The Epistemological concept of Space, Time, Motion and ... · The Epistemological concept of Space,...

International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 1 ISSN 2250-3153

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The Epistemological concept of Space, Time, Motion and

Transposition as a key philosophical aspect of Z-Theory

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

* Postdoctoral fellow, Department of Mathematics, Kuvempu University. Karnataka, India

Abstract: We study and examine key aspects of Z-Theory. Those are Motion and Transposition. Each aspect has its own

relationship with Space and Time (in common understanding). Z-Theory tends to use the notion of Duration instead of Time to

separate understanding of Time as a fourth dimension of the Universe and uncomplicated duration of any physical process without any

reference to a higher dimension. Difference between space trajectories and out-of-space trajectories (Z-Trajectories) is investigated

and explained. Relation between Z-Trajectories and Time-Shift Effect is analyzed and explained using concrete examples. Ipso facto a

time-independent system is shown to be the solution of the one in question. This of course is a desideratum with Synecdochal

syncretism for all the systems that fall under the categorization that has been enucleated herein below.

Index Terms- Space, Time, Motion, Transposition, Duration, RW-Trajectory, Z-Trajectory, Z-Theory

I. INTRODUCTION

here is an old question about motion. For many centuries, it was thought that phenomenon of motion relates to change of location

between two or more bodies. Subsequently, that point of view became a widely accepted one for the human mind and physical

science. In other words, motion was understood as a process happens relatively to a motionless object.

For many centuries, they believe that the Earth can be used for such an object. The same point of view was in force until the late

middle Ages when many experiments and observations overpowered that old point of view and the Earth became movable in human

philosophy. That was incredible fight for the truth between the progressive and regressive forces. Each group used its own

argumentation to defend their point of view. They used many arguments for and against of the idea of the movable Earth.

II. THE MOVABLE EARTH

Ironically, the Earth itself kept deep silence because it knew one more circle of the human delusion that was created some centuries

later. As soon as the Earth becomes movable any experiment on its surface should be understood as an experiment on a moving body.

Many experiments confirm motion and rotation of the Earth directly. For example, Mr. Foucault had conducted his remarkable

experiment that shown rotation of the Earth in a closed laboratory.

“Foucault pendulum is a relatively large mass suspended from a long line mounted so that its perpendicular plane of swing is not

confined to a particular direction and, in fact, rotates in relation to the Earth's surface. In 1851 the French physicist Jean-Bernard-Léon

Foucault assembled in Paris the first pendulums of this type, one of which consisted of a 28-kg (62-pound) iron ball suspended from

inside the dome of the Panthéon by a steel wire 67 metres (220 feet) long and set in motion by drawing the ball to one side and

carefully releasing it to start it swinging in a plane. The rotation of the plane of swing of Foucault's pendulums was the first laboratory

demonstration of the Earth's spin on its axis.

While a Foucault pendulum swings back and forth in a plane, the Earth rotates beneath it, so that relative motion exists between them.

At the North Pole, latitude 90° N, the relative motion as viewed from the plane of the pendulum's suspension is a counterclockwise

rotation of the Earth once approximately every 24 hours (more precisely, once every 23 hours 56 minutes 4 seconds, the length of a

sidereal day). Correspondingly, the plane of the pendulum as viewed from the Earth looking upward rotates in a clockwise direction

once a day. A Foucault pendulum always rotates clockwise in the Northern Hemisphere with a rate that becomes slower as the

pendulum's location approaches the Equator. Foucault's original pendulums at Paris rotated clockwise at a rate of more than 11° per

hour, or with a period of about 32 hours per complete rotation. The rate of rotation depends on the latitude. At the Equator, 0° latitude,

a Foucault pendulum does not rotate. In the Southern Hemisphere, rotation is counterclockwise.

The rate of rotation of a Foucault pendulum can be stated mathematically as equal to the rate of rotation of the Earth times the sine of

the number of degrees of latitude. Because the Earth rotates once a sidereal day, or 360° approximately every 24 hours, its rate of

rotation may be expressed as 15° per hour, which corresponds to the rate of rotation of a Foucault pendulum at the North or South

T

International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 2

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Pole. At latitude 30° N—for example, at Cairo or New Orleans—a Foucault pendulum would rotate at the rate of 7.5° per hour, for the

sine of 30° is equal to one-half. The rate of rotation of a Foucault pendulum at any given point is, in fact, numerically equal to the

component of the Earth's rate of rotation perpendicular to the Earth's surface at that point.”1

However, theoretical and practical support for that point of view caused one serious problem. If the Erath moves and rotates, what is

the frame of reference for that motion? They believed for some time that the Sun can be used as that frame of reference; but further

examination showed that obviously the Sun moves around the galaxy core as many other stars that form the galaxy. Moreover, the

galaxy core is not at rest. It moves relatively to other galaxies. Hence, they have faced the same question again. It was complete

impossibility to find any frame of reference at rest.

They assumed later that light can be used to find that frame of reference. The idea based on the possibility to make measurement of

the sped of light moving in different directions. According to that idea, famous Michelson-Morley experiment was conducted with so

called null result. That “problem” was described and discussed in the source [3].

The source shows clearly that each motion (even propagation of the light beam) appears as motion relatively to the space. Motion of

the Erath appears the same way. It moves relatively to the space. As a result, some variation in velocities of a beam of light moving

relatively to the Earth can be measures by the device called TSVD. Obviously, TSVD will use a different way of light speed detection

than Michelson-Morley interferometer (planar detector, see source [3]).

The same point of view is applicable to any form of motion without any exception. Figure 1 shows that aspect of motion.

Fig. 1

The figure shows schematically motion of the Earth and its few consequent locations at the points A, B and C. The process of motion

has duration DAC. The planet moves from point A to the point C with that duration.

Suppose, there is an observer on the Earth surface at some point H and a satellite located above the observer at the point D. As soon as

the process of motion occurs in reality, the Erath follows its path on the trajectory around the Sun (A-C) and the satellite mover around

the Earth according to the observer’s point of view.

However, motion of the satellite relatively to the Sun and its gravitational field appears as trajectory D-P-Q-R-E. The same trajectory

becomes true in the frame of reference bound to the Sun. As it clearly seen, each trajectory becomes different in a different frame of

reference. Hence, each trajectory depends on the frame of reference that is used as the frame at rest. Moreover, each trajectory

changes its form going from one frame of reference to another one until an observer reaches the frame of reference at rest. The speed

of motion of an object relatively to each frame of reference changes as soon as an observer changes the frame of reference.

1 Foucault pendulum. (2008). Encyclopædia Britannica. Encyclopaedia Britannica 2008 Deluxe Edition. Chicago: Encyclopædia Britannica

A

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Z2

D


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For example, a rock sitting on the ground next to the observer located at the point H has zero speed of motion in the frame of reference

bound to the planet. At the same time, it has a speed of motion equal to the orbital speed of the planet in the frame of reference bound

to the Sun.

The main problem appears here. Each frame of reference has a weakness because it uses this or that celestial body as a point of origin

for its axes and bounds whole frame of reference to a celestial body. Each celestial body moves and each frame of reference bound to

it moves, as well. As a result, the frame of reference at rest cannot be found that way.

They discussed that problem for some decades at the end of 19-th century until Theory of Relativity was established to avoid that

problem. The theory declares absence of any preferred frame at rest. That is its philosophical and logical conclusion based on the

wrong interpretation of so called null result of Michelson-Morley experiment2.

Unlike that point of view, there is another interpretation of motion. The space itself can be used as the frame of reference at rest. Any

other motion can be understood that way as some relative motion between an object (or celestial body) and the physical space. To

detect that motion, an experimenter should detect some variation in its motion relatively to a beam of light because light moves

relatively to the space with the constant speed3. That was the purpose of the Michelson-Morley experiment, but crucial

misunderstanding of relative motion and its experimental support led them to so called null result.

There is one more problem bound to that experiment. That is matter of time and its understanding by humans. The problem was

eliminated in the essay entitled ‘Human’s delusion of time’4 published later than Michelson-Morley experiment was conducted.

Michelson and Morley, as well as any other scientist from 19-th century, believed in the physical reality of time. From their point of

view, there is some time that the planet used to move from point A to the point C (see fig. 1). The satellite uses the same time to move

from the point D to the point E by the trajectory D-P-Q-R-E. They understand time as the fourth dimension of the Universe. However,

nobody was able to conduct any experiment that shows physical appearance of time. The key aspect of that impossibility arises from

the easiest question. How is it possible for a tree dimensional object to make its interaction with the fourth dimension of the Universe?

That question was left unanswered, as well5.

III. MOTION IN Z-THEORY

According to the Z-Theory any motion appears relatively to the space. As a result, in the case of fig. 1 all objects use the same law of

motion according their rate of motion. Hence, duration of motion of each object can be written the following way:

(1)

In the equation, the variables have the following meaning. DAC is the Duration of motion of the planet from point A to the point C; ST

is the length of any trajectory in the spatial dimensions of the physical space; VS is the speed of motion of an object (an observer, a

planet a star, and etc…) relatively to the space; SE is the length of motion of the Earth relatively to the space; VE is the speed of the

Earth relatively to the space (SAC/VE has the same meaning); SSAT is the length of trajectory of the satellite relatively to the space; VSAT

is the speed of the satellite relatively to the space (SDPQRE/VSAT has the same meaning).

That duration has the same value for each object moving through space. As a result, previously synchronized specific devices called

watches and clocks show the same indications after that duration because each of them calculates only duration of its internal recurrent

physical process6. That process uses the same trajectory as the device itself and obviously has the same duration as duration of motion

of the device relatively to the space7. Hence, an indication of a satellite-bound clock has the same indication as an Earth-bound clock

at the points of beginning and the end of their trajectories in the space (points A-C and D-E).

According theory of Relativity, there is not any preferable frame at rest and each frame can be used by an observer as a frame at rest.

In other words, motion of an observer by A-B-C trajectory (an Earth-bound trajectory) and D-P-Q-R-E trajectory has not any

difference in observation of any physical process.

2 See source [3] for more details 3 See source [3] for more details.

4 Source [4] 5 See source [4] for more details

6 See source [4] for more details 7 See source [5] for more details


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There is some variation in that process according to Z-Theory. At least, indication of an Earth-bound TSVD and the Satellite-bound

TSVD shows some variation. Those indications show a difference in rate of motion of the satellite and the Earth relatively to the

physical space. For that reason, authors require to make TSVD and launch the device to learn its indications. Those indications will

show motion of the Earth in space as well as Foucault experiment shows rotation of the Earth in a closed laboratory.

IV. PHYSICAL SPACE AND DURATION OF PHYSICAL PROCESSES

Using that aspect of motion relatively to the physical space one more way of research becomes possible. If any number of

previously synchronized clocks keeps their indications equally to each other, duration of any period on their motion can be described

by the equation 1. That is a typical situation for all Earth-bound clocks. For the same reason, they thought “flow” of time as a constant

for many centuries. For the same reason, idea of so called arrow of time was established and existed ever in human history. The

following statement explains that phenomenon.

As soon as any object (including a clock) moves forward with the planet using the same speed and direction of motion, duration of

any physical process (including internal recurrent physical process of a clock) between any two points of planetary motion relatively

to the space become equal to duration of any other physical process..

As a result, any phenomenon that leads to different readings between previously synchronized clocks shows certain prove for an

object to use different trajectory to move between the same points of the physical space. For example, imagine an object moves from

the point H to the point O (see fig. 1). It carries a clock that was previously synchronized with other Earth-bound clocks. Under usual

circumstances that object spends the same duration to reach the point O. That duration is equal to duration of motion for any other

Earth-bound object and clock. It coincides with duration of motion of the Earth between points A and C and RW-Trajectory between

the same points in terms of Z-Theory.

Suppose an object uses different trajectory to reach the point O from the point H. As soon as it happened duration of that process will

be calculated the same way. As a result, any object that uses RW-Trajectory spends the same duration for motion by that trajectory.

Otherwise, in case of different duration of that motion, an object uses different length of some different trajectory (Z-Trajectory in

terms of Z-Theory). However, duration itself is still calculable the same way that was shown by the equation 1.

(2)

Difference between equations 1 and 2 is that. An object that used Z-trajectory spends different duration to the process of relocation

between the first and the last points of that trajectory. Theoretically, minimal duration of that process can be calculated the following

way8:

(3)

In the equation (3) variables have the following meaning: DZ (or TZ) is the Duration (or time) of relocation by Z-Trajectory

(Transposition) according to the clock of the observer that uses the Z-Trajectory; L is the length of the moving object in the direction

of motion (length of a vehicle inside which the observer uses the Z-Trajectory); VS is the relative speed between the object and the

frame of reference to which points H and O are bound.

Under usual circumstances, DZ is extremely lesser than DHO (DZ << DHO). As a result, a clock bound to the object counts only

duration DZ unlike an Earth-bound clock that counts duration DHO. Hence, indications of those already synchronized clocks become

different as soon as an object appears after the last point of Z-Trajectory (point O in that case). Any other Z-Trajectory follows the

same law. One example of that trajectory is shown in the figure 1 as the trajectory D-Z1-Z2-E.

V. Z-TRAJECTORY VERSUS RW-TRAJECTORY

The key difference between any two objects that use RW-Trajectory and Z-Trajectory is that. An object that uses Z-Trajectory

makes not any interaction with surrounding conservative fields and becomes undetectable by any electromagnetic waves by any

mean9.

8 Source [5]

9 See source [5] for more details.


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That coincides with the idea of “now”. “’Now’ is a point in the Universe from where an observer (object, body, etc.) makes interaction

with surrounding Universe”10

. As soon as the interaction between an object and the Universe has broken down, the object becomes

out of “now” of the Universe. That object keeps that condition from the point of view of a Universe-bound observer as long as the

object has not interaction with the Universe. As soon as the object restores its interaction it appears in different “now” relatively to its

previous location, but in the same “now” that is actual to a Universe-bound observer. As it mentioned above, that happens because

both observers use different trajectories to reach the same point located in the physical space.

Relocation for an object by Z-Trajectory has not any physical difference in relocation of the same object between any two points of

the physical space. The observable difference for an Earth-bound observer appears only in relation to so called time. If an object uses

Z-Trajectory that connects two points that can be reached by the Earth-bound observer only by relocation with the planet by its orbital

motion, the Erath-bound observer sees and understands that Transposition as relocation “through time”11

. The best example of that

Transposition is Boeing 727 incident12

. In that case, all onboard clocks and watches showed the same 10 minutes left indications. That

happened because all of them used the same Z-Trajectory like D-Z1-Z2-E trajectory (Z-Trajectory) shown in the figure 1.

Otherwise, an Earth-bound observer understands the same process of Transposition as relocation through space. Example of that

relocation is shown in the figure 1 as Z-Trajectory between points T and U. Transposition between those points by Z-Trajectory means

relocation in case of the same location of the Earth at the point A. The best example of that Transposition is an experience of Bruce

Gernon13

.

NOTATION :

: Category one of relocation by Z-Trajectory (Transposition)

: Category two of relocation by Z-Trajectory (Transposition)

: Category three of relocation by Z-Trajectory (Transposition)

: Category one of relocation between any two points of the physical space

: Category two of relocation between any two points of the physical space

: Category three of relocation between any two points of the physical space

: Category one observable difference for an Earth-bound observer (systemic categorization based on characteristics’)

: Category two observable difference for an Earth-bound observer (systemic categorization based on characteristics’)

: Category three observable difference for an Earth-bound observer (systemic categorization based on characteristics’)

: Category one of appearance in relation to so called time (corresponding to Gs’ classification)

: Category two of appearance in relation to so called time (corresponding to Gs’ classification)

: Category three of appearance in relation to so called time (corresponding to Gs’ classification)

: Category one of using Z-Trajectory that connects two points that can be reached by the Earth-bound observer only by

relocation with the planet by its orbital motion, the Earth-bound observer sees and understands that relocation as relocation

“through time”

: Category two of using Z-Trajectory that connects two points that can be reached by the Earth-bound observer only by


“through time”

: Category three of using Z-Trajectory that connects two points that can be reached by the Earth-bound observer only by


“through time”

: Category one of relocation “through time”.

: Category two of relocation “through time”.

: Category three of relocation “through time”.

: Category one of all watches which used the same Z-Trajectory

: Category two of all watches which used the same Z-Trajectory

: Category one of all watches which used the same Z-Trajectory

: Category one of (The best known example of that relocation is Boeing 727 incident (We assume that there are many such

incidents which are recorded or have gone unrecorded. Moon doth exist even if we do not see it). In that case, all onboard clocks

and watches showed the same 10 minutes left indications.

10 Source [4] 11 The observer loses his mind that way because physics of 20-th century denies any possibility of such relocation. 12 See source [6] 13 See source [6]


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: Category two of (The best known example of that relocation is Boeing 727 incident (We assume that there are many such



: Category three of (The best known example of that relocation is Boeing 727 incident (We assume that there are many such



: Category one of relocation through space. Example of that relocation is shown in the figure 1 as Z-Trajectory between

points T and U.

: Category two of relocation through space. Example of that relocation is shown in the figure 1 as Z-Trajectory between

points T and U.

: Category three of relocation through space. Example of that relocation is shown in the figure 1 as Z-Trajectory between

points T and U.

: category one of Earth-bound observer’s understanding of the same process of relocation

: category two of Earth-bound observer’s understanding of the same process of relocation

: category three of Earth-bound observer’s understanding of the same process of relocation

: Category one of likely cases like the case of relocation in case of the same location of the Earth at the point A. The best

example of that relocation is experience of Bruce Gernon.

: Category two of likely cases like the case of relocation in case of the same location of the Earth at the point A. The best


: Category three of likely cases like the case of relocation in case of the same location of the Earth at the point A. The best


: Category one of Relocation between those points by Z-Trajectory (Such cases are legion albeit unrecorded. That is being

stressed all through)

: Category two of Relocation between those points by Z-Trajectory (Such cases are legion albeit unrecorded. That is being


: Category three of Relocation between those points by Z-Trajectory (Such cases are legion albeit unrecorded. That is being


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)( )( ) , ( )( )( ) are second detrition coefficients for category 1,2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third detrition coefficients for category 1,2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for category 1,2 and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are fifth detrition coefficients for category 1,2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are sixth detrition coefficients for category 1,2 and 3

( )

( ) [

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )( )( ) , (

)( )( ) , ( )( )( ) are first augmentation coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) , ( )( )( ) are second augmentation coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are third augmentation coefficients for category 1, 2 and 3

( )( )( ) , (

)( )( ) ( )( )( ) are fourth augmentation coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation coefficients for category 1, 2 and 3

( )

( ) [ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )( )( ) (

)( )( ) ( )( )( ) are 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


ISSN 2250-3153

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( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition coefficients for category 1, 2 and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth detrition coefficients for category 1, 2 and 3

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth augmentation coefficients for category 1, 2,and 3

( )( )( ) , (

)( )( ) ( )( )( ) are fifth augmentation coefficients for category 1, 2,and 3

( )( )( ) , (

)( )( ) , ( )( )( ) are sixth augmentation coefficients for category 1, 2,and 3

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

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

( )( )( ) , (

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

– ( )( )( ) – (

)( )( ) – ( )( )( )

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [

( )( ) (

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

)( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth augmentation coefficients for category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth augmentation coefficients for category 1,2,and 3

( )( )( ) (

)( )( ) ( )( )( ) are sixth augmentation coefficients for category 1,2, 3


ISSN 2250-3153

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( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )

( ) [ (

)( ) ( )( )( ) (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

– ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for category 1,2, and 3

( )( )( ) (

)( )( ) ( )( )( ) are fifth detrition coefficients for category 1,2, and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition coefficients for category 1,2, and 3

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

]

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) - are fourth augmentation coefficients

( )( )( ) (

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

( )( )( ) , (

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

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )

( ) [(

)( ) ( )( )( ) – (

)( )( ) – ( )( )( )

( )( )( ) (

)( )( ) – ( )( )( )

]

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( )

( )( )( ) (

)( )( ) ( )( )( ) are fourth detrition coefficients for category 1, 2, and 3

( )( )( ) , (

)( )( ) ( )( )( ) are fifth detrition coefficients for category 1, 2, and 3

– ( )( )( ) , – (

)( )( ) – ( )( )( ) are sixth detrition coefficients for category 1, 2, and 3

Where we suppose


ISSN 2250-3153

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(A) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(B) The functions ( )( ) (

)( ) are positive continuous increasing and bounded.

Definition of ( )( ) ( )

( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

(C) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( ) Definition of ( )

( ) ( )( ) :

Where ( )( ) ( )

( ) ( )( ) ( )

( ) are positive constants and

They satisfy Lipschitz condition:

( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )( ) (

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

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

) and( )( )( ) (

) and

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

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

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

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

absolutely continuous.


( ) :

(D) ( )( ) ( )

( ) are positive constants

( )

( )

( )( ) ( )

( )

( )( )


( ) :

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

( ) which together with ( )( ) ( )

( ) ( )( ) and ( )

( )

and the constants ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( )

satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

Where we suppose

(F) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(G) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

(H) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


) and( )( )( ) . (

) and


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

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



ISSN 2250-3153

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( ) :

(I) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )


( ) :

There exists two constants ( )( ) and ( )


( ) ( )( ) ( )

( ) and the

constants ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )

( ) satisfy the inequalities

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

Where we suppose

(J) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( ) Definition of ( )

( ) ( )( ) :

Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(

) ( )( )( ) ( )

( ) ( )( )


) and( )( )( ) . (

) And


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

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




( ) :

(K) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )

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

( ) which together with

( )( ) ( )

( ) ( )( ) ( )

( ) and the constants ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

Where we suppose

(L) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(M) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

(N) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


ISSN 2250-3153

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( ) :

Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


) and( )( )( ) . (

) and







( ) :

(O) ( )( ) ( )


(P) ( )

( )

( )( ) ( )

( )

( )( )


( ) :

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


( ) ( )( ) ( )

( ) and

the constants ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

Where we suppose

(R) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(S) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

(T) ( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


) and ( )( )( ) . (

) and







( ) :

(U) ( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )


( ) :

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


( ) ( )( ) ( )

( ) and the

constants ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

Where we suppose


ISSN 2250-3153

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( )( ) (

)( ) ( )( ) ( )

( ) ( )( ) (

)( )

(W) The functions ( )( ) (



( ):

( )( )( ) ( )

( ) ( )( )

( )( )(( ) ) ( )

( ) ( )( ) ( )

( )

(X) ( )( ) ( ) ( )

( )

( )( ) (( ) ) ( )

( )


( ) :

Where ( )( ) ( )

( ) ( )( ) ( )



( )( )(

) ( )( )( ) ( )

( ) ( )( )

( )( )(( )

) ( )( )(( ) ) ( )

( ) ( ) ( ) ( )( )


) and( )( )( ) . (

) and







( ) :

( )( ) ( )


( )

( )

( )( ) ( )

( )

( )( )


( ) :

There exists two constants ( )( ) and ( )


( ) ( )( ) ( )

( ) and the

constants ( )( ) (

)( ) ( )( ) (

)( ) ( )( ) ( )


( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

( )( ) ( )( ) (

)( ) ( )( ) ( )

( ) ( )( )

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

Definition of ( ) ( ) :

( ) ( )( )

( )( ) , ( )

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


Definition of ( ) ( )

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

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


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

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



( ) ( )( )

( )( ) , ( )

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




ISSN 2250-3153

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( ) ( )( )

( )( ) , ( )

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



( ) ( )( )

( )( ) , ( )

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

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

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

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

Proof:

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

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


Proof:


( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )


ISSN 2250-3153

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( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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



( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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



( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


Proof:


( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )( )

By

( ) ∫ [( )

( ) ( ( )) (( )( )

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

( ) ∫ [( )

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

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


ISSN 2250-3153

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( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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

( ) ∫ [( )

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

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


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )

From which it follows that

( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]

( ) is as defined in the statement of theorem 1

Analogous inequalities hold also for

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( ) ( ( )

( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]


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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]



ISSN 2250-3153

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

( ) ∫ [( )

( ) ( ( )

( ) ( )( ) ( ))]

( )

( ( )( ) )

( )( )( )( )

( )( ) ( ( )( ) )


( ( ) ) ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )]



It is now sufficient to take ( )

( )

( )( ) ( )

( )

( )( ) and to choose

( )( ) ( )

( ) large to have

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )

In order that the operator ( ) transforms the space of sextuples of functions satisfying 34,35,36 into itself

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

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

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of :

( ) ( )( )

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval From the hypotheses on 25,26,27,28 and 29 it follows

| ( ) ( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

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

And analogous inequalities for . Taking into account the hypothesis (34,35,36) the result follows

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

)( ) depending also on can be considered as not conformal with the

reality, however we have put this hypothesis ,in order that we can postulate condition necessary to prove the uniqueness of the

solution bounded by ( )( ) ( )( ) ( )

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

( )( ) (

)( ) depend only on and respectively on ( ) and hypothesis can replaced by a

usual Lipschitz condition.

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

From 19 to 24 it results

( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for

Definition of (( )( ))

(( )

( )) (( )

( )) :

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

( )( ) it follows

(( )

( )) (

)( ) and by integrating


ISSN 2250-3153

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(( )( ))

( )( )(( )

( )) (

)( )

In the same way , one can obtain

(( )( ))

( )( )(( )

( )) (

)( )

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

Remark 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.

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

)( ) then

Definition of ( )( ) :

Indeed let be so that for

( )( ) (

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

Then

( )

( )( )( ) which leads to

(( )( )( )( )

) ( )

If we take such that

it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded. The same property

holds for if ( )( ) ( ( ) ) (

)( )

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 42


( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

Definition of : ( ) ( )( )

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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


|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))







( )( ) (

)( ) depend only on and respectively on ( )( ) and hypothesis can replaced by a

usual Lipschitz condition.



ISSN 2250-3153

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( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )




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

)( ) then



( )( ) (

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

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)

By taking now sufficiently small one sees that is unbounded. The same property holds

for if ( )( ) (( )( ) ) (

)( )



( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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

Where ( ) represents integrand that is integrated over the interval

From the hypotheses on 25,26,27,28 and 29 it follows

| ( ) ( )| ( )( )


ISSN 2250-3153

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( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))







( )( ) (

)( ) depend only on and respectively on ( )( ) and hypothesis can replaced by

a usual Lipschitz condition.



( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )





)( ) then



( )( ) (

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

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


holds for if ( )( ) (( )( ) ) (

)( )



( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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


ISSN 2250-3153

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∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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


|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))







( )( ) (





( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )





)( ) then

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

( )( ) (

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

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


holds for if ( )( ) (( )( ) ) (

)( )




( )

( )( ) ( )

( )


( )( ) ( )

( ) large to have

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )



ISSN 2250-3153

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((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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



|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))







( )( ) (





( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )





)( ) then


( )( ) (

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

Then

( )


(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


holds for if ( )( ) (( )( ) ) (

)( )




( )

( )( ) ( )

( )



ISSN 2250-3153

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( )( ) ( )

( ) large to have

( )( )

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

( ) )

(( )( )

)

] ( )( )

( )( )

( )( ) [(( )( )

) (

( )( )

)

( )( )] ( )

( )



((( )( ) ( )

( )) (( )( ) ( )

( )))

| ( )( )

( )( )| ( )( )

| ( )( )

( )( )| ( )( )

Indeed if we denote

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

It results

| ( )

( )

| ∫ ( )( )

|

( )

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

∫ ( )( )|

( )

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

( )( )(

( ) ( ))|

( )

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

( )

( )( )(

( ) ( )) (

)( )( ( )

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



|( )( ) ( )

( )| ( )( )

( )( ) (( )( ) (

)( ) ( )( ) ( )

( )( )( )) ((( )

( ) ( )( ) ( )

( ) ( )( )))







( )( ) (





( ) [ ∫ {(

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

]

( ) ( (

)( ) ) for


(( )

( )) (( )

( )) :


( )( ) it follows

(( )

( )) (


(( )( ))

( )( )(( )

( )) (

)( )


(( )( ))

( )( )(( )

( )) (

)( )





)( ) then


( )( ) (

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

Then

( )



ISSN 2250-3153

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(( )( )( )( )

) ( )


it results

(( )( )( )( )

)


holds for if ( )( ) (( )( ) ( ) ) (

)( )


Behavior of the solutions of equation 37 to 42

Theorem 2: If we denote and define


( ) ( )( ) ( )

( ) :

(a) )( ) ( )

( ) ( )( ) ( )

( ) four constants satisfying

( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(b) By ( )( ) ( )

( ) and respectively ( )( ) ( )

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

( )( ) ( ) ( )

( ) and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) :

By ( )( ) ( )



( )( ) ( ) ( )

( ) and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

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

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )

are defined by 59 and 61 respectively

Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities

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

( )( )

where ( )( ) is defined by equation 25

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

( )( ) ( )( )

( ( )( )

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

( )( ) ( ) ( )( )

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

( )( )

( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )


ISSN 2250-3153

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( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

Behavior of the solutions of equation 37 to 42 Theorem 2: If we denote and define


( ) ( )( ) ( )

( ) :

(d) )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the roots

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the

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

( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) :-

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

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )


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

( )( )

( )( ) is defined by equation 25

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

( )( ) ( )( )

( ( )( )

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

( )( ) ( ) ( )( )

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

( )( )

( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )



ISSN 2250-3153

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( ) ( )( ) ( )

( ) :

(a) )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) :

(b) By ( )( ) ( )



( )( ) ( ) ( )

( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) :-

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

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) and ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities

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

( )( )

( )( ) is defined by equation 25

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

( )( ) ( )( )

( ( )( )

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

( )( ) ( ) ( )( )

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

( )( )

( )( )

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )




( ) ( )( ) ( )

( ) :


ISSN 2250-3153

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(d) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(e) By ( )( ) ( )



( )( ) ( ) ( )

( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

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

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )



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

( )( )


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

( )( ) ( )( )

(( )( )

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

( )( ) ( ) ( )( )

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

[ ( )( )


ISSN 2250-3153

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( )( ) ]

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )




( ) ( )( ) ( )

( ) :

(g) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(h) By ( )( ) ( )



( )( ) ( ) ( )

( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

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

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )


ISSN 2250-3153

www.ijsrp.org

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( )



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

( )( )


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

( )( ) ( )( )

(( )( )

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

( )( ) ( ) ( )( )

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

[ ( )( )

( )( ) ]

( )( ) )

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )




( ) ( )( ) ( )

( ) :


ISSN 2250-3153

www.ijsrp.org

(j) ( )( ) ( )

( ) ( )( ) ( )


( )( ) (

)( ) ( )( ) (

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

( )

( )( ) (

)( ) ( )( ) (

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

( )


( ) ( )( ) ( )

( ) ( ) ( ) :

(k) By ( )( ) ( )



( )( ) ( ) ( )

( )

and ( )( )( ( ))

( )

( ) ( ) ( )( ) and


( ) ( )( ) ( )

( ) :

By ( )( ) ( )


( ) the


( )

( ) ( ) ( )( )

and ( )( )( ( ))

( )

( ) ( ) ( )( )


( ) ( )( ) ( )

( ) ( )( ) :-

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

( ) ( )( ) ( )

( ) by

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

and analogously

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( )

and ( )( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) where ( )( ) ( )

( ) are defined by 59 and 66 respectively


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

( )( )


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

( )( ) ( )( )

(( )( )

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

( )( ) ( ) ( )( )

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

[ ( )( )

( )( ) ]

( )( ) )


ISSN 2250-3153

www.ijsrp.org

( )( ) ( )

(( )( ) ( )( ))

( )( ) ( )( ) ( )

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

( )( )

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

[ ( )( ) ( )( ) ]

( )( ) ( )

( )( )

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

( )( )


( ) ( )( ) ( )

( ):-

Where ( )( ) ( )

( )( )( ) (

)( )

( )( ) ( )

( ) ( )( )

( )( ) ( )

( )( )( ) (

)( )

( )( ) (

)( ) ( )( )

Proof : From 19,20,21,22,23,24 we obtain ( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )

Definition of ( ) :- ( )

It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))

From which one obtains


( ) :-

(a) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )

In the same manner , we get

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

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

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

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

( )( ) ( )( )

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

( )

(b) If ( )( ) ( )

( )

( )

( ) we find like in the previous case,

( )( )

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

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

( )( )

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

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

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

( )( )

(c) If ( )( ) ( )

( ) ( )( )

, we obtain


ISSN 2250-3153

www.ijsrp.org

( )( ) ( )( )

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

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

( )( )

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 19, 20,21,22,23, and 24 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 ( )( )


( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(d) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )


( )

(e) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

(f) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )




ISSN 2250-3153

www.ijsrp.org

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


Particular case :

If ( )( ) (

)( ) ( )( ) ( )



( ) then ( )( )


( ) ( )


)( ) ( )( ) ( )

( ) and then

( )( ) ( )




( )


( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))


(a) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )



( )

(b) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

(c) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


Particular case :

If ( )( ) (

)( ) ( )( ) ( )



( ) then ( )( )


( ) ( )


)( ) ( )( ) ( )

( ) and then


ISSN 2250-3153

www.ijsrp.org

( )( ) ( )




( )

Proof : From 19,20,21,22,23,24 we obtain

( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(d) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )


( )

(e) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

(f) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )




ISSN 2250-3153

www.ijsrp.org

( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


Particular case :

If ( )( ) (

)( ) ( )( ) ( )



( ) then ( )( )


( ) ( ) this also defines ( )( ) for the special case .


)( ) ( )( ) ( )

( ) and then

( )( ) ( )






( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(g) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


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

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

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

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

( )( ) ( )( )


( )

(h) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )


ISSN 2250-3153

www.ijsrp.org

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

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

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

( )( )

(i) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


Particular case :

If ( )( ) (

)( ) ( )( ) ( )



( ) then ( )( )


( ) ( ) this also defines ( )( ) for the special case .


)( ) ( )( ) ( )

( ) and then

( )( ) ( )






( )

( )

( ) (( )( ) (

)( ) ( )( )( )) (

)( )( ) ( ) ( )

( ) ( )


It follows

(( )( )( ( ))

( )

( ) ( ) ( )( ))

( )

(( )

( )( ( )) ( )

( ) ( ) ( )( ))



( ) :-

(j) For ( )( )

( )

( ) ( )( )

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

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

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

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

( )( ) ( )( )

( )( ) ( )( ) ( )

( )


ISSN 2250-3153

www.ijsrp.org


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

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

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

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

( )( ) ( )( )


( )

(k) If ( )( ) ( )

( )

( )


( )( )

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

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

( )( )

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

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

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

( )( )

(l) If ( )( ) ( )

( ) ( )( )

, we obtain

( )( ) ( )( )

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

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

( )( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )



( )( ) ( )( ) ( )

( ), ( )( ) ( )

( )


Particular case :

If ( )( ) (

)( ) ( )( ) ( )



( ) then ( )( )


( ) ( ) this also defines ( )( ) for the special case.


)( ) ( )( ) ( )

( ) and then

( )( ) ( )





We can prove the following

Theorem 3: If ( )( ) (

)( ) are independent on , and the conditions (with the notations 25,26,27,28)

( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )

( ) as defined by equation 25 are satisfied , then the system



( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,


ISSN 2250-3153

www.ijsrp.org

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )




( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


We can prove the following



( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )




( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )




( )( )(

)( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( )(

)( ) ( )( )( )

( ) ,

( )( )(

)( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( )

( )( ) ( )


( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

has a unique positive solution , which is an equilibrium solution for the system (19 to 24)

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

has a unique positive solution , which is an equilibrium solution for (19 to 24)

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

has a unique positive solution , which is an equilibrium solution for (19 to 24)

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]


ISSN 2250-3153

www.ijsrp.org

( )( ) (

)( ) ( )( )(( ))

( )( ) (

)( ) ( )( )(( ))

( )( ) (

)( ) ( )( )(( ))


( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )


( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) [(

)( ) ( )( )( )]

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )

( )( ) (

)( ) ( )( )( )


Proof:

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

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

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

)( )( )

Proof:


( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

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

)( )( )

Proof:



ISSN 2250-3153

www.ijsrp.org

( ) ( )( )(

)( ) ( )( )( )

( ) ( )( )(

)( )( ) ( )( )(

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

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

)( )( )( ) ( )

( )( )( ) (

)( )( )( )( ) (

)( )( )( )( ) (

)( )( )( )( )( )

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

)( )( )( ) ( )

( )( )( ) (

)( )( )( )( ) (

)( )( )( )( ) (

)( )( )( )( )( )

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

)( )( )( ) ( )

( )( )( ) (

)( )( )( )( ) (

)( )( )( )( ) (

)( )( )( )( )( )

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

( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]




( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]




( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]




( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]




( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]




( )( )

[( )( ) (

)( )( )]

, ( )( )

[( )( ) (

)( )( )]

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

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )

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

taking into account the hypothesis ( ) ( ) it follows that there exists a unique such that ( )

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

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )

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

in taking into account the hypothesis ( ) ( ) it follows that there exists a unique such that (( )

)

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

( ) ( )( )(

)( ) ( )( )( )

( ) [(

)( )( )( )( ) (

)( )( )( )( )] (

)( )( )( )( )( )

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


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taking into account the hypothesis ( ) ( ) it follows that there exists a unique such that (( )

)

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

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )



)

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

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )



)

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

( ) ( )( )(

)( ) ( )( )( )

( )

[( )( )(

)( )( ) ( )( )(

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

)( )( )


in taking into account the hypothesis ( ) ( ) it follows that there exists a unique such that ( )

Finally we obtain the unique solution of 89 to 94

( ) ,

( ) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )] ,

( )( )

[( )( ) (

)( )( )]

Obviously, these values represent an equilibrium solution of 19,20,21,22,23,24


(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]



(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]



( ) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]



(( )

) , (

) and

( )( )

[( )( ) (

)( )( )]

,

( )( )

[( )( ) (

)( )( )]

( )( )

[( )( ) (

)( )(( ) )] ,

( )( )

[( )( ) (

)( )(( ) )]



(( )

) , (

) and


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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 :-

,

(

)( )

(

) ( )( ) ,

( )( )

( )

Then taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )



)( ) Belong to ( )( )

then the above equilibrium point is asymptotically stable

Proof: Denote

Definition of :-

,

( )( )

(

) ( )( ) ,

( )( )

( ( )

)

taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )



)( ) Belong to

( )( ) then the above equilibrium point is asymptotically stable.

Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)


((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )


ISSN 2250-3153

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)( ) Belong to ( )( )

then the above equilibrium point is asymptotically stable.

Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

(( )

)


((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )



)( ) Belong to ( )( )


Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)


((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )



)( ) Belong to ( )( )


Proof: Denote

Definition of :-

,

(

)( )

(

) ( )( ) ,

( )( )

( ( )

)


((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ( )( )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

((

)( ) ( )( )) ( )

( ) ∑ ( ( )( ) )

The characteristic equation of this system is

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]


ISSN 2250-3153

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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


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((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+ (( )( ) (

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

)( ) ( )( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

+

(( )( ) ( )( ) ( )

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

( ))

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

( ))( )( )

( )( )( )

( ) )]

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

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

( ))( )( )

( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

((( )( )) ( (

)( ) ( )( ) ( )

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

( )

(( )( ) ( )( ) ( )

( )) (( )( )( )

( ) ( )

( )( )( )( )

( ) )

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

( )) ( ) ( ) ( )

( ) ( ) ( ) )

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and this proves the theorem.

REFERENCES

[1] A. Zade, Z-Theory and Its Applications. AuthorHouse, 2011, ISBN 978-1452018935

[2] Encyclopaedia Britannica 2008 Deluxe Edition. Chicago: Encyclopædia Britannica. (Electronic edition)

[3] Allan Zade - The Epistemological concept of the True Space-Velocity Detector - published at: "International Journal of Scientific and Research Publications (IJSRP), Volume 2, Issue 11, November 2012 Edition".

[4] Allan Zade - Human’s Delusion of Time - published at: "International Journal of Scientific and Research Publications (IJSRP), Volume 2, Issue 10, October 2012 Edition".

[5] Allan Zade - Motion and Transposition in conservative fields - published at: "International Journal of Scientific and Research Publications (IJSRP), Volume 2, Issue 8, August 2012 Edition".

[6] Allan Zade - Matter of Navigation - published at: "International Journal of Scientific and Research Publications (IJSRP), Volume 2, Issue 9, September 2012 Edition".

AUTHORS

First Author – Allan Zade [email protected].

http://www.ijsrp.org/research-paper-1112.php?rp=P11383




mailto:[email protected]


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Second Author – DR. K.N.PrasannnaKumar, Post Doctoral fellow, Department Of mathematics, Kuvempu University, Karnataka,

India. [email protected]

Correspondence Author – Allan Zade [email protected].



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