 6 β(π32 ) 6 π‘ β πβ(π2) 6 π‘ + πΊ34
0 πβ(π2) 6 π‘ β€ πΊ34 π‘ β€
(π34 ) 6 πΊ320
(π2) 6 (π1) 6 β(π34β² ) 6
π(π1) 6 π‘ β πβ(π34β² ) 6 π‘ + πΊ34
0 πβ(π34β² ) 6 π‘
π320 π(π 1) 6 π‘ β€ π32(π‘) β€ π32
0 π (π 1) 6 +(π32 ) 6 π‘
1
(π1) 6 π320 π(π 1) 6 π‘ β€ π32 (π‘) β€
1
(π2) 6 π320 π (π 1) 6 +(π32 ) 6 π‘
(π34 ) 6 π320
(π1) 6 (π 1) 6 β(π34β² ) 6
π(π 1) 6 π‘ β πβ(π34β² ) 6 π‘ + π34
0 πβ(π34β² ) 6 π‘ β€ π34 (π‘) β€
(π34 ) 6 π32
0
(π2) 6 (π 1) 6 +(π32 ) 6 +(π 2) 6 π (π 1) 6 +(π32 ) 6 π‘ β πβ(π 2) 6 π‘ + π34
0 πβ(π 2) 6 π‘
Definition of (π1) 6 , (π2) 6 , (π 1) 6 , (π 2) 6 :-
Where (π1) 6 = (π32 ) 6 (π2) 6 β (π32β² ) 6
(π2) 6 = (π34) 6 β (π34) 6
(π 1) 6 = (π32) 6 (π2) 6 β (π32β² ) 6
(π 2) 6 = (π34β² ) 6 β (π34) 6
Proof : From GLOBAL EQUATIONS we obtain
ππ 1
ππ‘= (π13 ) 1 β (π13
β² ) 1 β (π14β² ) 1 + (π13
β²β² ) 1 π14 , π‘ β (π14β²β² ) 1 π14 , π‘ π 1 β (π14 ) 1 π 1
Definition of π 1 :- π 1 =πΊ13
πΊ14
It follows
β (π14 ) 1 π 1 2
+ (π2) 1 π 1 β (π13) 1 β€ππ 1
ππ‘β€ β (π14 ) 1 π 1
2+ (π1) 1 π 1 β (π13) 1
From which one obtains
Definition of (π 1) 1 , (π0) 1 :-
(a) For 0 < (π0) 1 =πΊ13
0
πΊ140 < (π1) 1 < (π 1) 1
π 1 (π‘) β₯(π1) 1 +(πΆ) 1 (π2) 1 π
β π14 1 (π1) 1 β(π0) 1 π‘
1+(πΆ) 1 π β π14 1 (π1) 1 β(π0) 1 π‘
, (πΆ) 1 =(π1) 1 β(π0) 1
(π0) 1 β(π2) 1
it follows (π0) 1 β€ π 1 (π‘) β€ (π1) 1
In the same manner , we get
π 1 (π‘) β€(π 1) 1 +(πΆ ) 1 (π 2) 1 π
β π14 1 (π 1) 1 β(π 2) 1 π‘
1+(πΆ ) 1 π β π14 1 (π 1) 1 β(π 2) 1 π‘
, (πΆ ) 1 =(π 1) 1 β(π0) 1
(π0) 1 β(π 2) 1
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From which we deduce (π0) 1 β€ π 1 (π‘) β€ (π 1) 1
(b) If 0 < (π1) 1 < (π0) 1 =πΊ13
0
πΊ140 < (π 1) 1 we find like in the previous case,
(π1) 1 β€(π1) 1 + πΆ 1 (π2) 1 π
β π14 1 (π1) 1 β(π2) 1 π‘
1+ πΆ 1 π β π14 1 (π1) 1 β(π2) 1 π‘
β€ π 1 π‘ β€
(π 1) 1 + πΆ 1 (π 2) 1 π
β π14 1 (π 1) 1 β(π 2) 1 π‘
1+ πΆ 1 π β π14 1 (π 1) 1 β(π 2) 1 π‘
β€ (π 1) 1
(c) If 0 < (π1) 1 β€ (π 1) 1 β€ (π0) 1 =πΊ13
0
πΊ140 , we obtain
(π1) 1 β€ π 1 π‘ β€(π 1) 1 + πΆ 1 (π 2) 1 π
β π14 1 (π 1) 1 β(π 2) 1 π‘
1+ πΆ 1 π β π14 1 (π 1) 1 β(π 2) 1 π‘
β€ (π0) 1
And so with the notation of the first part of condition (c) , we have
Definition of π 1 π‘ :-
(π2) 1 β€ π 1 π‘ β€ (π1) 1 , π 1 π‘ =πΊ13 π‘
πΊ14 π‘
In a completely analogous way, we obtain
Definition of π’ 1 π‘ :-
(π2) 1 β€ π’ 1 π‘ β€ (π1) 1 , π’ 1 π‘ =π13 π‘
π14 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the
theorem.
Particular case :
If (π13β²β² ) 1 = (π14
β²β² ) 1 , π‘πππ (π1) 1 = (π2) 1 and in this case (π1) 1 = (π 1) 1 if in addition (π0) 1 =
(π1) 1 then π 1 π‘ = (π0) 1 and as a consequence πΊ13(π‘) = (π0) 1 πΊ14(π‘) this also defines (π0) 1 for the
special case
Analogously if (π13β²β² ) 1 = (π14
β²β² ) 1 , π‘πππ (π1) 1 = (π2) 1 and then
(π’1) 1 = (π’ 1) 1 if in addition (π’0) 1 = (π’1) 1 then π13(π‘) = (π’0) 1 π14 (π‘) This is an important
consequence of the relation between (π1) 1 and (π 1) 1 , and definition of (π’0) 1 .
we obtain
dπ 2
dt= (π16) 2 β (π16
β² ) 2 β (π17β² ) 2 + (π16
β²β² ) 2 T17 , t β (π17β²β² ) 2 T17 , t π 2 β (π17) 2 π 2
Definition of π 2 :- π 2 =G16
G17
It follows
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β (π17 ) 2 π 2 2
+ (Ο2) 2 π 2 β (π16 ) 2 β€dπ 2
dtβ€ β (π17 ) 2 π 2
2+ (Ο1) 2 π 2 β (π16 ) 2
From which one obtains
Definition of (π 1) 2 , (π0) 2 :-
(d) For 0 < (π0) 2 =G16
0
G170 < (π1) 2 < (π 1) 2
π 2 (π‘) β₯(π1) 2 +(C) 2 (π2) 2 π
β π17 2 (π1) 2 β(π0) 2 π‘
1+(C) 2 π β π17 2 (π1) 2 β(π0) 2 π‘
, (C) 2 =(π1) 2 β(π0) 2
(π0) 2 β(π2) 2
it follows (π0) 2 β€ π 2 (π‘) β€ (π1) 2
In the same manner , we get
π 2 (π‘) β€(π 1) 2 +(C ) 2 (π 2) 2 π
β π17 2 (π 1) 2 β(π 2) 2 π‘
1+(C ) 2 π β π17 2 (π 1) 2 β(π 2) 2 π‘
, (C ) 2 =(π 1) 2 β(π0) 2
(π0) 2 β(π 2) 2
From which we deduce (π0) 2 β€ π 2 (π‘) β€ (π 1) 2
(e) If 0 < (π1) 2 < (π0) 2 =G16
0
G170 < (π 1) 2 we find like in the previous case,
(π1) 2 β€(π1) 2 + C 2 (π2) 2 π
β π17 2 (π1) 2 β(π2) 2 π‘
1+ C 2 π β π17 2 (π1) 2 β(π2) 2 π‘
β€ π 2 π‘ β€
(π 1) 2 + C 2 (π 2) 2 π
β π17 2 (π 1) 2 β(π 2) 2 π‘
1+ C 2 π β π17 2 (π 1) 2 β(π 2) 2 π‘
β€ (π 1) 2
(f) If 0 < (π1) 2 β€ (π 1) 2 β€ (π0) 2 =G16
0
G170 , we obtain
(π1) 2 β€ π 2 π‘ β€(π 1) 2 + C 2 (π 2) 2 π
β π17 2 (π 1) 2 β(π 2) 2 π‘
1+ C 2 π β π17 2 (π 1) 2 β(π 2) 2 π‘
β€ (π0) 2
And so with the notation of the first part of condition (c) , we have
Definition of π 2 π‘ :-
(π2) 2 β€ π 2 π‘ β€ (π1) 2 , π 2 π‘ =πΊ16 π‘
πΊ17 π‘
In a completely analogous way, we obtain
Definition of π’ 2 π‘ :-
(π2) 2 β€ π’ 2 π‘ β€ (π1) 2 , π’ 2 π‘ =π16 π‘
π17 π‘
.
Particular case :
If (π16β²β² ) 2 = (π17
β²β² ) 2 , π‘πππ (Ο1) 2 = (Ο2) 2 and in this case (π1) 2 = (π 1) 2 if in addition (π0) 2 =
(π1) 2 then π 2 π‘ = (π0) 2 and as a consequence πΊ16(π‘) = (π0) 2 πΊ17(π‘)
Analogously if (π16β²β² ) 2 = (π17
β²β² ) 2 , π‘πππ (Ο1) 2 = (Ο2) 2 and then
(π’1) 2 = (π’ 1) 2 if in addition (π’0) 2 = (π’1) 2 then π16(π‘) = (π’0) 2 π17 (π‘) This is an important
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consequence of the relation between (π1) 2 and (π 1) 2
From GLOBAL EQUATIONS we obtain
ππ 3
ππ‘= (π20 ) 3 β (π20
β² ) 3 β (π21β² ) 3 + (π20
β²β² ) 3 π21 , π‘ β (π21β²β² ) 3 π21 , π‘ π 3 β (π21) 3 π 3
Definition of π 3 :- π 3 =πΊ20
πΊ21
It follows
β (π21 ) 3 π 3 2
+ (π2) 3 π 3 β (π20) 3 β€ππ 3
ππ‘β€ β (π21) 3 π 3
2+ (π1) 3 π 3 β (π20 ) 3
From which one obtains
(a) For 0 < (π0) 3 =πΊ20
0
πΊ210 < (π1) 3 < (π 1) 3
π 3 (π‘) β₯(π1) 3 +(πΆ) 3 (π2) 3 π
β π21 3 (π1) 3 β(π0) 3 π‘
1+(πΆ) 3 π β π21 3 (π1) 3 β(π0) 3 π‘
, (πΆ) 3 =(π1) 3 β(π0) 3
(π0) 3 β(π2) 3
it follows (π0) 3 β€ π 3 (π‘) β€ (π1) 3
In the same manner , we get
π 3 (π‘) β€(π 1) 3 +(πΆ ) 3 (π 2) 3 π
β π21 3 (π 1) 3 β(π 2) 3 π‘
1+(πΆ ) 3 π β π21 3 (π 1) 3 β(π 2) 3 π‘
, (πΆ ) 3 =(π 1) 3 β(π0) 3
(π0) 3 β(π 2) 3
Definition of (π 1) 3 :-
From which we deduce (π0) 3 β€ π 3 (π‘) β€ (π 1) 3
(b) If 0 < (π1) 3 < (π0) 3 =πΊ20
0
πΊ210 < (π 1) 3 we find like in the previous case,
(π1) 3 β€(π1) 3 + πΆ 3 (π2) 3 π
β π21 3 (π1) 3 β(π2) 3 π‘
1+ πΆ 3 π β π21 3 (π1) 3 β(π2) 3 π‘
β€ π 3 π‘ β€
(π 1) 3 + πΆ 3 (π 2) 3 π
β π21 3 (π 1) 3 β(π 2) 3 π‘
1+ πΆ 3 π β π21 3 (π 1) 3 β(π 2) 3 π‘
β€ (π 1) 3
(c) If 0 < (π1) 3 β€ (π 1) 3 β€ (π0) 3 =πΊ20
0
πΊ210 , we obtain
(π1) 3 β€ π 3 π‘ β€(π 1) 3 + πΆ 3 (π 2) 3 π
β π21 3 (π 1) 3 β(π 2) 3 π‘
1+ πΆ 3 π β π21 3 (π 1) 3 β(π 2) 3 π‘
β€ (π0) 3
And so with the notation of the first part of condition (c) , we have
Definition of π 3 π‘ :-
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(π2) 3 β€ π 3 π‘ β€ (π1) 3 , π 3 π‘ =πΊ20 π‘
πΊ21 π‘
In a completely analogous way, we obtain
Definition of π’ 3 π‘ :-
(π2) 3 β€ π’ 3 π‘ β€ (π1) 3 , π’ 3 π‘ =π20 π‘
π21 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the
theorem.
Particular case :
If (π20β²β² ) 3 = (π21
β²β² ) 3 , π‘πππ (π1) 3 = (π2) 3 and in this case (π1) 3 = (π 1) 3 if in addition (π0) 3 =
(π1) 3 then π 3 π‘ = (π0) 3 and as a consequence πΊ20(π‘) = (π0) 3 πΊ21(π‘)
Analogously if (π20β²β² ) 3 = (π21
β²β² ) 3 , π‘πππ (π1) 3 = (π2) 3 and then
(π’1) 3 = (π’ 1) 3 if in addition (π’0) 3 = (π’1) 3 then π20(π‘) = (π’0) 3 π21(π‘) This is an important
consequence of the relation between (π1) 3 and (π 1) 3
: From GLOBAL EQUATIONS we obtain
ππ 4
ππ‘= (π24 ) 4 β (π24
β² ) 4 β (π25β² ) 4 + (π24
β²β² ) 4 π25 , π‘ β (π25β²β² ) 4 π25 , π‘ π 4 β (π25) 4 π 4
Definition of π 4 :- π 4 =πΊ24
πΊ25
It follows
β (π25 ) 4 π 4 2
+ (π2) 4 π 4 β (π24) 4 β€ππ 4
ππ‘β€ β (π25) 4 π 4
2+ (π4) 4 π 4 β (π24 ) 4
From which one obtains
Definition of (π 1) 4 , (π0) 4 :-
(d) For 0 < (π0) 4 =πΊ24
0
πΊ250 < (π1) 4 < (π 1) 4
π 4 π‘ β₯(π1) 4 + πΆ 4 (π2) 4 π
β π25 4 (π1) 4 β(π0) 4 π‘
4+ πΆ 4 π β π25 4 (π1) 4 β(π0) 4 π‘
, πΆ 4 =(π1) 4 β(π0) 4
(π0) 4 β(π2) 4
it follows (π0) 4 β€ π 4 (π‘) β€ (π1) 4
In the same manner , we get
π 4 π‘ β€(π 1) 4 + πΆ 4 (π 2) 4 π
β π25 4 (π 1) 4 β(π 2) 4 π‘
4+ πΆ 4 π β π25 4 (π 1) 4 β(π 2) 4 π‘
, (πΆ ) 4 =(π 1) 4 β(π0) 4
(π0) 4 β(π 2) 4
From which we deduce (π0) 4 β€ π 4 (π‘) β€ (π 1) 4
(e) If 0 < (π1) 4 < (π0) 4 =πΊ24
0
πΊ250 < (π 1) 4 we find like in the previous case,
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(π1) 4 β€(π1) 4 + πΆ 4 (π2) 4 π
β π25 4 (π1) 4 β(π2) 4 π‘
1+ πΆ 4 π β π25 4 (π1) 4 β(π2) 4 π‘
β€ π 4 π‘ β€
(π 1) 4 + πΆ 4 (π 2) 4 π
β π25 4 (π 1) 4 β(π 2) 4 π‘
1+ πΆ 4 π β π25 4 (π 1) 4 β(π 2) 4 π‘
β€ (π 1) 4
(f) If 0 < (π1) 4 β€ (π 1) 4 β€ (π0) 4 =πΊ24
0
πΊ250 , we obtain
(π1) 4 β€ π 4 π‘ β€(π 1) 4 + πΆ 4 (π 2) 4 π
β π25 4 (π 1) 4 β(π 2) 4 π‘
1+ πΆ 4 π β π25 4 (π 1) 4 β(π 2) 4 π‘
β€ (π0) 4
And so with the notation of the first part of condition (c) , we have
Definition of π 4 π‘ :-
(π2) 4 β€ π 4 π‘ β€ (π1) 4 , π 4 π‘ =πΊ24 π‘
πΊ25 π‘
In a completely analogous way, we obtain
Definition of π’ 4 π‘ :-
(π2) 4 β€ π’ 4 π‘ β€ (π1) 4 , π’ 4 π‘ =π24 π‘
π25 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π24β²β² ) 4 = (π25
β²β² ) 4 , π‘πππ (π1) 4 = (π2) 4 and in this case (π1) 4 = (π 1) 4 if in addition (π0) 4 =
(π1) 4 then π 4 π‘ = (π0) 4 and as a consequence πΊ24(π‘) = (π0) 4 πΊ25(π‘) this also defines (π0) 4 for the special case .
Analogously if (π24β²β² ) 4 = (π25
β²β² ) 4 , π‘πππ (π1) 4 = (π2) 4 and then
(π’1) 4 = (π’ 4) 4 if in addition (π’0) 4 = (π’1) 4 then π24 (π‘) = (π’0) 4 π25 (π‘) This is an important
consequence of the relation between (π1) 4 and (π 1) 4 , and definition of (π’0) 4 .
From GLOBAL EQUATIONS we obtain
ππ 5
ππ‘= (π28 ) 5 β (π28
β² ) 5 β (π29β² ) 5 + (π28
β²β² ) 5 π29 , π‘ β (π29β²β² ) 5 π29 , π‘ π 5 β (π29) 5 π 5
Definition of π 5 :- π 5 =πΊ28
πΊ29
It follows
β (π29) 5 π 5 2
+ (π2) 5 π 5 β (π28) 5 β€ππ 5
ππ‘β€ β (π29) 5 π 5
2+ (π1) 5 π 5 β (π28 ) 5
From which one obtains
Definition of (π 1) 5 , (π0) 5 :-
(g) For 0 < (π0) 5 =πΊ28
0
πΊ290 < (π1) 5 < (π 1) 5
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π 5 (π‘) β₯(π1) 5 +(πΆ) 5 (π2) 5 π
β π29 5 (π1) 5 β(π0) 5 π‘
5+(πΆ) 5 π β π29 5 (π1) 5 β(π0) 5 π‘
, (πΆ) 5 =(π1) 5 β(π0) 5
(π0) 5 β(π2) 5
it follows (π0) 5 β€ π 5 (π‘) β€ (π1) 5
In the same manner , we get
π 5 (π‘) β€(π 1) 5 +(πΆ ) 5 (π 2) 5 π
β π29 5 (π 1) 5 β(π 2) 5 π‘
5+(πΆ ) 5 π β π29 5 (π 1) 5 β(π 2) 5 π‘
, (πΆ ) 5 =(π 1) 5 β(π0) 5
(π0) 5 β(π 2) 5
From which we deduce (π0) 5 β€ π 5 (π‘) β€ (π 5) 5
(h) If 0 < (π1) 5 < (π0) 5 =πΊ28
0
πΊ290 < (π 1) 5 we find like in the previous case,
(π1) 5 β€(π1) 5 + πΆ 5 (π2) 5 π
β π29 5 (π1) 5 β(π2) 5 π‘
1+ πΆ 5 π β π29 5 (π1) 5 β(π2) 5 π‘
β€ π 5 π‘ β€
(π 1) 5 + πΆ 5 (π 2) 5 π
β π29 5 (π 1) 5 β(π 2) 5 π‘
1+ πΆ 5 π β π29 5 (π 1) 5 β(π 2) 5 π‘
β€ (π 1) 5
(i) If 0 < (π1) 5 β€ (π 1) 5 β€ (π0) 5 =πΊ28
0
πΊ290 , we obtain
(π1) 5 β€ π 5 π‘ β€(π 1) 5 + πΆ 5 (π 2) 5 π
β π29 5 (π 1) 5 β(π 2) 5 π‘
1+ πΆ 5 π β π29 5 (π 1) 5 β(π 2) 5 π‘
β€ (π0) 5
And so with the notation of the first part of condition (c) , we have
Definition of π 5 π‘ :-
(π2) 5 β€ π 5 π‘ β€ (π1) 5 , π 5 π‘ =πΊ28 π‘
πΊ29 π‘
In a completely analogous way, we obtain
Definition of π’ 5 π‘ :-
(π2) 5 β€ π’ 5 π‘ β€ (π1) 5 , π’ 5 π‘ =π28 π‘
π29 π‘
Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π28β²β² ) 5 = (π29
β²β² ) 5 , π‘πππ (π1) 5 = (π2) 5 and in this case (π1) 5 = (π 1) 5 if in addition (π0) 5 =
(π5) 5 then π 5 π‘ = (π0) 5 and as a consequence πΊ28(π‘) = (π0) 5 πΊ29(π‘) this also defines (π0) 5 for the special case .
Analogously if (π28β²β² ) 5 = (π29
β²β² ) 5 , π‘πππ (π1) 5 = (π2) 5 and then
(π’1) 5 = (π’ 1) 5 if in addition (π’0) 5 = (π’1) 5 then π28 (π‘) = (π’0) 5 π29(π‘) This is an important
consequence of the relation between (π1) 5 and (π 1) 5 , and definition of (π’0) 5 .
we obtain
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ππ 6
ππ‘= (π32 ) 6 β (π32
β² ) 6 β (π33β² ) 6 + (π32
β²β² ) 6 π33 , π‘ β (π33β²β² ) 6 π33 , π‘ π 6 β (π33) 6 π 6
Definition of π 6 :- π 6 =πΊ32
πΊ33
It follows
β (π33 ) 6 π 6 2
+ (π2) 6 π 6 β (π32) 6 β€ππ 6
ππ‘β€ β (π33) 6 π 6
2+ (π1) 6 π 6 β (π32 ) 6
From which one obtains
Definition of (π 1) 6 , (π0) 6 :-
(j) For 0 < (π0) 6 =πΊ32
0
πΊ330 < (π1) 6 < (π 1) 6
π 6 (π‘) β₯(π1) 6 +(πΆ) 6 (π2) 6 π
β π33 6 (π1) 6 β(π0) 6 π‘
1+(πΆ) 6 π β π33 6 (π1) 6 β(π0) 6 π‘
, (πΆ) 6 =(π1) 6 β(π0) 6
(π0) 6 β(π2) 6
it follows (π0) 6 β€ π 6 (π‘) β€ (π1) 6
In the same manner , we get
π 6 (π‘) β€(π 1) 6 +(πΆ ) 6 (π 2) 6 π
β π33 6 (π 1) 6 β(π 2) 6 π‘
1+(πΆ ) 6 π β π33 6 (π 1) 6 β(π 2) 6 π‘
, (πΆ ) 6 =(π 1) 6 β(π0) 6
(π0) 6 β(π 2) 6
From which we deduce (π0) 6 β€ π 6 (π‘) β€ (π 1) 6
(k) If 0 < (π1) 6 < (π0) 6 =πΊ32
0
πΊ330 < (π 1) 6 we find like in the previous case,
(π1) 6 β€(π1) 6 + πΆ 6 (π2) 6 π
β π33 6 (π1) 6 β(π2) 6 π‘
1+ πΆ 6 π β π33 6 (π1) 6 β(π2) 6 π‘
β€ π 6 π‘ β€
(π 1) 6 + πΆ 6 (π 2) 6 π
β π33 6 (π 1) 6 β(π 2) 6 π‘
1+ πΆ 6 π β π33 6 (π 1) 6 β(π 2) 6 π‘
β€ (π 1) 6
(l) If 0 < (π1) 6 β€ (π 1) 6 β€ (π0) 6 =πΊ32
0
πΊ330 , we obtain
(π1) 6 β€ π 6 π‘ β€(π 1) 6 + πΆ 6 (π 2) 6 π
β π33 6 (π 1) 6 β(π 2) 6 π‘
1+ πΆ 6 π β π33 6 (π 1) 6 β(π 2) 6 π‘
β€ (π0) 6
And so with the notation of the first part of condition (c) , we have
Definition of π 6 π‘ :-
(π2) 6 β€ π 6 π‘ β€ (π1) 6 , π 6 π‘ =πΊ32 π‘
πΊ33 π‘
In a completely analogous way, we obtain
Definition of π’ 6 π‘ :-
(π2) 6 β€ π’ 6 π‘ β€ (π1) 6 , π’ 6 π‘ =π32 π‘
π33 π‘
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Now, using this result and replacing it in GLOBAL EQUATIONS we get easily the result stated in the theorem. Particular case :
If (π32β²β² ) 6 = (π33
β²β² ) 6 , π‘πππ (π1) 6 = (π2) 6 and in this case (π1) 6 = (π 1) 6 if in addition (π0) 6 =
(π1) 6 then π 6 π‘ = (π0) 6 and as a consequence πΊ32(π‘) = (π0) 6 πΊ33(π‘) this also defines (π0) 6 for the special case .
Analogously if (π32β²β² ) 6 = (π33
β²β² ) 6 , π‘πππ (π1) 6 = (π2) 6 and then
(π’1) 6 = (π’ 1) 6 if in addition (π’0) 6 = (π’1) 6 then π32 (π‘) = (π’0) 6 π33 (π‘) This is an important
consequence of the relation between (π1) 6 and (π 1) 6 , and definition of (π’0) 6 .
We can prove the following
Theorem 3: If (ππβ²β² ) 1 πππ (ππ
β²β² ) 1 are independent on π‘ , and the conditions
(π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 < 0
(π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 + π13 1 π13
1 + (π14β² ) 1 π14
1 + π13 1 π14
1 > 0
(π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 > 0 ,
(π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 β (π13β² ) 1 π14
1 β (π14β² ) 1 π14
1 + π13 1 π14
1 < 0
π€ππ‘π π13 1 , π14
1 as defined, then the system
If (ππβ²β² ) 2 πππ (ππ
β²β² ) 2 are independent on t , and the conditions
(π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 < 0
(π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 + π16 2 π16
2 + (π17β² ) 2 π17
2 + π16 2 π17
2 > 0
(π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 > 0 ,
(π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 β (π16β² ) 2 π17
2 β (π17β² ) 2 π17
2 + π16 2 π17
2 < 0
π€ππ‘π π16 2 , π17
2 as defined are satisfied , then the system
If (ππβ²β² ) 3 πππ (ππ
β²β² ) 3 are independent on π‘ , and the conditions
(π20β² ) 3 (π21
β² ) 3 β π20 3 π21
3 < 0
(π20β² ) 3 (π21
β² ) 3 β π20 3 π21
3 + π20 3 π20
3 + (π21β² ) 3 π21
3 + π20 3 π21
3 > 0
(π20β² ) 3 (π21
β² ) 3 β π20 3 π21
3 > 0 ,
(π20β² ) 3 (π21
β² ) 3 β π20 3 π21
3 β (π20β² ) 3 π21
3 β (π21β² ) 3 π21
3 + π20 3 π21
3 < 0
π€ππ‘π π20 3 , π21
3 as defined are satisfied , then the system
If (ππβ²β² ) 4 πππ (ππ
β²β² ) 4 are independent on π‘ , and the conditions
(π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 < 0
(π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 + π24 4 π24
4 + (π25β² ) 4 π25
4 + π24 4 π25
4 > 0
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(π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 > 0 ,
(π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 β (π24β² ) 4 π25
4 β (π25β² ) 4 π25
4 + π24 4 π25
4 < 0
π€ππ‘π π24 4 , π25
4 as defined are satisfied , then the system
If (ππβ²β² ) 5 πππ (ππ
β²β² ) 5 are independent on π‘ , and the conditions
(π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 < 0
(π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 + π28 5 π28
5 + (π29β² ) 5 π29
5 + π28 5 π29
5 > 0
(π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 > 0 ,
(π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 β (π28β² ) 5 π29
5 β (π29β² ) 5 π29
5 + π28 5 π29
5 < 0
π€ππ‘π π28 5 , π29
5 as defined satisfied , then the system
If (ππβ²β² ) 6 πππ (ππ
β²β² ) 6 are independent on π‘ , and the conditions
(π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 < 0
(π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 + π32 6 π32
6 + (π33β² ) 6 π33
6 + π32 6 π33
6 > 0
(π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 > 0 ,
(π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 β (π32β² ) 6 π33
6 β (π33β² ) 6 π33
6 + π32 6 π33
6 < 0
π€ππ‘π π32 6 , π33
6 as defined are satisfied , then the system
π13 1 πΊ14 β (π13
β² ) 1 + (π13β²β² ) 1 π14 πΊ13 = 0
π14 1 πΊ13 β (π14
β² ) 1 + (π14β²β² ) 1 π14 πΊ14 = 0
π15 1 πΊ14 β (π15
β² ) 1 + (π15β²β² ) 1 π14 πΊ15 = 0
π13 1 π14 β [(π13
β² ) 1 β (π13β²β² ) 1 πΊ ]π13 = 0
π14 1 π13 β [(π14
β² ) 1 β (π14β²β² ) 1 πΊ ]π14 = 0
π15 1 π14 β [(π15
β² ) 1 β (π15β²β² ) 1 πΊ ]π15 = 0
has a unique positive solution , which is an equilibrium solution for the system
π16 2 πΊ17 β (π16
β² ) 2 + (π16β²β² ) 2 π17 πΊ16 = 0
π17 2 πΊ16 β (π17
β² ) 2 + (π17β²β² ) 2 π17 πΊ17 = 0
π18 2 πΊ17 β (π18
β² ) 2 + (π18β²β² ) 2 π17 πΊ18 = 0
π16 2 π17 β [(π16
β² ) 2 β (π16β²β² ) 2 πΊ19 ]π16 = 0
π17 2 π16 β [(π17
β² ) 2 β (π17β²β² ) 2 πΊ19 ]π17 = 0
π18 2 π17 β [(π18
β² ) 2 β (π18β²β² ) 2 πΊ19 ]π18 = 0
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has a unique positive solution , which is an equilibrium solution for
π20 3 πΊ21 β (π20
β² ) 3 + (π20β²β² ) 3 π21 πΊ20 = 0
π21 3 πΊ20 β (π21
β² ) 3 + (π21β²β² ) 3 π21 πΊ21 = 0
π22 3 πΊ21 β (π22
β² ) 3 + (π22β²β² ) 3 π21 πΊ22 = 0
π20 3 π21 β [(π20
β² ) 3 β (π20β²β² ) 3 πΊ23 ]π20 = 0
π21 3 π20 β [(π21
β² ) 3 β (π21β²β² ) 3 πΊ23 ]π21 = 0
π22 3 π21 β [(π22
β² ) 3 β (π22β²β² ) 3 πΊ23 ]π22 = 0
has a unique positive solution , which is an equilibrium solution
π24 4 πΊ25 β (π24
β² ) 4 + (π24β²β² ) 4 π25 πΊ24 = 0
π25 4 πΊ24 β (π25
β² ) 4 + (π25β²β² ) 4 π25 πΊ25 = 0
π26 4 πΊ25 β (π26
β² ) 4 + (π26β²β² ) 4 π25 πΊ26 = 0
π24 4 π25 β [(π24
β² ) 4 β (π24β²β² ) 4 πΊ27 ]π24 = 0
π25 4 π24 β [(π25
β² ) 4 β (π25β²β² ) 4 πΊ27 ]π25 = 0
π26 4 π25 β [(π26
β² ) 4 β (π26β²β² ) 4 πΊ27 ]π26 = 0
has a unique positive solution , which is an equilibrium solution for the system
π28 5 πΊ29 β (π28
β² ) 5 + (π28β²β² ) 5 π29 πΊ28 = 0
π29 5 πΊ28 β (π29
β² ) 5 + (π29β²β² ) 5 π29 πΊ29 = 0
π30 5 πΊ29 β (π30
β² ) 5 + (π30β²β² ) 5 π29 πΊ30 = 0
π28 5 π29 β [(π28
β² ) 5 β (π28β²β² ) 5 πΊ31 ]π28 = 0
π29 5 π28 β [(π29
β² ) 5 β (π29β²β² ) 5 πΊ31 ]π29 = 0
π30 5 π29 β [(π30
β² ) 5 β (π30β²β² ) 5 πΊ31 ]π30 = 0
has a unique positive solution , which is an equilibrium solution for the system
π32 6 πΊ33 β (π32
β² ) 6 + (π32β²β² ) 6 π33 πΊ32 = 0
π33 6 πΊ32 β (π33
β² ) 6 + (π33β²β² ) 6 π33 πΊ33 = 0
π34 6 πΊ33 β (π34
β² ) 6 + (π34β²β² ) 6 π33 πΊ34 = 0
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π32 6 π33 β [(π32
β² ) 6 β (π32β²β² ) 6 πΊ35 ]π32 = 0
π33 6 π32 β [(π33
β² ) 6 β (π33β²β² ) 6 πΊ35 ]π33 = 0
π34 6 π33 β [(π34
β² ) 6 β (π34β²β² ) 6 πΊ35 ]π34 = 0
has a unique positive solution , which is an equilibrium solution for the system
(a) Indeed the first two equations have a nontrivial solution πΊ13 , πΊ14 if
πΉ π = (π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 + (π13β² ) 1 (π14
β²β² ) 1 π14 + (π14β² ) 1 (π13
β²β² ) 1 π14 +
(π13β²β² ) 1 π14 (π14
β²β² ) 1 π14 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ16 , πΊ17 if
F π19 = (π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 + (π16β² ) 2 (π17
β²β² ) 2 π17 + (π17β² ) 2 (π16
β²β² ) 2 π17 +
(π16β²β² ) 2 π17 (π17
β²β² ) 2 π17 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ20 , πΊ21 if
πΉ π23 = (π20β² ) 3 (π21
β² ) 3 β π20 3 π21
3 + (π20β² ) 3 (π21
β²β² ) 3 π21 + (π21β² ) 3 (π20
β²β² ) 3 π21 +
(π20β²β² ) 3 π21 (π21
β²β² ) 3 π21 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ24 , πΊ25 if
πΉ π27 =
(π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 + (π24β² ) 4 (π25
β²β² ) 4 π25 + (π25β² ) 4 (π24
β²β² ) 4 π25 + (π24β²β² ) 4 π25 (π25
β²β² ) 4 π25 =
0
(a) Indeed the first two equations have a nontrivial solution πΊ28 , πΊ29 if
πΉ π31 =
(π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 + (π28β² ) 5 (π29
β²β² ) 5 π29 + (π29β² ) 5 (π28
β²β² ) 5 π29 + (π28β²β² ) 5 π29 (π29
β²β² ) 5 π29 =
0
(a) Indeed the first two equations have a nontrivial solution πΊ32 , πΊ33 if
πΉ π35 =
(π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 + (π32β² ) 6 (π33
β²β² ) 6 π33 + (π33β² ) 6 (π32
β²β² ) 6 π33 + (π32β²β² ) 6 π33 (π33
β²β² ) 6 π33 =
0
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Definition and uniqueness of T14β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β² ) 1 π14 being increasing, it follows that there
exists a unique π14β for which π π14
β = 0. With this value , we obtain from the three first equations
πΊ13 = π13 1 πΊ14
(π13β² ) 1 +(π13
β²β² ) 1 π14β
, πΊ15 = π15 1 πΊ14
(π15β² ) 1 +(π15
β²β² ) 1 π14β
Definition and uniqueness of T17β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β² ) 2 π17 being increasing, it follows that there
exists a unique T17β for which π T17
β = 0. With this value , we obtain from the three first equations
πΊ16 = π16 2 G17
(π16β² ) 2 +(π16
β²β² ) 2 T17β
, πΊ18 = π18 2 G17
(π18β² ) 2 +(π18
β²β² ) 2 T17β
Definition and uniqueness of T21β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β²) 1 π21 being increasing, it follows that there
exists a unique π21β for which π π21
β = 0. With this value , we obtain from the three first equations
πΊ20 = π20 3 πΊ21
(π20β² ) 3 +(π20
β²β² ) 3 π21β
, πΊ22 = π22 3 πΊ21
(π22β² ) 3 +(π22
β²β² ) 3 π21β
Definition and uniqueness of T25β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β² ) 4 π25 being increasing, it follows that
there exists a unique π25β for which π π25
β = 0. With this value , we obtain from the three first
equations
πΊ24 = π24 4 πΊ25
(π24β² ) 4 +(π24
β²β² ) 4 π25β
, πΊ26 = π26 4 πΊ25
(π26β² ) 4 +(π26
β²β² ) 4 π25β
Definition and uniqueness of T29β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β² ) 5 π29 being increasing, it follows that
there exists a unique π29β for which π π29
β = 0. With this value , we obtain from the three first
equations
πΊ28 = π28 5 πΊ29
(π28β² ) 5 +(π28
β²β² ) 5 π29β
, πΊ30 = π30 5 πΊ29
(π30β² ) 5 +(π30
β²β² ) 5 π29β
Definition and uniqueness of T33β :-
After hypothesis π 0 < 0, π β > 0 and the functions (ππβ²β² ) 6 π33 being increasing, it follows that
there exists a unique π33β for which π π33
β = 0. With this value , we obtain from the three first
equations
πΊ32 = π32 6 πΊ33
(π32β² ) 6 +(π32
β²β² ) 6 π33β
, πΊ34 = π34 6 πΊ33
(π34β² ) 6 +(π34
β²β² ) 6 π33β
(e) By the same argument, the equations 92,93 admit solutions πΊ13 , πΊ14 if
π πΊ = (π13β² ) 1 (π14
β² ) 1 β π13 1 π14
1 β
(π13β² ) 1 (π14
β²β² ) 1 πΊ + (π14β² ) 1 (π13
β²β² ) 1 πΊ +(π13β²β² ) 1 πΊ (π14
β²β² ) 1 πΊ = 0
Where in πΊ πΊ13 , πΊ14 , πΊ15 , πΊ13 , πΊ15 must be replaced by their values from 96. It is easy to see that Ο is a
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decreasing function in πΊ14 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there
exists a unique πΊ14β such that π πΊβ = 0
(f) By the same argument, the equations 92,93 admit solutions πΊ16 , πΊ17 if
Ο πΊ19 = (π16β² ) 2 (π17
β² ) 2 β π16 2 π17
2 β
(π16β² ) 2 (π17
β²β² ) 2 πΊ19 + (π17β² ) 2 (π16
β²β² ) 2 πΊ19 +(π16β²β² ) 2 πΊ19 (π17
β²β² ) 2 πΊ19 = 0
Where in πΊ19 πΊ16 , πΊ17 , πΊ18 , πΊ16 , πΊ18 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ17 taking into account the hypothesis Ο 0 > 0 , π β < 0 it follows that there
exists a unique G14β such that Ο πΊ19
β = 0
(g) By the same argument, the concatenated equations admit solutions πΊ20 , πΊ21 if
π πΊ23 = (π20β² ) 3 (π21
β² ) 3 β π20 3 π21
3 β
(π20β² ) 3 (π21
β²β² ) 3 πΊ23 + (π21β² ) 3 (π20
β²β² ) 3 πΊ23 +(π20β²β² ) 3 πΊ23 (π21
β²β² ) 3 πΊ23 = 0
Where in πΊ23 πΊ20 , πΊ21 , πΊ22 , πΊ20 , πΊ22 must be replaced by their values from 96. It is easy to see that Ο is a
decreasing function in πΊ21 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there
exists a unique πΊ21β such that π πΊ23
β = 0
(h) By the same argument, the equations of modules admit solutions πΊ24 , πΊ25 if
π πΊ27 = (π24β² ) 4 (π25
β² ) 4 β π24 4 π25
4 β
(π24β² ) 4 (π25
β²β² ) 4 πΊ27 + (π25β² ) 4 (π24
β²β² ) 4 πΊ27 +(π24β²β² ) 4 πΊ27 (π25
β²β² ) 4 πΊ27 = 0
Where in πΊ27 πΊ24 , πΊ25 , πΊ26 , πΊ24 , πΊ26 must be replaced by their values from 96. It is easy to see that Ο is
a decreasing function in πΊ25 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there
exists a unique πΊ25β such that π πΊ27
β = 0
(i) By the same argument, the equations (modules) admit solutions πΊ28 , πΊ29 if
π πΊ31 = (π28β² ) 5 (π29
β² ) 5 β π28 5 π29
5 β
(π28β² ) 5 (π29
β²β² ) 5 πΊ31 + (π29β² ) 5 (π28
β²β² ) 5 πΊ31 +(π28β²β² ) 5 πΊ31 (π29
β²β² ) 5 πΊ31 = 0
Where in πΊ31 πΊ28 , πΊ29 , πΊ30 , πΊ28 , πΊ30 must be replaced by their values from 96. It is easy to see that Ο is
a decreasing function in πΊ29 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there
exists a unique πΊ29β such that π πΊ31
β = 0
(j) By the same argument, the equations (modules) admit solutions πΊ32 , πΊ33 if
π πΊ35 = (π32β² ) 6 (π33
β² ) 6 β π32 6 π33
6 β
(π32β² ) 6 (π33
β²β² ) 6 πΊ35 + (π33β² ) 6 (π32
β²β² ) 6 πΊ35 +(π32β²β² ) 6 πΊ35 (π33
β²β² ) 6 πΊ35 = 0
Where in πΊ35 πΊ32 , πΊ33 , πΊ34 , πΊ32 , πΊ34 must be replaced by their values It is easy to see that Ο is a
decreasing function in πΊ33 taking into account the hypothesis π 0 > 0 , π β < 0 it follows that there
exists a unique πΊ33β such that π πΊβ = 0
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Finally we obtain the unique solution of 89 to 94
πΊ14β given by π πΊβ = 0 , π14
β given by π π14β = 0 and
πΊ13β =
π13 1 πΊ14β
(π13β² ) 1 +(π13
β²β² ) 1 π14β
, πΊ15β =
π15 1 πΊ14β
(π15β² ) 1 +(π15
β²β² ) 1 π14β
π13β =
π13 1 π14β
(π13β² ) 1 β(π13
β²β² ) 1 πΊβ , π15
β = π15 1 π14
β
(π15β² ) 1 β(π15
β²β² ) 1 πΊβ
Obviously, these values represent an equilibrium solution
Finally we obtain the unique solution
G17β given by Ο πΊ19
β = 0 , T17β given by π T17
β = 0 and
G16β =
a16 2 G17β
(a16β² ) 2 +(a16
β²β² ) 2 T17β
, G18β =
a18 2 G17β
(a18β² ) 2 +(a18
β²β² ) 2 T17β
T16β =
b16 2 T17β
(b16β² ) 2 β(b16
β²β² ) 2 πΊ19 β , T18
β = b18 2 T17
β
(b18β² ) 2 β(b18
β²β² ) 2 πΊ19 β
Obviously, these values represent an equilibrium solution
Finally we obtain the unique solution
πΊ21β given by π πΊ23
β = 0 , π21β given by π π21
β = 0 and
πΊ20β =
π20 3 πΊ21β
(π20β² ) 3 +(π20
β²β² ) 3 π21β
, πΊ22β =
π22 3 πΊ21β
(π22β² ) 3 +(π22
β²β² ) 3 π21β
π20β =
π20 3 π21β
(π20β² ) 3 β(π20
β²β² ) 3 πΊ23β
, π22β =
π22 3 π21β
(π22β² ) 3 β(π22
β²β² ) 3 πΊ23β
Obviously, these values represent an equilibrium solution
Finally we obtain the unique solution
πΊ25β given by π πΊ27 = 0 , π25
β given by π π25β = 0 and
πΊ24β =
π24 4 πΊ25β
(π24β² ) 4 +(π24
β²β² ) 4 π25β
, πΊ26β =
π26 4 πΊ25β
(π26β² ) 4 +(π26
β²β² ) 4 π25β
π24β =
π24 4 π25β
(π24β² ) 4 β(π24
β²β² ) 4 πΊ27 β , π26
β = π26 4 π25
β
(π26β² ) 4 β(π26
β²β² ) 4 πΊ27 β
Obviously, these values represent an equilibrium solution
Finally we obtain the unique solution
πΊ29β given by π πΊ31
β = 0 , π29β given by π π29
β = 0 and
πΊ28β =
π28 5 πΊ29β
(π28β² ) 5 +(π28
β²β² ) 5 π29β
, πΊ30β =
π30 5 πΊ29β
(π30β² ) 5 +(π30
β²β² ) 5 π29β
π28β =
π28 5 π29β
(π28β² ) 5 β(π28
β²β² ) 5 πΊ31 β , π30
β = π30 5 π29
β
(π30β² ) 5 β(π30
β²β² ) 5 πΊ31 β
Obviously, these values represent an equilibrium solution
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Finally we obtain the unique solution
πΊ33β given by π πΊ35
β = 0 , π33β given by π π33
β = 0 and
πΊ32β =
π32 6 πΊ33β
(π32β² ) 6 +(π32
β²β² ) 6 π33β
, πΊ34β =
π34 6 πΊ33β
(π34β² ) 6 +(π34
β²β² ) 6 π33β
π32β =
π32 6 π33β
(π32β² ) 6 β(π32
β²β² ) 6 πΊ35 β , π34
β = π34 6 π33
β
(π34β² ) 6 β(π34
β²β² ) 6 πΊ35 β
Obviously, these values represent an equilibrium solution
ASYMPTOTIC STABILITY ANALYSIS
Theorem 4: If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 1 πππ (ππ
β²β² ) 1
Belong to πΆ 1 ( β+) then the above equilibrium point is asymptotically stable.
Proof: Denote
Definition of πΎπ , ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π14
β²β² ) 1
ππ14 π14
β = π14 1 ,
π(ππβ²β² ) 1
ππΊπ πΊβ = π ππ
Then taking into account equations (global) and neglecting the terms of power 2, we obtain
ππΎ13
ππ‘= β (π13
β² ) 1 + π13 1 πΎ13 + π13
1 πΎ14 β π13 1 πΊ13
β π14
ππΎ14
ππ‘= β (π14
β² ) 1 + π14 1 πΎ14 + π14
1 πΎ13 β π14 1 πΊ14
β π14
ππΎ15
ππ‘= β (π15
β² ) 1 + π15 1 πΎ15 + π15
1 πΎ14 β π15 1 πΊ15
β π14
ππ13
ππ‘= β (π13
β² ) 1 β π13 1 π13 + π13
1 π14 + π 13 π π13β πΎπ
15π=13
ππ14
ππ‘= β (π14
β² ) 1 β π14 1 π14 + π14
1 π13 + π 14 (π )π14β πΎπ
15π=13
ππ15
ππ‘= β (π15
β² ) 1 β π15 1 π15 + π15
1 π14 + π 15 (π )π15β πΎπ
15π=13
If the conditions of the previous theorem are satisfied and if the functions (aπβ²β² ) 2 and (bπ
β²β² ) 2 Belong to
C 2 ( β+) then the above equilibrium point is asymptotically stable
Denote
Definition of πΎπ , ππ :-
Gπ = Gπβ + πΎπ , Tπ = Tπ
β + ππ
β(π17β²β² ) 2
βT17 T17
β = π17 2 ,
β(ππβ²β² ) 2
βGπ πΊ19
β = π ππ
taking into account equations (global)and neglecting the terms of power 2, we obtain
dπΎ16
dt= β (π16
β² ) 2 + π16 2 πΎ16 + π16
2 πΎ17 β π16 2 G16
β π17
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dπΎ17
dt= β (π17
β² ) 2 + π17 2 πΎ17 + π17
2 πΎ16 β π17 2 G17
β π17
dπΎ18
dt= β (π18
β² ) 2 + π18 2 πΎ18 + π18
2 πΎ17 β π18 2 G18
β π17
dπ16
dt= β (π16
β² ) 2 β π16 2 π16 + π16
2 π17 + π 16 π T16β πΎπ
18π =16
dπ17
dt= β (π17
β² ) 2 β π17 2 π17 + π17
2 π16 + π 17 (π )T17β πΎπ
18π =16
dπ18
dt= β (π18
β² ) 2 β π18 2 π18 + π18
2 π17 + π 18 (π )T18β πΎπ
18π =16
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 3 πππ (ππ
β²β² ) 3 Belong to
πΆ 3 ( β+) then the above equilibrium point is asymptotically stabl
Denote
Definition of πΎπ , ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π21
β²β² ) 3
ππ21 π21
β = π21 3 ,
π(ππβ²β² ) 3
ππΊπ πΊ23
β = π ππ
Then taking into account equations (global) and neglecting the terms of power 2, we obtain
ππΎ20
ππ‘= β (π20
β² ) 3 + π20 3 πΎ20 + π20
3 πΎ21 β π20 3 πΊ20
β π21
ππΎ21
ππ‘= β (π21
β² ) 3 + π21 3 πΎ21 + π21
3 πΎ20 β π21 3 πΊ21
β π21
ππΎ22
ππ‘= β (π22
β² ) 3 + π22 3 πΎ22 + π22
3 πΎ21 β π22 3 πΊ22
β π21
ππ20
ππ‘= β (π20
β² ) 3 β π20 3 π20 + π20
3 π21 + π 20 π π20β πΎπ
22π =20
ππ21
ππ‘= β (π21
β² ) 3 β π21 3 π21 + π21
3 π20 + π 21 (π )π21β πΎπ
22π =20
ππ22
ππ‘= β (π22
β² ) 3 β π22 3 π22 + π22
3 π21 + π 22 (π )π22β πΎπ
22π =20
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 4 πππ (ππ
β²β² ) 4 Belong to
πΆ 4 ( β+) then the above equilibrium point is asymptotically stabl
Denote
Definition of πΎπ , ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π25
β²β² ) 4
ππ25 π25
β = π25 4 ,
π(ππβ²β² ) 4
ππΊπ πΊ27
β = π ππ
Then taking into account equations (global) and neglecting the terms of power 2, we obtain
ππΎ24
ππ‘= β (π24
β² ) 4 + π24 4 πΎ24 + π24
4 πΎ25 β π24 4 πΊ24
β π25
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ππΎ25
ππ‘= β (π25
β² ) 4 + π25 4 πΎ25 + π25
4 πΎ24 β π25 4 πΊ25
β π25
ππΎ26
ππ‘= β (π26
β² ) 4 + π26 4 πΎ26 + π26
4 πΎ25 β π26 4 πΊ26
β π25
ππ24
ππ‘= β (π24
β² ) 4 β π24 4 π24 + π24
4 π25 + π 24 π π24β πΎπ
26π =24
ππ25
ππ‘= β (π25
β² ) 4 β π25 4 π25 + π25
4 π24 + π 25 π π25β πΎπ
26π =24
ππ26
ππ‘= β (π26
β² ) 4 β π26 4 π26 + π26
4 π25 + π 26 (π )π26β πΎπ
26π =24
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 5 πππ (ππ
β²β² ) 5 Belong to
πΆ 5 ( β+) then the above equilibrium point is asymptotically stable
Denote
Definition of πΎπ , ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π29
β²β² ) 5
ππ29 π29
β = π29 5 ,
π(ππβ²β² ) 5
ππΊπ πΊ31
β = π ππ
Then taking into account equations (global) and neglecting the terms of power 2, we obtain
ππΎ28
ππ‘= β (π28
β² ) 5 + π28 5 πΎ28 + π28
5 πΎ29 β π28 5 πΊ28
β π29
ππΎ29
ππ‘= β (π29
β² ) 5 + π29 5 πΎ29 + π29
5 πΎ28 β π29 5 πΊ29
β π29
ππΎ30
ππ‘= β (π30
β² ) 5 + π30 5 πΎ30 + π30
5 πΎ29 β π30 5 πΊ30
β π29
ππ28
ππ‘= β (π28
β² ) 5 β π28 5 π28 + π28
5 π29 + π 28 π π28β πΎπ
30π=28
ππ29
ππ‘= β (π29
β² ) 5 β π29 5 π29 + π29
5 π28 + π 29 π π29β πΎπ
30π=28
ππ30
ππ‘= β (π30
β² ) 5 β π30 5 π30 + π30
5 π29 + π 30 (π )π30β πΎπ
30π=28
If the conditions of the previous theorem are satisfied and if the functions (ππβ²β² ) 6 πππ (ππ
β²β² ) 6 Belong to
πΆ 6 ( β+) then the above equilibrium point is asymptotically stable
Denote
Definition of πΎπ , ππ :-
πΊπ = πΊπβ + πΎπ , ππ = ππ
β + ππ
π(π33
β²β² ) 6
ππ33 π33
β = π33 6 ,
π(ππβ²β² ) 6
ππΊπ πΊ35
β = π ππ
Then taking into account equations(global) and neglecting the terms of power 2, we obtain
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ππΎ32
ππ‘= β (π32
β² ) 6 + π32 6 πΎ32 + π32
6 πΎ33 β π32 6 πΊ32
β π33
ππΎ33
ππ‘= β (π33
β² ) 6 + π33 6 πΎ33 + π33
6 πΎ32 β π33 6 πΊ33
β π33
ππΎ34
ππ‘= β (π34
β² ) 6 + π34 6 πΎ34 + π34
6 πΎ33 β π34 6 πΊ34
β π33
ππ32
ππ‘= β (π32
β² ) 6 β π32 6 π32 + π32
6 π33 + π 32 π π32β πΎπ
34π =32
ππ33
ππ‘= β (π33
β² ) 6 β π33 6 π33 + π33
6 π32 + π 33 π π33β πΎπ
34π =32
ππ34
ππ‘= β (π34
β² ) 6 β π34 6 π34 + π34
6 π33 + π 34 (π )π34β πΎπ
34π =32
The characteristic equation of this system is
π 1 + (π15β² ) 1 β π15
1 { π 1 + (π15β² ) 1 + π15
1
π 1 + (π13β² ) 1 + π13
1 π14 1 πΊ14
β + π14 1 π13
1 πΊ13β
π 1 + (π13β² ) 1 β π13
1 π 14 , 14 π14β + π14
1 π 13 , 14 π14β
+ π 1 + (π14β² ) 1 + π14
1 π13 1 πΊ13
β + π13 1 π14
1 πΊ14β
π 1 + (π13β² ) 1 β π13
1 π 14 , 13 π14β + π14
1 π 13 , 13 π13β
π 1 2
+ (π13β² ) 1 + (π14
β² ) 1 + π13 1 + π14
1 π 1
π 1 2
+ (π13β² ) 1 + (π14
β² ) 1 β π13 1 + π14
1 π 1
+ π 1 2
+ (π13β² ) 1 + (π14
β² ) 1 + π13 1 + π14
1 π 1 π15 1 πΊ15
+ π 1 + (π13β² ) 1 + π13
1 π15 1 π14
1 πΊ14β + π14
1 π15 1 π13
1 πΊ13β
π 1 + (π13β² ) 1 β π13
1 π 14 , 15 π14β + π14
1 π 13 , 15 π13β } = 0
+
π 2 + (π18β² ) 2 β π18
2 { π 2 + (π18β² ) 2 + π18
2
π 2 + (π16β² ) 2 + π16
2 π17 2 G17
β + π17 2 π16
2 G16β
π 2 + (π16β² ) 2 β π16
2 π 17 , 17 T17β + π17
2 π 16 , 17 T17β
+ π 2 + (π17β² ) 2 + π17
2 π16 2 G16
β + π16 2 π17
2 G17β
π 2 + (π16β² ) 2 β π16
2 π 17 , 16 T17β + π17
2 π 16 , 16 T16β
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π 2 2
+ (π16β² ) 2 + (π17
β² ) 2 + π16 2 + π17
2 π 2
π 2 2
+ (π16β² ) 2 + (π17
β² ) 2 β π16 2 + π17
2 π 2
+ π 2 2
+ (π16β² ) 2 + (π17
β² ) 2 + π16 2 + π17
2 π 2 π18 2 G18
+ π 2 + (π16β² ) 2 + π16
2 π18 2 π17
2 G17β + π17
2 π18 2 π16
2 G16β
π 2 + (π16β² ) 2 β π16
2 π 17 , 18 T17β + π17
2 π 16 , 18 T16β } = 0
+
π 3 + (π22β² ) 3 β π22
3 { π 3 + (π22β² ) 3 + π22
3
π 3 + (π20β² ) 3 + π20
3 π21 3 πΊ21
β + π21 3 π20
3 πΊ20β
π 3 + (π20β² ) 3 β π20
3 π 21 , 21 π21β + π21
3 π 20 , 21 π21β
+ π 3 + (π21β² ) 3 + π21
3 π20 3 πΊ20
β + π20 3 π21
1 πΊ21β
π 3 + (π20β² ) 3 β π20
3 π 21 , 20 π21β + π21
3 π 20 , 20 π20β
π 3 2
+ (π20β² ) 3 + (π21
β² ) 3 + π20 3 + π21
3 π 3
π 3 2
+ (π20β² ) 3 + (π21
β² ) 3 β π20 3 + π21
3 π 3
+ π 3 2
+ (π20β² ) 3 + (π21
β² ) 3 + π20 3 + π21
3 π 3 π22 3 πΊ22
+ π 3 + (π20β² ) 3 + π20
3 π22 3 π21
3 πΊ21β + π21
3 π22 3 π20
3 πΊ20β
π 3 + (π20β² ) 3 β π20
3 π 21 , 22 π21β + π21
3 π 20 , 22 π20β } = 0
+
π 4 + (π26β² ) 4 β π26
4 { π 4 + (π26β² ) 4 + π26
4
π 4 + (π24β² ) 4 + π24
4 π25 4 πΊ25
β + π25 4 π24
4 πΊ24β
π 4 + (π24β² ) 4 β π24
4 π 25 , 25 π25β + π25
4 π 24 , 25 π25β
+ π 4 + (π25β² ) 4 + π25
4 π24 4 πΊ24
β + π24 4 π25
4 πΊ25β
π 4 + (π24β² ) 4 β π24
4 π 25 , 24 π25β + π25
4 π 24 , 24 π24β
π 4 2
+ (π24β² ) 4 + (π25
β² ) 4 + π24 4 + π25
4 π 4
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π 4 2
+ (π24β² ) 4 + (π25
β² ) 4 β π24 4 + π25
4 π 4
+ π 4 2
+ (π24β² ) 4 + (π25
β² ) 4 + π24 4 + π25
4 π 4 π26 4 πΊ26
+ π 4 + (π24β² ) 4 + π24
4 π26 4 π25
4 πΊ25β + π25
4 π26 4 π24
4 πΊ24β
π 4 + (π24β² ) 4 β π24
4 π 25 , 26 π25β + π25
4 π 24 , 26 π24β } = 0
+
π 5 + (π30β² ) 5 β π30
5 { π 5 + (π30β² ) 5 + π30
5
π 5 + (π28β² ) 5 + π28
5 π29 5 πΊ29
β + π29 5 π28
5 πΊ28β
π 5 + (π28β² ) 5 β π28
5 π 29 , 29 π29β + π29
5 π 28 , 29 π29β
+ π 5 + (π29β² ) 5 + π29
5 π28 5 πΊ28
β + π28 5 π29
5 πΊ29β
π 5 + (π28β² ) 5 β π28
5 π 29 , 28 π29β + π29
5 π 28 , 28 π28β
π 5 2
+ (π28β² ) 5 + (π29
β² ) 5 + π28 5 + π29
5 π 5
π 5 2
+ (π28β² ) 5 + (π29
β² ) 5 β π28 5 + π29
5 π 5
+ π 5 2
+ (π28β² ) 5 + (π29
β² ) 5 + π28 5 + π29
5 π 5 π30 5 πΊ30
+ π 5 + (π28β² ) 5 + π28
5 π30 5 π29
5 πΊ29β + π29
5 π30 5 π28
5 πΊ28β
π 5 + (π28β² ) 5 β π28
5 π 29 , 30 π29β + π29
5 π 28 , 30 π28β } = 0
+
π 6 + (π34β² ) 6 β π34
6 { π 6 + (π34β² ) 6 + π34
6
π 6 + (π32β² ) 6 + π32
6 π33 6 πΊ33
β + π33 6 π32
6 πΊ32β
π 6 + (π32β² ) 6 β π32
6 π 33 , 33 π33β + π33
6 π 32 , 33 π33β
+ π 6 + (π33β² ) 6 + π33
6 π32 6 πΊ32
β + π32 6 π33
6 πΊ33β
π 6 + (π32β² ) 6 β π32
6 π 33 , 32 π33β + π33
6 π 32 , 32 π32β
π 6 2
+ (π32β² ) 6 + (π33
β² ) 6 + π32 6 + π33
6 π 6
π 6 2
+ (π32β² ) 6 + (π33
β² ) 6 β π32 6 + π33
6 π 6
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+ π 6 2
+ (π32β² ) 6 + (π33
β² ) 6 + π32 6 + π33
6 π 6 π34 6 πΊ34
+ π 6 + (π32β² ) 6 + π32
6 π34 6 π33
6 πΊ33β + π33
6 π34 6 π32
6 πΊ32β
π 6 + (π32β² ) 6 β π32
6 π 33 , 34 π33β + π33
6 π 32 , 34 π32β } = 0
And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and
this proves the theorem.
Acknowledgments:
The introduction is a collection of information from various articles, Books, News Paper reports, Home Pages Of
authors, Journal Reviews, Nature βs L:etters,Article Abstracts, Research papers, Abstracts Of Research Papers,
Stanford Encyclopedia, Web Pages, Ask a Physicist Column, Deliberations with Professors, the internet including
Wikipedia. We acknowledge all authors who have contributed to the same. In the eventuality of the fact that there
has been any act of omission on the part of the authors, we regret with great deal of compunction, contrition,
regret, trepidiation and remorse. As Newton said, it is only because erudite and eminent people allowed one to piggy
ride on their backs; probably an attempt has been made to look slightly further. Once again, it is stated that the
references are only illustrative and not comprehensive
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First Author: 1Mr. K. N.Prasanna Kumar has three doctorates one each in Mathematics, Economics,
Political Science. Thesis was based on Mathematical Modeling. He was recently awarded D.litt. for his work on
βMathematical Models in Political Scienceβ--- Department of studies in Mathematics, Kuvempu University,
Shimoga, Karnataka, India Corresponding Author:[email protected]
Second Author: 2Prof. B.S Kiranagi is the Former Chairman of the Department of Studies in Mathematics,
Manasa Gangotri and present Professor Emeritus of UGC in the Department. Professor Kiranagi has guided
over 25 students and he has received many encomiums and laurels for his contribution to Co homology Groups
and Mathematical Sciences. Known for his prolific writing, and one of the senior most Professors of the
country, he has over 150 publications to his credit. A prolific writer and a prodigious thinker, he has to his credit
several books on Lie Groups, Co Homology Groups, and other mathematical application topics, and excellent
publication history.-- UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri,
University of Mysore, Karnataka, India
Third Author: 3Prof. C.S. Bagewadi is the present Chairman of Department of Mathematics and Department
of Studies in Computer Science and has guided over 25 students. He has published articles in both national and
international journals. Professor Bagewadi specializes in Differential Geometry and its wide-ranging
ramifications. He has to his credit more than 159 research papers. Several Books on Differential Geometry,
Differential Equations are coauthored by him--- Chairman, Department of studies in Mathematics and Computer
science, Jnanasahyadri Kuvempu University, Shankarghatta, Shimoga district, Karnataka, India