Clear Event Space and Conservative Fields, Space Time ... · Clear Event Space and Conservative...

International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 1

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Clear Event Space and Conservative

Fields, Space Time Transposition,

Energy Mass Equivalence, and

Buckshot effect: Predicational

Interiorities and Ontological

Phenomenology thereof

Sir Allan Zade* and Dr. K.N.Prasanna Kumar**

*Author of famous Z-Theory

**Post-doctoral Scholar, Department of mathematics, Kuvempu University, India

Abstract: Systems of Z-Theory and Quantum Field Theory are investigated. It is shown that

the time independence of the contributions portrays another system by itself and constitutes

the equilibrium solution of the original time independent system. Further papers extensively

draw inferences upon such concatenation process, ipso facto fait accompli. One work that

relates Structural stability, electronic properties, and quantum conductivity of small-diameter

silicon nanowires is that of Inna Ponomareva, Madhu Menon, Ernst Richter, and Antonis N.

Andriotis, wherein they study structures and energetics of various types of silicon nanowires

have been investigated using quantum molecular dynamics simulations to determine the most

stable forms. The tetrahedral type nanowires oriented in the ⟨111⟩ direction are found to be the

most stable. The stability of the cage like nanowires is determined to lie somewhere between

this and tetrahedral nanowires oriented in other directions. Furthermore, their electrical

conducting properties are found to be better than those of tetrahedral nanowires, suggesting

useful molecular electronic applications. quantum dots which we shall later concatenate and

with Z-Theory is Stability of quantum dots in live cells by Zheng-Jiang Zhu, etal. Quantum

dots are highly fluorescent and photostable, making them excellent tools for imaging. When

using these quantum dots in cells and animals, however, intracellular biothiols (such as

glutathione and cysteine) can degrade the quantum dot monolayer, compromising function.

Here, we describe a label-free method to quantify the intracellular stability of monolayers on

quantum dot surfaces that couples laser desorption/ionization mass spectrometry with

inductively coupled plasma mass spectrometry.

INTRODUCTION :

We have to acknowledge that the Introduction is prepared by Google Search results and

Wikipedia, for which we humbly pay homage.

Following systems are taken in to consideration towards the end of consummation and

consolidation and consubstantiation of a holistic Model. we study the stability analysis,

Solution behaviour and asymptotic behaviour of the system

(1) Intrusion Area and Maximal Transportation Time

(2) Hidden Event Space and Passing Systems

(3) Half Time Decay and Positive Time Shift Effect


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(4) E is less than mc^2

(5) Space Time Transposition and Z Phenomenon

(6) Path integral formalism and canonical quantization

(7) Cross Shield Interaction and Space Time Transposition

(8) (1)

In the equation (1) variable have the following meaning: Vr is relative speed between a

moving observer and a beam of light; Vc is the speed of light relatively to the space; Vo is

the speed of the observer relatively to the space

(9) (1)

In the equation (1) variable have the following meaning: Vr is relative speed between a moving

observer and a beam of light; Vc is the speed of light relatively to the space; Vo is the speed of

the observer relatively to the space.

—VARIABLES USED

NOTATION

Module One

Intrusion Area and Maximal Transportation Time

: Category one of Intrusion Area

: Category two of Intrusion Area

: Category three of Intrusion Area

: Category one of Maximal Transportation Time

: Category two of Maximal Transportation Time

: Category three of Maximal Transportation Time

Module Two

Hidden Event Space and Passing Systems

: Category one of Hidden Event Space

: Category two of Hidden Event Space

: Category three of Hidden Event Space

: Category one of Passing Systems

: Category two of Passing Systems

: Category three of Passing Systems

Module three


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Half Time Decay and Positive Time Shift Effect

: Category one of Half Time Decay

: Category two of Half Time Decay

: Category three of Half Time Decay

: Category one of Positive Time Shift Effect

: Category two of Positive Time Shift Effect

: Category three of Positive Time Shift Effect

Module four

E is less than mc^2

: Category one of mc^2: here we are talking of the total energy of the universe. And the

systems to which Einstein’s Theory could be applied Classification is based on systemic

properties

: Category two of mc^2: here we are talking of the total energy of the universe. And the

systems to which Einstein’s Theory could be applied Classification is based on systemic

properties

: Category three of Clear Event Space mc^2: here we are talking of the total energy of

the universe. And the systems to which Einstein’s Theory could be applied Classification is

based on systemic properties

: Category one of Energy. Total energy in the universe. Stratification is based upon the

characteristics’ of the systems to which Mass energy equivalence can be applied.

: Category two of Energy. Total energy in the universe. Stratification is based upon the

characteristics’ of the systems to which Mass energy equivalence can be applied Note that

there is no sancrosanctness or sacrilege associated with the classification methodology.

Parametricization and concomitant characteristics’ which are higher parameters of the systems

are taken in to consideration.

: Category three of Energy. Total energy in the universe. Stratification is based upon the

characteristics’ of the systems to which Mass energy equivalence can be applied Note that

there is no sancrosanctness or sacrilege associated with the classification methodology.

Parametricization and concomitant characteristics’ which are higher parameters of the systems

are taken in to consideration.

Module five

Space Time Transposition and Z Phenomenon

: Category one of Space Time Transposition


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: Category two of Space Time Transposition

: Category three of Space Time Transposition

: Category one of Z Phenomenon

: Category two of Z Phenomenon

: Category three of Z Phenomenon

Module six

Surge effect and Buckshot effect

: Category one of Quantum Phase Transition

: Category two of Quantum Phase Transition

: Category three of Quantum Phase Transition

: Category one of Buckshot effect

: Category two of Buckshot effect

: Category three of Buckshot effect

Module seven

Cross Shield Interaction and Space Time Transposition

: Category one of Cross Shield Interaction

: Category two of Cross Shield Interaction

: Category three of Cross Shield Interaction

: Category one of Space Time Transposition

: Category two of Space Time Transposition

: Category three of Space Time Transposition

Module eight

(1) (1)




: Category one of lhs of the equation (10( sytemal classification based on the

characteristics’ of the systems to which the equation becomes applicable)

: Category two of lhs of the equation (10( sytemal classification based on the


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: Category three of lhs of the equation (10( sytemal classification based on the


: Category one of first term on RHS

: Category two of first term on RHS

: Category three of first term on RHS

Module Nine

(1) (1)




: Category one of Term on LHS of the equation (1)

: Category two of Term on LHS of the equation (1)

: Category three of Term on LHS of the equation (1)

: Category one of second term on the RHS

: Category two of second term on the RHS

: Category three of second term on the RHS

In perturbative quantum field theory, the forces between particles are mediated by other particles. The

electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector

bosons mediate the weak force and gluons mediate the strong force. There is currently no complete

quantum theory of the remaining fundamental force, gravity, but many of the proposed theories postulate

the existence of a graviton particle that mediates it. These force-carrying particles are virtual particles

and, by definition, cannot be detected while carrying the force, because such detection will imply that the

force is not being carried. In addition, the notion of "force mediating particle" comes from perturbation

theory, and thus does not make sense in a context of bound states.

In QFT, photons are not thought of as "little billiard balls" but are rather viewed as field quanta –

necessarily chunked ripples in a field, or "excitations", that "look like" particles. Fermions, like the

electron, can also be described as ripples/excitations in a field, where each kind of fermion has its own

field. In summary, the classical visualization of "everything is particles and fields", in quantum field

theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end,

particles are regarded as excited states of a field (field quanta). The gravitational field and the


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electromagnetic field are the only two fundamental fields in Nature that have infinite range and a

corresponding classical low-energy limit, which greatly diminishes and hides their "particle-like"

excitations. Albert Einstein, in 1905, attributed "particle-like" and discrete exchanges of momenta and

energy, characteristic of "field quanta", to the electromagnetic field. Originally, his principal motivation

was to explain the thermodynamics of radiation. Although it is often claimed that the photoelectric and

Compton effects require a quantum description of the EM field, this is now understood to be untrue, and

proper proof of the quantum nature of radiation is now taken up into modern quantum optics as in the

antibunching effect. The word "photon" was coined in 1926 by physical chemist Gilbert Newton Lewis

(see also the articles photon antibunching and laser).

Prigogine developed the concept of “dissipative structures” to describe the coherent space-time structures

that form in open systems in which an exchange of matter and energy occurs between a system and its

environment. Ilya Prigogine received the Nobel Prize in Chemistry in 1977 for “his contributions to

nonequilibrium thermodynamics, particularly the theories of dissipative structures.” Prigogine’s primary

interest was in nonequilibrium irreversible phenomena because in these systems the arrow of time

becomes manifest. Prigogine viewed the arrow of time and irreversibility as playing a constructive role

in nature. For him the arrow of time was essential to the existence of biological systems, which contain

highly organized irreversible structures. Prigogine’s first major work on irreversible systems was his

theorem of minimum entropy production which was applicable to nonequilibrium stationary states near

equilibrium. Prigogine next began to work on far-from-equilibrium irreversible phenomena, both in

hydrodynamic systems and chemical systems. Such systems, because of nonlinear interactions, can form

spatial and temporal structures (dissipative structures) that can exist as long as the system is held far

from equilibrium due to a continual flow of energy or matter through the system.

Irreversible systems have an arrow of time which appears to be incompatible with Newtonian and

quantum dynamics, which are reversible theories. This incompatibility of the reversible foundations of

science with the irreversible behavior that is actually observed in chemical, hydrodynamic, and

biological systems remains one of the great mysteries of science. What is the origin of the arrow of time?

Is it a fundamental property of nature, or is it only an illusion? Prigogine’s view was that it must be a

fundamental property of nature. In the 1950s, he began to work on the foundations of statistical

mechanics and the question of how to reconcile Newtonian mechanics with an irreversible world. The

Newtonian foundations of equilibrium statistical mechanics require that the Newtonian dynamics be

chaotic. Prigogine’s early work at UT was focused on the problem of dissipative structures, but in later

years he became more and more involved with the problem of reconciling the arrow of time with

Newtonian and quantum dynamics. While working with non-equilibrium chemical systems, it was a

natural step to extend concepts found in these systems to complex social and economic systems.

Prigogine is considered one of the founders of complexity science.

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ):

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ),

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )


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are Accentuation coefficients

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) , (

)( ) ( )( ) (

)( ) ( )( ) (

)( ) ( )( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) (

)( )

are Dissipation coefficients Module Numbered One The differential system of this model is now (Module Numbered one)

( )

( ) [( )( ) (

)( )( )] 1

( )

( ) [( )( ) (

)( )( )] 2

( )

( ) [( )( ) (

)( )( )] 3

( )

( ) [( )( ) (

)( )( )] 4

( )

( ) [( )( ) (

)( )( )] 5

( )

( ) [( )( ) (

)( )( )] 6

( )( )( ) First augmentation factor

( )( )( ) First detritions factor

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

( )

( ) [( )( ) (

)( )( )] 7

( )

( ) [( )( ) (

)( )( )] 8

( )

( ) [( )( ) (

)( )( )] 9

( )

( ) [( )( ) (

)( )(( ) )] 10

( )

( ) [( )( ) (

)( )(( ) )] 11

( )

( ) [( )( ) (

)( )(( ) )] 12


( )( )(( ) ) First detritions factor

Module Numbered Three

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


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( )

( ) [( )( ) (

)( )( )] 13

( )

( ) [( )( ) (

)( )( )] 14

( )

( ) [( )( ) (

)( )( )] 15

( )

( ) [( )( ) (

)( )( )] 16

( )

( ) [( )( ) (

)( )( )] 17

( )

( ) [( )( ) (

)( )( )] 18


( )( )( ) First detritions factor

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

( )

( ) [( )( ) (

)( )( )] 19

( )

( ) [( )( ) (

)( )( )] 20

( )

( ) [( )( ) (

)( )( )] 21

( )

( ) [( )( ) (

)( )(( ) )] 22

( )

( ) [( )( ) (

)( )(( ) )] 23

( )

( ) [( )( ) (

)( )(( ) )] 24



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

( )

( ) [( )( ) (

)( )( )] 25

( )

( ) [( )( ) (

)( )( )] 26

( )

( ) [( )( ) (

)( )( )] 27

( )

( ) [( )( ) (

)( )(( ) )] 28

( )

( ) [( )( ) (

)( )(( ) )] 29

( )

( ) [( )( ) (

)( )(( ) )] 30



Module Numbered Six


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

( )

( ) [( )( ) (

)( )( )] 31

( )

( ) [( )( ) (

)( )( )] 32

( )

( ) [( )( ) (

)( )( )] 33

( )

( ) [( )( ) (

)( )(( ) )] 34

( )

( ) [( )( ) (

)( )(( ) )] 35

( )

( ) [( )( ) (

)( )(( ) )] 36


Module Numbered Seven: The differential system of this model is now (SEVENTH MODULE)

( )

( ) [( )( ) (

)( )( )] 37

( )

( ) [( )( ) (

)( )( )] 38

( )

( ) [( )( ) (

)( )( )] 39

( )

( ) [( )( ) (

)( )(( ) )] 40

( )

( ) [( )( ) (

)( )(( ) )] 41

( )

( ) [( )( ) (

)( )(( ) )] 42


Module Numbered Eight

GOVERNING EQUATIONS:

The differential system of this model is now

( )

( ) [( )( ) (

)( )( )] 43

( )

( ) [( )( ) (

)( )( )] 44

( )

( ) [( )( ) (

)( )( )] 45

( )

( ) [( )( ) (

)( )(( ) )] 46

( )

( ) [( )( ) (

)( )(( ) )] 47

( )

( ) [( )( ) (

)( )(( ) )] 48

Module Numbered Nine

GOVERNING EQUATIONS:

The differential system of this model is now

( )

( ) [( )( ) (

)( )( )] 49


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( )

( ) [( )( ) (

)( )( )]

50

( )

( ) [( )( ) (

)( )( )]

51

( )

( ) [( )( ) (

)( )(( ) )]

52

( )

( ) [( )( ) (

)( )(( ) )]

53

( )

( ) [( )( ) (

)( )(( ) )]

54


( )( )(( ) ) First detrition factor

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

55

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

56

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

57

Where ( )( )( ) (

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

category 1, 2 and 3

( )( )( ) , (

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

for category 1, 2 and 3

( )( )( ) (

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


( )( )( ) , (

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

coefficient for category 1, 2 and 3

( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) (

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

1,2,3

( )( )( ) (

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

1,2,3

( )( )( ) , (

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

augmentation coefficient for 1,2,3


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( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

58

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

59

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

60

Where ( )( )( ) (

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

category 1, 2 and 3

( )( )( ) (

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

category 1, 2 and 3

( )( )( ) (

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

category 1, 2 and 3

( )( )( ) (

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

coefficients for category 1, 2 and 3

( )( )( ) , (

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


( )( )( ) , (

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


– ( )( )( ) – (

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

category 1, 2 and 3

– ( )( )( ) – (

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

category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth

detrition coefficients for category 1, 2 and 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

61

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

62

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

63

Where ( )( )( ) (

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



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( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) (

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


( )( )( ) , (

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

category 1, 2 and 3

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

64

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

65

( )

( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

66

( )( )( ) , (

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

category 1, 2 and 3

( )( )( ) (

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

category 1,2 and 3

( )( )( ) (

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

for category 1,2 and 3

( )( )( ) (


coefficients for category 1,2 and 3

( )( )( ) , (

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


( )( )( ) (

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


– ( )( )( ) , – (

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

for category 1,2 and 3

– ( )( )( ) , – (

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


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category 1,2 and 3

– ( )( )( ) , – (

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

category 1,2 and 3

( )( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

67

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

68

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

69

( )( )( ) , (

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

category 1, 2 and 3

( )( )( ) (

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


( )( )( ) (

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


( )( )( ) , (

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

augmentation coefficients for category 1, 2 and 3

( )( )( ) (



( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


( )( )( ) , (

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


( )( )

[

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

70

71


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( )( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[

( )( ) (

)( )( ) – ( )( )( ) – (

)( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

72

( )( )( ) (


category 1, 2 and 3

( )( )( ) , (

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


( )( )( ) (

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

category 1,2 and 3

( )( )( ) (



( )( )( ) (

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


( )( )( ) (

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


– ( )( )( ) , – (

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


– ( )( )( ) , – (

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


– ( )( )( ) , – (

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

category 1, 2 and 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

73

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

74

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

75

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )


ISSN 2250-3153

www.ijsrp.org

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) , (

)( )( ) ( )( )( )

( )( )( ) , (

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

( )( )( ) , (

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

( )( )( ) , (

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

( )( )( ) , (

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

for category 1 2 3

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

76

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

77

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

78

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )( )( ) , (

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

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) , – (

)( )( ) , – ( )( )( )

– ( )( )( ) , – (

)( )( ) – ( )( )( )


ISSN 2250-3153

www.ijsrp.org

– ( )( )( ) , – (

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

for category 1 2 3

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

79

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

80

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

81

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

coefficients for category 1,2, and 3

( )( )( ) (


coefficients for category 1,2,and 3

( )( )( ) (

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

coefficients for category 1,2, 3

( )( )( ) , (

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

augmentation coefficients for category 1,2, 3

( )( )( ) , (

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


( )( )( ) , (

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


( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

82

( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

83


ISSN 2250-3153

www.ijsrp.org

( )

( )

[

(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

84

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

for category 1,2, and 3

( )( )( ) (



– ( )( )( ) , – (

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


– ( )( )( ) , – (

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


– ( )( )( ) , – (

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


– ( )( )( ) , – (

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


( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

85

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

86

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

87


ISSN 2250-3153

www.ijsrp.org

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

augmentation coefficients

( )( )( ) (

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

coefficients

( )( )( ) , (

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

coefficients

( )( )( ) , (

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

( )( )( ) , (

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

Eighth augmentation coefficients

( )( )( ) , (

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

coefficients

( )( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

88

( )( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

89

( )( )

[ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

90

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (



ISSN 2250-3153

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coefficients for category 1, 2, and 3

( )( )( ) , (



– ( )( )( ) , – (

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


– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are seventh

detrition coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( )

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

– ( )( )( ) , – (



( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

91

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

92

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

93

Where ( )( )( ) (

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

category 1, 2 and 3

( )( )( ) , (

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

augmentation coefficient for category 1, 2 and 3

( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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



ISSN 2250-3153

www.ijsrp.org

( )( )( ) , (

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

are eighth augmentation coefficient for 1,2,3

( )( )( ) , (

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


( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

94

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (


category 1, 2 and 3

( )( )( ) (

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


( )( )( ) (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


– ( )( )( ) , – (

)( )( ) , – ( )( )( )

are seventh detrition coefficients for category 1, 2 and 3

– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth


– ( )( )( ) , – (




ISSN 2250-3153

www.ijsrp.org

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

95

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )

( )

[ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

Where ( )( )( ) (



( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]


ISSN 2250-3153

www.ijsrp.org

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (


category 1, 2 and 3

( )( )( ) (

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


( )( )( ) (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


– ( )( )( ) – (



– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are eighth


– ( )( )( ) , – (



( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

96

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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


ISSN 2250-3153

www.ijsrp.org

( )

( )

[

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

Where ( )( )( ) (



( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )

[

( )( ) (

)( )( ) ( )( )( ) – (

)( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

( )( )

[ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( )

– ( )( )( ) – (

)( )( ) – ( )( )( ) ]

Where ( )( )( ) (



ISSN 2250-3153

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category 1, 2 and 3

( )( )( ) (

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


( )( )( ) (

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


( )( )( ) (

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


( )( )( ) , (

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


( )( )( ) , (

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


– ( )( )( ) , – (



– ( )( )( ) – (

)( )( ) , – ( )( )( ) are eighth


– ( )( )( ) , – (

)( )( ) , – ( )( )( ) are ninth


Where we suppose

(A) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(B) The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

97

(C) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

98

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )( ) (

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

99

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

( )( )(

) and( )( )( ) (

) and ( ) are points belonging to the interval

[( )( ) ( )

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

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

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


( ) : 100


ISSN 2250-3153

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(D) ( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )


( ) :

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

( ) which together

with ( )( ) ( )

( ) ( )( ) and ( )

( ) and the constants

( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

101

Where we suppose

(F) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(G) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

102

( )( )( ) ( )

( ) ( )( ) ( )

( ) 103

(H) ( )( ) ( ) ( )

( ) 104

( )( ) (( ) ) ( )

( ) 105


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


106


( )( )(

) ( )( )( ) ( )

( ) ( )( ) 107

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( ) 108

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

)

and( )( )( ) . (

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

( )] . It

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

( )( ) then the function (

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


( ) :

(I) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

109


( ) :

There exists two constants ( )( ) and ( )

( ) which together

with ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )


ISSN 2250-3153

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satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 110

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) 111

Where we suppose

(J) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

112

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


113


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(

) ( )( )( ) ( )

( ) ( )( )

114


)

and( )( )( ) . (

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

( )] . It



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


( ) :

(K) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

115

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

( ) which together with

( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

116

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(L) The functions ( )( ) (


117


ISSN 2250-3153

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( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

(M) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


118


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

119


)

and( )( )( ) . (


( )] . It



)( )( ) , the first augmentation coefficient would be

absolutely continuous.


( ) :

( )( ) ( )


( )( )

( )( ) ( )

( )

( )( )

120


( ) :

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


( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

121

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(O) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

122


ISSN 2250-3153

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(P) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


123


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

124


)

and( )( )( ) . (


( )] . It






( ) :

( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

125


( ) :



( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

126

Where we suppose

( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(Q) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

127

(R) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


( ) :

128


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Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


)

and( )( )( ) . (


( )] . It






( ) :

( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

129


( ) :



( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

130

Where we suppose

(S) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(T) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

131

(U) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )

Definition of ( )

( ) ( )( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


and

132


( )( )(

) ( )( )( ) ( )

( ) ( )( )

133


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( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


)

and( )( )( ) . (


( )] . It



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


( ) :

(V) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

134


( ) :

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


( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

135

Where we suppose

A. ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

136

B. The functions ( )( ) (

)( ) are positive continuous increasing and bounded


( ):

137

( )( )( ) ( )

( ) ( )( )

138

( )( )(( ) ) ( )

( ) ( )( ) ( )

( ) 139

C. ( )( ) ( ) ( )

( )

140

( )( ) (( ) ) ( )

( ) 141


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

142

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )

143


)


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and ( )( )( ) . (


( )] . It



)( )( ) , the eighth augmentation coefficient , would be absolutely continuous.


( ) :

D. ( )( ) ( )


( )( )

( )( ) ( )

( )

( )( )

144


( ) : E. There exists two constants ( )

( ) and ( )( ) which together with

( )( ) ( )

( ) ( )( )

( )( ) and the constants ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

Satisfy the inequalities

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

145

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

146

Where we suppose

(X) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(Y) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

146A

(Z) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )


and


( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


)

and( )( )( ) . (


( )] . It



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


ISSN 2250-3153

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( ) :

(AA) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )


( ) : (BB) There exists two constants ( )

( ) and ( )( ) which together with

( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( )

( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

147

Theorem 2 : if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions

Definition of ( ) ( )

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

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

148

Theorem 3 : if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions

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

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

149

Theorem 4 : if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the

conditions


( ) ( )( )

( )( ) , ( )

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

150


conditions

151


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( ) ( )( )

( )( ) , ( )

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


conditions


( ) ( )( )

( )( ) , ( )

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

152

Theorem 7: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

153

Theorem 8: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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

153

A

Theorem 9: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the

conditions


( ) ( )( )

( )( ) , ( )

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

153

B

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

154

( ) ( )

( )

( ) ( )

( ) 155

( ) ( )

( ) ( )( ) 156

( ) ( )

( ) ( )( ) 157


ISSN 2250-3153

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By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

158

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

Proof:

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

159

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

160

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


Proof:

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

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By 161


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( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

162

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

163


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( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


Proof:

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

which satisfy

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

164

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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



( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )


ISSN 2250-3153

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By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

165

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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



( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

166

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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



166A


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( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

167

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 1

168

Analogous inequalities hold also for


( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( ) ( ( )

( ) )

( )( )( )( )

( )( ) ( ( )( ) )

169


ISSN 2250-3153

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( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

170



( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

171


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

172


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

173


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


174


( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


175


( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

176


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( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]



177

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

178


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

180


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]



181

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]




ISSN 2250-3153

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It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

182

( )( )

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

( ) )

(( )( )

)

] ( )( )

183

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

184

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

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

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

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

185

Indeed if we denote

Definition of : ( ) ( )( )

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses it follows

| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

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

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

186

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

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

( )( ) ( )( ) ( )

( ) ( )( ) respectively of

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

then it suffices to consider that ( )( ) (

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

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


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From 19 to 24 it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

Definition of (( )( ))

(( )

( )) (( )

( ))

:

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

( )( ) it follows

(( )

( )) (

)( ) and by integrating

(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

187

Remark 4: If bounded, from below, the same property holds for The proof is analogous with the preceding one. An analogous property is true if is bounded from below.

188

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

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

189

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is

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

)( )

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


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

190

( )( )

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

( ) )

(( )( )

)

] ( )( )

191

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

192


193

The operator ( ) is a contraction with respect to the metric 194


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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of : ( ) ( )( )

195

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

196



197

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))

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



( )( ) ( )( ) ( )




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

199


it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

200


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )

201


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202

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

)( ) then



( )( ) (

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

203

Then

( )


(( )( )( )( )

) ( )


it results

204

(( )( )( )( )

)


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

)( )


205

206


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

207

( )( )

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

( ) )

(( )( )

)

] ( )( )

208

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

209


210


((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

211

Indeed if we denote

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

212

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

213


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( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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



| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))


214



( )( ) ( )( ) ( )





215


it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

216


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


217


218


)( ) then



( )( ) (

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

219

Then

( )

( )( )( ) which leads to 220


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(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )



( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

221

( )( )

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

( ) )

(( )( )

)

] ( )( )

222

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

223

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

Equations into itself

224


((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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


From the hypotheses on Equations it follows

225

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))


226


ISSN 2250-3153

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)( ) depending also on can be considered

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

postulate condition necessary to prove the uniqueness of the solution bounded by

( )( ) ( )( ) ( )




)( ) depend only on and

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

227


it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

228


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


229

Remark 19: If bounded, from below, the same property holds for The proof is

analogous with the preceding one. An analogous property is true if is bounded from below.

230


)( ) then



( )( ) (

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

231

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )


37 to 42

232


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( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

233

( )( )

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

( ) )

(( )( )

)

] ( )( )

234

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

235




((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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


From the hypotheses on it follows

236

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))


237





( )( ) ( )( ) ( )



238


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it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

239


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


240



241


)( ) then



( )( ) (

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

242

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )



243


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

244


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( )( )

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

( ) )

(( )( )

)

] ( )( )

245

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

246




((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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



247

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))


248





( )( ) ( )( ) ( )






249


it results

250


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( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


251



252


)( ) then



( )( ) (

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

253

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


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

)( )


254



( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

255

( )( )

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

( ) )

(( )( )

)

] ( )( )

256

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

257


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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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


From the hypotheses on it follows

258

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))


259



( )( ) ( )( ) ( )

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



260

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

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

261


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )

( ))

( )

( )(( )( ))

(

)( )


262


ISSN 2250-3153

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(( )( ))

( )( )(( )

( )) (

)( )

If is bounded, the same property follows for and respectively. Remark 34: If bounded, from below, the same property holds for The proof is analogous with the preceding one. An analogous property is true if is bounded from below.

263


)( ) then

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

( )( ) (

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

264

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )


265


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

266

( )( )

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

( ) )

(( )( )

)

] ( )( )

267

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

268



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

269

Indeed if we denote

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

270

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

271


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( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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



272

|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))


273



( )( ) ( )( ) ( )




274

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

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

275


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )


276


277


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

( )( ) (

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

278


ISSN 2250-3153

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Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


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

)( )

279


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

279A

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

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


((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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


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

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

( )( ) (( )( ) (

)( ) ( )( )

( )( )( )

( )) ((( )( ) ( )

( ) ( )( ) ( )

( )))

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




ISSN 2250-3153

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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

( )( ) ( )( ) ( )





From 99 to 44 it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( ))

:


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )




)( ) then

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

( )( ) (

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

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)



)( )

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


ISSN 2250-3153

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Behavior of the solutions of equation

Theorem If we denote and define


( ) ( )( ) ( )

( ) :

(a) )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

280


( ) ( )( ) ( )

( ) ( ) ( ) :

(b) By ( )( ) ( )

( ) and respectively ( )( ) ( )

( ) the roots of the

equations ( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( )

( )( )

281


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( ) and ( )

( )( ( )) ( )

( ) ( ) ( )( )

282


( ) ( )( ) ( )

( ) ( )( ) :-

(c) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

283

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

are defined

284

Then the solution of global equations satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation

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

( )( ) ( )( )

285

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

286


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( )( ) ( )

(( )( ) ( )( )) 287

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

289


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

290

Behavior of the solutions of equation

Theorem 2: If we denote and define

291


( ) ( )( ) ( )

( ) :

(d) )( ) ( )

( ) ( )( ) ( )


292

( )( ) (

)( ) ( )( ) (

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

( ) 293

( )( ) (

)( ) ( )( ) (

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

( ) 294


( ) ( )( ) ( )

( ) : 295

By ( )( ) ( )


( ) the roots 296

(e) of the equations ( )( )( ( ))

( )

( ) ( ) ( )( ) 297

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and 298


( ) ( )( ) ( )

( ) : 299

By ( )( ) ( )


( ) the 300

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

( )

( ) ( ) ( )( ) 301

and ( )( )( ( ))

( )

( ) ( ) ( )( ) 302


( ) ( )( ) ( )

( ) :- 303

(f) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by 304

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 305

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

306

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 307


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and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

308

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) 309


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

( )( )

310

( )( ) is defined by equation

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

( )( ) ( )( ) 311

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

312

( )( ) ( )

(( )( ) ( )( )) 313

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

315


( ) ( )( ) ( )

( ):- 316

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

317

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

318

Behavior of the solutions

Theorem 3: If we denote and define


( ) ( )( ) ( )

( ) :

(a) )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

319


( ) ( )( ) ( )

( ) :

(b) By ( )( ) ( )




( )

( ) ( ) ( )( )

320


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and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) :-

(c) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

321

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) and ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )


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

( )( )

( )( ) is defined by equation

322

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

( )( ) ( )( ) 323

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

324

( )( ) ( )

(( )( ) ( )( )) 325

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

327


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

328


ISSN 2250-3153

www.ijsrp.org

( )( ) (

)( ) ( )( )

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

( ) ( )( ) ( )

( ) ( )( ) :

(d) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(e) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

329


( ) ( )( ) ( )

( ) : By ( )

( ) ( )( ) and respectively ( )

( ) ( )( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(f) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

330

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

331


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

( )( )


332

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

( )( ) ( )( )

333


ISSN 2250-3153

www.ijsrp.org

(( )( )

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

( )( ) ( )

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) )

334

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

335

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

336


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )

( ) ( )( )( )

( ) ( )( )

( )( ) (

)( ) ( )( )

337

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

( ) ( )( ) ( )

( ) ( )( ) :

(g) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

338


( ) ( )( ) ( )

( ) ( ) ( ) : (h) By ( )


( ) ( )( ) the roots of the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

339


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(i) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

340


ISSN 2250-3153

www.ijsrp.org

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

341


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

( )( )


342

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

( )( ) ( )( )

343

(( )( )

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

( )( ) ( )

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) )

344

( )( ) ( )

(( )( ) ( )( ))

345

( )( ) ( )( ) ( )

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

346

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

347


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )

( ) ( )( )( )

( ) ( )( )

( )( ) (

)( ) ( )( )

348


( ) ( )( ) ( )

( ) ( )( ) :

(j) ( )( ) ( )

( ) ( )( ) ( )


349


ISSN 2250-3153

www.ijsrp.org

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( ) Definition of ( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) :

(k) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

350


( ) ( )( ) ( )

( ) : By ( )


( ) ( )( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(l) If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

351

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

352


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

( )( )


353

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

( )( ) ( )( )

354

(( )( )

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

( )( ) ( )

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) )

355

( )( ) ( )

(( )( ) ( )( ))

356

( )( ) ( )( ) ( )

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

357


ISSN 2250-3153

www.ijsrp.org

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

358


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

359

360


( ) ( )( ) ( )

( ) ( )( ) :

( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( ) ( )

( ) ( )( ) (

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

)( )(( ) ) ( )( )


( ) ( )( ) ( )

( ) ( ) ( ) : By ( )


( ) ( )( ) the roots of the equations

( )( )( ( ))

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

361


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

If we define ( )( ) ( )

( ) ( )( ) ( )

( ) by ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

362

and analogously ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

363


ISSN 2250-3153

www.ijsrp.org

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )


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

( )( ) where ( )

( ) is defined by equation

364

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

( )( ) ( )( )

365

(( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

366

( )( ) ( )

(( )( ) ( )( ))

367

( )( ) ( )( ) ( )

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

368

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

369


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

370

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


( ) ( )( ) ( )

( ) : (m) ( )

( ) ( )( ) ( )

( ) ( )( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( ) ( )

( ) ( )( ) (

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

)( )(( ) ) ( )( )

371


( ) ( )( ) ( )

( ) ( ) ( ) :

(n) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

372


ISSN 2250-3153

www.ijsrp.org


( ) ( )( ) ( )

( ) : By ( )


( ) ( )( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(o) If we define ( )

( ) ( )( ) ( )

( ) ( )( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and analogously ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

374


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

( )( ) where ( )

( ) is defined by equation

375

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

( )( ) ( )( )

376

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

377

( )( ) ( )

(( )( ) ( )( ))

378

( )( ) ( )( ) ( )

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

379

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )

380


( ) ( )( ) ( )

( ):- 381


ISSN 2250-3153

www.ijsrp.org

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

Behavior of the solutions of equation 37 to 92 Theorem 2: If we denote and define Definition of ( )

( ) ( )( ) ( )

( ) ( )( ) :

(p) )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )

382


( ) ( )( ) ( )

( ) ( ) ( ) :

(q) By ( )( ) ( )




( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and


( ) ( )( ) ( )

( ) : By ( )


( ) ( )( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

(r) If we define ( )

( ) ( )( ) ( )

( ) ( )( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and ( )( )

( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined by 59 and 69 respectively

Then the solution of 99,20,44,22,23 and 44 satisfies the inequalities


ISSN 2250-3153

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

( )( )

where ( )( ) is defined by equation 45

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

( )( ) ( )( )

( ( )( )

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

( )( ) ( )

( )( )

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

( )( ) ( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )

( ) ( )( )( )

( ) ( )( )

( )( ) (

)( ) ( )( )

Proof : From global equations we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains


( ) :-

(a) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

383


ISSN 2250-3153

www.ijsrp.org

In the same manner , we get

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

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

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

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

( )( ) ( )( )

From which we deduce ( )( ) ( )( ) ( )

( )

384

(b) If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

385

(c) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

And so with the notation of the first part of condition (c) , we have

Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain


( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the theorem.

Particular case :

If ( )( ) (

)( ) ( )( ) ( )

( ) and in this case ( )( ) ( )

( ) if in addition

( )( ) ( )

( ) then ( )( ) ( )( ) and as a consequence ( ) ( )

( ) ( ) this also

defines ( )( ) for the special case

Analogously if ( )( ) (

)( ) ( )( ) ( )

( ) and then

( )( ) ( )

( )if in addition ( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important

consequence of the relation between ( )( ) and ( )

( ) and definition of ( )( )

386


( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

387


388


ISSN 2250-3153

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It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

389



( ) :-

(d) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

390


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

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

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

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

( )( ) ( )( )

391


( ) 392

(e) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

393

(f) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )


394


( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

395



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

396


Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( )

397


ISSN 2250-3153

www.ijsrp.org


)( ) ( )( ) ( )

( ) and then

( )( ) ( )




( )


( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

398


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

399


(a) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

400


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

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

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

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

( )( ) ( )( )



( )

401

(b) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

402

(c) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

403


ISSN 2250-3153

www.ijsrp.org



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( )


)( ) ( )( ) ( )

( ) and then

( )( ) ( )




( )


( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains Definition of ( )

( ) ( )( ) :-

(d) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )

( ) ( )( ) ( )( )

404


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

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

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

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

( )( ) ( )( )


( )

405

(e) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

406


ISSN 2250-3153

www.ijsrp.org

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

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

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

( )( )

(f) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the theorem. Particular case : If (

)( ) ( )( ) ( )

( ) ( )( ) and in this case ( )

( ) ( )( ) if in addition

( )( ) ( )


( ) ( ) this also

defines ( )( ) for the special case .

Analogously if (

)( ) ( )( ) ( )

( ) ( )( ) and then

( )( ) ( )





407


( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))


Definition of ( )

( ) ( )( ) :-

(g) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

408


ISSN 2250-3153

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( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )


( )

409

(h) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

410

(i) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


)( ) ( )( ) ( )



( )( ) ( )


( ) ( ) this also



)( ) ( )( ) ( )

( ) and then

( )( ) ( )





411


( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

412


ISSN 2250-3153

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(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(j) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )

From which we deduce ( )

( ) ( )( ) ( )( )

413

(k) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

414

(l) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


)( ) ( )( ) ( )



( )( ) ( )


( ) ( ) this also



)( ) ( )( ) ( )

( ) and then

415


ISSN 2250-3153

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( )( ) ( )





Proof : From global equations we obtain ( )

( )

( ) (( )( ) (

)( ) ( )( )( ))

( )( )( )

( ) ( )( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )( )( ( ))

( )

( ) ( ) ( )( ))



( ) :-

For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

416


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

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

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

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

( )( ) ( )( )

From which we deduce ( )

( ) ( )( ) ( )( )

417

If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

418

If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )

And so with the notation of the first part of condition (c) , we have Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

419


ISSN 2250-3153

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( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


)( ) ( )( ) ( )



( )( ) ( )


( ) ( ) this also



)( ) ( )( ) ( )

( ) and then ( )( ) ( )

( )if in addition

( )( ) ( )

( ) then ( ) ( )( ) ( ) This is an important consequence of the relation

between ( )( ) and ( )


420

Proof : From global equations we obtain ( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(m) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

421


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

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

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

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

( )( ) ( )( )


( )

422

(n) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

423


ISSN 2250-3153

www.ijsrp.org

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

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

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

( )( )

(o) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in global equations we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( ) this also


Analogously if (

)( ) ( )( ) ( )

( ) ( )( ) and then

( )( ) ( )





424

Proof : From 99,20,44,22,23,44 we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( ))

( )( )( )

( ) ( )( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )( )( ( ))

( )

( ) ( ) ( )( ))



( ) :-

(p) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

424A


ISSN 2250-3153

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( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )


( )

(q) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

(r) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

In a completely analogous way, we obtain Definition of ( )( ) :-

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )

Now, using this result and replacing it in 99, 20,44,22,23, and 44 we get easily the result stated in the theorem. Particular case :

If ( )( ) (

)( ) ( )( ) ( )


( ) if in addition

( )( ) ( )


( ) ( ) this also



)( ) ( )( ) ( )

( ) and then

( )( ) ( )





We can prove the following

Theorem : If ( )( ) (

)( ) are independent on , and the conditions with the notations

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

425


ISSN 2250-3153

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( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation are satisfied , then the system


)( ) are independent on , and the conditions with the notations 426

( )( )(

)( ) ( )( )( )

( ) 427

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) 428

( )( )(

)( ) ( )( )( )

( ) , 429

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


430



( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


431

We can prove the following



( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


432



( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


433

Theorem If ( )( ) (


( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

434


ISSN 2250-3153

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( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )




( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) (

)( )( )( ) ( )

( )( )( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )

( ) ( )( ) as defined by equation are satisfied , then the system

435



( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) (

)( )( )( ) ( )

( )( )( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


436


)( ) are independent on , and the conditions (with the notations

45,46,27,28)

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation 45 are satisfied , then the system

436A

( )( ) [(

)( ) ( )( )( )] 437

( )( ) [(

)( ) ( )( )( )] 438

( )( ) [(

)( ) ( )( )( )] 439

( )( ) (

)( ) ( )( )( ) 440

( )( ) (

)( ) ( )( )( ) 441

( )( ) (

)( ) ( )( )( ) 442

has a unique positive solution , which is an equilibrium solution for the system


ISSN 2250-3153

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( )( ) [(

)( ) ( )( )( )] 443

( )( ) [(

)( ) ( )( )( )] 444

( )( ) [(

)( ) ( )( )( )] 445

( )( ) (

)( ) ( )( )( ) 446

( )( ) (

)( ) ( )( )( ) 447

( )( ) (

)( ) ( )( )( ) 448

has a unique positive solution , which is an equilibrium solution

( )( ) [(

)( ) ( )( )( )] 449

( )( ) [(

)( ) ( )( )( )] 450

( )( ) [(

)( ) ( )( )( )] 451

( )( ) (

)( ) ( )( )( ) 452

( )( ) (

)( ) ( )( )( ) 453

( )( ) (

)( ) ( )( )( ) 454


( )( ) [(

)( ) ( )( )( )]

455

( )( ) [(

)( ) ( )( )( )] 456

( )( ) [(

)( ) ( )( )( )]

457

( )( ) (

)( ) ( )( )(( ))

458

( )( ) (

)( ) ( )( )(( ))

459

( )( ) (

)( ) ( )( )(( ))

460


( )( ) [(

)( ) ( )( )( )]

461

( )( ) [(

)( ) ( )( )( )]

462

( )( ) [(

)( ) ( )( )( )]

463

( )( ) (

)( ) ( )( )( )

464

( )( ) (

)( ) ( )( )( )

465


ISSN 2250-3153

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( )( ) (

)( ) ( )( )( )

466


( )( ) [(

)( ) ( )( )( )]

467

( )( ) [(

)( ) ( )( )( )]

468

( )( ) [(

)( ) ( )( )( )]

469

( )( ) (

)( ) ( )( )( )

470

( )( ) (

)( ) ( )( )( )

471

( )( ) (

)( ) ( )( )( )

472


( )( ) [(

)( ) ( )( )( )]

473

( )( ) [(

)( ) ( )( )( )]

474

( )( ) [(

)( ) ( )( )( )]

475

( )( ) (

)( ) ( )( )( )

476

( )( ) (

)( ) ( )( )( )

477

( )( ) (

)( ) ( )( )( )

478

( )( ) [(

)( ) ( )( )( )]

479

( )( ) [(

)( ) ( )( )( )]

480

( )( ) [(

)( ) ( )( )( )]

481

( )( ) (

)( ) ( )( )( )

482

( )( ) (

)( ) ( )( )( )

483

( )( ) (

)( ) ( )( )( )

484

( )( ) [(

)( ) ( )( )( )]

484A

( )( ) [(

)( ) ( )( )( )]


ISSN 2250-3153

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( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

Proof:

(a) Indeed the first two equations have a nontrivial solution if

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

485

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

486

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

487

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

488

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

489

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

490

Proof: (a) Indeed the first two equations have a nontrivial solution if

491


ISSN 2250-3153

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( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

Proof: (a) Indeed the first two equations have a nontrivial solution if ( ) (

)( )( )( ) ( )

( )( )( ) (

)( )( )( )( ) (

)( )( )( )( )

( )( )( )(

)( )( )

492

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

)( )( )

( )( )( )(

)( )( )

492

A

Definition and uniqueness of :-

After hypothesis ( ) ( ) and the functions ( )( )( ) being increasing, it follows

that there exists a unique for which (

) . With this value , we obtain from the three first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

493




) . With this value , we obtain from the three

first equations

494

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

495





first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

496





first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

497





498


ISSN 2250-3153

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first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]





first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

499





( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

500





( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

501





first equations

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

501

A

(c) By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )

Where in ( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it follows that there exists a unique

such that ( )

502

(d) By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )

503


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Where in ( )( ) must be replaced by their values from 96. It is easy to see

that is a decreasing function in taking into account the hypothesis ( ) ( ) it

follows that there exists a unique such that (( )

)

504

(a) By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )

Where in ( ) must be replaced by their values from 96. It is easy to see that

is a decreasing function in taking into account the hypothesis ( ) ( ) it follows

that there exists a unique such that (( )

)

505

(b) By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )




)

506

(c) By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )




)

507

(d) By the same argument, the equations admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )



follows that there exists a unique such that ( )

508

(e) By the same argument, the equations admit solutions if ( ) (

)( )( )( ) ( )

( )( )( )

[( )( )(

)( )( ) ( )( )(

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

)( )( ) Where in ( )( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it follows that there exists a unique

such that ( )

509

(f) By the same argument, the equations admit solutions if

510


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( ) ( )( )(

)( ) ( )( )( )

( ) [(

)( )( )( )( ) (

)( )( )( )( )] (

)( )( )( )( )( )

Where in ( )( ) must be replaced by their values from 96. It is easy to see that is a decreasing function in taking into account the hypothesis ( ) ( ) it follows that there exists a unique

such that ( )

(g) By the same argument, the equations 92,93 admit solutions if

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )




)

Finally we obtain the unique solution

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )] ,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution

511


(( )

) , (

) and 512

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

513

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )] 514

Obviously, these values represent an equilibrium solution


(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution of global equations

515


( ) ,

( ) and

516


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( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]


517


(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

518

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]


519


(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

520

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]


521


(( ) ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

522


(( ) ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

523

Finally we obtain the unique solution of 89 to 99

(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

523

A


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( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]

ASYMPTOTIC STABILITY ANALYSIS

Theorem 4: If the conditions of the previous theorem are satisfied and if the functions

( )( ) (

)( ) Belong to ( )( ) then the above equilibrium point is asymptotically stable.

Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( )

524

Then taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

525

((

)( ) ( )( )) ( )

( ) ( )( )

526

((

)( ) ( )( )) ( )

( ) ( )( )

527

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

528

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

529

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

530



( )( ) (

)( ) Belong to ( )( ) then the above equilibrium point is asymptotically stable

531

Proof: Denote

Definition of :-

,

532

( )( )

(

) ( )( ) ,

( )( )

( ( )

) 533

taking into account equations and neglecting the terms of power 2, we obtain

((

)( ) ( )( )) ( )

( ) ( )( )

534

((

)( ) ( )( )) ( )

( ) ( )( )

535

((

)( ) ( )( )) ( )

( ) ( )( )

536


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((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

537

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

538

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

539



( )( ) (


Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

540


((

)( ) ( )( )) ( )

( ) ( )( )

541

((

)( ) ( )( )) ( )

( ) ( )( )

542

((

)( ) ( )( )) ( )

( ) ( )( )

543

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

544

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

545

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

546



( )( ) (


Proof: Denote

547

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

(( )

)

548


((

)( ) ( )( )) ( )

( ) ( )( )

549

((

)( ) ( )( )) ( )

( ) ( )( )

550


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((

)( ) ( )( )) ( )

( ) ( )( )

551

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

552

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

553

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

554



( )( ) (


Proof: Denote

555

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

556


((

)( ) ( )( )) ( )

( ) ( )( )

557

((

)( ) ( )( )) ( )

( ) ( )( )

558

((

)( ) ( )( )) ( )

( ) ( )( )

559

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

560

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

561

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

562



( )( ) (


Proof: Denote

563

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

564


((

)( ) ( )( )) ( )

( ) ( )( )

565


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((

)( ) ( )( )) ( )

( ) ( )( )

566

((

)( ) ( )( )) ( )

( ) ( )( )

567

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

568

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

569

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

570

ASYMPTOTIC STABILITY ANALYSIS Theorem 7: If the conditions of the previous theorem are satisfied and if the functions

( )( ) (

)( ) Belong to ( )( ) then the above equilibrium point is asymptotically stable. Proof: Denote

571

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

572

Then taking into account equations and neglecting the terms of power 2, we obtain from

((

)( ) ( )( )) ( )

( ) ( )( )

573

((

)( ) ( )( )) ( )

( ) ( )( )

574

((

)( ) ( )( )) ( )

( ) ( )( )

575

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

576

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

578

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

579

Obviously, these values represent an equilibrium solution ASYMPTOTIC STABILITY ANALYSIS Theorem 8: If the conditions of the previous theorem are satisfied and if the functions

( )( ) (

)( ) Belong to ( )( ) then the above equilibrium point is asymptotically stable. Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

580


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((

)( ) ( )( )) ( )

( ) ( )( )

581

((

)( ) ( )( )) ( )

( ) ( )( )

582

((

)( ) ( )( )) ( )

( ) ( )( )

583

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

584

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

585

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586

ASYMPTOTIC STABILITY ANALYSIS Theorem 9: If the conditions of the previous theorem are satisfied and if the functions

( )( ) (


Proof: Denote

586A

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)

Then taking into account equations 89 to 99 and neglecting the terms of power 2, we obtain from 99 to 44

((

)( ) ( )( )) ( )

( ) ( )( )

586B

((

)( ) ( )( )) ( )

( ) ( )( )

586 C

((

)( ) ( )( )) ( )

( ) ( )( )

586 D

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586 E

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586 F

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

586

G

The characteristic equation of this system is 587


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(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )


ISSN 2250-3153

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )


ISSN 2250-3153

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((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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


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((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )


ISSN 2250-3153

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part,

and this proves the theorem.

References

(1) Z-Theory and its applications by Sir Allan Zade

(2) ^ Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World ISBN 0-19-

851997-4. Pais recounts how his astonishment at the rapidity with which Feynman could

calculate using his method. Feynman's method is now part of the standard methods for

physicists.

(3) ^ Newton, T.D.; Wigner, E.P. (1949). "Localized states for elementary particles". Reviews of

Modern Physics 21 (3): 400–

406. Bibcode 1949RvMP...21..400N.doi:10.1103/RevModPhys.21.400.

(4) Weinberg, S. Quantum Field Theory, Vols. I to III, 2000, Cambridge University Press:

Cambridge, UK.

(5) Feynman, R.P. (2001) [1964]. The Character of Physical Law. MIT Press. ISBN 0-262-56003-

8.

(6) Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton

University Press. ISBN 0-691-12575-9.

(7) Gribbin, J. (1998). Q is for Quantum: Particle Physics from A to Z. Weidenfeld &

Nicolson. ISBN 0-297-81752-3.

(8) Schumm, Bruce A. (2004) Deep Down Things. Johns Hopkins Univ. Press. Chpt. 4.

(9) Bogoliubov, N.; Shirkov, D. (1982). Quantum Fields. Benjamin-Cummings. ISBN 0-8053-

0983-7.

(10) Frampton, P.H. (2000). Gauge Field Theories. Frontiers in Physics (2nd ed.). Wiley.

(11) Greiner, W; Müller, B. (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-

67672-4.

(12) Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory. McGraw-Hill. ISBN 0-07-032071-3.

(13) Kane, G.L. (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.

(14) Kleinert, H.; Schulte-Frohlinde, Verena (2001). Critical Properties of φ4-Theories. World

Scientific. ISBN 981-02-4658-7.

(15) Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and

Gravitation. World Scientific. ISBN 978-981-279-170-2.

(16) Loudon, R (1983). The Quantum Theory of Light. Oxford University Press. ISBN 0-19-

851155-8.

(17) Mandl, F.; Shaw, G. (1993). Quantum Field Theory. John Wiley & Sons. ISBN [[Special:

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(18) Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview

Press. ISBN 0-201-50397-2.

(19) Ryder, L.H. (1985). Quantum Field Theory. Cambridge University Press. ISBN 0-521-33859-

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(20) Srednicki, Mark (2007) Quantum Field Theory. Cambridge Univ. Press.

(21) Yndurain, F.J. (1996). Relativistic Quantum Mechanics and Introduction to Field Theory (1st

Ed.). Springer. ISBN 978-3-540-60453-2.

(22) Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press. ISBN 0-691-

01019-6.


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