Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 428963, 16 pages
doi:10.1155/2010/428963
Research Article
Nonoscillatory Solutions for Higher-Order Neutral
Dynamic Equations on Time Scales
Taixiang Sun,1 Hongjian Xi,2 Xiaofeng Peng,1 and Weiyong Yu1
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, stx1963@163.com
Received 16 May 2010; Revised 4 July 2010; Accepted 16 July 2010
Academic Editor: D. Anderson
Copyright q 2010 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.
α Δ
We study the higher-order neutral dynamic equation {atxt − ptxτtΔ } ft,
xδt 0 for t ∈ t0 , ∞T and obtain some necessary and sufficient conditions for the existence
of nonoscillatory bounded solutions for this equation.
m
1. Introduction
A time scale T is an arbitrary nonempty closed subset of the real numbers. Thus, R, Z, and N,
that is, the real numbers, the integers, and the natural numbers, are examples of time scales.
We assume throughout that the time scale T has the topology that it inherits from the real
numbers with the standard topology.
The theory of time scale, which has recently received a lot of attention, was introduced
by Hilger’s landmark paper 1, a rapidly expanding body of literature has sought to
unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous
calculus to arbitrary time scale calculus, where a time scale is a nonempty closed subset of the
real numbers, and the cases when this time scale is equal to the real numbers or to the integers
represent the classical theories of differential and 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 socalled q-difference equations when T {1, q, q2 , . . .}, which has important applications in
quantum theory see 3.
2
Abstract and Applied Analysis
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,
1.1
respectively. We refer the reader to 2, 4 for further results on time scale calculus.
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 the
papers of 5–20.
In 21 Zhu and Wang studied the existence of nonoscillatory solutions to neutral
dynamic equation
xt ptxgt
Δ
ft, xht 0.
1.2
Karpuz 22 studied the asymptotic behavior of delay dynamic equations having the
following form:
xt AtxαtΔ BtF x βt − CtG x γt ϕt.
1.3
Furthermore, Karpuz in 23 obtained necessary and sufficient conditions for the asymptotic
behaviour of all bounded solutions of the following higher-order nonlinear forced neutral
dynamic equation
n
xt AtxαtΔ f t, x βt , x γt ϕt
1.4
and also studied in 24 oscillation of unbounded solutions of a similar type of equations.
Li et al. 25 considered the existence of nonoscillatory solutions to the second-order
neutral delay dynamic equation of the form
ΔΔ
xt ptxτ0 t
q1 txτ1 t − q2 txτ2 t et.
1.5
In 26, 27, Zhang et al. obtained some sufficient conditions for the existence of
nonoscillatory solutions for the following higher-order equations:
Δn
ft, xt − τ1 t, . . . , xt − τk t 0 ,
xt ptxτt
xt ptxτt
Δn
ft, xτ1 t, . . . , xτk t 0,
1.6
respectively.
Motivated by the above studies, in this paper, we investigate the existence of
nonoscillatory solutions of the following higher order neutral dynamic equation:
m α Δ
ft, xδt 0 for t ∈ t0 , ∞T ,
at xt − ptxτtΔ
1.7
Abstract and Applied Analysis
3
where m ∈ N, α is the quotient of odd positive integers, t0 ∈ T, the time scale interval
t0 , ∞T {t ∈ T : t ≥ t0 }, a ∈ Crd t0 , ∞T , 0, ∞, p ∈ Ct0 , ∞T , R, τ, δ ∈ CT, T with
limt → ∞ τt limt → ∞ δt ∞ and f ∈ Ct0 , ∞T × R, R satisfying the following conditions:
i uft, u > 0 for any t ∈ t0 , ∞T and u /
0.
ii ft, u is nondecreasing in u for any t ∈ t0 , ∞T .
Since we are interested in the oscillatory behavior of solutions near infinity, we
assume that sup T ∞. By a solution of 1.7 we mean a nontrivial real-valued function
m α
1
Tx , ∞T , R and
x ∈ Crd Tx , ∞T , R, Tx ≥ t0 , such that atxt − ptxτtΔ ∈ Crd
satisfies 1.7 on Tx , ∞. The solutions vanishing in some neighborhood of infinity will be
excluded from our consideration. A solution x of 1.7 is said to be oscillatory if it is neither
eventually positive nor eventually negative, otherwise it is called nonoscillatory.
2. Auxiliary Results
We state the following conditions, which are needed in the sequel:
∞
H1 t0 at−1/α Δt ∞;
H2 there exist constants a1 , b1 ∈ 0, 1 with a1 b1 < 1 such that −a1 ≤ pt ≤ b1 for
all t ∈ t0 , ∞T ;
H3 there exist constants a2 , b2 ∈ 1, ∞ such that −a2 ≤ pt ≤ −b2 for all t ∈ t0 , ∞T ;
H4 there exist constants a3 , b3 ∈ 1, ∞ such that a3 ≤ pt ≤ b3 for all t ∈ t0 , ∞T .
Let k be a nonnegative integer and s, t ∈ T; we define two sequences of functions
hk t, s and gk t, s as follows:
⎧
⎪
⎪
⎨1
hk t, s t
⎪
⎪
⎩ hk−1 τ, sΔτ
if k 0,
if k ≥ 1,
s
⎧
⎪
⎪
⎨1
gk t, s t
⎪
⎪
⎩ gk−1 στ, sΔτ
2.1
if k 0,
if k ≥ 1.
s
By Theorems 1.112 and 1.60 of 2, we have
hk t, s −1k gk s, t
t
hΔ
t, s
k
gkΔt t, s for all t, s ∈ T,
⎧
⎨0
if k 0,
⎩h
k−1 t, s
⎧
⎨0
⎩g
if k ≥ 1,
if k 0,
k−1 σt, s
if k ≥ 1,
2.2
4
Abstract and Applied Analysis
t
t, s denote for each fixed s the derivatisve of gk t, s and
where gkΔt t, s and hΔ
k
hk t, s with respect to t, respectively.
Lemma 2.1 see 23, 24. Assume that s, t ∈ T and g ∈ Crd T × T, R, then
t t
s
t σζ
g η, ζ Δζ Δη g η, ζ Δη Δζ.
η
s
2.3
s
Lemma 2.2 see 23, 24. Let n be a nonnegative integer, h ∈ Crd T, 0, ∞, and s ∈ T. Then each
of the following is true:
∞
∞
i s gn σθ, shθΔθ < ∞ implies that t gn σθ, thθΔθ < ∞ for all t ∈ T;
∞
∞
ii s gn σθ, shθΔθ ∞ implies that t gn σθ, thθΔθ ∞ for all t ∈ T.
Lemma 2.3 see 23. Let n be a nonnegative integer, h ∈ Crd T, 0, ∞ and s ∈ T. Then
∞
gn σθ, shθΔθ < ∞
2.4
s
implies that each of the following is true:
∞
i t gj σθ, thθΔθ is decreasing for all t ∈ T and all 0 ≤ j ≤ n.
∞
ii limt → ∞ t gj σθ, thθΔθ 0 for all 0 ≤ j ≤ n.
∞
iii t gj σθ, thθΔθ < ∞ for all t ∈ T and all 0 ≤ j ≤ n − 1.
Lemma 2.4 see 28. Let m ∈ N and f Δ ∈ Crd t0 , ∞T , R. Then
m
1 lim inft → ∞ f Δ t > 0 implies limt → ∞ f Δ t ∞ for all 0 ≤ i ≤ m − 1.
m
i
2 lim supt → ∞ f Δ t < 0 implies limt → ∞ f Δ t −∞ for all 0 ≤ i ≤ m − 1.
m
i
Lemma 2.5 see 29. Let zt be bounded for t ∈ t0 , ∞T with zΔ t > 0, where n ∈ N. Then
i
−1n−i zΔ t > 0 for 1 ≤ i ≤ n and
n
lim zΔ t 0
i
t→∞
for 1 ≤ i ≤ n − 1.
2.5
In the sequel, write
yt xt − ptxτt.
2.6
Lemma 2.6. Assume that pt is bounded and H1 holds. If xt is a bounded nonoscillatory solution
m
of 1.7, then xtyΔ t > 0 eventually.
Proof. Without loss of generality, assume that there is some t1 ≥ t0 such that xt > 0 and
xδt > 0 for t ≥ t1 . From 1.7 we have
α Δ
{atyΔ t } −ft, xδt < 0
m
for t ≥ t1 .
2.7
Abstract and Applied Analysis
5
Thus, Rt atyΔ tα is strictly decreasing on t1 , ∞T . If there exists t2 ≥ t1 such that
Rt2 < 0, then
m
yΔ t ≤
m
Rt2 at
1/α
2.8
for t ≥ t2 .
Therefore, we have
yΔ
m−1
t − yΔ
m−1
t2 ≤
t t2
Rt2 as
1/α
Δs.
2.9
By condition H1 , we obtain limt → ∞ yΔ t −∞. Then it follows from Lemma 2.4 that
m
limt → ∞ yt −∞, which is a contradiction since xt and pt are bounded. Hence, yΔ t Rt/at1/α > 0 for all t ≥ t1 . The proof is completed.
Let BCrd t0 , ∞T , R be the Banach space of all bounded rd-continuous functions on
t0 , ∞T with sup norm x supt≥t0 |xt|. Let X ⊂ BCrd t0 , ∞T , R, we say that X is
uniformly Cauchy if for any given ε > 0, there exists t1 > t0 such that for any x ∈ X,
|xu − xv| < ε for all u, v ∈ t1 , ∞T . X is said to be equicontinuous on a, bT if, for any
given ε > 0, there exists δ > 0 such that, for any x ∈ X and u, v ∈ a, bT with |u − v| < δ,
|xu−xv| < ε. S : X → BCrd t0 , ∞T , R is called completely continuous if it is continuous
and maps bounded sets into relatively compact sets.
m−1
Lemma 2.7 see 21. Suppose that X ⊂ BCrd t0 , ∞T , R is bounded and uniformly Cauchy.
Further, suppose that X is equi-continuous on t0 , t1 T for any t1 ∈ t0 , ∞T . Then X is relatively
compact.
Lemma 2.8 see 21. Suppose that X is a Banach space and Ω is a bounded, convex, and closed
subset of X. Suppose further that there exist two operators U and S : Ω → X such that
i Ux Sy ∈ Ω for all x, y ∈ Ω,
ii U is a contraction mapping,
iii S is completely continuous.
Then U S has a fixed point in Ω.
3. Main Results and Examples
Now, we state and prove our main results.
Theorem 3.1. Assume that H1 and H2 hold. Then 1.7 has a nonoscillatory bounded solution
xt with lim inft → ∞ |xt| > 0 if and only if there exists some constant K / 0 such that
∞
t0
1
gm−1 σs, t0 as
∞
1/α
fθ, |K|Δθ
Δs < ∞.
3.1
s
Proof. Sufficiency. Assume that 1.7 has a nonoscillatory bounded solution xt on t0 , ∞T
with lim inft → ∞ |xt| > 0. Without loss of generality, we assume that there is a constant K > 0
6
Abstract and Applied Analysis
and some t1 ≥ t0 such that xt > K and xδt > K for t ≥ t1 . It follows from Lemma 2.6 that
m
yΔ t > 0 for t ≥ t1 . By assumption that xt is bounded and condition H2 , we see that yt
is bounded. Thus, by Lemma 2.5 we get that there exists t2 ≥ t1 such that
−1m−k yΔ t > 0 for t ≥ t ≥ t2 , 1 ≤ k ≤ m.
k
3.2
Integrating 1.7 from t≥ t2 to ∞, we have
at y
Δm
t
α
≥
∞
∞
fs, xδsΔs ≥
t
fs, KΔs for t ≥ t2 .
3.3
t
Therefore, it follows from 3.2 and 3.3 that, for t ≥ t2 ,
∞
1/α
1
gm−1 σθ, t
fs, KΔs
Δθ
aθ θ
t
∞
m
≤
yΔ θgm−1 σθ, tΔθ
∞
t
y
Δm−1
∞ ∞ m−1
θgm−1 θ, t −
yΔ θgm−2 σθ, tΔθ
t
− lim −1m−m−1 yΔ
t
m−1
θ→∞
≤−
∞
θgm−1 θ, t −
yΔ
m−2
≤ −12
yΔ
m−1
θgm−2 σθ, tΔθ
t
m−1
θgm−2 σθ, tΔθ
t
−yΔ
∞
∞
θgm−2 θ, t −12
t
∞
yΔ
m−2
∞
3.4
yΔ
m−2
θgm−3 σθ, tΔθ
t
θgm−3 σθ, tΔθ
t
············
∞
m−1
yΔ θΔθ
≤ −1
t
∞
−1m−1 yθ < ∞.
t
By Lemma 2.2, we see that 3.1 holds.
Necessity. Suppose that there exists some constant K > 0 such that
∞
t0
gm−1 σs, t0 AsΔs < ∞,
3.5
Abstract and Applied Analysis
where As ∞
s
fθ, KΔθ
1/α
7
/as. Then by Lemma 2.3 there exists t1 ≥ t0 such that
∞
gm−1 σs, tAsΔs <
t
1 − b1
K
n
3.6
and min{δt, τt} ≥ t0 for t ≥ t1 , where n > 21 − b1 /1 − b1 − a1 is a constant. Let
Ω
n1 − b1 − a1 − 21 − b1 K ≤ xt ≤ K
x ∈ BCrd t0 , ∞T , R :
n
for t ≥ t0 .
3.7
It is easy to verify that Ω is a bounded, convex, and closed subset of BCrd t0 , ∞T , R.
Now we define two operators S and T : Ω → BCrd t0 , ∞T , R as follows:
Sxt pt∗ xτt∗ ,
∞
n − 11 − b1 m
K
gm−1 σs, t∗ As, xΔs,
xt
−1
T
n
∗
t
3.8
∞
1/α
where u∗ max{u, t1 } for any u ∈ t0 , ∞T and As, z s fθ, zδθΔθ /as for
any z ∈ Ω. Now we show that S and T satisfy the conditions in Lemma 2.8.
1 We will show that Sx T y ∈ Ω for any x, y ∈ Ω. In fact, for any x, y ∈ Ω and t ≥ t0 ,
xt, yt ∈ n1 − b1 − a1 − 21 − b1 K/n, K,
n − 11 − b1 K pt∗ xτt∗ Sxt T y t n
∞
m
−1
gm−1 σs, t∗ As, xΔs
t∗
≤
1 − b1
n − 11 − b1 K b1 K K
n
n
K,
n − 11 − b1 K pt∗ xτt∗ Sxt T y t n
∞
−1m
gm−1 σs, t∗ As, xΔs
t∗
≥
1 − b1
n − 11 − b1 K − a1 K −
K
n
n
n1 − b1 − a1 − 21 − b1 K,
n
which implies that Sx T y ∈ Ω for any x, y ∈ Ω.
3.9
8
Abstract and Applied Analysis
2 We will show that S is a contraction mapping. Indeed, for any x, y ∈ Ω and t ≥ t0 ,
we have
Sxt − Sy t pt∗ xτt∗ − pt∗ yτt∗ ≤ max{a1 , b1 }x − y.
3.10
Sx − Sy ≤ max{a1 , b1 }x − y,
3.11
Therefore, we have
which implies that S is a contraction mapping.
3 We will show that T is a completely continuous mapping.
i By the proof of 1, we see that n1 − b1 − a1 − 21 − b1 K/n ≤ T xt ≤ K for
t ∈ t0 , ∞T . That is, T Ω ⊂ Ω.
ii We consider the continuity of T . Let xn ∈ Ω and xn − x → 0 as n → ∞, then
x ∈ Ω and |xn t − xt| → 0 as n → ∞ for any t ∈ t0 , ∞T . Consequently, for any
s ∈ t1 , ∞T , we have
lim gm−1 σs, t1 As, xn − As, x 0.
3.12
gm−1 σs, t1 As, xn − As, x ≤ 2gm−1 σs, t1 As
3.13
n→∞
Since
and, for any t ∈ t0 , ∞T ,
|T xn t − T xt| ≤
∞
gm−1 σs, t1 |As, xn − As, x|Δs,
3.14
t1
we have
T xn − T x ≤
∞
gm−1 σs, t1 |As, xn − As, x|Δs.
3.15
t1
By Chapter 5 in 4, we see that the Lebesgue dominated convergence theorem
holds for the integral on time scales. Then
lim T xn − T x 0,
n→∞
which implies that T is continuous on Ω.
3.16
Abstract and Applied Analysis
9
iii We show that T Ω is uniformly Cauchy. In fact, for any ε > 0, take t2 > t1 so that
∞
gm−1 σs, t2 AsΔs < ε.
3.17
t2
Then for any x ∈ Ω and u, v ∈ t2 , ∞T , we have
|T xu − T xv| < 2ε,
3.18
which implies that T Ω is uniformly Cauchy.
iv We show that T Ω is equicontinuous on t0 , t2 T for any t2 ∈ t0 , ∞T . Without
of generality, we assume t2 ≥ t1 . For any ε > 0, choose δ ε/1 loss
∞
g
σs, t0 AsΔs, then when u, v ∈ t0 , t2 with |u − v| < δ, we have by
t0 m−2
Lemma 2.1 that for any x ∈ Ω,
∞
∞
∗
∗
gm−1 σs, u As, xΔs −
gm−1 σs, v As, xΔs
|T xu − T xv| u∗
v∗
∞
∞
hm−1 u∗ , σsAs, xΔs −
hm−1 v∗ , σsAs, xΔs
u∗
v∗
∗
∞ u
h θ, σsAs, xΔθ Δs
u∗ σs m−2
∞ v ∗
hm−2 θ, σsAs, xΔθ Δs
−
∗
v
σs
∞ ∞
hm−2 θ, σsAs, xΔs Δθ
u∗
θ
∞ ∞
hm−2 θ, σsAs, xΔs Δθ
v∗
θ
∞ ∞
gm−2 σs, θAs, xΔs Δθ
−
u∗
−
θ
∞ ∞
v∗
θ
gm−2 σs, θAs, xΔs Δθ
∗ ∞
v
g σs, θAs, xΔs Δθ
u∗ θ m−2
∗ ∞
v
≤
gm−2 σs, t0 As, xΔs Δθ
u∗ t0
∞
|u∗ − v∗ |
gm−2 σs, t0 As, xΔs
≤δ
∞
t0
gm−2 σs, t0 AsΔs < ε,
t0
which implies that T Ω is equi-continuous on t0 , t2 T for any t2 ∈ t0 , ∞T .
3.19
10
Abstract and Applied Analysis
By Lemma 2.7, we see that T is a completely continuous mapping. It follows from
Lemma 2.8 that there exists x ∈ Ω such that U Sx x, which is the desired bounded
solution of 1.7 with lim inft → ∞ |xt| > 0. The proof is completed.
Theorem 3.2. Assume that H1 and H3 hold, and that τ has the inverse τ −1 ∈ CT, T. Then
1.7 has a nonoscillatory bounded solution xt with lim inft → ∞ |xt| > 0 if and only if there exists
some constant K /
0 such that 3.1 holds.
Proof. The proof of sufficiency is similar to that of Theorem 3.1.
Necessity. Suppose that there exists some constant K > 0 such that
∞
gm−1 σs, t0 AsΔs < ∞,
3.20
t0
where As ∞
s
fθ, KΔθ
1/α
/as. Then by Lemma 2.3 there exists t1 ≥ t0 such that
∞
b2
gm−1 σs, τ −1 t AsΔs < K,
n
−1
τ t
3.21
and min{δτ −1 t, τ −1 t} ≥ t0 for t ≥ t1 , where n > 2b2 /b2 − 1 is a constant. Let
Ω
x ∈ BCrd t0 , ∞T , R :
n − 2b2 − n
K ≤ xt ≤ K
a2 n
for t ≥ t0 .
3.22
It is easy to verify that Ω is a bounded, convex and closed subset of BCrd t0 , ∞T , R.
Now we define two operators S and T : Ω → BCrd t0 , ∞T , R as follows:
x τ −1 t∗ n − 1b2 K
Sxt −1 ∗ p τ t −np τ −1 t∗ 1
T xt −1 ∗ −1m−1
p τ t ∞
τ −1 t∗ 3.23
gm−1 σs, τ −1 t∗ As, xΔs,
∞
1/α
where u∗ max{u, t1 } for any u ∈ t0 , ∞T and As, z s fθ, zδθΔθ /as for
any z ∈ Ω. Now we show that S and T satisfy the conditions in Lemma 2.8.
Abstract and Applied Analysis
11
We will show that Sx T y ∈ Ω for any x, y ∈ Ω. In fact, for any x, y ∈ Ω and t ≥ t0 ,
xt, yt ∈ n − 2b2 − nK/a2 n, K,
x τ −1 t∗ 1
n − 1b2
K
−1 ∗ Sxt T y t −1 ∗ n
p τ t −p τ t m
−1
∞
τ −1 t∗ −1
∗
gm−1 σs, τ t As, xΔs
1 n − 1b2
b2
≤
K K
b2
n
n
K,
x τ −1 t∗ 1
n − 1b2
K
−1 ∗ Sxt T y t −1 ∗ n
p τ t −p τ t −1
m
∞
τ −1 t∗ −1
∗
3.24
gm−1 σs, τ t As, xΔs
1 n − 1b2
b2
≥
K−K− K
a2
n
n
n − 2b2 − n
K,
a2 n
and |T xt| ≤ b2 K/n, which implies that Sx T y ∈ Ω for any x, y ∈ Ω and T Ω is uniformly
bounded.
Now we show that T Ω is equicontinuous on t0 , t2 T for any t2 ∈ t0 , ∞T . Without
loss of generality, we assume that t2 ≥ t1 . Since 1/pτ −1 t, τ −1 t are continuous on t0 , t2 T ,
so they are uniformly continuous on t0 , t2 T . For any ε > 0, choose δ > 0 such that when
u, v ∈ t0 , t2 T with |u − v| < δ, we have
1
ε
1
∞
−1 − −1 <
p τ u
p τ v
1 t0 gm−1 σs, t0 AsΔs
−1
τ u − τ −1 v <
1
∞
t0
ε
.
gm−2 σs, t0 AsΔs
Then, it follows from Lemma 2.1 that, for any x ∈ Ω,
∞
1
gm−1 σs, τ −1 u∗ As, xΔs
|T xu − T xv| −1 ∗ p τ u τ −1 u∗ ∞
1
−1 ∗
− −1 ∗ gm−1 σs, τ v As, xΔs
p τ v τ −1 v∗ 3.25
12
Abstract and Applied Analysis
∞
1
1
gm−1 σs, τ −1 u∗ As, xΔs
≤ −1 ∗ − −1 ∗ p τ u −1
∗
p τ v τ u ∞
1
−1 ∗ gm−1 σs, τ −1 u∗ As, xΔs
p τ v τ −1 u∗ −
∞
τ −1 v∗ gm−1
σs, τ v As, xΔs −1
∗
∞
1
1
−1 ∗
−1 ∗ − −1 ∗ gm−1 σs, τ u As, xΔs
p τ u p τ v τ −1 u∗ τ −1 v∗ ∞
1
−1 ∗ g σs, θAs, xΔs Δθ
p τ v τ −1 u∗ θ m−2
∞
1
1
≤ −1 ∗ − −1 ∗ gm−1 σs, t0 AsΔs
p τ u p τ v t0
τ −1 v∗ ∞
1
−1 ∗ gm−2 σs, t0 AsΔs Δθ
p τ v τ −1 u∗ t0
∞
−1 ∗
−1 ∗ ≤ ε τ u − τ v gm−2 σs, t0 AsΔs
t0
< 2ε,
3.26
which implies that T Ω is equi-continuous on t0 , t2 T for any t2 ∈ t0 , ∞T .
The rest of the proof is similar to that of Theorem 3.1. The proof is completed.
Theorem 3.3. Assume that H1 and H4 hold and that τ has the inverse τ −1 ∈ CT, T. Then 1.7
has a nonoscillatory bounded solution xt with lim inft → ∞ |xt| > 0 if and only if there exists some
constant K /
0 such that 3.1 holds.
Proof. The proof of sufficiency is similar to that of Theorem 3.1.
Necessity. Suppose that there exists some constant K > 0 such that
∞
gm−1 σs, t0 AsΔs < ∞,
3.27
t0
where As ∞
s
fθ, KΔθ
1/α
/as. Then by Lemma 2.3 there exists t1 ≥ t0 such that
∞
a3 − 1
K
gm−1 σs, τ −1 t AsΔs <
n
−1
τ t
3.28
Abstract and Applied Analysis
13
and min{δτ −1 t, τ −1 t} ≥ t0 for t ≥ t1 , where n > 2 is a constant. Let
Ω
x ∈ BCrd t0 , ∞T , R :
n − 2a3 − 1
K ≤ xt ≤ K
nb3 − 1
for t ≥ t0 .
3.29
It is easy to verify that Ω is a bounded, convex, and closed subset of BCrd t0 , ∞T , R.
Now we define two operators S and T : Ω → BCrd t0 , ∞T , R as follows:
x τ −1 t∗ n − 1a3 − 1K
,
Sxt −1 ∗ p τ t np τ −1 t∗ ∞
1
m−1
gm−1 σs, τ −1 t∗ As, xΔs,
T xt −1 ∗ −1
p τ t τ −1 t∗ 3.30
∞
1/α
where u∗ max{u, t1 } for any u ∈ t0 , ∞T and As, z s fθ, zδθΔθ /as for
any z ∈ Ω. Now we show that S and T satisfy the conditions in Lemma 2.8.
We will show that Sx T y ∈ Ω for any x, y ∈ Ω. In fact, for any x, y ∈ Ω and t ≥ t0 ,
xt, yt ∈ n − 2a3 − 1K/nb3 − 1, K,
x τ −1 t∗ 1
Sxt T y t −1 ∗ −1 ∗ p τ t p τ t ∞
n − 1a3 − 1
m−1
−1 ∗
K −1
×
gm−1 σs, τ t As, xΔs
n
τ −1 t∗ 1 n − 1a3 − 1
a3 − 1
≤
KK
K
a3
n
n
K,
x τ −1 t∗ 1
Sxt T y t −1 ∗ −1 ∗ p τ t p τ t ∞
n − 1a3 − 1
m−1
−1 ∗
K −1
×
gm−1 σs, τ t As, xΔs
n
τ −1 t∗ 1 n − 1a3 − 1
n − 2a3 − 1K a3 − 1
≥
K
−
K
b3
n
nb3 − 1
n
n − 2a3 − 1K
nb3 − 1
3.31
and |T xt| ≤ a3 − 1K/n, which implies that Sx T y ∈ Ω for any x, y ∈ Ω and T Ω
is uniformly bounded. The rest of the proof is similar to that of Theorem 3.2. The proof is
completed.
According to the proofs of Theorem 3.1, Theorem 3.2, and Theorem 3.3 in 23, we
have
14
Abstract and Applied Analysis
Lemma 3.4. Assume that H1 holds. Suppose that one of the following holds:
i pt satisfies condition H2 ,
ii pt satisfies condition H3 and τ has the inverse τ −1 ∈ CT, T,
iii pt satisfies condition H4 and τ has the inverse τ −1 ∈ CT, T.
If xt is a bounded nonoscillatory solution of 1.7, then lim inft → ∞ |xt| 0 implies that
limt → ∞ xt 0.
By Lemma 3.4, Theorem 3.1,Theorem 3.2 and Theorem 3.3, we have.
Corollary 3.5. Assume that H1 holds. Suppose that one of the following holds:
i pt satisfies condition H2 ,
ii pt satisfies condition H3 and τ has the inverse τ −1 ∈ CT, T,
iii pt satisfies condition H4 and τ has the inverse τ −1 ∈ CT, T.
Then every bounded solution of 1.7 is oscillation or converges to zero at infinity if and only if there
exists some constant K / 0 such that
∞
t0
1
gm−1 σs, t0 as
∞
1/α
fθ, |K|Δθ
Δs ∞.
3.32
s
Example 3.6. Let T {qn : n ∈ Z} ∪ {0} with q > 1. Consider the following higher order
dynamic equation:
m α Δ
tα xt−pk txqtΔ
2 α α
qgm−1
q t, 0 −gm−1 qt, 0
2 α 2 α x2r1 qr3 t 0
q−1 qt gm−1 q t, 0 gm−1 qt, 0
3.33
for t ∈ 1, ∞T ,
where m, r ∈ N, α is the quotient of odd positive integers, pk t −2−1k k2 −1logq t /5,,
α
α
q2 t, 0 − gm−1
qt, 0/q −
where k 1, 2, 3, at tα , τt qt, δt q3r t and ft, u qgm−1
α
2 α
2
2r1
.
1qt gm−1 q t, 0gm−1 qt, 0u
It is easy to verify that pk t satisfies condition Hk1 and τ −1 ∈ CT, T. On the other
hand, we have
∞ 1
1
at
1/α
Δt ∞ 1
1
tα
1/α
Δt ∞
3.34
Abstract and Applied Analysis
15
andfor any constant K > 0,
∞
gm−1 σs, 1
1
K 2r1/α
∞
1
K
2r1/α
∞
1
K
2r1/α
∞
1
1
as
∞
1/α
fθ, KΔθ
Δs
s
⎧
Δ ⎫1/α
⎬
⎨ 1 ∞
−1
gm−1 σs, 1
Δs
α
⎩ sα s θgm−1
σθ, 0 ⎭
1
1
gm−1 σs, 1 α α
s sgm−1 σs, 0
1/α
Δs
3.35
1
Δs
s11/α
2r 1 1/α q
q−1
α
K
< ∞.
q1/α − 1
That is, conditions H1 and 3.1 hold. By Theorem 3.1, Theorem 3.2, and Theorem 3.3, we
see that 3.33 has a nonoscillatory bounded solution xt with lim inft → ∞ |xt| > 0.
Remark 3.7. Results of this paper can be extended to the case with several delays easily.
Acknowledgment
Project Supported by NSF of China 10861002 and NSF of Guangxi 2010GXNSFA013106,
2011GXNSFA014781 and SF of Education Department of Guangxi 200911MS212 and
Innovation Project of Guangxi Graduate Education 2010105930701M43.
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