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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1977 | Page
1Dr. K. N. Prasanna Kumar,
2Prof. B. S. Kiranagi,
3 Prof. C. S. Bagewadi
1Post doctoral researcher, Dr KNP Kumar has three PhDβs, one each in Mathematics, Economics and Political science
and a D.Litt. in Political Science, Department of studies in Mathematics, Kuvempu University, Shimoga, Karnataka, India 2UGC Emeritus Professor (Department of studies in Mathematics), Manasagangotri, University of Mysore, Karnataka, India
3Chairman, Department of studies in Mathematics and Computer science, Jnanasahyadri Kuvempu university,
Shankarghatta, Shimoga district, Karnataka, India
ABSTRACT: Von Neumann Entropy and computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according
to their inherent difficulty, and relating those classes to each other. In this context, a computational problem is understood to
be a task that is in principle amenable to being solved by a computer (which basically means that the problem can be stated
by a set of mathematical instructions). Informally, a computational problem consists of problem instances and solutions
to these problem instances. For example, primality testing is the problem of determining whether a given number is prime or
not. The instances of this problem are natural numbers, and the solution to an instance is yes or no based on whether the
number is prime or not.A problem is regarded as inherently difficult if its solution requires significant resources, whatever
the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these
problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity
measures are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of
computational complexity theory is to determine the practical limits on what computers can and cannot do. Closely
related fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction
between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of
resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all
possible algorithms that could be used to solve the same problem. More precisely, it tries to classify problems that can or
cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what
distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in
principle, be solved algorithmically. Low-energy excitations of one-dimensional spin-orbital models which consist of spin
waves, orbital waves, and joint spin-orbital excitations. Among the latter we identify strongly entangled spin-orbital bound
states which appear as peaks in the von Neumann entropy (vNE) spectral function introduced in this work. The strong
entanglement of bound states is manifested by a universal logarithmic scaling of the vNE with system size, while the vNE of other spin-orbital excitations saturates. We suggest that spin-orbital entanglement can be experimentally explored by the
measurement of the dynamical spin-orbital correlations using resonant inelastic x-ray scattering, where strong spin-orbit
coupling associated with the core hole plays a role. Distinguish ability of States and von Neumann Entropy have been
studied by Richard Jozsa, Juergen Schlienz.Consider an ensemble of pure quantum states |\psi_j>, j=1,...,n taken with
prior probabilities p_j respectively. It has been shown that it is possible to increase all of the pair wise overlaps
|<\psi_j|\psi_j>| i.e. make each constituent pair of the states more parallel (while keeping the prior probabilities the same),
in such a way that the von Neumann entropy S is increased, and dually, make all pairs more orthogonal while decreasing S.
This phenomenon cannot occur for ensembles in two dimensions but that it is a feature of almost all ensembles of three
states in three dimensions. It is known that the von Neumann entropy characterizes the classical and quantum information
capacities of the ensemble and we argue that information capacity in turn, is a manifestation of the distinguish ability of the
signal states. Hence our result shows that the notion of distinguish ability within an ensemble is a global property that cannot be reduced to considering distinguish ability of each constituent pair of states.
Key words: Von Neumann entropy, Quantum computation, Governing equations
Introduction
Von Neumann entropy In quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of
classical entropy concepts to the field of quantum mechanics. John von Neumann rigorously established the mathematical
framework for quantum mechanics in his work Mathematische Grundlagen der Quantenmechanik In it, he provided a theory
of measurement, where the usual notion of wave-function collapse is described as an irreversible process (the so-called von
Neumann or projective measurement).
The density matrix was introduced, with different motivations, by von Neumann and by Lev Landau. The motivation that
inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector. On the
Von Neumann Entropy in Quantum Computation and Sine qua
non Relativistic Parameters- a Gesellschaft-Gemeinschaft Model
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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1978 | Page
other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory
of quantum measurements. The density matrix formalism was developed to extend the tools of classical statistical mechanics
to the quantum domain. In the classical framework, we compute the partition function of the system in order to evaluate all
possible thermodynamic quantities. Von Neumann introduced the density matrix in the context of states and operators in a Hilbert space. The knowledge of the statistical density matrix operator would allow us to compute all average quantities in a
conceptually similar, but mathematically different way. Let us suppose we have a set of wave functions |Ξ¨ β© which depend
parametrically on a set of quantum numbers . The natural variable which we have is the amplitude with
which a particular wavefunction of the basic set participates in the actual wavefunction of the system. Let us denote the
square of this amplitude by . The goal is to turn this quantity p into the classical density function in
phase space. We have to verify that p goes over into the density function in the classical limit, and that it
hasergodic properties. After checking that is a constant of motion, an ergodic assumption for the
probabilities makes p a function of the energy only .
After this procedure, one finally arrives at the density matrix formalism when seeking a form where
is invariant with respect to the representation used. In the form it is written, it will only yield the correct expectation values for quantities which are diagonal with respect to the quantum numbers .
Expectation values of operators which are not diagonal involve the phases of the quantum amplitudes. Suppose we encode
the quantum numbers into the single index or . Then our wave function has the form
The expectation value of an operator which is not diagonal in these wave functions, so
The role, which was originally reserved for the quantities, is thus taken over by the density matrix of the system S.
Therefore reads as
The invariance of the above term is described by matrix theory. A mathematical framework was described where the
expectation value of quantum operators, as described by matrices, is obtained by taking the trace of the product of the
density operator and an operator (Hilbert scalar product between operators). The matrix formalism here is in the statistical mechanics framework, although it applies as well for finite quantum systems, which is usually the case, where the
state of the system cannot be described by a pure state, but as a statistical operator of the above form. Mathematically, is a positive, semi definite Hermitian matrix with unit trace
Given the density matrix Ο, von Neumann defined the entropy as
Which is a proper extension of the Gibbs entropy (up to a factor ) and the Shannon entropy to the quantum case. To
compute S(Ο) it is convenient (see logarithm of a matrix) to compute the Eigen decomposition of The von Neumann entropy is then given by
Since, for a pure state, the density matrix is idempotent, Ο=Ο2, the entropy S(Ο) for it vanishes. Thus, if the system is finite
(finite dimensional matrix representation), the entropy (Ο) quantifies the departure of the system from a pure state. In other words, it codifies the degree of mixing of the state describing a given finite system. Measurement decohere a quantum
system into something noninterfering and ostensibly classical; so, e.g., the vanishing entropy of a pure state |Ξ¨β© =
(|0β©+|1β©)/β2, corresponding to a density matrix
increases to S=ln 2 =0.69 for the measurement outcome mixture
As the quantum interference information is erased.
Properties
Some properties of the von Neumann entropy:
S(Ο) is only zero for pure states.
S (Ο) is maximal and equal to for a maximally mixed state, being the dimension of the Hilbert space.
S (Ο) is invariant under changes in the basis of , that is, , with U a unitary transformation.
S (Ο) is concave, that is, given a collection of positive numbers which sum to unity ( ) and density
operators , we have
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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1979 | Page
S (Ο) is additive for independent systems. Given two density matrices describing independent systems A and B,
we have .
S(Ο) strongly sub additive for any three systems A, B, and C:
.
This automatically means that S(Ο) is sub additive:
Below, the concept of subadditivity is discussed, followed by its generalization to strong Subadditivity.
Subadditivity
If are the reduced density matrices of the general state , then
This right hand inequality is known as subadditivity. The two inequalities together are sometimes known as the triangle
inequality. They were proved in 1970 by Huzihiro Araki andElliott H. Lieb While in Shannon's theory the entropy of a
composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case, i.e., it is
possible that while and .
Intuitively, this can be understood as follows: In quantum mechanics, the entropy of the joint system can be less than the sum of the entropy of its components because the components may be entangled. For instance, the Bell state of two spin-
1/2's, , is a pure state with zero entropy, but each spin has maximum entropy when considered
individually. The entropy in one spin can be "cancelled" by being correlated with the entropy of the other. The left-hand
inequality can be roughly interpreted as saying that entropy can only be canceled by an equal amount of entropy.
If system and system have different amounts of entropy, the lesser can only partially cancel the greater, and some
entropy must be left over. Likewise, the right-hand inequality can be interpreted as saying that the entropy of a composite
system is maximized when its components are uncorrelated, in which case the total entropy is just a sum of the sub-
entropies. This may be more intuitive in the phase space, instead of Hilbert space, representation, where the Von Neumann
entropy amounts to minus the expected value of the β-logarithm of the Wigner function up to an offset shift.
Strong Subadditivity The von Neumann entropy is also strongly sub additive. Given three Hilbert spaces, ,
This is a more difficult theorem and was proved in 1973 by Elliott H. Lieb and Mary Beth Ruskai using a matrix inequality
of Elliott H. Lieb proved in 1973. By using the proof technique that establishes the left side of the triangle inequality above,
one can show that the strong subadditivity inequality is equivalent to the following inequality.
When , etc. are the reduced density matrices of a density matrix . If we apply ordinary subadditivity to the left
side of this inequality, and consider all permutations of , we obtain the triangle inequality for : Each of
the three numbers is less than or equal to the sum of the other two.
Uses
The von Neumann entropy is being extensively used in different forms (conditional entropies, relative entropies, etc.) in the
framework of quantum information theory. Entanglement measures are based upon some quantity directly related to the von
Neumann entropy. However, there have appeared in the literature several papers dealing with the possible inadequacy of
the Shannon information measure, and consequently of the von Neumann entropy as an appropriate quantum generalization
of Shannon entropy. The main argument is that in classical measurement the Shannon information measure is a natural
measure of our ignorance about the properties of a system, whose existence is independent of measurement.
Conversely, quantum measurement cannot be claimed to reveal the properties of a system that existed before the
measurement was made. This controversy has encouraged some authors to introduce the non-additivity property of Tsallis
entropy (a generalization of the standard BoltzmannβGibbs entropy) as the main reason for recovering a true quantal
information measure in the quantum context, claiming that non-local correlations ought to be described because of the
particularity of Tsallis entropy.
THE SYSTEM IN QUESTION IS:
1. Von Neumann Entropy And Quantum Entanglement
2. Velocity Field Of The Particle And Wave Function
3. Matter Presence In Abundance And Break Down Of Parity Conservation
4. Dissipation In Quantum Computation And Efficiency Of Quantum Algorithms 5. Decoherence And Computational Complexity
6. Coherent Superposition Of Outputs And Different Possible Inputs In The Form Of Qubits
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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1980 | Page
VON NEUMANN ENTROPY AND QUANTUM ENTANGLEMENT: MODULE ONE
NOTATION :
πΊ13 : Category One Of Quantum Entanglement
πΊ14 : Category Two Of Quantum Entanglement
πΊ15 : Category Three Of Quantum Entanglement
π13 : Category One Of Von Neumann Entropy
π14 : Category Two Of Von Neumann Entropy π15 : Category Three Of Von Neumann Entropy
- WAVE FUNCTIONS AND VELOCITY FIELD OF THE PARTICLES: MODULE TWO
πΊ16 : Category One Of Velocity Field Of The Particles
πΊ17 : Category Two Of The Velocity Field Of The Particles
πΊ18 : Category Three Of The Velocity Field Of The Particles
π16 : Category One Of Wave Functions Concomitant To The Velocity Fields
π17 :Category Two Of Wave Functions Corresponding To Category Two Of Velocity Field
π18 : Category Three Of Wave Functions-
BREAK DOWN OF PARITY CONSERVATION AND ABUNDANCE OF MATTER PRESCENCE: MODULE
THREE:
πΊ20 : Category One Of Systems Where There Is Break Down Of Parity Conservation
πΊ21 : Category Two Of Systems Where There Is Break Down In Parity Conservation
πΊ22 : Category Three Of Systems Where There Is Break Down Of Parity Conservation
π20 : Category Three Of Systems Where There Is Break Down Of Parity Conservation
π21 : Category One Of Systems Where There Is Abundance Of Matter
π22 : Category Two Of Systems Where There Is Abundance Of Matter
πΊ24 : Category Three Of Systems Where There Is Abundance Of Matter
EFFICIENCY OF QUANTUM ALGORITHMS AND DISSIPATION IN QUANTUM COMPUTATION MODULE
NUMBERED FOUR:
πΊ25 : Category Two Of Efficiency Of Quantum Algorithms
πΊ26 : Category Three Of efficiency Of Quantum Algorithms
πΊ24 : Category One Of Efficiency Of Quantum Algorithms
π24 : Category Three Of Dissipation In Quantum Computation
π25 : Category One Of Systems With Efficiency In Quantum Algorithm
π26 : Category Two Of Systems With Quantum Algorithm Of Efficiency (Different From Category One)
COMPUTATIONAL COMPLEXITY AND DECOHERENCE MODULE NUMBERED FIVE
πΊ28 : Category One Of Computational Complexity
πΊ29 :Category Two Of Computational Complexity
πΊ30 : Category Three Of Computational Complexity
π28 : Category One Of Decoherence
π29 : Category Two Of Decoherence π30 : Category Three Of Decoherence
DIFFERENT POSSSIBLE INPUTS (QUBITS) AND QUANTUM SUPERPOSITION OF OUTPUTS MODULE
NUMBERED SIX
πΊ32 : Category One Of Different Possible Qubits Inputs
πΊ33 : Category Two Of Different Possible Qubits Inputs
πΊ34 : Category Three Of Different Possible Qubits Inputs
T32 : Category One Of Coherent Superposition Of Outputs
π33 : Category Two Of Coherent Superposition Of Outputs T34 : Category Three Of Coherent Superposition Of Outputs
ACCENTUATION COEFFCIENTS OF THE HOLISTIC SYSTEM
Von Neumann Entropy And Quantum Entanglement
Velocity Field Of The Particle And Wave Function
Matter Presence In Abundance And Break Down Of Parity Conservation Dissipation In Quantum Computation And Efficiency Of Quantum Algorithms
Decoherence And Computational Complexity
Coherent Superposition Of Outputs And Different Possible Inputs In The Form Of Qubits
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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1981 | Page
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 , π32
6 , π33 6 , π34
6 , π32 6 , π33
6 , π34 6
DISSIPATION COEFFCIENTS:
π13β² 1 , π14
β² 1 , π15β²
1 , π13
β² 1 , π14β² 1 , π15
β² 1
, π16β² 2 , π17
β² 2 , a18β² 2 , π16
β² 2 , π17β² 2 , π18
β² 2
, π20β² 3 , π21
β² 3 , π22β² 3 , π20
β² 3 , π21β² 3 , π22
β² 3
π24β² 4 , π25
β² 4
, π26β² 4 , π24
β² 4 , π25β²
4 , π26
β² 4 , π28β² 5 , π29
β² 5 , π30β² 5 π28
β² 5 , π29β² 5 , π30
β² 5 ,
π32β² 6 , π33
β² 6 , π34β² 6 , π32
β² 6 , π33β² 6 , π34
β² 6
- GOVERNING EQUATIONS:OF THE SYSTEM VONNEUMANN ENTROPY AND QUANTUM
ENTANGLEMENT
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 -
GOVERNING EQUATIONS OF THE SYSTEM VELOCITY FIELD OF THE PARTICLE AND WAVE
FUNCTION:
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 - dT16
dt= b16
2 T17 β b16β² 2 β π16
β²β² 2 πΊ19 , π‘ π16 -
dT17
dt= b17
2 T16 β b17β² 2 β π17
β²β² 2 πΊ19 , π‘ T17 - dT18
dt= b18
2 T17 β b18β² 2 β b18
β²β² 2 G19 , t T18 -
+ a16β²β² 2 T17 , t = First augmentation factor -
β b16β²β² 2 G19 , t = First detritions factor -
GOVERNING EQUATIONS:OF BREAK DOWN OF PARITY CONSERVATION AND MATTER ABUNDANCE:
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 -
GOVERNING EQUATIONS:OF DISSIPATION IN QUANTUM COMPUTATION AND EFFICIENCY OF
QUANTUM ALGORITHMS:
The differential system of this model is now - ππΊ24
ππ‘= π24
4 πΊ25 β π24β² 4 + π24
β²β² 4 π25 , π‘ πΊ24 -
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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1982 | Page
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 -
GOVERNING EQUATIONS:OF THE SYSTEM DECOHERENCE AND COMPUTATIONAL COMPLEXITY:
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 - 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 -
GOVERNING EQUATIONS:COHERENT SUPERPOSITION OF OUTPUTS AND DIFFERENT POSSIBILITIES
OF QUBIT INPUTS
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 -
CONCATENATED GOVERNING SYSTEMS OF THE HOLISTIC GLOBAL SYSTEM:
(1) Von Neumann Entropy And Quantum Entanglement
(2) Velocity Field Of The Particle And Wave Function
(3) Matter Presence In Abundance And Break Down Of Parity Conservation
(4) Dissipation In Quantum Computation And Efficiency Of Quantum Algorithms
(5) Decoherence And Computational Complexity
Coherent Superposition Of Outputs And Different Possible Inputs In The Form Of Qubits-
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 -
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
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International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.4, July-Aug. 2012 pp-1977-2016 ISSN: 2249-6645
www.ijmer.com 1983 | Page
+ 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 -
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 -
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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 π16β²β² 2 πΊ19 , π‘ , β π17
β²β² 2 πΊ19 , π‘ , β π18β²β² 2 πΊ19 , π‘ are first detrition coefficients for category 1, 2 and 3
β b13β²β² 1,1, G, t , 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 -
+ 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
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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 -
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
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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
+ 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
+ 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 -
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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‘ -
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-
ππ 2 , ππ
β² 2
, ππβ²β²
2 , ππ
2 , ππβ²
2 , ππ
β²β² 2
> 0, π, π = 16,17,18-
The functions ππβ²β²
2 , ππ
β²β² 2
are positive continuous increasing and bounded.-
Definition of (pi) 2 , (ri)
2 :-
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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) -
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) :-
(F) ( 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
( π 16 )(2) [ (ππ) 2 + (ππ
β² ) 2 + ( π΄ 16 )(2) + ( π 16 )(2) ( π 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-
(G) 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 first
augmentation coefficient would be absolutely continuous. -
Definition of ( π 20 )(3), ( π 20 )(3) :
(H) ( 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, 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 -
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Where we suppose-
(I) ππ 4 , ππ
β² 4
, ππβ²β²
4 , ππ
4 , ππβ²
4 , ππ
β²β² 4
> 0, π, π = 24,25,26
(J) 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)-
(K) 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) :
(L) ( π 24 )(4), ( π 24 )(4), are positive constants (ππ)
4
( π 24 )(4) ,(ππ) 4
( π 24 )(4) < 1 -
Definition of ( π 24 )(4), ( π 24 )(4) :
(M) 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
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-
(N) ππ 5 , ππ
β² 5 , ππβ²β² 5 , ππ
5 , ππβ² 5 , ππ
β²β² 5 > 0, π, π = 28,29,30
(O) 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)-
(P) 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‘ , the FIFTH augmentation
coefficient would be absolutely continuous. -
Definition of ( π 28 )(5), ( π 28 )(5) :
(Q) ( 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) :
(R) 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
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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-
ππ 6 , ππ
β² 6 , ππβ²β² 6 , ππ
6 , ππβ² 6 , ππ
β²β² 6 > 0, π, π = 32,33,34
(S) 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)-
(T) 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‘
. π33β² , π‘ and π33 , π‘ are points belonging to the interval ( π 32 )(6), ( π 32 )(6) . It is to be noted that (ππ
β²β² ) 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 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
ππ(π‘) β€ ( π 13 )(1)π( π 13 )(1)π‘ , ππ 0 = ππ0 > 0 -
if the conditions above are fulfilled, there exists a solution satisfying the conditions
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-
if the conditions above are fulfilled, there exists a solution satisfying the conditions
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-
if the conditions above are fulfilled, there exists a solution satisfying the conditions
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 -
if the conditions above are fulfilled, there exists a solution satisfying the conditions
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 -
if the conditions above are fulfilled, there exists a solution satisfying the conditions
Definition of πΊπ 0 , ππ 0 :
πΊπ π‘ β€ π 32 6
π π 32 6 π‘ , πΊπ 0 = πΊπ0 > 0
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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‘ -
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‘ -
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‘ -
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‘ -
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 -
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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‘ -
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
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-
The operator π(2) maps the space of functions satisfying GLOBAL EQATIONS 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‘ πΊ17
0 +(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 =
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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 -
(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 -
(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 1
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 =
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 +
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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
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 GLOBAL EQUATIONS into itself-
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 CONCATENATED SYTEM OF GLOBAL EQUATIONS it results
Gπ t β₯ Gπ0e
β (ππβ² ) 2 β(ππ
β²β² ) 2 T17 π 16 ,π 16 dπ 16 t
0 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
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 satisfying GLOBAL EQUATIONS 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
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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 (34,35,36) 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 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 (ππβ² ) 3 β(ππ
β²β² ) 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-
(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 GLOBAL EQUATIONS into
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
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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
From THE CONCATENATED SYTEM OF GLOBAL EQUATIONS 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-
(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 satisfying GLOBAL EQUATIONS into itself-
The operator π(5) is a contraction with respect to the metric
π πΊ31 1 , π31
1 , πΊ31 2 , π31
2 =
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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
From 19 to 28 it results
πΊπ π‘ β₯ πΊπ0π
β (ππβ² ) 5 β(ππ
β²β² ) 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-
(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) -
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In order that the operator π(6) transforms the space of sextuples of functions πΊπ ,ππ satisfying GLOBAL EQUATIONS 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
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 of the system-
Behavior of the solutions
Theorem 2: 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
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 :
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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 :-
(b) 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 of CONCATENATED GLOBAL EQUATIONS 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 by equation 25
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 πΊ13
0
(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€
(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 :
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(π16
β² ) 2 + (π17β² ) 2 β (π16
β²β² ) 2 πΊ19 , π‘ β (π17β²β² ) 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-
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 :--
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 ,
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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
(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 of CONCATENATED GLOBAL EQUATIONS 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 G16
0
(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
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-
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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 πΊ20
0
(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
(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
Then the solution of CONCATENATED GLOBAL EQUATIONS satisfies the inequalities
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)4πΊ240(π2)4(π1)4β(π26β²)4π(π1)4π‘βπβ(π26β²)4π‘+ πΊ260πβ(π26β²)4π‘
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 π240
(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‘
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Definition of (π1) 4 , (π2) 4 , (π 1) 4 , (π 2) 4 :-
(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 and (π 2) 4 = (π26
β² ) 4 β (π26 ) 4
Behavior of the solutions Theorem 2: 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
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
Then the solution of CONCATENATED SYSTEM OF GLOBAL EQUATIONS 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)5πΊ280(π2)5(π1)5β(π30β²)5π(π1)5π‘βπβ(π30β²)5π‘+ πΊ300πβ(π30β²)5π‘
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 π280
(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
Behavior of the solutions Theorem 2: 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(π32
β² ) 6 + (π33β² ) 6 β (π32
β²β² ) 6 πΊ35 , π‘ β (π33β²β² ) 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
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(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 of CONCATENATED SYSTEM OF GLOBAL EQUATIONS 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‘
(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)6πΊ320(π2)6(π1)6β(π34β²)6π(π1)6π‘βπβ(π34β²)6π‘+ πΊ340πβ(π34β²)6π‘
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 π320
(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
From which we deduce (π0 ) 1 β€ π 1 (π‘) β€ (π 1 ) 1
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(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 CONCATENATED 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 .
From CONCATENATED SYSTEM OF GLOBAL EQUATIONS 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
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‘
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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‘
Now, using this result and replacing it in CONCATENATED SYSTEM OF GLOBAL EQUATIONS we get easily the
result stated in the theorem.
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 consequence of the
relation between (π1) 2 and (π 1 ) 2
: From CONCATENATED 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‘ :-
(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
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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,
(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
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
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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 CONCATENATED GOVERNING EQUATIONS OF THE GLOBAL
SYSTEM 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 .
From GLOBAL EQUATIONS we obtain 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‘ :-
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(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‘
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 (with the notations 25,26,27,28)
(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 are satisfied , 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 by equation 25 are satisfied , then the system
If (ππβ²β² ) 4 πππ (ππ
β²β² ) 4 are independent on π‘ , and the conditions WE CAN UNMISTAKABLY PROVE THAT:
(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
(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 are 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
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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
has a unique positive solution , which is an equilibrium solution for THE SYSTEM
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 for THE HOLISTIC SYSTEM
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 HOLISTIC 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 (HOLISTIC 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
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 (GLOBAL)
Proof:
(a) Indeed the first two equations have a nontrivial solution πΊ13 ,πΊ14 if πΉ π =(π13
β² ) 1 (π14β² ) 1 β π13
1 π14 1 + (π13
β² ) 1 (π14β²β² ) 1 π14 + (π14
β² ) 1 (π13β²β² ) 1 π14 + (π13
β²β² ) 1 π14 (π14β²β² ) 1 π14 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ16 , πΊ17 if F π19 =(π16
β² ) 2 (π17β² ) 2 β π16
2 π17 2 + (π16
β² ) 2 (π17β²β² ) 2 π17 + (π17
β² ) 2 (π16β²β² ) 2 π17 + (π16
β²β² ) 2 π17 (π17β²β² ) 2 π17 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ20 ,πΊ21 if πΉ π23 =(π20
β² ) 3 (π21β² ) 3 β π20
3 π21 3 + (π20
β² ) 3 (π21β²β² ) 3 π21 + (π21
β² ) 3 (π20β²β² ) 3 π21 + (π20
β²β² ) 3 π21 (π21β²β² ) 3 π21 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ24 ,πΊ25 if
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πΉ π27 =(π24
β² ) 4 (π25β² ) 4 β π24
4 π25 4 + (π24
β² ) 4 (π25β²β² ) 4 π25 + (π25
β² ) 4 (π24β²β² ) 4 π25 + (π24
β²β² ) 4 π25 (π25β²β² ) 4 π25 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ28 ,πΊ29 if πΉ π31 =(π28
β² ) 5 (π29β² ) 5 β π28
5 π29 5 + (π28
β² ) 5 (π29β²β² ) 5 π29 + (π29
β² ) 5 (π28β² β² ) 5 π29 + (π28
β²β² ) 5 π29 (π29β²β² ) 5 π29 = 0
(a) Indeed the first two equations have a nontrivial solution πΊ32 ,πΊ33 if πΉ π35 =(π32
β² ) 6 (π33β² ) 6 β π32
6 π33 6 + (π32
β² ) 6 (π33β²β² ) 6 π33 + (π33
β² ) 6 (π32β²β² ) 6 π33 + (π32
β²β² ) 6 π33 (π33β²β² ) 6 π33 = 0
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(SOLUTIONALOF THE GLOBAL EQUATIONS) 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 It is easy to see that Ο is a 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 equations 92,93 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 SOLUTIONAL SYSTEM OF THE GLOBAL EQUATIONS 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
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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 SOLUTIONAL SYSTEM OF THE GLOBAL EQUATIONS 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 SOLUTIONAL SYSTEM OF THE GLOBAL EQUATIONS 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
Finally we obtain the unique solution of THE HOLISTIC SYSTEM
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 of THE GLOBAL SYSTEM OF GOVERNING
EQUATIONS
Finally we obtain the unique solution of THE HOLISTIC SYSTEM
G17β given by Ο πΊ19
β = 0 , T17β given by π T17
β = 0 and
πΊ16β =
π16 2 πΊ17β
(π16β² ) 2 +(π16
β²β² ) 2 π17β
, πΊ18β =
π18 2 πΊ17β
(π18β² ) 2 +(π18
β²β² ) 2 π17β
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 of THE HOLISTIC SYSTEM
Finally we obtain the unique solution of SOLUTIONAL EQUATIONS OF THE GLOBAL SYSTEM
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 of THE GLOBAL GOVERNING EQUATIONS
Finally we obtain the unique solution of SOLUTIONS FOR THE GLOBAL GOVERNING EQUATIONS
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 of GLOBAL GOVERNING EQUATIONS
Finally we obtain the unique solution of THE HOLISTIC SYSTEM
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 of THE HOLISTIC SYSTEM
Finally we obtain the unique solution of SLOUTIONAL EQUATIONS OF THE CONCATENATED EQUATIONS
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 of the GLOBAL SYSTEM
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.
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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 OF SOLUTIONALEQUATIONS OF THE GLOBAL SYSTEM 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(π17
β²β² ) 2
βT17 T17
β = π17 2 ,
β(ππβ²β² ) 2
βGπ πΊ19
β = π ππ
dπΎ16
dt= β (π16
β² ) 2 + π16 2 πΎ16 + π16
2 πΎ17 β π16 2 G16
β π17
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 stable
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
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
ππΎ24
ππ‘= β (π24
β² ) 4 + π24 4 πΎ24 + π24
4 πΎ25 β π24 4 πΊ24
β π25
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
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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
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 stabl
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
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
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
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+ 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
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, 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, 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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[7] R Wood βThe rate of loss of cloud droplets by coalescence in warm cloudsβ J.Geophys. Res., 111, doi:
10.1029/2006JD007553, 2006
[8] H. Rund, βThe Differential Geometry of Finsler Spacesβ, Grund. Math. Wiss. Springer-Verlag, Berlin, 1959
[9] A. Dold, βLectures on Algebraic Topologyβ, 1972, Springer-Verlag
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American Mathematical society, Providence , Rhode island 1976
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
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