15 / 23
its neighboring points of local extremum,
x
0
j
and
x
0
j
+1
,
ultimately satisfies
the estimates
m
≤
(
x
0
j
+1
−
x
0
j
)
F
y
(
x
0
j
)
k
−
1
≤
M.
(16)
J
Put
E
+
= Φ
E
Ω
1
+
.
It is a compact subset of the set
E
defined
by (6) and lying inside
Ω
+
. Put
m
= inf
{
ξ
1
(
a
) :
a
∈
E
+
, a
1
= 0
}
>
0;
M
= sup
{
ξ
1
(
a
) :
a
∈
E
+
, a
1
= 0
}
<
∞
.
Let
y
(
x
)
be a typical solution to equation (1),
x
0
j
and
x
0
j
+1
be
neighboring points of its local extremum. We can choose positive constants
A
and
B
such that the function
z
(
x
) =
Ay
(
Bx
+
x
0
j
)
is a solution to equation
(1) with
p
0
= 1
and its data at zero correspond to some point in
E,
i.e.
F
z
(0) = 1
.
It is sufficient for this to find a positive solution to the system
A
k
−
1
=
B
4
p
0
;
3
X
m
=0
AB
m
y
(
m
)
(
x
0
j
)
1
m
(
k
−
1)+4
= 1
,
namely
A
=
3
X
m
=0
y
(
m
)
(
x
0
j
)
p
m
4
0
1
m
(
k
−
1)+4
−
4
;
B
=
3
X
m
=0
ρy
(
m
)
(
x
0
j
)
1
m
(
k
−
1)+4
!
−
(
k
−
1)
=
F
y
(
x
j
)
−
(
k
−
1)
.
Moreover, for local extrema with sufficiently large numbers, the point
defined in
R
4
by the data of the function
z
(
x
)
at zero belongs to
E
+
.
Hence
the first positive point L of local extremum of
z
(
x
)
belongs to
[
m, M
]
,
whence the difference
x
0
j
+1
−
x
0
j
is equal to
LB
and satisfies (16).
I
Lemma 11.
For any
k >
1
and
p
0
>
0
there exists a constant
θ >
0
such that local extrema of any typical solution
y
(
x
)
to equation
(1), ultimately satisfy the inequality
y
(
x
0
j
)
≥
θF
y
(
x
0
j
)
4
.
J
Let
y
(
x
)
be a typical solution to equation (1) and
x
0
j
be its
local extremum point with sufficiently large number. Put
θ
= inf
{|
a
0
|
:
a
∈
E
+
, a
1
= 0
}
>
0
and choose a constant
λ >
0
such that the data
at zero for the solution
z
(
x
) =
λ
4
y
(
λ
k
−
1
x
+
x
0
j
)
correspond to some
point in
E
+
.
Then
F
z
(0) = 1
and
|
z
(0)
| ≥
θ.
Since
z
(0) =
λ
4
y
(
x
0
j
)
and
F
z
(0) =
λF
y
(
x
0
j
)
,
the lemma is proved.
I
Remark 1.
For typical solutions to
(1)
with their corresponding curves
lying completely in
Ω
+
,
the statements of Lemmas 8, 10, and 11 hold in
the whole domain, not only ultimately.
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2
17