14 / 23
that
|
A
|
k
−
1
=
B
4
p
0
;
X
m
=0
,
2
,
3
AB
m
y
(
m
)
(
x
0
j
)
2
= 1
,
which follows from existence of a root
A
to the equation
y
(
x
0
j
)
2
A
2
+
y
00
(
x
0
j
)
2
p
−
1
0
|
A
|
k
+1
+
y
000
(
x
0
j
)
2
p
−
3
2
0
|
A
|
3
k
+1
2
= 1
.
The value
y
(
x
000
j
+1
)
y
(
x
0
j
)
1
4
is equal to this for
z
(
x
)
at
ξ
3
(
a
)
and
0
,
where
a
0
=
|
A
|
, a
1
= 0
, a
2
=
|
A
|
B
2
, a
3
=
|
A
|
B
3
,
i.e. equal to
ψ
103
(
a
)
−
1
4
.
Put
q
= sup
K
1
ψ
103
(
a
)
−
1
4
, Q
= inf
K
1
ψ
103
(
a
)
−
1
4
and obtain the statement of
the lemma for the first ratio. The same procedure can be used for others.
Then we just choose the minimum of 12 values of
q
and the maximum of
12 values of
Q.
I
Lemma 9.
The domain of any typical
(
to the right
)
solution
y
(
x
)
to
equation
(1)
is right-bounded. If
x
∗
is its right boundary, then
lim
x
→
x
∗
y
(
n
)
(
x
) = +
∞
,
n
= 0
,
1
,
2
,
3
.
(14)
J
It follows from Lemma 8 that the absolute values of the neighboring
local extrema of any typical solution for sufficiently large number, say for
j
≥
J,
satisfy the inequality
y
(
x
0
j
+1
)
≥
q
12
y
(
x
0
j
)
with some
q >
1
,
whence
y
(
x
0
j
)
≥
q
12(
j
−
J
)
|
y
(
x
0
J
)
|
.
(15)
In particular, this yields (14) for
n
= 0
.
Other
n
are treated similarly.
It is proved in [7] that there exists a constant
C >
0
depending only
on
k
and
p
0
such that all positive solutions to equation (1) defined on
a segment
[
a, b
]
satisfy the inequality
|
y
(
x
)
| ≤
C
|
b
−
a
|
−
4
k
−
1
.
The same
holds for negative ones. Hence the local extrema satisfy the estimate
y
(
x
0
j
)
≤
C
(
x
j
−
x
j
−
1
)
−
4
k
−
1
,
which yields, together with (15), the
inequality
(
x
j
−
x
j
−
1
)
≤
Q
−
3(
k
−
1)(
j
−
J
)
C
y
(
x
0
J
)
k
−
1
4
.
It follows from
Q >
1
that
Q
−
3(
k
−
1)
<
1
,
∞
X
j
=
J
(
x
j
−
x
j
−
1
)
<
∞
,
and
the domain is right-bounded.
I
Lemma 10.
For any
k >
1
there exists positive constants
m
≤
M
such that for any typical solution
y
(
x
)
to equation
(1)
the distance between
16
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2