Nonoscillation for second order sublinear dynamic equations on time scales
Consider the Emden–Fowler sublinear dynamic equation (0.1) x Δ Δ ( t ) + p ( t ) f ( x ( σ ( t ) ) ) = 0 , where p ∈ C ( T , R ) , T is a time scale, f ( x ) = ∑ i = 1 m a i x β i , where a i > 0 , 0 < β i < 1 , with β i the quotient of odd positive integers, 1 ≤ i ≤ m . When m = 1 , and T...
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Published in | Journal of computational and applied mathematics Vol. 232; no. 2; pp. 594 - 599 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Kidlington
Elsevier B.V
15.10.2009
Elsevier |
Subjects | |
Online Access | Get full text |
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Summary: | Consider the Emden–Fowler sublinear dynamic equation
(0.1)
x
Δ
Δ
(
t
)
+
p
(
t
)
f
(
x
(
σ
(
t
)
)
)
=
0
,
where
p
∈
C
(
T
,
R
)
,
T
is a time scale,
f
(
x
)
=
∑
i
=
1
m
a
i
x
β
i
, where
a
i
>
0
,
0
<
β
i
<
1
, with
β
i
the quotient of odd positive integers,
1
≤
i
≤
m
. When
m
=
1
, and
T
=
[
a
,
∞
)
⊂
R
,
(0.1) is the usual sublinear Emden–Fowler equation which has attracted the attention of many researchers. In this paper, we allow the coefficient function
p
(
t
)
to be negative for arbitrarily large values of
t
. We extend a nonoscillation result of Wong for the second order sublinear Emden–Fowler equation in the continuous case to the dynamic equation
(0.1). As applications, we show that the sublinear difference equation
Δ
2
x
(
n
)
+
b
(
−
1
)
n
n
−
c
x
α
(
n
+
1
)
=
0
,
0
<
α
<
1
,
has a nonoscillatory solution, for
b
>
0
,
c
>
α
, and the sublinear q-difference equation
x
Δ
Δ
(
t
)
+
b
(
−
1
)
n
t
−
c
x
α
(
q
t
)
=
0
,
0
<
α
<
1
,
has a nonoscillatory solution, for
t
=
q
n
∈
T
=
q
0
N
,
q
>
1
,
b
>
0
,
c
>
1
+
α
. |
---|---|
Bibliography: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0377-0427 1879-1778 |
DOI: | 10.1016/j.cam.2009.06.039 |