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5. Defined on the whole axis periodic oscillatory solutions. All of them
can be received from one, say
z
(
x
)
,
by the relation
y
(
x
) =
λ
4
z
(
λ
k
−
1
x
+
+
x
0
)
with arbitrary
λ >
0
and
x
0
.
So, there exists such a solution with any
maximum
h >
0
and with any period
T >
0
,
but not with any pair
(
h, T
)
.
6–9. Defined on bounded intervals
(
b
0
, b
00
)
solutions with the power
asymptotic behavior near the boundaries of the domain (with the independent
signs
±
):
y
(
x
)
∼ ±
C
4
k
(
p
(
b
0
))(
x
−
b
0
)
−
4
k
−
1
,
x
→
b
0
+ 0;
y
(
x
)
∼ ±
C
4
k
(
p
(
b
00
)) (
b
00
−
x
)
−
4
k
−
1
, x
→
b
00
−
0
.
10–11. Defined on semi-axes
(
−∞
, b
)
solutions which are oscillatory
as
x
→ −∞
and have the power asymptotic behavior near the right
boundary of the domain:
y
(
x
)
∼ ±
C
4
k
(
p
(
b
))(
b
−
x
)
−
4
k
−
1
, x
→
b
−
0
.
For
each solution a finite limit of the absolute values of its local extrema exists
as
x
→ −∞
.
12–13. Defined on semi-axes
(
b,
+
∞
)
solutions which are oscillatory as
x
→
+
∞
and have the power asymptotic behavior near the left boundary of
the domain:
y
(
x
)
∼ ±
C
4
k
(
p
(
b
))(
x
−
b
)
−
4
k
−
1
, x
→
b
+ 0
.
For each solution
a finite limit of the absolute values of its local extrema exists as
x
→
+
∞
.
Asymptotic classification of the solutions to the third-order equa-
tion (3).
In this section previously obtained results on the asymptotic
behavior of solutions to equation
(3)
are formulated
[7
,
28]
.
Theorem 4.
Suppose
k >
1
,
and
p
(
x
)
is a globally defined positive
continuous function with positive limits
p
∗
and
p
∗
as
x
→ ±∞
.
Then any
nontrivial non-extensible solution to
(3)
is either
(
Fig. 5
):
1–2
)
a Kneser solution on a semi-axis
(
b,
+
∞
)
satisfying
y
(
x
) =
±
C
3
k
(
p
(
b
)) (
x
−
b
)
−
3
k
−
1
(1 +
o
(1))
as
x
→
b
+ 0
,
y
(
x
) =
±
C
3
k
(
p
∗
)
x
−
3
k
−
1
(1 +
o
(1))
as
x
→
+
∞
,
where
C
3
k
(
p
) =
3(
k
+ 2)(2
k
+ 1)
p
(
k
−
1)
3
1
k
−
1
;
3
)
an oscillating, in both directions, solution on a semi-axis
(
−∞
, b
)
satisfying
,
at its local extremum points
,
|
y
(
x
0
)
|
=
|
x
0
|
−
3
k
−
1
+
o
(1)
as
x
0
→ −∞
,
|
y
(
x
0
)
|
=
|
b
−
x
0
|
−
3
k
−
1
+
o
(1)
as
x
0
→
b
+ 0;
4–5
)
an oscillating near the right boundary and non-vanishing near the
left one solution on a bounded interval
(
b
0
, b
00
)
satisfying
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2
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