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JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1529
examples presented in [8] to illustrate the main results are
valid only when T= and cannot be applied when T = .
Agarwal, O′Regan and Saker [3]considered (2) where γ ≥
1 is an odd positive integer and α Δ () t > 0, and established
some new oscillation criteria by employing the Riccati
transformation technique which can be applied on any
time scale T and improved the results in [6, 8].
Bohner and Saker [9] considered perturbed nonlinear
dynadynamic equation
Δ
{ ( t)(( x( t)) ) γ Δ
} F( t, x σ ) G( t, x σ Δ
α
+ = , x ). (3)
on a time scales T . Where γ > 0 is an odd positive
integer, using Riccati transformation techniques, they
obtained some sufficient conditions for the solution to be
oscillatory or converge to zero.
Following this trend, we shall study the oscillation for
the second-order neutral nonlinear perturbed dynamic
equations of the form
and
Δ γ
{ α( t)(( x( t) + c( t) x( τ( t))) ) }
Δ
+ F( tx , ( δ( t))) = Gtx ( , ( δ( t)), x),
Δ γ
{ α( t)(( x( t) − c( t) x( τ( t))) ) }
Δ
+ F( tx , ( δ( t))) = Gtx ( , ( δ( t)), x).
on an arbitrary time scales T , where γ is a quotient of
positive odd integer, α, c is a positive real-valued rdcontinuous
function defined on a time scales T and the
following conditions are satisfied:
+∞
1 γ
(H1) 0 ≤ct
( ) ≤ c < 1,
t ( α( t))
0 ∫ Δ =∞, for all t ∈ T ;
t
0
(H2) τ , δ : T → T satisfies τ () t ≤ t,
for all t ∈ T , either
δ () t ≥ t or δ () t ≤ t for all suffici-ently large t , and
lim τ ( t)
= lim δ ( t)
=∞;
t→∞
t→∞
(H3) pq , : T → are rd-continuous function, such that
qt () − pt () > 0, for all t ∈ T ;
2
(H4) F : T× →
and G : T× →
are functions
such that uF(, t u ) > 0and uG(, t u, v ) > 0, for all u ∈ −
{0} , v ∈ , t ∈ T ;
(H5) F(, tu) u γ
≥ qt (), and Gtuv (, , ) u γ
≤ pt () for all
uv∈ , −{0}
, t ∈ T .
We note that in all the above results the conditions
0 ≤ ct ( ) < 1, γ ≥ 1 and δ () t ≤ t are required. And some
authors utilized the kernel function ( t−
s
) m
Δ
Δ
(4)
(5)
or the general
class of functions H (, ts)
and obtained some oscillation
Δs
criteria, but the condition H ( ts , ) ≤ 0 is required. In this
paper the study is free of these restrictions and contains
the cases when 0< γ < 1, δ ( t) ≥t,
and − 1 < ct ( ) ≤ 0 . In
particular, by utilizing the general class of functions
H (, ts ), we shall derive some sufficient conditions for
the solutions of (4) and (5) to be oscillatory or converge
Δs
to zero when the condition H (, t s) ≤ 0is relaxed. Our
results are different from the existing results for neutral
equations on time scales that were established in [6-11,
13-17]. Also, we give some examples to illustrate the
main results.
Since we are interested in the oscillatory and asymptotic
behavior of solutions near infinity, we assume that
sup T = ∞ , and define the time scale interval [ t 0
, ∞)
T
by
[ t , ∞ ) : = [ t , ∞)
∩ T . By a solution of (4), we mean a
0 T 0
nontrivial real-valued function x (t) satisfying (4)
for t ≥ t . A solution x (t) of (4) is said to be oscillatory if
0
it is neither eventually positive nor eventually negative,
otherwise it is called nonoscillatory. Equation (4) is said
to be oscillatory if all its solutions are oscillatory. Our
attention is restricted to those solutions of (4) which exist
on some half line[ t 0
, ∞)
and satisfy sup{| xt ( ) |: t≥ t x
} > 0 ,
for any t ≥ t .
x 0
The paper is organized as follows. In next section, we
present some basic formula and lemma concerning the
calculus on time scales. In Section 3, we will use Riccati
transformation techniques and the general class of
functions H (, ts)
and give some sufficient conditions for
the oscillatory behavior of solutions of (4) and (5). In last
section, we give some examples to illustrate our main
results.
Through this paper, we let
γ
d () t = max[0, d()], t Q() t = ( q() t − p())(1 t −c( δ ())), t
+
∫ Δs
α () s
d () t = max[0, − d()], t ρ(, t u): =
,
−
() s
and for sufficiently largeT ∗ ,
δ () t 1 γ
u
t 1 γ
∫ Δs
α
u
1, δ ( t) t,
∗
⎧
≥
β (, tT ) = ⎨ γ ∗
⎩ρ
(, tT ), δ() t ≤ t.
II. SOME PRELIMINARIES ON TIME SCALES
A time scales T is an arbitrary nonempty closed subset
of the real numbers . In this paper, we only consider
time scales interval of form [ t 0
, ∞)
T
, on T we define the
forward jump operatorσ and the graininess μ by
{ }
σ (): t = inf s∈ T : s > t and μ(): t = σ () t − t.
A point t ∈ T with σ () t = tis called right-dense, while t
is referred to as being right-scattered if σ () t > t . A
function f : T → is said to be rd-continuous if it is
continu-ous at each right-dense point and if there exists a
left limit in all left-dense points. The ( Δ derivative) f Δ of
f is defined by
Δ f ( σ ( t)) − f( s)
f () t = lim , where Ut () = T \{ σ ()} t .
σ () t − s
s→t
sU ∈ () t
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