Abstract
UDC 517.9
We consider the balanced pantograph equation (BPE)
y
′
(
x
)
+
y
(
x
)
=
∑
k
=
1
m
p
k
y
(
a
k
x
)
,
where
a
k
,
p
k
>
0
and
∑
k
=
1
m
p
k
=
1.
It is known that if
K
=
∑
k
=
1
m
p
k
ln
a
k
≤
0
then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for
K
>
0
these solutions exist. In the present paper, we deal with a BPE of <em>mixed type</em>, i.e.,
a
1
<
1
<
a
m
,
and prove that, in this case, the BPE has a nonconstant solution
y
and that
y
(
x
)
∼
c
x
σ
as
x
→
∞
,
where
c
>
0
and
σ
is the unique positive root of the characteristic equation
P
(
s
)
=
1
-
∑
k
=
1
m
p
k
a
k
-
s
=
0.
We also show that
y
is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as
x
→
∞
.
Publisher
SIGMA (Symmetry, Integrability and Geometry: Methods and Application)
Reference16 articles.
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