6 / 11
O.G. Styrt, A.P. Krishchenko
20
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
◄
We have
def
2
*
*
( ( )) = 2( ( ( )) ( ( )))
= 2(
( )).
t
y t
h t
G x t
d h G x
Since the
function
( )
y t
is of constant sign,
*
( )
( ;
).
t
y t
y
Consequently, the
trajectory
tends to the point
def
* *
2
= ( , )
.
p x y
Note that
0
{ }.
p
Ox
Besides,
U
implying
,
p U
{ } = .
p U Ox
Thus,
( )
= { }
t
t
p
Ox
and, therefore,
*
( ) = .
h p G
Also,
*
*
= ( ( )) = ( ) = .
lim
t
h
h t
h p G
►
We see that, if
is completely located outside the localizing set
and
*
x
is a
finite number, then the first case of Theorem 1 takes place.
Now suppose that
*
| |= .
x
Statement 3.
If
2
( , )
x y
and
def
*
= ( , )
0,
C h x y G then
2 ( ).
y
C x
◄
We have
2
2
2
= 2( ( , ) ( )) = 2 ( ( ))
2 ( ) ,
y h x y G x C x
C x
2
y
C
( ).
x
►
Proposition 1.
If
is completely located outside the localizing set
,
then
*
( ) <| |
2(
) ( )
x y
h G x
on
.
◄
On
,
we have
> ( )
y x
and, also,
*
> ,
h G
*
> 0,
h G
and, by Statement 3,
*
2(
) ( ).
y
h G x
►
Corollary 3.
If
is completely located outside the localizing set
,
then there
exists a number
> 0
C
such that
| |< ( )
y x C
on
once
0.
t
◄
The proof follows from Proposition 1 and decreasing of the function
h
on
.
►
Lemma 2.
If
is completely located outside the localizing set
and
*
=
x
*
( = ),
x
then there exist numbers
0
0
t
and
> 0
C
such that, for each
0
,
t t
we
have
( )
( )
t U C
( ( )
( )).
t U C
◄
By Corolla
r
y 3, there exists a number
> 0
C
such that
( ) < < ( )
x C y x C
on
once
0.
t
Suppose that
*
= .
x
Then
U
and, thus,
> ( )
y x
on
.
Besides, there
exists a number
0
0
t
such that > 0
x
on
once
0
.
t t
If
( , ) = ( ),
x y t
0
,
t t
then
> 0
x
and
( )< < ( )
x y x C
implying
( , )
( ).
x y U C
Now assume that
*
= .
x
Then
U
and, thus,
< ( )
y
x
on
.
Besides,
there exists a number
0
0
t
such that < 0
x
on
once
0
.
t t
If
( , ) = ( ),
x y t
0
,
t t
then < 0
x
and
( ) < < ( )
x C y
x
implying
( , )
( ).
x y U C
►
Lemma 3.
If the condition (6) holds, the trajectory
is completely located outside
the localizing set
,
and
*
= ,
x
then
*
*
= ,
h G
and the integral
*
*
0
( )
x
G G x dx
is
convergent.