Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 562634, 20 pages
doi:10.1155/2010/562634
Research Article
Asymptotic Dichotomy in a Class of Third-Order
Nonlinear Differential Equations with Impulses
Kun-Wen Wen,1 Gen-Qiang Wang,1 and Sui Sun Cheng2
1
Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou,
Guangdong 510665, China
2
Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan
Correspondence should be addressed to Sui Sun Cheng, sscheng@math.nthu.edu.tw
Received 15 November 2009; Revised 26 January 2010; Accepted 25 February 2010
Academic Editor: Ağacik Zafer
Copyright q 2010 Kun-Wen Wen 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.
Solutions of quite a few higher-order delay functional differential equations oscillate or converge to
zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear
differential equation with impulses.
1. Introduction
It has been observed that the solutions of quite a few higher-order delay functional
differential equations oscillate or converge to zero see, e.g., the recent paper 1 in which
a third order nonlinear delay differential equation with damping is considered. Such a
dichotomy may yield useful information in real problems see, e.g., 2 in which implications
of this dichotomy are applied to the deflection of an elastic beam. Thus it is of interest to see
whether similar dichotomies occur in different types of functional differential equations.
One such type consists of impulsive differential equations which are important in
simulation of processes with jump conditions see, e.g., 3–22. But papers devoted to the
study of asymptotic behaviors of third-order equations with impulses are quite rare. For this
reason, we study here the third-order nonlinear differential equation with impulses of the
form
tk ,
rtx t ft, x 0, t ≥ t0 , t /
i
xi tk gk xi tk , i 0, 1, 2; k 1, 2, . . . ,
i
xi t0 x0 ,
i 0, 1, 2,
1.1
2
Abstract and Applied Analysis
where x0 t xt, 0 ≤ t0 < t1 < · · · < tk < · · · such that limk → ∞ tk ∞,
xi tk lim xi t,
xi t−k lim− xi t
t → tk
1.2
t → tk
i
i
for i 0, 1, 2. Here gk , i 0, 1, 2 and k 1, 2, . . . , are real functions and x0 , i 0, 1, 2, are real
numbers.
By a solution of 1.1, we mean a real function x xt defined on t0 , ∞ such that
i
i xi t0 x0 for i 0, 1, 2;
ii xi t, i 0, 1, 2, and rtx t are continuous on t0 , ∞ \ {tk }; for i i
0, 1, 2, xi tk and xi t−k exist, xi t−k xi tk and xi tk gk xi tk for any
tk ;
iii xt satisfies rtx t ft, x 0 at each point t ∈ t0 , ∞ \ {tk }.
A solution of 1.1 is said to be nonoscillatory if it is eventually positive or eventually
negative. Otherwise, it is said to be oscillatory.
We will establish dichotomous criteria that guarantee solutions of 1.1 that are either
oscillatory or zero convergent based on combinations of the following conditions.
A rt is positive and continuous on t0 , ∞, ft, x is continuous on t0 , ∞ ×
R, xft, x > 0 for x /
0, and ft, x/ϕx pt, where pt is positive and
continuous on t0 , ∞, and ϕ is differentiable in R such that ϕ x ≥ 0 for x ∈ R.
i
B For each k 1, 2, . . . , gk x is continuous in R and there exist positive numbers
i
i
i
i
i
ak , bk such that ak ≤ gk x/x ≤ bk for x /
0 and i 0, 1, 2.
C One has
∞ t0 t0 <tk <s
∞
t0
⎛
⎝
1
ak
0
bk
⎛
⎞
⎠ds ∞,
⎞
2
1 ⎝ ak ⎠
ds ∞.
rs t0 <tk <s b1
1.3
k
In the next section, we state four theorems to ensure that every solution of 1.1 either
oscillates or tends to zero. Examples will also be given. Then in Section 3, we prove several
preparatory lemmas. In the final section, proofs of our main theorems will be given.
2. Main Results
The main results of the paper are as follows.
Abstract and Applied Analysis
3
Theorem 2.1. Assume that the conditions (A)–(C) hold. Suppose further that there exists a positive
0
integer k0 such that for k ≥ k0 , ak ≥ 1,
∞ 0
bk − 1 < ∞,
k1
⎞
1
⎝
⎠psds ∞.
2
bk
t0 <tk <s
∞ t0
2.1
⎛
2.2
Then every solution of 1.1 either oscillates or tends to zero.
Theorem 2.2. Assume that the conditions (A)–(C) hold. Suppose further that there exists a positive
0
1
integer k0 such that for k ≥ k0 , bk ≤ 1, ak ≥ 1,
0
ak ≥ σ > 0,
t0 ≤tk <∞
∞
t0
⎛
⎞
1 ⎝ ∞ 1
pudu⎠ds ∞.
2
rs
s s<tk <u b
2.3
2.4
k
Then every solution of 1.1 either oscillates or tends to zero.
Theorem 2.3. Assume that the conditions (A)–(C) hold and that ϕab ≥ ϕaϕb for any ab > 0.
Suppose further that there exists a positive integer k0 such that for
0
2
k ≥ k0 , bk ≤ 1,
bk ≤ 1,
∞
2
0
bk ≤ ϕ ak ,
psds ∞.
2.5
2.6
t0
Then every solution of 1.1 either oscillates or tends to zero.
Theorem 2.4. Assume that the conditions (A)–(C) hold and that ϕab ≥ ϕaϕb for any ab > 0.
2
0 0
Suppose further that bk ≤ ak , { nk1 bk } is bounded, that
∞
0
0
max ak − 1, bk − 1 < ∞,
k1
∞ 2
bk − 1 < ∞,
2.7
k1
∞
psds ∞.
t0
Then every solution of 1.1 either oscillates or tends to zero.
2.8
4
Abstract and Applied Analysis
Before giving proofs, we first illustrate our theorems by several examples.
Example 2.5. Consider the equation
tx t et xt 0,
t≥
1
, t/
k,
2
1
xi k, i 0, 1, 2; k 1, 2, . . . ,
k2
1
1
1
0
1
2
x0 ,
x0 ,
x0 ,
x
x
x
2
2
2
xi k i
1
2.9
i
where ak bk 1 1/k2 ≥ 1 for i 0, 1, 2; pt et , rt t, tk k, ϕx x. It is not
difficult to see that conditions A–C are satisfied. Furthermore,
∞ ∞ t0
k1
∞
1
0
bk − 1 < ∞,
2
k
k1
∞ k2 s
e ds ∞.
psds
2
2
1/2 1/2<tk <s k 1
t0 <tk <s b
1
2.10
k
Thus by Theorem 2.1, every solution of 2.9 either oscillates or tends to zero.
Example 2.6. Consider the equation
√
t2 − sin tgtx t
k
xk,
k1
1
0
x0 ,
x
2
xk 0
0
i
t−3/2 x3 t 0,
xi k xi k,
x
1
1
x0 ,
2
t≥
1
, t/
k,
2
i 1, 2; k 1, 2, . . . ,
x
2.11
1
2
x0 ,
2
i
where ak bk k/k 1, ak bk 1 for i 1, 2; pt t−3/2 , tk k, ϕx x3 , and
√
rt t2 − sin tgt,
here gt t − k −
1 1, t ∈ k, k 1, k 1, 2, . . . .
2
2.12
Abstract and Applied Analysis
5
Here, we do not assume that rt is bounded, monotonic, or differential. It is not difficult to
see that conditions A–C are satisfied. Furthermore,
∞
t0
⎛
⎞
∞
∞
1 ⎝ ∞ 1
1
−3/2
⎠ds pudu
u
du
ds
√
2
rs
s s<tk <u b
1/2 s2 − sin sgs
s
k
∞
1
u−3/2 du ds
√
1/2 3 sgs
s
∞
∞
2
−3/2
≥
u
du
ds
√
1/2 9 s
s
∞
4
ds ∞.
1/2 9s
≥
∞
2.13
Thus by Theorem 2.2, every solution of 2.11 either oscillates or tends to zero.
Example 2.7. Consider the equation
e−2t x t e−2t xt 0,
xk xk,
x
i
i
2
1
, t
/ k,
2
k
x k, k 1, 2, . . . ,
k1
1
1
1
2
x
x
x0 ,
x0 ,
2
2
x k x k,
1
0
x0 ,
2
t≥
x k 2.14
2
where ak bk 1 for i 0, 1, ak bk k/k 1; pt e−2t , rt e−2t , tk k; ϕx x.
It is not difficult to see that conditions A–C are satisfied. Furthermore,
∞
t0
⎛
⎞
∞ ∞ ∞ 1 ⎝
1
k
1
2s
−2u
e du ds
pudu⎠ds e
2
rs
s s<tk <u b
1/2
s s<tk <u k
k
∞ ∞
2s
−2u
≥
e
e du ds
1/2
∞
2.15
s
1
ds ∞.
1/2 2
Thus, by Theorem 2.2, every solution of 2.14 either oscillates or tends to zero.
Note that the ordinary differential equation
e−2t x t
e−2t xt 0
2.16
6
Abstract and Applied Analysis
has a nonnegative solution xt et → ∞ as t → ∞. This example shows that impulses
play an important role in oscillatory and asymptotic behaviors of equations under perturbing
impulses.
3. Preparatory Lemmas
To prove our theorems, we need the following lemmas.
Lemma 3.1 Lakshmikantham et al. 3. Assume the following.
H0 m ∈ P C R , R and mt is left-continuous at tk , k 1, 2, . . ..
H1 For tk , k 1, 2, . . . and t ≥ t0 ,
m t ≤ ptmt qt, t /
tk ,
m tk ≤ dk mtk bk ,
3.1
where p, q ∈ P CR , R, dk ≥ 0, and bk are real constants. Then for t ≥ t0 ,
mt ≤ mt0 t
dk exp
t0
psds
t0
t0 <tk <t
t dk
s<tk <t
⎛
⎝
t0 <tk <t
t
exp
t
dj exp
tk <tj <t
tk
⎞
psds ⎠bk
3.2
pσdσ qsds.
s
Lemma 3.2. Suppose that conditions (A)–(C) hold and xt is a solution of 1.1. One has the
following statements.
a If there exists some T ≥ t0 such that x t > 0 and rtx t ≥ 0 for t ≥ T , then there
exists some T1 ≥ T such that x t > 0 for t ≥ T1 .
b If there exists some T ≥ t0 such that x t > 0 and x t ≥ 0 for t ≥ T , then there exists
some T1 ≥ T such that xt > 0 for t ≥ T1 .
Proof. First of all, we will prove that a is true. Without loss of generality, we may assume
that x t > 0 and rtx t ≥ 0 for t ≥ t0 . We assert that there exists some j such that
x tj > 0 for tj ≥ t0 . If this is not true, then for any tk ≥ t0 , we have x tk ≤ 0. Since x t is
increasing on intervals of the form tk , tk1 , we see that x t ≤ 0 for t ≥ t0 . Since rtx t is
increasing on intervals of the form tk , tk1 , we see that for t1 , t2 ,
rtx t ≥ rt1 x t1 ,
3.3
that is,
x t ≥
rt1 x t1 .
rt
3.4
Abstract and Applied Analysis
7
In particular,
rt1 x t1 .
rt2 3.5
rt2 rt2 2 rt1 2 x t2 ≥
a2 x t2 ≥
a x t1 .
rt
rt
rt 2
3.6
x t2 ≥
Similarly, for t2 , t3 , we have
x t ≥
By induction, we know that for t > t1 ,
x t ≥
rt1 2 a x t1 ,
rt t1 <tk <t k
t/
tk .
3.7
From condition B, we have
1
x tk ≥ bk x tk ,
3.8
k 2, 3, . . . .
Set mt −x t. Then from 3.7 and 3.8, we see that for t > t1 ,
m t ≤ −
rt1 2 a x t1 ,
rt t1 <tk <t k
1
m tk ≤ bk mtk ,
t/
tk
3.9
k 2, 3, . . . .
It follows from Lemma 3.1 that
t
1
mt ≤ m t1
bk − x t1 rt1 1 1 2
bk
ak ds
t1 rs s<tk <t
t1 <tk <t
t1 <tk <s
⎫
⎧
t
⎬
1 ⎨ a2
1
k
.
m t1 − x t1 rt1 bk
ds
1
⎭
⎩
t1 rs t <t <s b
t <t <t
1
k
1
k
3.10
k
That is,
x t ≥
i
i
⎧
⎨
1
x t1
bk
⎩
t1 <tk <t
x t1 rt1 ⎫
2
⎬
1 ak
ds
.
1
⎭
t1 rs t1 <tk <s b
t
3.11
k
Note that ak > 0, bk > 0, and the second equality of condition C holds. Thus we get
x t > 0 for all sufficiently large t. The relation x t ≤ 0 leads to a contradiction. Thus, there
8
Abstract and Applied Analysis
exists some j such that tj ≥ t0 and x tj > 0. Since x t is increasing on intervals of the form
tjλ , tjλ1 for λ 0, 1, 2, . . . , thus for t ∈ tj , tj1 , we have
1 x t ≥ x tj ≥ aj x tj > 0.
3.12
1 1 1 x t ≥ x tj1 ≥ aj1 x tj1 ≥ aj aj1 x tj > 0.
3.13
Similarly, for t ∈ tj1 , tj2 ,
We can easily prove that, for any positive integer λ ≥ 2 and t ∈ tjλ , tjλ1 ,
1 1
1
x t ≥ aj aj1 · · · ajλ x tj > 0.
3.14
Therefore, x t > 0 for t ≥ tj . Thus, a is true.
Next, we will prove that b is true. Without loss of generality, we may assume that
x t > 0 and x t ≥ 0 for t ≥ t0 . We assert that there exists some j such that xtj > 0 for
tj ≥ t0 . If this is not true, then for any tk ≥ t0 , we have xtk ≤ 0. Since xt is increasing on
intervals of the form tk , tk1 , we see that xt ≤ 0 for t ≥ t0 . By x t > 0, x t ≥ 0, t ∈ tk , tk1 ,
we have that x t is nondecreasing on tk , tk1 . For t ∈ t1 , t2 , we have
x t ≥ x t1 .
3.15
x t2 ≥ x t1 .
3.16
1
1 x t ≥ x t2 ≥ a2 x t2 ≥ a2 x t1 .
3.17
In particular,
Similarly, for t ∈ t2 , t3 , we have
By induction, we know that for t > t1 ,
x t ≥
1 ak x t1 ,
t
/ tk .
3.18
k 2, 3, . . . .
3.19
t1 <tk <t
From condition B, we have
0
x tk ≥ bk xtk ,
Abstract and Applied Analysis
9
Set ut −xt. Then from 3.18 and 3.19, we see that for t > t1 ,
u t ≤ −
1 ak x t1 ,
t/
tk ,
t1 <tk <t
0
u tk ≤ bk utk ,
3.20
k 2, 3, . . . .
It follows from Lemma 3.1 that
t 0 1
0
bk − x t 1
bk
ak ds
ut ≤ u t1
t1 s<tk <t
t1 <tk <t
t1 <tk <s
⎫
⎧
t 1 ⎬
0 ⎨ a
k
u t − x t1
bk
ds .
0
⎭
⎩ 1
t1 t <t <s b
t <t <t
1
k
1
k
3.21
k
That is,
xt ≥
i
⎧
⎨
0
x t1
bk
⎩
t1 <tk <t
t 1
ak
x t1
t1 t1 <tk <s b
0
k
⎫
⎬
ds .
⎭
3.22
i
Note that ak > 0, bk > 0, and the first equality of condition C holds. Thus we get xt > 0
for all sufficiently large t. The relation xt ≤ 0 leads to a contradiction. So there exists some j
such that tj ≥ t0 and xtj > 0. Then
0 x tj ≥ aj x tj > 0.
3.23
Since x t > 0, we see that xt is strictly monotonically increasing on tjm , tjm1 for m 0, 1, 2, . . . . For t ∈ tj , tj1 , we have
xt ≥ x tj > 0.
3.24
x tj1 ≥ x tj > 0.
3.25
0 xt ≥ x tj1 ≥ aj1 x tj1 > 0.
3.26
In particular,
Similarly, for t ∈ tj1 , tj2 , we have
By induction, we have xt > 0 for t ∈ tjm , tjm1 . Thus, we know that xt > 0, for t ≥ tj .
The proof of Lemma 3.2 is complete.
10
Abstract and Applied Analysis
Remark 3.3. We may prove in similar manners the following statements.
a If we replace the condition a in Lemma 3.2 “x t > 0 and rtx t ≥ 0 for
t ≥ T ” with “x t < 0 and rtx t ≤ 0 for t ≥ T ”, then there exists some T1 ≥ T
such that x t < 0 for t ≥ T1 .
b If we replace the condition b in Lemma 3.2 “x t > 0 and x t ≥ 0 for t ≥ T ” with
“x t < 0 and x t ≤ 0 for t ≥ T ”, then there exists some T1 ≥ T such that xt < 0
for t ≥ T1 .
Lemma 3.4. Suppose that conditions (A)–(C) hold and xt is a solution of 1.1 such that xt > 0
for t ≥ T, where T ≥ t0 . Then there exists T ≥ T such that either (a) x t > 0, x t < 0 for t ≥ T or
(b) x t > 0, x t > 0 for t ≥ T .
Proof. Without loss of generality, we may assume that xt > 0 for t ≥ t0 . By 1.1 and
condition A, we have for t ≥ t0 .
rtx t −ft, x ≤ −ptϕx < 0.
3.27
We assert that for any tk ≥ t0 , x tk > 0. If this is not true, then there exists some j such
2
that x tj ≤ 0, so x tj ≤ aj x tj ≤ 0. Since rtx t is decreasing on tjk−1 , tjk for
k 1, 2, . . ., we see that for t ∈ tj , tj1 ,
r tj x tj ≤ 0.
x t <
rt
3.28
In particular,
x tj1
r tj
< x tj ≤ 0.
r tj1
3.29
Similarly, for t ∈ tj1 , tj2 , we have
r tj1 r tj1 2 r tj 2 x tj1 ≤
a x tj1 ≤
a x tj ≤ 0.
x t <
rt
rt j1
rt j1
3.30
In particular,
x tj2
r tj
2
< aj1 x tj ≤ 0.
r tj2
3.31
By induction, for any t ∈ tjn−1 , tjn for n 2, 3, . . . , we have
n−1
r tj 2
x t <
ajk x tj ≤ 0.
rt k1
3.32
Abstract and Applied Analysis
11
Hence, x t < 0 for t ≥ tj . By Remark 3.3a , there exists T1 ≥ tj such that x t < 0 for t ≥ T1 ;
by Remark 3.3b , we get xt < 0 for t ≥ T1 , which is contrary to xt > 0 for t ≥ t0 . Hence,
for any tk ≥ t0 , x tk > 0, since rtx t is decreasing on tjk−1 , tjk for k 1, 2, . . ., therefore
x t > 0 for t ≥ t0 . It follows that x t is strictly increasing on tk , tk1 for k 1, 2, . . ..
1
Furthermore, note that ak > 0, k 1, 2, . . . . we see that if for any tk , x tk < 0, then x t < 0
for t ≥ t0 . If there exists some tj such that x tj ≥ 0, then x t > 0 for t > tj . The proof of
Lemma 3.4 is complete.
Lemma 3.5 see 12. Suppose that xt is continuous at t > 0 and t /
tk , it is left-continuous at
t tk and limt → tk xt exists for k 1, 2, . . . . Further assume that
H2 there exists t ∈ R , such that xt > 0< 0 for t ≥ t;
H3 xt is nonincreasing (resp., nondecreasing) on tk , tk1 for k 1, 2, . . . ;
H4 ∞
k1 xtk − xtk is convergent.
Then limt → ∞ xt r exists and r ≥ 0 (resp., ≤ 0).
4. Proofs of Main Theorems
We now turn to the proof of Theorem 2.1. Without loss of generality, we may assume that
k0 1. If 1.1 has a nonoscillatory solution x xt, we first assume that xt > 0 for t ≥ t0 .
By 1.1 and the condition A, for t ≥ T ≥ t0 , we get
rtx t −ft, xt ≤ −ptϕxt,
t/
tk .
4.1
From the condition B, we know that
2
r tk x tk ≤ bk rtk x tk .
4.2
By Lemma 3.4, there exists a T ≥ t0 such that either a x t > 0, x t < 0 for t ≥ T or b
x t > 0, x t > 0 for t ≥ T.
Suppose that a holds. Then we see that the conditions H2 and H3 of Lemma 3.5
0
0
0
are satisfied. Furthermore, note that ∞
k1 bk − 1 < ∞ and bk ≥ ak ≥ 1. Then we have
∞
0
bk < ∞.
4.3
k1
Since x t < 0, t ≥ T, we obtain for any tk > T,
xtk ≤
0
bj xT .
4.4
T <tj <tk
By 4.3 and 4.4, we know that the sequence {xtk } is bounded. Thus there exists M > 0
such that |xtk | ≤ M. It follows from the condition B that
x t − xtk ≤ b0 − 1|xtk | ≤ M b0 − 1 .
k
k
k
4.5
12
Abstract and Applied Analysis
∞
0
From 4.5 and the fact that ∞
k1 bk − 1 is convergent, we know that
k1 xtk − xtk is
convergent. Therefore, the condition H4 of Lemma 3.5 is also satisfied. By Lemma 3.5, we
know that limt → ∞ xt r ≥ 0. We assert that r 0. If r > 0, then there exists T1 ≥ T such
that for any t ≥ T1 , xt > r/2 > 0. Note further that ϕ x ≥ 0; so we obtain ϕxt ≥ ϕr/2
for t ≥ T1 . Let mt rtx t for t ≥ T1 . By 4.1 and 4.2, we have
m t ≤ qt,
tk ,
t ≥ T1 , t /
2
m tk ≤ bk mtk ,
4.6
4.7
tk ≥ T 1 ,
where qt −ϕr/2pt. From 4.6, 4.7, and Lemma 3.1, we get for t ≥ T1 ,
mt ≤ m
T1
2
bk
T1 <tk <t
t 2
bk
s<tk <t
T1
qsds
⎫
⎧
⎛
⎞
⎬
⎨ r t
1 ⎠
2
⎝
.
m T1 − ϕ
psds
bk
⎭
⎩
2 T1 T <t <s b2
<t
4.8
T1 <tk
1
k
k
It is easy to see from 2.2 and 4.8 that mt < 0 for sufficiently large t. This is contrary to
mt > 0 for t ≥ T1 . Thus r 0, that is, limt → ∞ xt 0.
Suppose that b holds. Let Ψt rtx t/ϕxt for t ≥ T. Then Ψt > 0 for t ≥ T .
By 1.1 and the condition A, we get, for t ≥ T,
Ψ t −ft, xt rtx tϕ xtx t −ft, xt
−
≤ −pt,
≤
ϕxt
ϕxt
ϕ2 xt
t
/ tk .
4.9
0
From the conditions A, B and ak ≥ 1, we know that
2
rtk b x tk rtk x tk
2 rtk x tk 2
Ψ tk ≤ k
≤ bk Ψtk ,
≤ bk
0
ϕxt
ϕ x tk
k
ϕ ak xtk tk ≥ T.
4.10
From 4.9, 4.10, and Lemma 3.1, we get, for t ≥ T,
Ψt ≤ ΨT
2
bk
T <tk <t
−
t T
2
bk
s<tk <t
psds
⎫
⎞
⎬
1 ⎠
2
psds .
ΨT − ⎝
bk
2
⎭
⎩
T
T <tk <t
T <tk <s bk
⎧
⎨
t
⎛
4.11
It is easy to see from 2.2 and 4.11 that Ψt < 0 for sufficiently large t. This is contrary
to Ψt > 0 for t ≥ T, and hence we obtain a contradiction. Thus in case b xt must be
oscillatory. The proof of Theorem 2.1 is complete.
Abstract and Applied Analysis
13
Next, we give the proof of Theorem 2.2. Without loss of generality, we may assume that
k0 1. If 1.1 has an eventually positive solution x xt for t ≥ t0 . By 1.1 and conditions
A and B, we have that 4.1 and 4.2 hold. By Lemma 3.4, there exists a T ≥ t0 such that
either a x t > 0, x t < 0 for t ≥ T or b x t > 0, x t > 0 for t ≥ T.
0
Suppose that a holds. Note that bk ≤ 1 and for tj ≥ T and each l 0, 1, 2, . . ., xt is
decreasing on tjl , tjl1 ; we have for t ∈ tj , tj1 0 xt < x tj ≤ bj x tj ≤ x tj .
4.12
Similarly, for t ∈ tj1 , tj2 , we have
0 xt < x tj1 ≤ bj1 x tj1 ≤ x tj1 ≤ x tj .
4.13
By induction, for each l 0, 1, 2, . . ., we have
xt < x tjl ≤ · · · ≤ x tj1 ≤ x tj ,
t ∈ tjl , tjl1
4.14
so that xt is decreasing on tj , ∞. We know that xt is convergent as t → ∞. Let
limt → ∞ xt r. Then r ≥ 0. We assert that r 0. If r > 0, then there exists T1 ≥ t0 , such
that for t ≥ T1 , xt > r/2 > 0. Since ϕ x ≥ 0, then ϕxt ≥ ϕr/2. Let mt rtx t
for t ≥ T1 . Then By 4.1 and 4.2, we have that 4.6 and 4.7 hold. From 4.6, 4.7, and
Lemma 3.1, we get for t ≥ T1 ,
m∞ ≤
2
mt
bk
t<tk <∞
r ∞ 2
−ϕ
b psds.
2 t s<tk <∞ k
4.15
That is,
0 ≤ lim rtx t ≤ rtx
t → ∞
2
bk
t
t<tk <∞
r ∞ 2
−ϕ
b psds.
2 t s<tk <∞ k
4.16
It is easy to see from 4.16 that the following inequality holds:
ϕr/2
x t ≥
rt
∞ 1
t
2
t<tk <s bk
psds,
t ≥ T1 .
4.17
14
Abstract and Applied Analysis
1
Note that ak ≥ 1; it follows from integrating 4.17 from t0 to t and by using the condition
B that
1
ak − 1 x tk x t − x t0 ≥ x t − x t0 t0 <tk <t
≥ x t − x t0 x tk − x tk t0 <tk <t
⎛
⎞
∞ r t 1
1
⎝
pudu⎠ds.
≥ϕ
2
2 t0 rs
s s<tk <u b
4.18
k
It is easy to see from 2.4 and 4.18 that x t > 0 for sufficiently large t. This is contrary to
x t < 0 for t ≥ T1 . Thus r 0, that is, limt → ∞ xt 0.
Suppose b holds. Without loss of generality, we may assume that T t0 . Then we see
that x t > 0, t ≥ t0 . Since xt is nondecreasing on tk , tk1 , for t ∈ t0 , t1 , we have
xt ≥ x t0 .
4.19
xt1 ≥ x t0 .
4.20
0
0 xt ≥ x t1 ≥ a1 xt1 ≥ a1 x t0 .
4.21
In particular,
Similarly, for t ∈ t1 , t2 , we have
By induction, we know that
xt ≥
0 ak x t0 ,
t > t0 .
4.22
t0 <tk <t
0
0
0
That is, xt ≥ t0 <tk <t ak xt0 for t > t0 . Note that bk ≤ 1 and t0 ≤tk <∞ ak ≥ σ > 0. From
the condition B, we have xt ≥ σxt0 . Since ϕ x ≥ 0, we have ϕxt ≥ ϕσxt0 . Let
mt rtx t; by 4.1 and 4.2, we have, for t ≥ t0 , that
m t ≤ −ϕ σx t0 pt,
2
m tk ≤ bk mtk ,
t/
tk ,
4.23
tk > t0 .
Similar to the proof of 4.17, we obtain
∞
1
ϕ σx t0
psds,
x t ≥
2
rt
t t<tk <s b
k
t ≥ t0 .
4.24
Abstract and Applied Analysis
15
Let st −x t for t ≥ t0 . Then st ≤ 0. By 4.24 and the condition B, and noting that
1
ak ≥ 1, we have for t ≥ t0 ,
s t ≤ −
∞
1
ϕ σx t0
psds,
2
rt
t t<tk <s b
t
/ tk ,
k
1
s tk ≤ ak stk ≤ stk ,
4.25
tk ≥ t0 .
By Lemma 3.1, we get
∞
0 ≤ s∞ ≤ st − ϕ σx t0
t
⎛
⎞
1 ⎝ ∞ 1
pudu⎠ds.
2
rs
s s<tk <u b
4.26
k
It follows that
∞
0 x t ϕ σx t0
t
⎞
⎛
1 ⎝ ∞ 1
pudu⎠ds.
2
rs
s s<tk <u b
4.27
k
In view of 4.27, we have, for t ≥ t0 ,
∞
x t ≤ −ϕ σx t0
t
⎞
⎛
1 ⎝ ∞ 1
pudu⎠ds.
2
rs
s s<tk <u b
4.28
k
It is easy to see from 2.4 and 4.28 that x t < 0. This is contrary to x t > 0 for t ≥ t0 . Thus
in case b xt must be oscillatory. The proof of Theorem 2.2 is complete.
We now give the proof of Theorem 2.3. Without loss of generality, we may assume that
k0 1. If 1.1 has an eventually positive solution, x xt for t ≥ t0 . By Lemma 3.4, there
exists a T ≥ t0 such that either a x t > 0, x t < 0, t ≥ T or b x t > 0, x t > 0, t ≥ T
holds.
0
Suppose that a holds. Note that bk ≤ 1, since for tj ≥ T and each l 0, 1, 2, . . . , xt
is decreasing on tjl , tjl1 ; then for t ∈ tj , tj1 , we have
0 xt < x tj ≤ bj x tj ≤ x tj .
4.29
Similarly, for t ∈ tj , tj1 , we have
0 xt < x tj1 ≤ bj1 x tj1 ≤ x tj1 ≤ x tj .
4.30
By induction, for any t ∈ tjl , tjl1 for l 0, 1, 2, . . . , we have
xt < x tjl ≤ · · · ≤ x tj1 ≤ x tj .
4.31
16
Abstract and Applied Analysis
So xt is decreasing and bounded on tj , ∞; we know that xt is convergent as t → ∞.
Let limt → ∞ xt r, then r ≥ 0. We assert that r 0. If r > 0, then there exists T1 ≥ T, such
that for t ≥ T1 , xt > r/2 > 0. Since ϕ x ≥ 0, then ϕxt ≥ ϕr/2. By 1.1 and condition
A, we have for t ≥ T1
r pt < 0,
rtx t −ft, x ≤ −ptϕxt ≤ −ϕ
2
t
/ tk ,
4.32
2
From condition B, and noting that bk ≤ 1, we have
2
r tk x tk ≤ bk rtk x tk ≤ rtk x tk ,
4.33
tk ≥ T 1 .
Let Φt rtx t. Then Φt > 0 for t ≥ T1 . By 4.32 and 4.33, we have for t ≥ T1 , that
r pt, t /
tk ,
Φ t ≤ −ϕ
2
Φ tk ≤ Φtk , tk ≥ T1 .
4.34
4.35
From 4.34, 4.35, and Lemma 3.1, we get, for t ≥ T1 , that
r t
Φt ≤ Φ T1 − ϕ
psds,
2 T1
4.36
It is easy to see from 2.6 and 4.36 that Φt ≤ 0 for sufficiently large t. This is contrary to
Φt > 0 for t ≥ T1 . Thus r 0, that is, limt → ∞ xt 0.
If b holds, let Ψt rtx t/ϕxt for t ≥ T. We see that Ψt > 0 for t ≥ T . By
1.1 and the condition A, we get for t ≥ T
Ψ t −ft, xt rtx tϕ xtx t −ft, xt
−
≤ −pt,
≤
ϕxt
ϕxt
ϕ2 xt
t
/ tk .
4.37
From the conditions A and B, we know that
2
2
rtk b x tk b
rtk x tk
rtk x tk Ψ tk ≤ k
≤ Ψtk ,
≤ k 0
0
ϕxtk ϕ x tk
ϕ ak xtk ϕ ak
tk ≥ T.
4.38
From 4.37, 4.38, and Lemma 3.1, we get for t ≥ T
Ψt ≤ ΨT −
t
psds.
4.39
T
It is easy to see from 2.6 and 4.39 that Ψt < 0 for sufficiently large t. This is contrary
to Ψt > 0 for t ≥ T. Thus in case b xt must be oscillatory. The proof of Theorem 2.3 is
complete.
Abstract and Applied Analysis
17
Finally, we give the proof of Theorem 2.4. Without loss of generality, we may assume
that k0 1. If 1.1 has an eventually positive solution, x xt for t ≥ t0 . By Lemma 3.4, there
exists a T ≥ t0 such that either a x t > 0, x t < 0, t ≥ T or b x t > 0, x t > 0, t ≥ T
holds.
Suppose that a holds. We may easily see that the conditions H2 , H3 of Lemma 3.5
are satisfied. Furthermore, since x t < 0, t ≥ T, then there exists some ti ≥ T , such that for
t ∈ ti , ti1 xt ≤ x ti .
4.40
xti1 ≤ x ti .
4.41
0
0 xt ≤ x ti1 ≤ bi1 xti1 ≤ bi1 x ti .
4.42
0 xti2 ≤ bi1 x ti .
4.43
In particular,
Similarly, we have for t ∈ ti1 , ti2 In particular,
By induction, we obtain for any tk > ti
xtk ≤
ti <tj <tk
0 bj x ti .
4.44
0
Since { nk1 bk } is bounded and 4.44 holds, we know that {xtk } is bounded. Thus there
exists M1 > 0, such that |xtk | ≤ M1 . It follows from the condition B that
x t − xtk ≤ max a0 − 1, b0 − 1 |xtk | ≤ M1 max a0 − 1, b0 − 1 .
k
k
k
k
k
4.45
By 4.45, we know that ∞
k1 xtk − xtk is convergent. Therefore, the condition H4 of
Lemma 3.5 is also satisfied. By Lemma 3.5, we know that limt → ∞ xt r ≥ 0. We assert that
r 0. If r > 0, then there exists T1 ≥ T, such that for t ≥ T1 , xt > r/2 > 0. Since ϕ x ≥ 0,
we have ϕxt ≥ ϕr/2. Since rtx t < 0, t ≥ T1 , there exists some ti ≥ T1 such that for
t ∈ ti , ti1 rtx t ≤ rti x ti .
4.46
rti1 x ti1 ≤ rti x ti .
4.47
In particular,
18
Abstract and Applied Analysis
Similarly, we have for t ∈ ti1 , ti2 2
2
rtx t ≤ rti1 x ti1 ≤ bi1 rti1 x ti1 ≤ bi1 rti x ti .
4.48
2
rti2 x ti2 ≤ bi1 rti x ti .
4.49
In particular,
By induction, we obtain for any tk > ti
rtk x tk ≤
ti <tj <tk
2
bj rti x ti .
4.50
2
0
2
By bk ≤ ak and the condition B, we know that { nk1 bk } is bounded, and from 4.50,
we see that {rtk x tk } is bounded. There then exists M2 > 0 such that |rtk x tk | ≤ M2 .
Therefore, we have
2
2
bk − 1 rtk x tk ≤ M2 bk − 1.
4.51
By 1.1 and the condition A, we have that 4.1 holds. Integrating 4.1 from T1 to t, it
follows from 4.51 and ϕxt ≥ ϕr/2 for t ≥ T1 that
t
rtx t − rT1 x T1 ≤
rtk x tk − x tk −
psϕxsds
T1
T1 <tk <t
r t
rtk x tk − x tk − ϕ
psds
≤
2 T1
T1 <tk <t
≤
T1 <tk <t
≤
r t
2
bk − 1 rtk x tk − ϕ
psds
2 T1
4.52
r t
2
b
−
ϕ
−
1
rt
psds
x
t
k
k
k 2 T1
T1 <tk <t
r t
2
M2 bk − 1 − ϕ
psds.
≤
2 T1
T1 <tk <t
2
Note that ∞
k1 |bk − 1| is convergent. Thus it is easy to see from 2.8 and 4.52 that x t < 0
for sufficiently large t. This is contrary to x t > 0 for t ≥ T. Thus r 0, that is, limt → ∞ xt 0.
Suppose that b holds. Let Ψt rtx t/ϕxt for t ≥ T. We see that Ψt > 0
for t ≥ T . Similar to the proof of 4.39, we also obtain
Ψt ≤ ΨT −
t
psds.
T
4.53
Abstract and Applied Analysis
19
It is easy to see from 2.8 and 4.53 that Ψt < 0 for sufficiently large t. This is contrary
to Ψt > 0 for t ≥ T . Thus in case b xt must be oscillatory. The proof of Theorem 2.4 is
complete.
Acknowledgments
This research is supported by the Natural Science Foundation of Guang Dong of China
under Grant 9151008002000012. The authors would also like to thank the reviewers for their
comments and corrections of their mistakes in the original version of this paper.
References
1 A. Tiryaki and M. F. Aktas, “Oscillation criteria of a certain class of third order nonlinear delay
differential equations with damping,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1,
pp. 54–68, 2007.
2 C. M. Hou and S. S. Cheng, “Asymptotic dichotomy in a class of fourth-order nonlinear delay
differential equations with damping,” Abstract and Applied Analysis, vol. 2009, Article ID 484158, 7
pages, 2009.
3 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
4 D. D. Bainov and M. B. Dimitrova, “Sufficient conditions for oscillations of all solutions of a class
of impulsive differential equations with deviating argument,” Journal of Applied Mathematics and
Stochastic Analysis, vol. 9, no. 1, pp. 33–42, 1996.
5 D. D. Bainov, M. B. Dimitrova, and P. S. Simeonov, “Sufficient conditions for oscillation of the solutions
of a class of impulsive differential equations with advanced argument,” Note di Matematica, vol. 14,
no. 1, pp. 139–145, 1994.
6 D. D. Bainov, M. B. Dimitrova, and A. B. Dishliev, “Oscillation of the solutions of impulsive differential
equations and inequalities with a retarded argument,” The Rocky Mountain Journal of Mathematics, vol.
28, no. 1, pp. 25–40, 1998.
7 D. D. Bainov and M. B. Dimitrova, “Oscillatory properties of the solutions of impulsive differential
equations with a deviating argument and nonconstant coefficients,” The Rocky Mountain Journal of
Mathematics, vol. 27, no. 4, pp. 1027–1040, 1997.
8 D. D. Bainov, M. B. Dimitrova, and V. A. Peter, “Oscillation properities of solutions of impulsive
differential equations and inequalities with several retarded arguments,” The Georgian Mathematical
Journal , vol. 5, no. 3, pp. 201–212, 1998.
9 D. Bainov, I. Domshlak Yu., and P. S. Simeonov, “On the oscillation properties of first-order impulsive
differential equations with a deviating argument,” Israel Journal of Mathematics, vol. 98, pp. 167–187,
1997.
10 D. D. Bainov and M. B. Dimitrova, “Oscillation of nonlinear impulsive differential equations with
deviating argument,” Boletim da Sociedade Paranaense de Matemática, vol. 16, no. 1-2, pp. 9–21, 1996.
11 D. D. Bainov, M. B. Dimitrova, and A. B. Dishliev, “Oscillating solutions of nonlinear impulsive
differential equations with a deviating argument,” Note di Matematica, vol. 15, no. 1, pp. 45–54, 1995.
12 J. H. Shen and J. S. Yu, “Nonlinear delay differential equations with impulsive perturbations,”
Mathematica Applicata, vol. 9, no. 3, pp. 272–277, 1996.
13 Y. S. Chen and W. Z. Feng, “Oscillations of second order nonlinear ODE with impulses,” Journal of
Mathematical Analysis and Applications, vol. 210, no. 1, pp. 150–169, 1997.
14 G. Q. Wang, “Oscillations of second order differential equations with impulses,” Annals of Differential
Equations, vol. 14, no. 2, pp. 295–306, 1998.
15 J .W. Luo and L. Debnath, “Oscillations of second-order nonlinear ordinary differential equations
with impulses,” Journal of Mathematical Analysis and Applications, vol. 240, no. 1, pp. 105–114, 1999.
16 H. J. Li and C. C. Yeh, “Oscillation and nonoscillation criteria for second order linear differential
equations,” Mathematische Nachrichten, vol. 194, pp. 171–184, 1998.
17 M. S. Peng and W. G. Ge, “Oscillation criteria for second-order nonlinear differential equations with
impulses,” Computers & Mathematics with Applications, vol. 39, no. 5-6, pp. 217–225, 2000.
20
Abstract and Applied Analysis
18 Z. M. He and W. G. Ge, “Oscillations of second-order nonlinear impulsive ordinary differential
equations,” Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 397–406, 2003.
19 X. L. Wu, S. Y. Chen, and H. Ji, “Oscillation of a class of second-order nonlinear ODE with impulses,”
Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 181–188, 2003.
20 Z. G. Luo and J. H. Shen, “Oscillation of second order linear differential equations with impulses,”
Applied Mathematics Letters, vol. 20, no. 1, pp. 75–81, 2007.
21 X. X. Liu and Z. T. Xu, “Oscillation of a forced super-linear second order differential equation with
impulses,” Computers & Mathematics with Applications, vol. 53, no. 11, pp. 1740–1749, 2007.
22 J. J. Jiao, L. S. Chen, and L. M. Li, “Asymptotic behavior of solutions of second-order nonlinear
impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp.
458–463, 2008.