Abstract
This paper is concerned with oscillation of the second-order half-linear dynamic equation
(
r
(
t
)
(
x
Δ
)
γ
)
Δ
+
p
(
t
)
x
γ
(
t
)
=
0
,
on a time scale
T
where
γ is the quotient of odd positive integers,
r
(
t
)
and
p
(
t
)
are positive rd-continuous functions on
T
. Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375–387] and our results in the special cases when
T
=
R
and
T
=
Z
involve and improve some oscillation results for second-order differential and difference equations; and when
T
=
h
Z
,
T
=
q
N
0
and
T
=
N
0
2
, etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.