Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 237219, 14 pages
doi:10.1155/2011/237219
Research Article
Asymptotic Behavior of Solutions of Higher-Order
Dynamic Equations on Time Scales
Taixiang Sun,1 Hongjian Xi,2 and Xiaofeng Peng1
1
2
College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
Department of Mathematics, Guangxi College of Finance and Economics, Nanning,
Guangxi 530003, China
Correspondence should be addressed to Taixiang Sun, stx1963@163.com
Received 18 November 2010; Accepted 23 February 2011
Academic Editor: Abdelkader Boucherif
Copyright q 2011 Taixiang Sun et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We investigate the asymptotic behavior of solutions of the following higher-order dynamic
n
n−1
equation xΔ t ft, xt, xΔ t, . . . , xΔ t 0, on an arbitrary time scale T, where the
n
function f is defined on T × R . We give sufficient conditions under which every solution x of
n−1
this equation satisfies one of the following conditions: 1 limt → ∞ xΔ t 0; 2 there exist
n−1
0, such that limt → ∞ xt/ i0 ai hn−i−1 t, t0 1, where
constants ai 0 ≤ i ≤ n − 1 with a0 /
hi t, t0 0 ≤ i ≤ n − 1 are as in Main Results.
1. Introduction
In this paper, we investigate the asymptotic behavior of solutions of the following higherorder dynamic equation
n
n−1
xΔ t f t, xt, xΔ t, . . . , xΔ t 0,
1.1
on an arbitrary time scale T, where the function f is defined on T × Rn .
Since we are interested in the asymptotic and oscillatory behavior of solutions near
infinity, we assume that sup T ∞, and define the time scale interval t0 , ∞T {t ∈ T :
t ≥ t0 }, where t0 ∈ T. By a solution of 1.1, we mean a nontrivial real-valued function
n
x ∈ Crd Tx , ∞T , R, Tx ≥ t0 , which has the property that xΔ t ∈ Crd Tx , ∞T , R and
satisfies 1.1 on Tx , ∞T , where Crd is the space of rd-continuous functions. The solutions
vanishing in some neighborhood of infinity will be excluded from our consideration.
A solution x of 1.1 is said to be oscillatory if it is neither eventually positive nor eventually
negative, otherwise it is called nonoscillatory.
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Advances in Difference Equations
The theory of time scales, which has recently received a lot of attention, was introduced
by Hilger’s landmark paper 1
in order to create a theory that can unify continuous
and discrete analysis. The cases when a time scale is equal to the real numbers or to the
integers represent the classical theories of differential and of difference equations. Many other
interesting time scales exist, and they give rise to many applications see 2
. Not only the
new theory of the so-called “dynamic equations” unifies the theories of differential equations
and difference equations but also extends these classical cases to cases “in between,” for
example, to the so-called q-difference equations when T qN0 , which has important
applications in quantum theory see 3
.
On a time scale T, the forward jump operator, the backward jump operator, and the
graininess function are defined as
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t},
μt σt − t,
1.2
respectively. We refer the reader to 2, 4
for further results on time scale calculus. Let p ∈
0, for all t ∈ T, then the delta exponential function ep t, t0 is
Crd T, R with 1 μtpt /
defined as the unique solution of the initial value problem
yΔ pty,
yt0 1.
1.3
In recent years, there has been much research activity concerning the oscillation and
nonoscillation of solutions of various equations on time scales, and we refer the reader to
5–18
.
Recently, Erbe et al. 19–21
considered the asymptotic behavior of solutions of the
third-order dynamic equations
Δ Δ
Δ
ptfxt 0,
at rtx t
xΔΔΔ t ptxt 0,
Δ γ Δ
Δ
ft, xt 0,
at rtx t
1.4
respectively, and established some sufficient conditions for oscillation.
Karpuz 22
studied the asymptotic nature of all bounded solutions of the following
higher-order nonlinear forced neutral dynamic equation
n
xt Atxαt
Δ f t, x βt , x γt ϕt.
1.5
Chen 23
derived some sufficient conditions for the oscillation and asymptotic
behavior of the nth-order nonlinear neutral delay dynamic equations
γ Δ
Δn−1 α−1
n−1
λFt, xδt 0,
atΨxt xt ptxτt
xt ptxτtΔ
1.6
Advances in Difference Equations
3
on an arbitrary time scale T. Motivated by the above studies, in this paper, we study 1.1 and
give sufficient conditions under which every solution x of 1.1 satisfies one of the following
n−1
0, such
conditions: 1 limt → ∞ xΔ t 0; 2 there exist constants ai 0 ≤ i ≤ n − 1 with a0 /
n−1
that limt → ∞ xt/ i0 ai hn−i−1 t, t0 1, where hi t, t0 0 ≤ i ≤ n − 1 are as in Section 2.
2. Main Results
Let k be a nonnegative integer and s, t ∈ T, then we define a sequence of functions hk t, s as
follows:
⎧
⎪
⎪
⎨1
hk t, s t
⎪
⎪
⎩ hk−1 τ, sΔτ
if k 0,
if k ≥ 1.
2.1
s
To obtain our main results, we need the following lemmas.
Lemma 2.1. Let n be a positive integer, then there exists Tn > t0 , such that
hk1 t, t0 − hk t, t0 ≥ 1 for t ≥ Tn , 0 ≤ k ≤ n − 1.
2.2
Proof. We will prove the above by induction. First, if k 0, then we take T1 ≥ t0 2. Thus,
h1 t, t0 − h0 t, t0 t − t0 − 1 ≥ 1 for t ≥ T1 .
2.3
Next, we assume that there exists Tm > t0 , such that hk1 t, t0 − hk t, t0 ≥ 1 for t ≥ Tm and
0 ≤ k ≤ m with 0 ≤ m < n − 1, then
hm1 t, t0 − hm t, t0 t
hm τ, t0 − hm−1 τ, t0 Δτ
t0
Tm
hm τ, t0 − hm−1 τ, t0 Δτ ≥
t0
Tm
hm τ, t0 − hm−1 τ, t0 Δτ
Tm
t0
Tm
t
hm τ, t0 − hm−1 τ, t0 Δτ t
2.4
Δτ
Tm
hm τ, t0 − hm−1 τ, t0 Δτ t − Tm ,
t0
from which it follows that there exists Tm1 > Tm , such that hk1 t, t0 −hk t, t0 ≥ 1 for t ≥ Tm1
and 0 ≤ k ≤ m 1. The proof is completed.
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Advances in Difference Equations
Lemma 2.2 see 24
. Let p ∈ Crd T, 0, ∞, then
1
t
t
psΔs ≤ ep t, t0 ≤ e t0
psΔs
2.5
.
t0
Lemma 2.3 see 2
. Let y, p ∈ Crd T, 0, ∞ and A ∈ 0, ∞, then
yt ≤ A t
yτpτΔτ,
2.6
∀t ∈ T
t0
implies
yt ≤ Aep t, t0 ,
∀t ∈ T.
2.7
Lemma 2.4 see 2
. Let n be a positive integer. Suppose that x is n times differentiable on T. Let
n−1
α ∈ T κ and t ∈ T, then
xt ρn−1 t
n−1
k
n
hk t, αxΔ α hn−1 t, στxΔ τΔτ.
2.8
α
k0
Lemma 2.5 see 2
. Assume that f and g are differentiable on T with limt → ∞ gt ∞. If there
exists T > t0 , such that
gt > 0,
g Δ t > 0,
2.9
∀t ≥ T,
then
f Δ t
ft
r or ∞.
r or ∞ implies lim
t → ∞ g Δ t
t → ∞ gt
lim
2.10
Lemma 2.6 see 23
. Let x be defined on t0 , ∞T , and xt > 0 with xΔ t ≤ 0 for t ≥ t0 and not
eventually zero. If x is bounded, then
n
1 limt → ∞ xΔ t 0 for 1 ≤ i ≤ n − 1,
i
2 −1i1 xΔ t > 0 for all t ≥ t0 and 1 ≤ i ≤ n − 1.
n−i
Now, one states and proves the main results.
Theorem 2.7. Assume that there exists t1 > t0 , such that the function ft, u0 , . . . , un−1 satisfies
n−1
ft, u0 , . . . , un−1 ≤
pi t|ui |,
i0
∀t, u0 , . . . , un−1 ∈ t1 , ∞T × Rn ,
2.11
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5
where pi t 0 ≤ i ≤ n − 1 are nonnegative functions on t1 , ∞T and
lim eq t, t1 < ∞,
2.12
t→∞
with qt conditions:
n−1
i0
pi thn−i−1 t, t0 t ≥ t1 , then every solution x of 1.1 satisfies one of the following
1 limt → ∞ xΔ t 0,
n−1
0, such that
2 there exist constants ai 0 ≤ i ≤ n − 1 with a0 /
lim n−1
t→∞
i0
xt
ai hn−i−1 t, t0 1.
2.13
Proof. Let x be a solution of 1.1, then it follows from Lemma 2.4 that for 0 ≤ m ≤ n − 1,
x
Δm
t n−m−1
hk t, t1 x
Δkm
ρn−m−1 t
t1 hn−m−1 t, στxΔ τΔτ
n
for t ≥ t1 .
2.14
t1
k0
By 2.11 and Lemma 2.1, we see that there exists T > t1 , such that for t ≥ T and 0 ≤ m ≤ n − 1,
m Δ
x t ≤ hn−m−1 t, t0 n−1
t i Δ
Δkm
pi τx τΔτ .
t1 x
n−m−1
2.15
t1 i0
k0
Then we obtain
m Δ
x t ≤ hn−m−1 t, t0 Ft for t ≥ T, 0 ≤ m ≤ n − 1,
2.16
t n−1
i Ft A pi τxΔ τΔτ,
2.17
where
T i0
with
T
n−1
i Δkm
pi τxΔ τΔτ.
t1 x
A max
0≤m≤n−1
n−m−1
2.18
t1 i0
k0
Using 2.16 and 2.17, it follows that
Ft ≤ A t n−1
T i0
pi τhn−i−1 τ, t0 FτΔτ
for t ≥ T.
2.19
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Advances in Difference Equations
By Lemma 2.3, we have
Ft ≤ Aeq t, T ∀t ≥ T,
2.20
with qt n−1
i0 pi thn−i−1 t, t0 . Hence from 2.12, there exists a finite constant c > 0, such
that Ft ≤ c for t ≥ T . Thus, inequality 2.20 implies that
m Δ
x t ≤ hn−m−1 t, t0 c
for t ≥ T, 0 ≤ m ≤ n − 1.
2.21
By 1.1, we see that if t ≥ T , then
x
Δn−1
t x
Δn−1
t n−1
T − f τ, xτ, xΔ τ, . . . , xΔ τ Δτ.
2.22
T
Since condition 2.12 and Lemma 2.2 implies that
t n−1
lim
t→∞
2.23
pi τhn−i−1 τ, t0 Δτ < ∞,
T i0
we find from 2.11 and 2.21 that the sum in 2.22 converges as t → ∞. Therefore,
n−1
n−1
0, then it follows
limt → ∞ xΔ t exists and is a finite number. Let limt → ∞ xΔ t a0 . If a0 /
from Lemma 2.5 that
xt
n−1
lim xΔ t a0 ,
t → ∞ hn−1 t, t0 t→∞
2.24
lim
and x has the desired asymptotic property. The proof is completed.
Theorem 2.8. Assume that there exist functions pi : t0 , ∞T → 0, ∞ 0 ≤ i ≤ n, and
nondecreasing continuous functions gi : 0, ∞ → 0, ∞ 0 ≤ i ≤ n − 1, and t1 > t0 such that
n−1
ft, u0 , . . . , un−1 ≤
pi tgi
i0
|ui |
hn−i−1 t, t0 pn t
for t ≥ t1 ,
2.25
with
∞
pi tΔt Pi < ∞
for 0 ≤ i ≤ n,
t1
∞
ε
ds
n−1
i0
gi s
2.26
∞ for any ε > 0,
Advances in Difference Equations
7
then every solution x of 1.1 satisfies one of the following conditions:
1 limt → ∞ xΔ t 0,
n−1
2 there exist constants ai 0 ≤ i ≤ n − 1 with a0 / 0 such that
lim n−1
t→∞
i0
xt
ai hn−i−1 t, t0 1.
2.27
Proof. Let x be a solution of 1.1, then it follows from Lemma 2.4 that for 0 ≤ m ≤ n − 1,
xΔ t m
n−m−1
hk t, t1 xΔ
km
t1 ρn−m−1 t
hn−m−1 t, στxΔ τΔτ
n
for t ≥ t1 .
2.28
t1
k0
By Lemma 2.1 and 2.25, we see that there exists T > t1 , such that for t ≥ T and 0 ≤ m ≤ n − 1,
⎛ i ⎞
⎡
⎤ ⎤
⎡
Δ
n−m−1
n−1
t km
m τ
x
Δ
Δ
⎠ pn τ⎦Δτ ⎦.
t1 ⎣ pi τgi ⎝
x
x t ≤ hn−m−1 t, t0 ⎣
hn−i−1 τ, t0 t1 i0
k0
2.29
Then, we obtain
m Δ
x t ≤ hn−m−1 t, t0 Ft,
for t ≥ T, 0 ≤ m ≤ n − 1,
2.30
where
⎛ i ⎞
t n−1
xΔ τ
⎝
⎠Δτ,
Ft A pi τgi
hn−i−1 τ, t0 T i0
2.31
with
A max
0≤m≤n−1
⎛ Δi ⎞
T
n−1
x τ
Δkm
⎠Δτ Pn .
pi τgi ⎝
t1 x
hn−i−1 τ, t0 t1 i0
n−m−1
k0
2.32
Using 2.30 and 2.31, it follows that
Ft ≤ A t n−1
pi τgi FτΔτ
T i0
for t ≥ T.
2.33
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Advances in Difference Equations
Write
ut A t n−1
pi τgi FτΔτ
T i0
G y y
A
2.34
for t ≥ T,
ds
,
n−1
i0 gi s
2.35
then
Δ
Δ
Gut
u t
1
G hut 1 − huσ tdh
0
$ 1
#
n−1
pi tgi Ft
n−1
0
i0
n−1
gi hut 1 − huσ t
i0 pi tgi ut
n−1
i0 gi ut
≤
≤
i0
dh
2.36
n−1
pi t,
i0
from which it follows that
Gut ≤ GuT t n−1
T i0
pi τΔτ ≤ GuT n−1
Pi .
2.37
i0
Since limy → ∞ Gy ∞ and Gy is strictly increasing, there exists a constant c > 0, such that
ut ≤ c for t ≥ T . By 2.30, 2.33, and 2.34, we have
m Δ
x t ≤ hn−m−1 t, t0 c
for t ≥ T, 0 ≤ m ≤ n − 1.
2.38
It follows from 1.1 that if t ≥ T , then
x
Δn−1
t x
Δn−1
t n−1
T − f τ, xτ, xΔ τ, . . . , xΔ τ Δτ.
T
2.39
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Since 2.38 and condition 2.25 implies that
t n−1
f τ, xτ, xΔ τ, . . . , xΔ τ Δτ
T
⎛ i ⎞
⎤
⎡
t n−1
xΔ τ
⎠ pn τ⎦Δτ
≤ ⎣ pi τgi ⎝
hn−i−1 τ, t0 T i0
≤
2.40
n−1
Pi gi c Pn
i0
M < ∞,
we see that the sum in 2.39 converges as t → ∞. Therefore, limt → ∞ xΔ t exists and is a
n−1
0, then it follows from Lemma 2.5 that
finite number. Let limt → ∞ xΔ t a0 . If a0 /
n−1
lim
xt
lim xΔ t a0 ,
n−1
t → ∞ hn−1 t, t0 2.41
t→∞
and x has the desired asymptotic property. The proof is completed.
Theorem 2.9. Assume that there exist positive functions p : t0 , ∞T → 0, ∞, and nondecreasing
continuous functions gi : 0, ∞ → 0, ∞ 0 ≤ i ≤ n − 1, and t1 > t0 , such that
n−1
%
ft, u0 , . . . , un−1 ≤ pt gi
i0
|ui |
hn−i−1 t, t0 for t ≥ t1 ,
2.42
with
∞
∞
ε
ptΔt P < ∞,
t1
2.43
ds
&n−1
i0
gi s
∞,
for any ε > 0,
then every solution x of 1.1 satisfies one of the following conditions:
1 limt → ∞ xΔ t 0,
n−1
2 there exist constants ai 0 ≤ i ≤ n − 1 with a0 /
0, such that
lim n−1
t→∞
i0
xt
ai hn−i−1 t, t0 1.
2.44
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Proof. Arguing as in the proof of Theorem 2.8, we see that there exists T > t1 , such that for
t ≥ T and 0 ≤ m ≤ n − 1,
⎛ Δi ⎞ ⎤
⎡
n−1
n−m−1
t %
km
m x τ
Δ
Δ
⎠Δτ ⎦,
pτgi ⎝
t1 x
x t ≤ hn−m−1 t, t0 ⎣
hn−i−1 τ, t0 t1 i0
k0
2.45
from which we obtain
m Δ
x t ≤ hn−m−1 t, t0 Ft for t ≥ T, 0 ≤ m ≤ n − 1,
2.46
where
Ft A T
n−m−1
A max
0≤m≤n−1
Δ
x
⎛ Δi ⎞
x τ
⎠,
pτgi ⎝
hn−i−1 τ, t0 i0
t %
n−1
km
k0
⎛ Δi ⎞
T n−1
x τ
%
⎠.
pτgi ⎝
t0 hn−i−1 τ, t0 t1 i0
2.47
2.48
Using 2.46 and 2.47, it follows that
Ft ≤ A t %
n−1
pτgi FτΔτ
for t ≥ T.
2.49
for t ≥ T,
2.50
T i0
Write
ut A t %
n−1
pτgi FτΔτ
T i0
G y y
A
ds
,
&n−1
i0 gi s
2.51
then
Gut
Δ uΔ t
# n−1
%
1
G hut 1 − huσ tdh
0
ptgi Ft
i0
&n−1
≤
i0 ptgi ut
&n−1
i0 gi ut
pt,
$ 1
0
&n−1
i0
dh
gi hut 1 − huσ t
2.52
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11
from which it follows that
Gut ≤ GuT t
2.53
pτΔτ ≤ GuT P.
T
The rest of the proof is similar to that of Theorem 2.8, and the details are omitted. The proof
is completed.
Theorem 2.10. Assume that the function ft, u0 , . . . , un−1 satisfies
1 ft, u0 , . . . , un−1 ptFu0 , . . . , un−1 for all t, u0 , . . . , un−1 ∈ t0 , ∞T × Rn ,
∞
2 pt ≥ 0 for t ≥ t0 and t0 hn−1 τ, t0 pτΔτ ∞,
3 u0 Fu0 , . . . , un−1 > 0 for u0 /
0 and Fu0 , . . . , un−1 is continuous at u0 , 0, . . . , 0 with
u0 /
0,
then (1) if n is even, then every bounded solution of 1.1 is oscillatory; (2) if n is odd, then every
bounded solution xt of 1.1 is either oscillatory or tends monotonically to zero together with
i
xΔ t 1 ≤ i ≤ n − 1.
Proof. Assume that 1.1 has a nonoscillatory solution x on t0 , ∞, then, without loss of
generality, there is a t1 ≥ t0 , sufficiently large, such that xt > 0 for t ≥ t1 . It follows from
n
1.1 that xΔ t ≤ 0 for t ≥ t1 and not eventually zero. By Lemma 2.6, we have
lim xΔ t 0,
i
t→∞
−1
i1 Δn−i
x
for 1 ≤ i ≤ n − 1,
2.54
t > 0 ∀t ≥ t1 , 1 ≤ i ≤ n − 1,
and xt is eventually monotone. Also xΔ t > 0 for t ≥ t1 if n is even and xΔ t < 0 for t ≥ t1
if n is odd. Since xt is bounded, we find limt → ∞ xt c ≥ 0. Furthermore, if n is even, then
c > 0.
We claim that c 0. If not, then there exists t2 > t1 , such that
Fc, 0, . . . , 0
n−1
>0
F xt, xΔ t, . . . , xΔ t >
2
for t ≥ t2 ,
2.55
since F is continuous at c, 0, . . . , 0 by the condition 3. From 1.1 and 2.55, we have
xΔ t pt
n
Fc, 0, . . . , 0
≤ 0,
2
2.56
for t ≥ t2 .
Multiplying the above inequality by hn−1 t, t0 , and integrating from t2 to t, we obtain
t
t2
hn−1 τ, t0 xΔ τΔτ n
t
t2
hn−1 τ, t0 pτ
Fc, 0, . . . , 0
Δτ ≤ 0,
2
for t ≥ t2 .
2.57
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Since
t
n
i1
Δn−i
hn−1 τ, t0 x τΔτ ≥
−1 hn−i τ, t0 x τ
t2
i1
t
Δn
t2
≥
n
2.58
−1i hn−i t2 , t0 xΔ t2 −1n1 xt,
n−i
i1
we get
n1
A −1
xt t
hn−1 τ, t0 pτ
t2
Fc, 0, . . . , 0
Δτ ≤ 0,
2
for t ≥ t2 ,
2.59
∞
n−i
where A ni1 −1i hn−i t2 , t0 xΔ t2 . Thus, t2 hn−1 τ, t0 pτΔτ < ∞ since xt is bounded,
which gives a contradiction to the condition 2. The proof is completed.
3. Examples
Example 3.1. Consider the following higher-order dynamic equation:
xΔ t n
i
n−1
1
xΔ t
0,
βi h
n−i−1 t, t0 i0 t
3.1
where t ≥ t1 > t0 > 0 and βi > 1 0 ≤ i ≤ n − 1. Let pi t 1/tβi hn−i−1 t, t0 0 ≤ i ≤ n − 1 and
ft, u0 , . . . , un−1 n−1
1
ui
,
β
i
hn−i−1 t, t0 i0 t
3.2
then we have
n−1
ft, u0 , . . . , un−1 ≤
pi t|ui |,
∀t, u0 , . . . , un−1 ∈ t1 , ∞T × Rn ,
i0
t n−1
t
en−1
t, t1 en−1
βi t, t1 ≤ e 1
i0 pi thn−i−1
i0 1/t
i0
1/τ βi Δτ
3.3
< ∞,
by Example 5.60 in 4
. Thus, it follows from Theorem 2.7 that if x is a solution of 3.1
n−1
0, then there exist constants ai 0 ≤ i ≤ n − 1 with a0 with limt → ∞ xΔ t / 0, such that
n−1 /
limt → ∞ xt/ i0 ai hn−i−1 t, t0 1.
Example 3.2. Consider the following higher-order dynamic equation:
n−1
1
x t βi
i0 t
Δn
#
xΔ t
hn−i−1 t, t0 i
$αi
1
0,
tβ n
3.4
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where t > t0 > 0, αi ∈ 0, 1 0 ≤ i ≤ n − 1, and βi > 1 0 ≤ i ≤ n. Let gi u uαi 0 ≤ i ≤ n − 1,
pi t 1/tβi 0 ≤ i ≤ n, and
ft, u0 , . . . , un−1 αi
n−1
1
1
ui
β .
β
i
hn−i−1 t, t0 tn
i0 t
3.5
It is easy to verify that ft, u0 , . . . , un−1 satisfies the conditions of Theorem 2.8. Thus, it follows
n−1
0, then there exist constants ai 0 ≤ i ≤
that if x is a solution of 3.4 with limt → ∞ xΔ t /
n−1
0, such that limt → ∞ xt/ i0 ai hn−i−1 t, t0 1.
n − 1 with a0 /
Example 3.3. Consider the following higher-order dynamic equation:
#
$αi
i
n−1
xΔ t
1%
x t β
0,
t i0 hn−i−1 t, t0 Δn
where t > t0 > 0, αi ∈ 0, 1 0 ≤ i ≤ n − 1 with 0 <
i ≤ n − 1, pt 1/tβ , and
ft, u0 , . . . , un−1 n−1
n−1
%
1
i0
tβ
i0
3.6
αi < 1 and β > 1. Let gi u uαi 0 ≤
ui
hn−i−1 t, t0 αi
.
3.7
It is easy to verify that ft, u0 , . . . , un−1 satisfies the conditions of Theorem 2.9. Thus, it follows
n−1
0, then there exist constants ai 0 ≤ i ≤
that if x is a solution of 3.6 with limt → ∞ xΔ t /
n−1
0, such that limt → ∞ xt/ i0 ai hn−i−1 t, t0 1.
n − 1 with a0 /
Acknowledgment
This paper was supported by NSFC no. 10861002 and NSFG no. 2010GXNSFA013106, no.
2011GXNSFA018135 and IPGGE no. 105931003060.
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