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1534 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
∗
t
ϕ s β s t g s q s p s s
t0 3
∫
() (, ) ()(() − ()) Δ 0on [ t
4
4
, ∞ ) . And then there are
two cases of Theorem 3. As in the proof of Theorem 3, if
the case 1 holds, we get a contradiction with (33) ; if the
case 2 holds, we obtain lim xt ( ) = 0 . This completes the
t→∞
proof.
In Theorem 4, let g (t) =1 and H ( ts , ) = ( t− s) m
, we
have the following result.
Corollary 6 Assume that (H1) - (H5) hold, and m ≥ 1 ,
for all sufficiently largeT ∗ , we have
1 t
m
∗
limsup ( ) ( , )( ( ) ( ))
m ∫ t−s β sT qs − ps Δ s=∞.
t→∞
t0
t
Then every solution of (5) is either oscillatory on [ t , ∞)
0 T
or tends to zero.
IV. EXAMPLES
In this section, we give some examples to illustrate our
main results. Define
⎧1 , δ ( t) ≥ t,
ξ () t = ⎨ γ
⎩ρ
(, tt), δ() t ≤ t.
0
∗
∞
1 γ
β (, tT )
Note that ∫ Δ t ( α( t))
=∞, implies lim = 1.
t0
t→∞
ξ () t
Example 1 Consider the nonlinear neutral perturbed
dynamic equation
( t (( x( t) ± x( τ ( t))) ) )
t + 1
Δ
+ F( tx , ( δ( t))) = Gtx ( , ( δ( t)), x),
γ−1 1
Δ γ Δ
(35)
for t ∈[1, ∞)
T
, where γ is the quotient of odd positive
integers. Let
α
k(1 + δ ( t)) 1
t δ () t ξ()
t t
γ
γ−1 2 γ
() t = t , F(, t u) = ( + + u ) u ,
2 γ
4
γ
γ+
2
1 k(1 + δ ( t))
u
ct () = , Gtuv (, , ) =
,
2 γ
2 2
t+ 1 2 t δ () t ξ()( t u + v + 1)
where k is a positive constant. Then
2
Qt () = k 2 t ξ () t .
t0 t0
1 γ γ 1 γ
Since ∫ ∞ Δ t ( α( t))
= ∫ ∞ Δ t t
−
=∞, hence the conditions
(H1) - (H5) are clearly satisfied. And,
t
∗
α()
s
limsup ∫ ( sβ
(, sT ) gsQs () () − ) Δs
t→∞
T
γ+
1 γ
( γ + 1) s
k 1
t Δs
= ( − )limsup ,
γ + 1 ∫ =∞
t→∞
T
2 ( γ + 1)
s
t
∗
α()
s
limsup ∫ ( sβ
( sT , ) gs ( )( qs ( ) − ps ( )) − ) Δs
t→∞
T
γ+
1 γ
( γ + 1) s
k 1
t Δs
= ( − )limsup ,
γ + 1 ∫ =∞
t→∞
T
2 ( γ + 1)
s
γ + 1
if k > 2( γ + 1) . Thus it follows from Corollary 1 that
every solution of (35) + is oscillatory on [1, ∞)
T
if k >
γ + 1
2( γ 1) ,
+ and it follows from Corollary 4 that every
solution of (35) − is either oscillatory on [1, ∞)
T
or tends
1
to zero if k > 2( γ 1) γ +
+ .
Example 2 Consider the nonlinear neutral perturbed
dynamic equation
1
t x t − x τ t
2 + sin t
Δ
+ F( t, x( δ( t))) = G( t, x( δ( t)), x ).
23 Δ 53 Δ
( (( ( ) ( ( ))) ) )
2
(36)
23 2
for t ∈[2, ∞)
T
, where α() t = t , γ=53, c() t = 1( 2+
sin t)
Let
Ftu 1 4 2 5 3
(, ) = ( + ) ,
tξ
() t
t + u u
and
11 3
1 u
Gtuv (, , ) =
.
2 4
2 tξ
( t) ( u + v + 2)
Then qt () − pt () = 12 tξ
() t . The conditions (H1) - (H5)
are clearly satisfied. For all t > s ≥ 2 , let m=2, we have
1 t
2
∗
limsup
2 ∫ ( t− s) β ( sT , )( qs ( ) − ps ( )) Δs
t→∞
2
t
2
1 t ( t−
s)
= limsup
t
2 ∫ Δs
→∞
2
t 2s
1 t s t 1 t−
2
= limsup[ s s ] .
t→∞
2 ∫ Δ +
2 ∫ Δ − =∞
2
t 2 2s t
Thus it follows from Corollary 6 that every solution of
(36) is either oscillatory on [2, ∞)
T
or tends to zero.
IV. CONCLUSIONS
To investigate the oscillatory and asymptotic behavior
for a certain class of second order nonlinear neutral
perturbed dynamic equations on time scales. This paper
proposed some new sufficient conditions for oscillation
of such dynamic equations on time scales were
established. The results not only improve and extend
some known results in the literature, but also unify the
oscillation of second order nonlinear neutral perturbed
differential equations and second order nonlinear neutral
perturbed difference equations. In particular, the results
are essentially new under the relaxed conditions for the
parameter function.
ACKNOWLEDGMENT
if
k
γ + 1
> 2( γ + 1) . Also,
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