Abstract
This paper concerns the oscillation of solutions to the second-order dynamic equation
(
r
(
t
)
x
Δ
(
t
)
)
Δ
+
p
(
t
)
x
Δ
(
t
)
+
q
(
t
)
f
(
x
σ
(
t
)
)
=
0
,
on a time scale
T
which is unbounded above. No sign conditions are imposed on
r
(
t
)
,
p
(
t
)
, and
q
(
t
)
. The function
f
∈
C
(
R
,
R
)
is assumed to satisfy
x
f
(
x
)
>
0
and
f
′
(
x
)
>
0
for
x
≠
0
. In addition, there is no need to assume certain restrictive conditions and also the both cases
∫
t
0
∞
Δ
t
r
(
t
)
=
∞
and
∫
t
0
∞
Δ
t
r
(
t
)
<
∞
are considered. Our results will improve and extend results in (Baoguo
et al.
in Can. Math. Bull. 54:580-592, 2011; Bohner
et al.
in J. Math. Anal. Appl. 301:491-507, 2005; Hassan
et al.
in Comput. Math. Anal. 59:550-558, 2010; Hassan
et al.
in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as
T
=
Z
,
T
=
h
Z
,
h
>
0
, or
T
=
{
t
:
t
=
q
k
,
k
∈
N
0
,
q
>
1
}
and the space of harmonic numbers
T
=
H
n
. Some examples illustrating the importance of our results are also included.
MSC:
34K11, 39A10, 39A99.