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International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 1 ISSN 2250-3153
www.ijsrp.org
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
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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
Z1
B
C
E
H O
P
Q
R
T U
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
relocation with the planet by its orbital motion, the Earth-bound observer sees and understands that relocation as relocation
“through time”
: Category three 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 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
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.
: Category three 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.
: 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
example of that relocation is experience of Bruce Gernon.
: 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
example of that relocation is experience of Bruce Gernon.
: 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
stressed all through)
: Category three of Relocation between those points by Z-Trajectory (Such cases are legion albeit unrecorded. That is being
stressed all through)
( )( ) ( )
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are Accentuation coefficients
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are Dissipation coefficients
GOVERNING EQUATIONS:
The differential system of this model is now
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( )( )( ) First augmentation factor attributable
( )( )( ) First detrition factor contributed
GOVERNING EQUATIONS: The differential system of this model is now
( )
( ) [( )( ) (
)( )( )]
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 7
ISSN 2250-3153
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( )
( ) [( )( ) (
)( )( )]
( )
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( )( )( ) First augmentation factor attributable
( )( )(( ) ) First detrition factor contributed
GOVERNING EQUATIONS: The differential system of this model is now
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( )( )( ) First augmentation factor attributable
( )( )( ) First detrition factor contributed
GOVERNING EQUATIONS: The differential system of this model is now
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( )( )( ) First augmentation factor attributable
( )( )(( ) ) First detrition factor contributed
GOVERNING EQUATIONS: The differential system of this model is now
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( )( )( ) First augmentation factor attributable
( )( )(( ) ) First detrition factor contributed
GOVERNING EQUATIONS: The differential system of this model is now
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( )
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)( )( )]
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 8
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( )
( ) [( )( ) (
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]
Where ( )( )( ) (
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( )( )( ) , (
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( )( )( ) (
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( )( )( ) , (
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( )( )( ) (
)( )( ) ( )( )( ) are fifth augmentation coefficient for category 1, 2 and 3
( )( )( ) , (
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( )
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Where ( )( )( ) (
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( )( )( ) (
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( )( )( ) (
)( )( ) ( )( )( ) 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
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]
Where ( )( )( ) (
)( )( ) ( )( )( ) are first augmentation coefficients for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are second augmentation coefficient for category 1, 2 and 3
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 9
ISSN 2250-3153
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( )( )( ) (
)( )( ) ( )( )( ) are third augmentation coefficient for category 1, 2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are fourth augmentation coefficient for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are fifth augmentation coefficient for category 1, 2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are sixth augmentation coefficient for category 1, 2 and 3
( )
( ) [ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
]
( )
( ) [ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
]
( )
( ) [ (
)( ) ( )( )( ) (
)( )( ) – ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
]
( )( )( ) , (
)( )( ) , ( )( )( ) are first detrition coefficients for category 1, 2 and 3
( )( )( ) (
)( )( ) , ( )( )( ) are second detrition coefficients for category 1,2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are third detrition coefficients for category 1,2 and 3
( )( )( ) (
)( )( ) ( )( )( ) are fourth detrition coefficients for category 1,2 and 3
( )( )( ) , (
)( )( ) , ( )( )( ) are fifth detrition coefficients for category 1,2 and 3
( )( )( ) (
)( )( ) , ( )( )( ) are sixth detrition coefficients for category 1,2 and 3
( )
( ) [
( )( ) (
)( )( ) ( )( )( ) (
)( )( )
( )( )( ) (
)( )( ) ( )( )( )
]
( )
( ) [(
)( ) ( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
]
( )
( ) [(
)( ) ( )( )( ) (
)( )( ) ( )( )( )
( )( )( ) (
)( )( ) ( )( )( )
]
( )( )( ) , (
)( )( ) , ( )( )( ) are 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
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 10
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
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 11
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
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 12
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.
Definition of ( )( ) ( )
( ) :
(D) ( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
Definition of ( )( ) ( )
( ) :
(E) There exists two constants ( )( ) and ( )
( ) which together with ( )( ) ( )
( ) ( )( ) and ( )
( )
and the constants ( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( )
satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
Where we suppose
(F) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
(G) The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )( ) ( )
( ) ( )( ) ( )
( )
(H) ( )( ) ( ) ( )
( )
( )( ) (( ) ) ( )
( )
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
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 13
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absolutely continuous.
Definition of ( )( ) ( )
( ) :
(I) ( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( ) which together with ( )( ) ( )
( ) ( )( ) ( )
( ) and the
constants ( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
Where we suppose
(J) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( ) ( )
( )
( )( ) ( ) ( )
( ) Definition of ( )
( ) ( )( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
( )( )(
) ( )( )( ) ( )
( ) ( )( )
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
) and( )( )( ) . (
) And
( ) are points belonging to the interval [( )( ) ( )
( )] . It is to be noted that ( )( )( ) is uniformly continuous.
In the eventuality of the fact, that if ( )( ) then the function (
)( )( ) , the first augmentation coefficient would be
absolutely continuous.
Definition of ( )( ) ( )
( ) :
(K) ( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
There exists two constants There exists two constants ( )( ) and ( )
( ) which together with
( )( ) ( )
( ) ( )( ) ( )
( ) and the constants ( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
Where we suppose
(L) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
(M) The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )(( ) ) ( )
( ) ( )( ) ( )
( )
(N) ( )( ) ( ) ( )
( )
( )( ) (( ) ) ( )
( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 14
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Definition of ( )( ) ( )
( ) :
Where ( )( ) ( )
( ) ( )( ) ( )
( ) are positive constants and
They satisfy Lipschitz condition:
( )( )(
) ( )( )( ) ( )
( ) ( )( )
( )( )(( )
) ( )( )(( ) ) ( )
( ) ( ) ( ) ( )( )
With the Lipschitz condition, we place a restriction on the behavior of functions ( )( )(
) and( )( )( ) . (
) and
( ) are points belonging to the interval [( )( ) ( )
( )] . It is to be noted that ( )( )( ) is uniformly continuous.
In the eventuality of the fact, that if ( )( ) then the function (
)( )( ) , the first augmentation coefficient would be
absolutely continuous.
Definition of ( )( ) ( )
( ) :
(O) ( )( ) ( )
( ) are positive constants
(P) ( )
( )
( )( ) ( )
( )
( )( )
Definition of ( )( ) ( )
( ) :
(Q) There exists two constants ( )( ) and ( )
( ) which together with ( )( ) ( )
( ) ( )( ) ( )
( ) and
the constants ( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
Where we suppose
(R) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
(S) The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )(( ) ) ( )
( ) ( )( ) ( )
( )
(T) ( )( ) ( ) ( )
( )
( )( ) ( ) ( )
( )
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.
Definition of ( )( ) ( )
( ) :
(U) ( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
Definition of ( )( ) ( )
( ) :
(V) There exists two constants ( )( ) and ( )
( ) which together with ( )( ) ( )
( ) ( )( ) ( )
( ) and the
constants ( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
Where we suppose
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 15
ISSN 2250-3153
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( )( ) (
)( ) ( )( ) ( )
( ) ( )( ) (
)( )
(W) The functions ( )( ) (
)( ) are positive continuous increasing and bounded.
Definition of ( )( ) ( )
( ):
( )( )( ) ( )
( ) ( )( )
( )( )(( ) ) ( )
( ) ( )( ) ( )
( )
(X) ( )( ) ( ) ( )
( )
( )( ) (( ) ) ( )
( )
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.
Definition of ( )( ) ( )
( ) :
( )( ) ( )
( ) are positive constants
( )
( )
( )( ) ( )
( )
( )( )
Definition of ( )( ) ( )
( ) :
There exists two constants ( )( ) and ( )
( ) which together with ( )( ) ( )
( ) ( )( ) ( )
( ) and the
constants ( )( ) (
)( ) ( )( ) (
)( ) ( )( ) ( )
( ) satisfy the inequalities
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
( )( ) ( )( ) (
)( ) ( )( ) ( )
( ) ( )( )
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( )
( ) ( )( ) ( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions
( ) ( )( ) ( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 16
ISSN 2250-3153
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( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions
Definition of ( ) ( ) :
( ) ( )( )
( )( ) , ( )
( ) ( )( ) ( )( ) , ( )
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
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
Proof:
Consider operator ( ) defined on the space of sextuples of continuous functions which satisfy
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 17
ISSN 2250-3153
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( ) ( )
( ) ( )( )
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
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
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
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
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
( ) ∫ [( )
( ) ( ( )) (( )( )
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 18
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( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
( ) ∫ [( )
( ) ( ( )) (( )( ) (
)( )( ( ( )) ( ))) ( ( ))] ( )
Where ( ) is the integrand that is integrated over an interval ( )
(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
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( ) ( ( )
( ) )
( )( )( )( )
( )( ) ( ( )( ) )
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
Analogous inequalities hold also for
(a) The operator ( ) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
Analogous inequalities hold also for
(b) 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 4
(c) The operator ( ) maps the space of functions satisfying 35,35,36 into itself .Indeed it is obvious that
( ) ∫ [( )
( ) ( ( )
( ) ( )( ) ( ))]
( )
( ( )( ) )
( )( )( )( )
( )( ) ( ( )( ) )
From which it follows that
( ( ) ) ( )( )
( )( )
( )( ) [(( )( )
) (
( )( )
)
( )( )]
( ) is as defined in the statement of theorem 1
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 19
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(d) 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 6
Analogous inequalities hold also for
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
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 20
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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
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 ( ) ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 21
ISSN 2250-3153
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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.
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
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
| ( ) ( )| ( )( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 22
ISSN 2250-3153
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( )( ) (( )( ) (
)( ) ( )( ) ( )
( )( )( )) ((( )
( ) ( )( ) ( )
( ) ( )( )))
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
(( )( ))
( )( )(( )
( )) (
)( )
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
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
| ( )
( )
| ∫ ( )( )
|
( )
( )| ( )( ) ( ) ( )( ) ( ) ( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 23
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∫ ( )( )|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )( )(
( ) ( ))|
( )
( )| ( )( ) ( ) ( )( ) ( )
( )
( )( )(
( ) ( )) (
)( )( ( )
( )) ( )( ) ( ) ( )( ) ( ) ( )
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
(( )( ))
( )( )(( )
( )) (
)( )
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
Analogous inequalities hold also for
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 35,35,36 into itself
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 24
ISSN 2250-3153
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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 (35,35,36) 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 28 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.
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
Analogous inequalities hold also for
It is now sufficient to take ( )
( )
( )( ) ( )
( )
( )( ) and to choose
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 25
ISSN 2250-3153
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( )( ) ( )
( ) 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 (36,35,36) 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 69 to 32 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.
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
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(( )( )( )( )
) ( )
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
Behavior of the solutions of equation 37 to 42
Theorem 2: If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
(a) )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) :
(b) By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations ( )( )( ( ))
( )( ) ( ) ( )
( ) and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations ( )( )( ( ))
( )( ) ( ) ( )
( ) and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) :-
(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
( )( ) (( )( ) ( )( )) ( )
( )( ) ( )( )
( ( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( ) ( ) ( )( )
( )( )(( )( ) ( )( ))
( )( )
( )( )
( )( ) )
( )( ) ( )
(( )( ) ( )( ))
( )( ) ( )( ) ( )
( )( ) (( )( ) ( )( ))
( )( )
( )( )(( )( ) ( )( ))
[ ( )( ) ( )( ) ]
( )( ) ( )
( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ):-
Where ( )( ) ( )
( )( )( ) (
)( )
( )( ) ( )
( ) ( )( )
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( )( ) ( )
( )( )( ) (
)( )
( )( ) (
)( ) ( )( )
Behavior of the solutions of equation 37 to 42 Theorem 2: If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
(d) )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )(( ) ) ( )( )(( ) ) ( )
( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots
(e) of the equations ( )( )( ( ))
( )
( ) ( ) ( )( )
and ( )( )( ( ))
( )
( ) ( ) ( )( ) and
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the
roots of the equations ( )( )( ( ))
( )
( ) ( ) ( )( )
and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :-
(f) 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
( )( ) (( )( ) ( )( )) ( )
( )( ) ( )( )
( ( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( ) ( ) ( )( )
( )( )(( )( ) ( )( ))
( )( )
( )( )
( )( ) )
( )( ) ( )
(( )( ) ( )( ))
( )( ) ( )( ) ( )
( )( ) (( )( ) ( )( ))
( )( )
( )( )(( )( ) ( )( ))
[ ( )( ) ( )( ) ]
( )( ) ( )
( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ):-
Where ( )( ) ( )
( )( )( ) (
)( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
( )( ) (
)( ) ( )( )
Behavior of the solutions of equation 37 to 42
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Theorem 2: If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
(a) )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )(( ) ) ( )
( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
(b) By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations ( )( )( ( ))
( )( ) ( ) ( )
( )
and ( )( )( ( ))
( )
( ) ( ) ( )( ) and
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the
roots of the equations ( )( )( ( ))
( )
( ) ( ) ( )( )
and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :-
(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
( )( ) (( )( ) ( )( )) ( )
( )( ) ( )( )
( ( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( ) ( ) ( )( )
( )( )(( )( ) ( )( ))
( )( )
( )( )
( )( ) )
( )( ) ( )
(( )( ) ( )( ))
( )( ) ( )( ) ( )
( )( ) (( )( ) ( )( ))
( )( )
( )( )(( )( ) ( )( ))
[ ( )( ) ( )( ) ]
( )( ) ( )
( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ):-
Where ( )( ) ( )
( )( )( ) (
)( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
( )( ) (
)( ) ( )( )
Behavior of the solutions of equation 37 to 12
Theorem 2: If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
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(d) ( )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )(( ) ) ( )( )(( ) ) ( )
( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) :
(e) By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations ( )( )( ( ))
( )( ) ( ) ( )
( )
and ( )( )( ( ))
( )
( ) ( ) ( )( ) and
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the
roots of the equations ( )( )( ( ))
( )
( ) ( ) ( )( )
and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) :-
(f) If we define ( )( ) ( )
( ) ( )( ) ( )
( ) by
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
and ( )( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
and analogously
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
and ( )( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) where ( )( ) ( )
( )
are defined by 59 and 64 respectively
Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities
(( )( ) ( )( )) ( )
( )( )
where ( )( ) is defined by equation 25
( )( ) (( )( ) ( )( )) ( )
( )( ) ( )( )
(( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( ) ( ) ( )( )
( )( )(( )( ) ( )( ))
[ ( )( )
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( )( ) ]
( )( ) )
( )( ) ( )
(( )( ) ( )( ))
( )( ) ( )( ) ( )
( )( ) (( )( ) ( )( ))
( )( )
( )( )(( )( ) ( )( ))
[ ( )( ) ( )( ) ]
( )( ) ( )
( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ):-
Where ( )( ) ( )
( )( )( ) (
)( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
( )( ) (
)( ) ( )( )
Behavior of the solutions of equation 37 to 42
Theorem 2: If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
(g) ( )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )(( ) ) ( )( )(( ) ) ( )
( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) :
(h) By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations ( )( )( ( ))
( )( ) ( ) ( )
( )
and ( )( )( ( ))
( )
( ) ( ) ( )( ) and
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the
roots of the equations ( )( )( ( ))
( )
( ) ( ) ( )( )
and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) :-
(i) If we define ( )( ) ( )
( ) ( )( ) ( )
( ) by
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
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and ( )( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) and analogously
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
and ( )( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) where ( )( ) ( )
( )
are defined by 59 and 65 respectively
Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities
(( )( ) ( )( )) ( )
( )( )
where ( )( ) is defined by equation 29
( )( ) (( )( ) ( )( )) ( )
( )( ) ( )( )
(( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( ) ( ) ( )( )
( )( )(( )( ) ( )( ))
[ ( )( )
( )( ) ]
( )( ) )
( )( ) ( )
(( )( ) ( )( ))
( )( ) ( )( ) ( )
( )( ) (( )( ) ( )( ))
( )( )
( )( )(( )( ) ( )( ))
[ ( )( ) ( )( ) ]
( )( ) ( )
( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ):-
Where ( )( ) ( )
( )( )( ) (
)( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
( )( ) (
)( ) ( )( )
Behavior of the solutions of equation 37 to 42
Theorem 2: If we denote and define
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
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(j) ( )( ) ( )
( ) ( )( ) ( )
( ) four constants satisfying
( )( ) (
)( ) ( )( ) (
)( )( ) ( )( )( ) ( )
( )
( )( ) (
)( ) ( )( ) (
)( )(( ) ) ( )( )(( ) ) ( )
( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) :
(k) By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the roots of the equations ( )( )( ( ))
( )( ) ( ) ( )
( )
and ( )( )( ( ))
( )
( ) ( ) ( )( ) and
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) :
By ( )( ) ( )
( ) and respectively ( )( ) ( )
( ) the
roots of the equations ( )( )( ( ))
( )
( ) ( ) ( )( )
and ( )( )( ( ))
( )
( ) ( ) ( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) :-
(l) If we define ( )( ) ( )
( ) ( )( ) ( )
( ) by
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
and ( )( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
and analogously
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
and ( )( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) where ( )( ) ( )
( ) are defined by 59 and 66 respectively
Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities
(( )( ) ( )( )) ( )
( )( )
where ( )( ) is defined by equation 25
( )( ) (( )( ) ( )( )) ( )
( )( ) ( )( )
(( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( ) ( ) ( )( )
( )( )(( )( ) ( )( ))
[ ( )( )
( )( ) ]
( )( ) )
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( )( ) ( )
(( )( ) ( )( ))
( )( ) ( )( ) ( )
( )( ) (( )( ) ( )( ))
( )( )
( )( )(( )( ) ( )( ))
[ ( )( ) ( )( ) ]
( )( ) ( )
( )( )
( )( )(( )( ) ( )( ) ( )( ))[ (( )( ) ( )( )) ( )( ) ]
( )( )
Definition of ( )( ) ( )
( ) ( )( ) ( )
( ):-
Where ( )( ) ( )
( )( )( ) (
)( )
( )( ) ( )
( ) ( )( )
( )( ) ( )
( )( )( ) (
)( )
( )( ) (
)( ) ( )( )
Proof : From 19,20,21,22,23,24 we obtain ( )
( )
( ) (( )( ) (
)( ) ( )( )( )) (
)( )( ) ( ) ( )
( ) ( )
Definition of ( ) :- ( )
It follows
(( )( )( ( ))
( )
( ) ( ) ( )( ))
( )
(( )
( )( ( )) ( )
( ) ( ) ( )( ))
From which one obtains
Definition of ( )( ) ( )
( ) :-
(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
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( )( ) ( )( )
( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
And so with the notation of the first part of condition (c) , we have
Definition of ( )( ) :-
( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
In a completely analogous way, we obtain
Definition of ( )( ) :-
( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
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 ( )( )
Proof : From 19,20,21,22,23,24 we obtain ( )
( )
( ) (( )( ) (
)( ) ( )( )( )) (
)( )( ) ( ) ( )
( ) ( )
Definition of ( ) :- ( )
It follows
(( )( )( ( ))
( )
( ) ( ) ( )( ))
( )
(( )
( )( ( )) ( )
( ) ( ) ( )( ))
From which one obtains
Definition of ( )( ) ( )
( ) :-
(d) For ( )( )
( )
( ) ( )( )
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) ( )
( )
In the same manner , we get
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
From which we deduce ( )( ) ( )( ) ( )
( )
(e) If ( )( ) ( )
( )
( )
( ) we find like in the previous case,
( )( )
( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
(f) 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
Definition of ( )( ) :-
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( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
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 ( ) ( )
( ) ( )
Analogously if ( )( ) (
)( ) ( )( ) ( )
( ) and then
( )( ) ( )
( )if in addition ( )( ) ( )
( ) then ( ) ( )( ) ( ) This is an important consequence of the relation
between ( )( ) and ( )
( )
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
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
Definition of ( )( ) :-
From which we deduce ( )( ) ( )( ) ( )
( )
(b) If ( )( ) ( )
( )
( )
( ) we find like in the previous case,
( )( )
( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
(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
Definition of ( )( ) :-
( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
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 ( ) ( )
( ) ( )
Analogously if ( )( ) (
)( ) ( )( ) ( )
( ) and then
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( )( ) ( )
( )if in addition ( )( ) ( )
( ) then ( ) ( )( ) ( ) This is an important consequence of the relation
between ( )( ) and ( )
( )
Proof : From 19,20,21,22,23,24 we obtain
( )
( )
( ) (( )( ) (
)( ) ( )( )( )) (
)( )( ) ( ) ( )
( ) ( )
Definition of ( ) :- ( )
It follows
(( )( )( ( ))
( )
( ) ( ) ( )( ))
( )
(( )
( )( ( )) ( )
( ) ( ) ( )( ))
From which one obtains
Definition of ( )( ) ( )
( ) :-
(d) For ( )( )
( )
( ) ( )( )
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) ( )
( )
In the same manner , we get
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
From which we deduce ( )( ) ( )( ) ( )
( )
(e) If ( )( ) ( )
( )
( )
( ) we find like in the previous case,
( )( )
( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
(f) 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
Definition of ( )( ) :-
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( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
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 ( )( )
Proof : From 19,20,21,22,23,24 we obtain
( )
( )
( ) (( )( ) (
)( ) ( )( )( )) (
)( )( ) ( ) ( )
( ) ( )
Definition of ( ) :- ( )
It follows
(( )( )( ( ))
( )
( ) ( ) ( )( ))
( )
(( )
( )( ( )) ( )
( ) ( ) ( )( ))
From which one obtains
Definition of ( )( ) ( )
( ) :-
(g) For ( )( )
( )
( ) ( )( )
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) ( )
( )
In the same manner , we get
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
From which we deduce ( )( ) ( )( ) ( )
( )
(h) If ( )( ) ( )
( )
( )
( ) we find like in the previous case,
( )( )
( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 38
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( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
(i) 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
Definition of ( )( ) :-
( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
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 ( )( )
Proof : From 19,20,21,22,23,24 we obtain
( )
( )
( ) (( )( ) (
)( ) ( )( )( )) (
)( )( ) ( ) ( )
( ) ( )
Definition of ( ) :- ( )
It follows
(( )( )( ( ))
( )
( ) ( ) ( )( ))
( )
(( )
( )( ( )) ( )
( ) ( ) ( )( ))
From which one obtains
Definition of ( )( ) ( )
( ) :-
(j) For ( )( )
( )
( ) ( )( )
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ) ( )
( )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 39
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In the same manner , we get
( )( ) ( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
, ( )( ) ( )( ) ( )( )
( )( ) ( )( )
From which we deduce ( )( ) ( )( ) ( )
( )
(k) If ( )( ) ( )
( )
( )
( ) we find like in the previous case,
( )( )
( )( ) ( )( )( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
( )( ) ( )( )( )( )
[ ( )( )(( )( ) ( )( )) ]
( )( ) [ ( )( )(( )( ) ( )( )) ]
( )( )
(l) 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
Definition of ( )( ) :-
( )( ) ( )( ) ( )
( ), ( )( ) ( )
( )
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 ( )( )
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
Theorem 3: If ( )( ) (
)( ) are independent on , and the conditions (with the notations 25,26,27,28)
( )( )(
)( ) ( )( )( )
( )
( )( )(
)( ) ( )( )( )
( ) ( )( )( )
( ) ( )( )( )
( ) ( )( )( )
( )
( )( )(
)( ) ( )( )( )
( ) ,
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 40
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( )( )(
)( ) ( )( )( )
( ) ( )( )( )
( ) ( )( )( )
( ) ( )( )( )
( )
( )( ) ( )
( ) as defined by equation 25 are satisfied , then the system
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
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
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
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
( )( ) [(
)( ) ( )( )( )]
( )( ) [(
)( ) ( )( )( )]
( )( ) [(
)( ) ( )( )( )]
( )( ) (
)( ) ( )( )( )
( )( ) (
)( ) ( )( )( )
( )( ) (
)( ) ( )( )( )
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)
( )( ) [(
)( ) ( )( )( )]
( )( ) [(
)( ) ( )( )( )]
( )( ) [(
)( ) ( )( )( )]
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 41
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( )( ) (
)( ) ( )( )(( ))
( )( ) (
)( ) ( )( )(( ))
( )( ) (
)( ) ( )( )(( ))
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 the system (19 to 28)
( )( ) [(
)( ) ( )( )( )]
( )( ) [(
)( ) ( )( )( )]
( )( ) [(
)( ) ( )( )( )]
( )( ) (
)( ) ( )( )( )
( )( ) (
)( ) ( )( )( )
( )( ) (
)( ) ( )( )( )
has a unique positive solution , which is an equilibrium solution for the system (19 to 24)
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
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( ) ( )( )(
)( ) ( )( )( )
( ) ( )( )(
)( )( ) ( )( )(
)( )( ) ( )( )( )(
)( )( ) 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
( )( )
[( )( ) (
)( )( )]
, ( )( )
[( )( ) (
)( )( )]
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
( )( )
[( )( ) (
)( )( )]
, ( )( )
[( )( ) (
)( )( )]
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
( )( )
[( )( ) (
)( )( )]
, ( )( )
[( )( ) (
)( )( )]
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
( )( )
[( )( ) (
)( )( )]
, ( )( )
[( )( ) (
)( )( )]
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
( )( )
[( )( ) (
)( )( )]
, ( )( )
[( )( ) (
)( )( )]
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
( ) ( )( )(
)( ) ( )( )( )
( )
[( )( )(
)( )( ) ( )( )(
)( )( )] ( )( )( )(
)( )( )
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 (( )
)
(i) 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 (( )
)
(j) 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 ( )
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
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
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
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
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
Finally we obtain the unique solution of 89 to 94
(( )
) , (
) and
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( )( )
[( )( ) (
)( )( )]
,
( )( )
[( )( ) (
)( )( )]
( )( )
[( )( ) (
)( )(( ) )] ,
( )( )
[( )( ) (
)( )(( ) )]
Obviously, these values represent an equilibrium solution of 69,20,32,22,23,32
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
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
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 :-
,
( )( )
(
) ( )( ) ,
( )( )
( ( )
)
taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
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
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 45
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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 49 to 24
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
ASYMPTOTIC STABILITY ANALYSIS
Theorem 5: 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
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
ASYMPTOTIC STABILITY ANALYSIS
Theorem 6: 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 69 to 32
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ( )( )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
((
)( ) ( )( )) ( )
( ) ∑ ( ( )( ) )
The characteristic equation of this system is
(( )( ) ( )( ) ( )
( )) (( )( ) ( )( ) ( )
( ))
[((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )]
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((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( )) ( )
( )
(( )( ) ( )( ) ( )
( )) (( )( )( )
( ) ( )
( )( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
+
(( )( ) ( )( ) ( )
( )) (( )( ) ( )( ) ( )
( ))
[((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )]
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( )) ( )
( )
(( )( ) ( )( ) ( )
( )) (( )( )( )
( ) ( )
( )( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
+
(( )( ) ( )( ) ( )
( )) (( )( ) ( )( ) ( )
( ))
[((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )]
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( )) ( )
( )
(( )( ) ( )( ) ( )
( )) (( )( )( )
( ) ( )
( )( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
+
(( )( ) ( )( ) ( )
( )) (( )( ) ( )( ) ( )
( ))
[((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )]
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( ) ( )( ) ( )
( ))( )( )
( )( )( )
( ) )
((( )( ) ( )( ) ( )
( )) ( ) ( ) ( )
( ) ( ) ( ) )
((( )( )) ( (
)( ) ( )( ) ( )
( ) ( )( )) ( )( ))
International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013 47
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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].
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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].