Affiliation:
1. College of Information and Statistics Guangxi University of Finance and Economics Nanning, Guangxi 530003 P.R. CHINA
Abstract
Abstract
In this paper, we study the oscillation criteria of the following higher order nonlinear delay dynamic equation
R
n
△
(
t
,
x
(
t
)
)
+
b
(
t
)
|
R
n
−
1
△
(
t
,
x
(
t
)
)
|
γ
−
1
R
n
−
1
△
(
t
,
x
(
t
)
)
+
q
(
t
)
f
(
|
x
(
τ
(
t
)
)
|
γ
−
1
x
(
τ
(
t
)
)
)
=
0
$$R_n^\triangle (t, x(t))+b(t)|R_{n-1}^\triangle(t, x(t))|^{\gamma-1} R_{n-1}^\triangle(t, x(t))+q(t)f(|x(\tau(t))|^{\gamma-1}x(\tau(t)))=0$$
on an arbitrary time scale T with sup T = ∞, where n ≥ 2, γ > 0 is a constant, τ: T → T with τ(t) ≤ t and
lim
t
→
∞
τ
(
t
)
=
∞
$\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $
and
R
k
(
t
,
x
(
t
)
)
=
x
(
t
)
,
if
k
=
0
,
r
k
(
t
)
R
k
−
1
△
(
t
,
x
(
t
)
)
,
if
1
≤
k
≤
n
−
1
,
r
n
(
t
)
|
R
n
−
1
△
(
t
,
x
(
t
)
)
|
γ
−
1
R
n
−
1
△
(
t
,
x
(
t
)
)
,
if
k
=
n
,
$$R_k(t, x(t))=
\begin{cases}
x(t),&\text{if}\ \;\;k=0,
\\
r_k(t)R^\triangle_{k-1}(t, x(t)),&\text{if}\ \;\;1\leq k\leq n-1,
\\
r_n(t)|R^\triangle_{n-1}(t, x(t))|^{\gamma-1}
R^\triangle_{n-1}(t, x(t)),&\text{if}\ \;\;k= n,
\end{cases}$$
with rk
(t) (1 ≤ k ≤ n) are positive rd-continuous functions. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.
Cited by
5 articles.
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