Abstract
It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation
(
a
(
t
)
{
[
r
(
t
)
x
Δ
(
t
)
]
Δ
}
γ
)
Δ
+
f
(
t
,
x
(
τ
(
t
)
)
)
=
0
,
on a time scale
T
, where
γ
≥
1
is the quotient of odd positive integers,
a
and
r
are positive
r
d
-continuous functions on
T
, and the so-called delay function
τ
:
T
→
T
satisfies
τ
(
t
)
≤
t
for
t
∈
T
and
lim
t
→
∞
τ
(
t
)
=
∞
and
f
∈
C
(
T
×
R
,
R
)
. Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when
T
=
R
and
T
=
N
involve and improve some oscillation results for third order delay differential and difference equations; when
T
=
h
N
,
T
=
q
N
0
and
T
=
N
2
our oscillation results are essentially new. Some examples are given to illustrate the main results.