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 Peng
1
1
College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
2
Department of Mathematics, Guangxi College of Finance and Economics, Nanning,
Guangxi 530003, China
Correspondence should be addressed to Taixiang Sun, [email protected]
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
equation x
Δ
n
tft, xt,x
Δ
t, ,x
h
i
t, t
0
0 ≤ i ≤ n − 1 are as in Main Results.
1. Introduction
In this paper, we investigate the asymptotic behavior of solutions of the following higher-
order dynamic equation
x
Δ
n
t
f
t, x
t
,x
Δ
t
, ,x
Δ
n−1
t
t ∈ C
rd
T
x
, ∞
T
, R and
satisfies 1.1 on T
x
, ∞
T
, where C
rd
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.
2 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,” f or
example, to the so-called q-difference equations when T q
N
0
, which has important
applications in quantum theory see 3.
T, R with 1 μtpt
/
0, for all t ∈ T, then the delta exponential function e
p
t, t
0
is
defined as the unique solution of the initial value problem
y
Δ
p
t
y,
y
t
0
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
a
t
p
t
x
t
0,
a
t
r
t
x
Δ
t
Δ
n
f
t, x
β
t
,x
γ
t
ϕ
t
.
1.5
Chen 23 derived some sufficient conditions for the oscillation and asymptotic
behavior of the nth-order nonlinear neutral delay dynamic equations
a
t
α−1
xtptxτt
Δ
n−1
γ
Δ
λF
t, x
δ
t
0,
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
conditions: 1 lim
t →∞
x
Δ
n−1
t0; 2 there exist constants a
i
0 ≤ i ≤ n − 1 with a
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1ifk 0,
t
s
h
k−1
τ,s
Δτ if k ≥ 1.
2.1
To obtain our main results, we need the following lemmas.
Lemma 2.1. Let n be a positive integer, then there exists T
n
>t
0
, such that
h
k1
t, t
0
1
. 2.3
Next, we assume that there exists T
m
>t
0
, such that h
k1
t, t
0
− h
k
t, t
0
≥ 1fort ≥ T
m
and
0 ≤ k ≤ m with 0 ≤ m<n− 1, then
h
m1
t, t
0
− h
m
t, t
0
0
− h
m−1
τ,t
0
Δτ
t
T
m
h
m
τ,t
0
− h
m−1
τ,t
0
Δτ
≥
T
m
τ,t
0
− h
m−1
τ,t
0
Δτ t − T
m
,
2.4
from which it follows that there exists T
m1
>T
m
, such that h
k1
t, t
0
−h
k
t, t
0
≥ 1fort ≥ T
m1
and 0 ≤ k ≤ m 1. The proof is completed.
y
t
≤ A
t
t
0
y
τ
p
τ
Δτ, ∀t ∈ T
2.6
implies
y
t
≤ Ae
p
t, t
0
α
h
n−1
t, σ
τ
x
Δ
n
τ
Δτ.
2.8
Lemma 2.5 see 2. Assume that f and g are differentiable on T with lim
t →∞
gt∞.Ifthere
exists T>t
0
, such that
g
t
> 0,g
Δ
t
r
or ∞
.
2.10
Lemma 2.6 see 23. Let x be defined on t
0
, ∞
T
, and xt > 0 with x
Δ
n
t ≤ 0 for t ≥ t
0
and not
eventually zero. If x is bounded, then
1 lim
t →∞
x
Δ
i
t0 for 1 ≤ i ≤ n − 1,
2−1
i1
x
Δ
n−i
t > 0 for all t ≥ t
0
|
u
i
|
, ∀
t, u
0
, ,u
n−1
∈
t
1
, ∞
T
× R
n
,
2.11
Advances in Difference Equations 5
where p
i
t0 ≤ i ≤ n − 1 are nonnegative functions on t
1
, ∞
T
t0,
2 there exist constants a
i
0 ≤ i ≤ n − 1 with a
0
/
0, such that
lim
t →∞
x
t
n−1
i0
a
i
h
n−i−1
t, t
0
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
Δ
m
t, σ
τ
x
Δ
n
τ
Δτ for t ≥ t
1
.
2.14
By 2.11 and Lemma 2.1, we see that there exists T>t
1
, such that for t ≥ T and 0 ≤ m ≤ n − 1,
x
Δ
m
t
≤ h
p
i
τ
x
Δ
i
τ
Δτ
.
2.15
Then we obtain
x
Δ
m
t
τ
x
Δ
i
τ
Δτ,
2.17
with
A max
0≤m≤n−1
n−m−1
k0
x
Δ
km
Δτ.
2.18
Using 2.16 and 2.17, it follows that
F
t
≤ A
t
T
n−1
i0
p
i
τ
h
n−i−1
τ,t
0
F
τ
x
Δ
m
t
≤ h
n−m−1
t, t
0
c for t ≥ T, 0 ≤ m ≤ n − 1. 2.21
By 1.1,weseethatift ≥ T, then
x
Δ
n−1
t
x
Δ
n−1
T
−
i0
p
i
τ
h
n−i−1
τ,t
0
Δτ<∞,
2.23
we find from 2.11 and 2.21 that the sum in 2.22 converges as t →∞. Therefore,
lim
t →∞
x
Δ
n−1
t exists and is a finite number. Let lim
t →∞
x
Δ
n−1
ta
0
.Ifa
0
: t
0
, ∞
T
→ 0, ∞0 ≤ i ≤ n, and
nondecreasing continuous functions g
i
: 0, ∞ → 0, ∞0 ≤ i ≤ n − 1, and t
1
>t
0
such that
f
t, u
0
, ,u
n−1
≤
n−1
i0
p
i
p
i
t
Δt P
i
< ∞ for 0 ≤ i ≤ n,
∞
ε
ds
n−1
i0
g
i
s
∞ for any ε>0,
2.26
Advances in Difference Equations 7
then every solution x of 1.1 satisfies one of the following conditions:
1 lim
t →∞
x
Δ
n−1
t0,
t
n−m−1
k0
h
k
t, t
1
x
Δ
km
t
1
ρ
n−m−1
t
t
1
h
n−m−1
t, σ
t, t
0
⎡
⎣
n−m−1
k0
x
Δ
km
t
1
t
t
1
⎡
⎣
n−1
⎠
p
n
τ
⎤
⎦
Δτ
⎤
⎦
.
2.29
Then, we obtain
x
Δ
m
t
≤ h
n−m−1
t, t
x
Δ
i
τ
h
n−i−1
τ,t
0
⎞
⎠
Δτ,
2.31
with
A max
0≤m≤n−1
n−m−1
k0
x
Δ
i
τ
h
n−i−1
τ,t
0
⎞
⎠
Δτ P
n
.
2.32
Using 2.30 and 2.31, it follows that
F
t
≤ A
t
i0
p
i
τ
g
i
F
τ
Δτ for t ≥ T,
2.34
G
y
y
A
ds
n−1
i0
g
i
1 − h
u
σ
t
dh
n−1
i0
p
i
t
g
i
F
t
1
0
dh
u
t
n−1
i0
g
i
u
t
≤
n−1
i0
p
i
t
,
2.36
from which it follows that
G
u
i0
P
i
.
2.37
Since lim
y →∞
Gy∞ and Gy is strictly increasing, there exists a constant c>0, such that
ut ≤ c for t ≥ T.By2.30, 2.33,and2.34, we have
x
Δ
m
t
≤ h
n−m−1
t, t
0
c for t ≥ T, 0 ≤ m ≤ n − 1. 2.38
It follows from 1.1 that if t ≥ T, then
x
τ
Δτ.
2.39
Advances in Difference Equations 9
Since 2.38 and condition 2.25 implies that
t
T
f
τ,x
τ
,x
Δ
τ
, ,x
Δ
n−1
τ
τ
h
n−i−1
τ,t
0
⎞
⎠
p
n
τ
⎤
⎦
Δτ
≤
n−1
i0
P
i
g
i
n−1
t, t
0
lim
t →∞
x
Δ
n−1
t
a
0
,
2.41
and x has the desired asymptotic property. The proof is completed.
Theorem 2.9. Assume that there exist positive functions p : t
0
, ∞
T
→ 0, ∞, and nondecreasing
continuous functions g
i
: 0, ∞ → 0, ∞0 ≤ i ≤ n − 1, and t
1
>t
0
, such that
for t ≥ t
1
,
2.42
with
∞
t
1
p
t
Δt P<∞,
∞
ε
ds
n−1
i0
g
i
s
∞, for any ε>0,
2.43
1.
2.44
10 Advances in Difference Equations
Proof. Arguing as in the proof of Theorem 2.8, we see that there exists T>t
1
, such that for
t ≥ T and 0 ≤ m ≤ n − 1,
x
Δ
m
t
≤ h
n−m−1
t, t
0
⎡
⎣
n−m−1
k0
x
Δ
i
τ
h
n−i−1
τ,t
0
⎞
⎠
Δτ
⎤
⎦
,
2.45
from which we obtain
x
Δ
p
τ
g
i
⎛
⎝
x
Δ
i
τ
h
n−i−1
τ,t
0
⎞
⎠
,
2.47
τ
g
i
⎛
⎝
x
Δ
i
τ
h
n−i−1
τ,t
0
⎞
⎠
.
2.48
Using 2.46 and 2.47, it follows that
F
t
T
n−1
i0
p
τ
g
i
F
τ
Δτ for t ≥ T,
2.50
G
y
y
A
ds
n−1
i0
1 − h
u
σ
t
dh
n−1
i0
p
t
g
i
F
t
1
i
u
t
n−1
i0
g
i
u
t
p
t
,
2.52
Advances in Difference Equations 11
from which it follows that
G
u
t
0
, ,u
n−1
ptFu
0
, ,u
n−1
for all t, u
0
, ,u
n−1
∈ t
0
, ∞
T
× R
n
,
2 pt ≥ 0 for t ≥ t
0
and
∞
t
0
h
n−1
τ,t
0
pτΔτ ∞,
1
≥ t
0
,sufficiently large, such that xt > 0fort ≥ t
1
. It follows from
1.1 that x
Δ
n
t ≤ 0fort ≥ t
1
and not eventually zero. By Lemma 2.6, we have
lim
t →∞
x
Δ
i
t
0, for 1 ≤ i ≤ n − 1,
−1
i1
x
Δ
n−i
t
t
, ,x
Δ
n−1
t
>
F
c, 0, ,0
2
> 0fort ≥ t
2
,
2.55
since F is continuous at c, 0, ,0 by the condition 3.From1.1 and 2.55, we have
x
Δ
n
t
p
t
τ
Δτ
t
t
2
h
n−1
τ,t
0
p
τ
F
c, 0, ,0
2
Δτ ≤ 0, for t ≥ t
2
.
2.57
12 Advances in Difference Equations
Since
t
t
2
≥
n
i1
−1
i
h
n−i
t
2
,t
0
x
Δ
n−i
t
2
0
p
τ
F
c, 0, ,0
2
Δτ ≤ 0, for t ≥ t
2
,
2.59
where A
n
i1
−1
i
h
n−i
t
2
,t
0
x
Δ
n−i
Δ
i
t
h
n−i−1
t, t
0
0,
3.1
where t ≥ t
1
>t
0
> 0andβ
i
> 1 0 ≤ i ≤ n − 1.Letp
i
t1/t
β
i
h
n−i−1
t, t
0
0 ≤ i ≤ n − 1 and
f
0
, ,u
n−1
≤
n−1
i0
p
i
t
|
u
i
|
, ∀
t, u
0
, ,u
n−1
∈
t
1
t
t
1
n−1
i0
1/τ
β
i
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
with lim
t →∞
x
Δ
n−1
t
/
0, then there exist constants a
i
0 ≤ i ≤ n − 1 with a
0
/
0, such that
lim
t →∞
xt/
t
h
n−i−1
t, t
0
α
i
1
t
β
n
0,
3.4
Advances in Difference Equations 13
where t>t
0
> 0, α
i
∈ 0, 10 ≤ i ≤ n− 1,andβ
i
> 1 0 ≤ i ≤ n.Letg
i
uu
α
i
α
i
1
t
β
n
.
3.5
It is easy to verify that ft, u
0
, ,u
n−1
satisfies the conditions of Theorem 2.8. Thus, it follows
that if x is a solution of 3.4 with lim
t →∞
x
Δ
n−1
t
/
0, then there exist constants a
i
0 ≤ i ≤
n − 1 with a
0
/
0, such that lim
t
h
n−i−1
t, t
0
α
i
0,
3.6
where t>t
0
> 0,α
i
∈ 0, 10 ≤ i ≤ n − 1 with 0 <
n−1
i0
α
i
< 1andβ>1. Let g
i
uu
α
i
0 ≤
3.7
It is easy to verify that ft, u
0
, ,u
n−1
satisfies the conditions of Theorem 2.9. Thus, it follows
that if x is a solution of 3.6 with lim
t →∞
x
Δ
n−1
t
/
0, then there exist constants a
i
0 ≤ i ≤
n − 1 with a
0
/
0, such that lim
t →∞
xt/
n−1
i0
a
i
h
n−i−1
t, t
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