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Suppose
x
+
<
+
∞
.
If either of the limits mentioned is finite, then all
other limits are finite, too, which is impossible for a maximally extended
solution. If all limits are infinite, they must have the same sign, which
contradicts to equation (1).
Now suppose
x
+
= +
∞
.
If either of the limits mentioned is non-
zero, then all limits must be infinite and have the same sign, which
contradicts to equation (1). If all these limits are zero, then
y
(
x
)
,
which is
ultimately positive, is decreasing to 0. Hence,
y
0
(
x
)
is ultimately negative
and increasing to 0. Similarly,
y
00
(
x
)
is ultimately positive and decreasing
to 0,
y
000
(
x
)
is ultimately negative and increasing to 0, which contradicts to
equation (1), since
y
(
x
)
is ultimately positive. These contradictions prove
the lemma.
I
Thus, no trajectory generated in
R
4
by a non-trivial solution to (1) can
ultimately rest in one of the sets
±
±
±
±
.
Corollary 1.
All maximally extended solutions to equation (1), as well
as their derivatives, are oscillatory near both boundaries of their domains.
Note that according to Lemma 3 we can distinguish two types of
asymptotic behavior of oscillatory solutions to equation (1), near the right
boundaries of their domains.
Definition 1.
An oscillatory solution to equation
(1)
is called typical
(
to the right
)
if ultimately this solution and its derivatives change their
signs according to scheme
(7)
, and non-typical if according to
(8)
.
Asymptotic Behavior of Typical Solutions.
This section is devoted
to the asymptotic behavior of typical (to the right) solutions to equation
(1), i.e. those generating trajectories ultimately lying inside
Ω
+
.
Since such a trajectory ultimately admits only the passages shown in (1),
there exists an increasing sequence of the points
x
000
0
< x
00
0
< x
0
0
< x
0
<
< x
000
1
< x
00
1
< x
0
1
< x
1
< . . .
such that
y
(
x
j
) =
y
0
(
x
0
j
) =
y
00
(
x
00
j
) =
=
y
000
(
x
000
j
) = 0 (
j
= 1
,
2
, . . .
)
,
and each point is a zero only for one of the
functions
y
(
x
)
, y
0
(
x
)
, y
00
(
x
)
, y
000
(
x
)
(Fig. 2). The points
x
j
, x
0
j
, x
00
j
, x
000
j
will
be called the
nodes
of the solution
y
(
x
)
.
For solutions generating trajectories completely lying inside
Ω
+
,
the
sequences of their nodes can be indexed by all integers (negative ones,
too).
Lemma 6.
Any typical solution
y
(
x
)
to equation
(1)
satisfies at its
nodes the following inequalities:
y
(
x
0
j
)
< y
(
x
000
j
+1
)
< y
(
x
00
j
+1
)
< y
(
x
0
j
+1
) ;
(9)
12
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2