Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 493927, 29 pages
doi:10.1155/2010/493927
Research Article
Oscillations of Second-Order Neutral Impulsive
Differential Equations
Jin-Fa Cheng1 and Yu-Ming Chu2
1
2
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 2 January 2010; Revised 26 February 2010; Accepted 28 February 2010
Academic Editor: Alberto Cabada
Copyright q 2010 J.-F. Cheng and Y.-M. Chu. 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.
Necessary and sufficient conditions are established for oscillation of second-order neutral
impulsive differential equation ytpyt−τ qyt−σ 0, t /
tk , Δy tk py tk −τq1 ytk −
σ 0, where the coefficients q, q1 > 0; p < 0, τ, σ < 0, and Δyi tk yi tk − yi t−k , i 0, 1.
1. Introduction
Oscillation theory is one of the directions which initiated the investigations of the qualitative
properties of differential equations. This theory started with the classical works of Sturm and
Kneser, and still attracts the attention of many mathematicians as much for the interesting
results obtained as for their various applications.
In 1989 the paper of Gopalsamy and Zhang 1 was published, where the first
investigation on oscillatory properties of impulsive differential equations was carried out.
The monograph 2 is the first book to present systematically the results known
up to 1998, and to demonstrate how well-know mathematical techniques and methods,
after suitable modification, can be applied in proving oscillatory theorems for impulsive
differential equations.
Recently, the oscillatory theory of differential equations has been the subject of
intensive research 3–8. In particular, many remarkable results for the oscillatory properties
of various classes of impulsive differential equations can be found in the literature 2, 9–11.
The notion of characteristic system was first introduced by Bainov and Simeonov 2;
it can be used in obtaining of various necessary and sufficient conditions for oscillation of
constant coefficients linear impulsive differential equations of first order with one or several
deviating arguments.
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Journal of Inequalities and Applications
As we know, on the case when we investigate constant coefficients linear neutral
differential equations without impulse, it is very significant to obtain necessary and sufficient
conditions for the oscillation corresponding to their characteristic equations; some necessary
and sufficient conditions in terms of the characteristic equation for the oscillation of all
solutions of first-or second-order neutral differential equations were established in 12–20.
However, the oscillation theory of the second order impulsive differential equations is not
yet perfect compared to the second order differential equations with deviating argument see
2. For example, due to some obstacles of theoretical and technical character in handling
with constant coefficients linear impulsive differential equations of second or higher order,
there are no results which studied the necessary and sufficient conditions in monograph 2.
How to establish the necessary and sufficient conditions for second order constant coefficients
linear impulsive differential equations corresponding to their characteristic systems? This is
also an important open problem see monograph 2. In this paper, we study and solve this
problem for a class of linear impulsive differential equations of second order with advanced
argument.
We shall restrict ourselves to the studying of impulsive differential equations for which
the impulse effects take part at fixed moments {tk }. Considering the second order neutral
impulsive differential equations with constant coefficients
tk ,
yt pyt − τ qyt − σ 0, t /
Δ y tk py tk − τ q1 ytk − σ 0,
1.1
where
q, q1 > 0; p < 0, τ, σ < 0,
Δyi tk yi tk − yi t−k ,
i 0, 1.
1.2
Throughout this paper, we assume that the sequence {tk } of the moments of impulse
effects has the following properties H:
H1.1 0 < t1 < t2 < · · · , and limk → ∞ tk ∞.
Here considering |τ|-periodic and |σ|-periodic equation with |τ|, |σ| > 0, we suppose
that the sequence {tk } satisfies the following conditions.
H1.2 There exist nonnegative integers n1 and n2 such that
tkn1 tk |τ|,
tkn2 tk |σ|,
k ∈ N.
1.3
This condition is equivalent to the next one.
H1.3 There exist nonnegative integers n1 and n2 such that
it, t |τ| n1 ,
it, t |σ| n2 ,
t ∈ R .
Here ia, b denotes the number of the points tk , lying in the interval a, b.
1.4
Journal of Inequalities and Applications
3
As customary, a solution of 1.1 is said to be oscillatory if it has arbitrarily large zeros.
Otherwise the solution is called nonoscillatory. As usual, we use the term “finally” to mean
“for sufficiently large t”. For other related notions, see monographs 2, 9–11 for details.
We clearly see that 1.1 is an equation without impulse and Δytk Δy tk 0
together with q1 0 if n1 n2 0 in H1.2 or H1.3. In this case, 1.1 reduces to
yt pyt − τ qyt − σ 0.
1.5
2. Asymptotic Behavior of the Solutions
In the sequence, we assume that conditions H1.1–H1.3 hold, and
vt − yt pyt − τ .
2.1
Lemma 2.1. Suppose that vt is defined by 2.1. If 1.1 has a finally positive solution yt, then
a vt is also a solution of 1.1, that is,
tk ,
vt pvt − τ qvt − σ 0, t /
Δ v tk pv tk − τ q1 vtk − σ 0.
2.2
b
−1ν vν t > 0,
lim vν t 0,
ν 0, 1, 2,
Δvtk < 0,
t→∞
ν 0, 1,
Δv tk > 0.
2.3
or
vν t > 0,
lim vν t ∞,
ν 0, 1, 2,
t→∞
Δvtk > 0,
Δv tk > 0.
ν 0, 1, 2,
2.4
c If 2.3 holds, then p < −1.
d If yt is a solution of 1.1, then
zt y t,
t/
tk ; ztk q
Δytk q1
2.5
is also a solution of 1.1.
Proof. a It follows from condition H1.2 that if tk is a moment of impulse effect, then tk − τ
is also a such moment. Thus, yt − τ is also a solution of 1.1 since 1.1 is a linear one
with constant coefficients. Therefore, vt is a solution of 1.1 which follows from the linear
combination of solutions.
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Journal of Inequalities and Applications
b We clearly see that
v t qyt − σ > 0,
t/
tk ,
Δv tk q1 ytk − σ > 0,
2.6
which imply that v t is strictly increasing and so either
lim v t ∞
2.7
lim v t l < ∞.
2.8
t→∞
or
t→∞
We clearly see that 2.7 implies inequality 2.4. Now, we assume that inequality 2.8 holds.
We first prove that l 0.
In fact, integrating 2.6 from t0 to t and letting t → ∞, we get
l − v t0 q
∞
ys − σds Δv tk ,
tk ≥t0
t0
2.9
hence
∞
ys − σds < ∞,
Δv tk < ∞,
tk ≥t0
t0
2.10
which implies that yt is integrable in t0 , ∞,
y ∈ L1 t0 , ∞.
2.11
v ∈ L1 t0 , ∞,
2.12
Then from 2.1 we get
and so l 0. Otherwise, by L’Hospital’s rule, we have limt → ∞ vt/t limt → ∞ v t l/
0. vt → ±∞ when t → ∞, then vt /
∈ L1 t0 , ∞. Thus v t increases to zero, which
implies that eventually
v t < 0.
2.13
Then vt decreases and in view of v ∈ L1 t0 , ∞, we have
lim vt 0,
t→∞
2.14
Journal of Inequalities and Applications
5
hence
2.15
vt > 0.
c For the sake of contradiction, assume that 2.3 holds and p ≥ −1. From parts a
and b, we know that wt −vt pvt − τ is also a positive solution of 1.1, therefore,
− vt pvt − τ > 0,
2.16
vt < −pvt − τ ≤ vt − τ;
2.17
which implies that
this contradicts with the fact that vt is a decreasing function.
d Let yt be a solution of 1.1, and
zt y t,
ztk t/
tk ,
q
Δytk .
q1
2.18
We prove that zt is also a solution of 1.1.
Clearly, zt satisfies
tk ,
zt pzt − τ qzt − σ 0, t /
Δz tk z tk − z t−k y tk − y t−k .
2.19
Since yt is a solution of 1.1, so
Δz tk pΔz tk − τ y tk − y t−k py tk − τ − py t−k − τ
−qy tk − σ qy t−k − σ −qΔytk − σ
2.20
q1
−q ztk − σ q1 ztk − σ.
q
That is,
Δ z tk pz tk − τ q1 ztk − σ 0,
2.21
which implies that zt is also a solution of 1.1.
Let W − and W be the set of all functions of the form
wt − vt pvt − τ > 0,
2.22
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Journal of Inequalities and Applications
where vt is a solution of 1.1 which satisfies 2.3 and 2.4, respectively. In view of
Lemma 2.1, either W − or W is nonempty. Also, an argument similar to that of Lemma 2.1
shows that each function w ∈ W − W is a solution of 1.1, and satisfies
tk ,
wt pwt − τ qwt − σ 0, t /
Δ w tk pw tk − τ q1 wtk − σ 0.
2.23
Also, there is a solution v of 1.1 which satisfies 2.3 if w ∈ W − or 2.4 if w ∈ W −
such that
w t qvt − σ,
t/
tk ,
2.24
Δw tk q1 vtk − σ.
Clearly, every function w ∈ W − satisfies
−1ν wν t > 0,
lim wν t 0,
ν 0, 1, 2,
Δwtk < 0,
t→∞
ν 0, 1,
Δw tk > 0,
2.25
while every function w ∈ W satisfies
wν t > 0,
ν 0, 1, 2,
Δwtk > 0,
lim ων t ∞,
t→∞
ν 0, 1, 2.
Δw tk > 0.
2.26
Furthermore,
wt ∈ W − ⇒ − wt pwt − τ ∈ W − ,
wt ∈ W ⇒ − wt pwt − τ ∈ W .
2.27
Finally, w1 , w2 ∈ W − ∈ W , resp. and a, b > 0 ⇒ aw1 bw2 ∈ W − ∈ W , resp..
3. Oscillation of the Unbounded Solutions
First, we will assume that W − ∅ i.e., W /
∅.
The present section is devoted to the characterizing of the oscillatory properties of
solutions of the periodic neutral impulsive differential equation 1.1.
We are looking for a positive solution of 1.1 having the form
i0,t
,
yt eλt 1 μ
where λ, μ > 0 are constants.
3.1
Journal of Inequalities and Applications
7
Substituting 3.1 in 1.1, and from condition H1.3, we obtain the characteristic
system of 1.1 as follows:
n
n
λ2 pλ2 e−λτ 1 μ 1 qe−λσ 1 μ 2 0,
n
n
λμ pλμe−λτ 1 μ 1 qe−λσ 1 μ 2 0.
3.2
As q, q1 > 0, the characteristic system is equivalent to
q1
λ,
q
q1 n1
q1 n2
2
2 −λτ
−λσ
Fλ ≡ λ pλ e
qe
0.
1 λ
1 λ
q
q
μ
3.3
Equation 3.3 is called a characteristic equation, corresponding to the 1.1.
Theorem 3.1. If the conditions (H1.1)–(H1.3) hold, then the following assertions are equivalent.
a Each unbounded regular solution of 1.1 is oscillatory.
b The characteristic equation 3.3 has no positive real roots.
The proof of a ⇒ b is obvious. In fact, if the characteristic equation 3.3 has a real
root λ > 0, then we clearly see that μ q1 /qλ > 0 and therefore the function
i0,t
yt eλt 1 μ
3.4
is an unbounded positive solution of 1.1.
The proof of b ⇒ a is quite complicated and will be accomplished by establishing
a series of lemmas.
We assume that 3.3 has no positive real roots and, for the sake of contradiction, we
assume that 1.1 has an unbounded finally positive solution yt.
Lemma 3.2. If 1.1 has an eventually positive solution, then
a σ < τ;
b there exists a positive constant m such that
q1 n1
q1 n2
Fλ ≡ λ2 pλ2 e−λτ 1 λ
qe−λσ 1 λ
≥ m,
q
q
∀λ ≥ 0.
3.5
Proof. a Otherwise, σ ≥ τ, therefore F∞ −∞, but F0 q > 0 is impossible because the
characteristic equation Fλ 0 has no positive real roots.
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Journal of Inequalities and Applications
b We have F0 q > 0, F∞ ∞ and so Fλ > 0 for all λ ≥ 0.
Hence there exists a positive constant m such that
Fλ ≥ m,
∀λ ≥ 0,
3.6
which completes the proof of Lemma 3.2.
For each function w ∈ W , define the set
⎧
⎪
⎨
⎫
−w t λ2 wt ≤ 0
finally ⎪
⎬
Λw λ ≥ 0
.
q
−Δw tk 1 λ2 wtk ≤ 0 finally ⎪
⎪
⎩
⎭
q
3.7
Clearly, 0 ∈ Λw and if λ ∈ Λw then 0, λ ⊆ Λw. Therefore, Λw is a nonempty
subinterval of R .
Lemma 3.3. For wt ∈ W and Λw, we have the folloeing.
a If w ∈ W and λ0 ≡ q/ − p1/2 , then λ0 ∈ Λw.
b Λw is bounded above by a positive constant γ, for all w ∈ W .
c If w ∈ W and λ ∈ Λw, then
w t − λwt ≥ 0,
Δwtk −
t/
tk ,
q1
λwtk ≥ 0.
q
3.8
Proof. a From 2.23, 2.26 and the fact that w t > 0 and Δw tk > 0, we have
pw t − τ qwt − σ < 0,
pΔw tk − τ q1 wtk − σ < 0.
3.9
That is,
w t q
wt τ − σ > 0,
p
q1
Δw tk wtk τ − σ < 0.
p
3.10
The increasing nature of wt and the fact that τ > σ imply that
w t q
wt > 0,
p
t/
tk ,
q1
Δw tk wtk < 0,
p
3.11
Journal of Inequalities and Applications
9
which shows that
λ0 ≡
q
−p
1/2
3.12
∈ Λw.
By integrating 3.10 from t − α to t with α > 0, we get
w t − w t − α q
p
t
ws τ − σds −
t−α
Δw tk > 0.
3.13
t≤tk <t−α
From the fact that w t − α > 0 and Δw tk > 0, we have
w t q
αwt − α τ − σ > 0.
p
3.14
By integrating again from t − β to t with β > 0 we get
q
wt − w t − β p
t
ws − α τ − σds −
t−β
Δwtk > 0,
3.15
t≤tk <t−β
from the fact that wt − β > 0 and Δwtk > 0 together with w t > 0, we have
wt q
αβw t − α β τ − σ > 0.
p
3.16
q τ − σ2
τ −σ
w t
> 0.
p 16
2
3.17
τ −σ
< Awt,
t
2
3.18
Let α β 1/4τ − σ > 0, then
wt That is,
where A −16p/qτ − σ2 > 0.
Now let k ∈ N such that −σ ≤ τ − σ/2k, then 3.18 and the increasing nature of wt
imply that
τ −σ
τ −σ
k < Aw t wt − σ ≤ w t k − 1 < · · · < Ak wt.
2
2
3.19
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Journal of Inequalities and Applications
By integrating 2.24 from t − α to t with α > 0, we get
w t > w t − w t − α t
qvs − σds t−α
Δw tk t−α≤tk <t
3.20
> qαvt − α − σ.
By integrating again from t − β to t with β > 0, we have
wt > wt − w t − β t
qαvs − α − σds t−β
Δwtk t−β≤tk <t
> qαβv t − α − β − σ .
3.21
Let α β −σ/2 > 0, then
wt >
qσ 2
vt.
4
3.22
Combining 2.24, 3.19, and 3.22, we have
w t qvt − σ <
4Ak
4
wt − σ < 2 wt,
2
σ
σ
3.23
which shows that
γ≡
2Ak/2
>0
−σ
3.24
and γ is not in the set Λw for all w ∈ W , that is, Λw is bounded above by the positive
constant γ, for all w ∈ W .
c Let
ϕt e
−λt
−i0,t
q1
1 λ
q
wt,
3.25
Journal of Inequalities and Applications
11
then
q1 −i0,t
q1 −i0,t wt e−λt 1 λ
w t
ϕ t −λe−λt 1 λ
q
q
q1 −i0,t −λt
1 λ
e
tk ,
w t − λwt , t /
q
q1 −i0,tk q1 −i0,tk −λtk
wtk e−λtk 1 λ
Δϕtk Δ e
Δwtk 1 λ
q
q
q1 −i0,tk q1
1 λ
Δwtk − λwtk ,
e
q
q
q1 −i0,t w t − 2λw t λ2 wt ,
ϕ t e−λt 1 λ
q
q1 −i0,tk −λtk
Δϕ tk Δ e
1 λ
w tk − λwtk q
−λtk
t
/ tk ,
3.26
q1 −i0,tk e−λtk 1 λ
Δw tk − λΔwtk q
q1 −i0,tk q
q1
1 λ
Δwtk − λwtk e−λtk − λ
q
q
q1
q1 −i0,tk e−λtk 1 λ
Δw tk − λΔwtk q
q1 −i0,tk −λtk
2 q1
e
wtk .
1 λ
Δw tk − 2λΔwtk λ
q
q
Therefore,
q1 −i0,t tk ,
ϕ t 2λϕ t e
w t − λ2 wt ≥ 0, t /
1 λ
q
q1
q1 −i0,tk Δϕ tk 2λΔϕtk e−λtk 1 λ
Δw tk − λ2 wtk ≥ 0.
q
q
−λt
3.27
3.28
We clearly see that ϕ te2λt is a nondecreasing function and so if the conclusion in part c
was false, then
ϕ t < 0,
Δϕtk < 0.
3.29
Δϕ tk > 0,
3.30
From 3.27, 3.28, and 3.29, we know that
ϕ t > 0,
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Journal of Inequalities and Applications
and so
w t − 2λw t λ2 wt > 0,
t/
tk ,
q1
wtk > 0,
q
Δw tk − 2λΔwtk λ2
3.31
which together with the hypothesis yield that
w t − λ2 wt ≥ 0,
Δw tk − λ2
t/
tk ,
q1
wtk ≥ 0.
q
3.32
Hence
w t − λw t > 0,
t/
tk ,
3.33
Δw tk − λΔwtk > 0.
3.34
tk ,
ut − w t − λwt , t /
q
Δwtk − λwtk ,
utk − w tk − λwtk −
q1
3.35
Let
Then by Lemma 2.1d and the linear combination of solutions, we know that ut is a
solution of 1.1. From 3.29, 3.33, and 3.34, we have
ut > 0,
u t < 0,
Δut < 0.
3.36
Now using u instead of y in 2.1 and the hypothesis that W − ∅ together with a
similar argument as in 2.4, we get
lim − ut put − τ ∞.
3.37
lim ut ∈ R,
3.38
t→∞
But 3.36 implies that
t→∞
and the contradiction completes the proof of Lemma 3.3.
Journal of Inequalities and Applications
13
By integrating both sides of 2.23 from t0 σ to t, we get
w t pw t − τ − w t0 σ pw t0 σ − τ q
−
Δw tk −
t0 σ≤tk <t
t
t0 σ
ws − σds
pΔw tk − τ 0.
3.39
t0 σ≤tk <t
That is,
− w t pw t − τ c q
cq
t
t0 σ
t−σ
ws − σds q1 wtk − σ
t0 σ≤tk <t
wsds 3.40
q1 wtk ,
t0 ≤tk <t−σ
t0
where
c − w t0 σ pw t0 σ − τ .
3.41
As w t is a solution of 1.1 and wt satisfying 2.24, it follows from 3.40 that if w ∈ W ,
then
cq
t−σ
wsds q1 wtk ∈ W ,
t0 ≤tk <t−σ
t0
3.42
where c is the constant given by 3.41.
Lemma 3.4. Suppose that m and γ are the constants defined in Lemma 3.2b and Lemma 3.3b,
respectively. If w ∈ W , λ ∈ Λw, and
m
N n2
n > 0,
−γσ
2 e
− pe−γτ 1 q1 /q γ 1
1 q1 /q γ
3.43
then
λ2 N
1/2
3.44
∈ Λz,
where
zt − wt pwt − τ λ
t−σ
t0
wsds q1
cλ
λ
.
wtk q t0 ≤tk <t−σ
q
3.45
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Journal of Inequalities and Applications
Proof. Clearly zt is an element of W . From Lemma 3.3c, we have
w t − λwt ≥ 0,
Δwtk −
t
/ tk ,
3.46
q1
λwtk ≥ 0.
q
Then from 3.45 we get
z t qwt − σ λw t − σ ≥ q λ2 wt − σ,
Δz tk z tk − z t−k q1 wtk − σ λΔwtk − σ ≥
t/
tk ,
q1
q1 λ2 wtk − σ.
q
3.47
By integrating 3.46 from t0 to t, we obtain
0 ≤ wt − λ
t
wsds −
t−σ
t0
Δwtk ≤ wt − λ
t
t0 ≤tk <t
t0
wt − λ
wsds −
t0
q1
λwtk λ
wsds −
q
t0 ≤tk <t−σ
t−σ
t
q1
λwtk q
t0 ≤tk <t
q1
λwtk wsds q
t≤tk <t−σ
3.48
and so
−wt λ
t−σ
t0
wsds q1
λwtk ≤ λ
q
t0 ≤tk <t−σ
t−σ
wsds t
q1
λwtk .
q
t≤tk <t−σ
3.49
Let
q1 i0,t
;
θt eλt 1 λ
q
3.50
then
θ t λθt,
Δθtk t/
tk ,
q1
λθtk .
q
3.51
By integrating it from t to t − σ, we have
θt − σ − θt λ
t−σ
t
θsds q1
λθtk .
q
t≤tk <t−σ
3.52
Journal of Inequalities and Applications
15
Set
q1 −i0,t
ϕt wte−λt 1 λ
,
q
3.53
as defined in the proof of Lemma 3.3c.
We clearly see that ϕt is a nondecreasing function, therefore,
t−σ
λ
wsds t
t−σ
q1
q1
λwtk λ
λϕtk θtk ϕsθsdt q
q
t
t≤tk <t−σ
t≤tk <t−σ
t−σ
q1
λθtk θsds ≤ ϕt − σ λ
q
t
t≤tk <t−σ
3.54
ϕt − σθt − σ − θt.
Let
cλ
.
c1 − λ2 N
q
3.55
Using 3.45, 3.47, 3.49, 3.54, the increasing nature of wt, ϕt, and the fact that
t0 < t < t − τ < t − σ,
we have
z t − λ2 N zt
t−σ
q1
2
λwtk − pwt − τ c1
≥ λ q wt − σ − λ N λ
wsds q
t
t≤tk <t−σ
2
q1 i0,t−σ
≥ λ2 q ϕt − σeλt−σ 1 λ
q
q1 i0,t−τ
2
λt−τ
c1
− λ N ϕt − σθt − σ − θt − pϕt − σe
1 λ
q
ϕt − σe
λt
q1 i0,t
1 λ
q
3.56
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Journal of Inequalities and Applications
×
q1 n2 2
− λ N
λ2 q e−λσ 1 λ
q
× e
−λσ
n q1 n2
q1 n1 c1 e−λσ 1 q1 /q λ 2
2
−λτ
1 λ
1 λ
− 1 λ N pe
q
q
wt − σ
q1 i0,t
ϕt − σeλt 1 λ
q
q1 n1
q1 n2
× λ2 pe−λτ 1 λ
qe−λσ 1 λ
− Ne−λσ
q
q
n q1 n2
q1 n1 c1 e−λσ 1 q1 /q λ 2
−λτ
× 1 λ
N Npe
1 λ
q
q
wt − σ
q1 i0,t
≥ ϕt − σeλt 1 λ
q
q1 n1
q1 n2
2
−λτ
−λσ
× λ pe
qe
1 λ
1 λ
q
q
n
c1 e−λσ 1 q1 /q λ 2
q1 n2
q1 n1
−λσ
−λτ
−N e
1 λ
1 λ
− pe
wt − σ
q
q
q1 i0,t
≥ ϕt − σe
1 λ
q
⎞
⎛
q1 n2
−λσ
λ
c
e
1
1
⎜
q
q1 n2
q1 n1 ⎟
⎟
⎜
− N e−λσ 1 λ
× ⎜m − pe−λτ 1 λ
⎟,
⎠
⎝
wt − σ
q
q
λt
t/
tk .
3.57
As
lim wt ∞,
3.58
n
c1 e−λσ 1 q1 /q λ 2
m
≥ ,
m
wt − σ
2
3.59
t→∞
we see that for sufficiently large t,
and as γ ≥ λ, so
e
−λσ
q1 n2
q1 n1
q1 n2
q1 n1
m
−λτ
−γ
−γτ
. 3.60
1 λ
1 λ
1 γ
1 γ
− pe
≤e
− pe
q
q
q
q
2N
Journal of Inequalities and Applications
17
Then
z t − λ N zt ≥ ϕt − σe
2
λt
q1 i0,t m m −
0,
1 λ
q
2
2
t/
tk .
3.61
Analogously,
Δz tk −
≥
q1 2
λ N ztk q
tk −σ
q1 2
q1 2
q1 q1
λ N λ
λwtl − pwtk − τ
q1 λ wtk − σ −
wsds q
q
q tk ≤tl <tk −σ q
tk
q1
q1 2
q1 i0,tk −σ
c1 ≥
λ q ϕtk − σeλtk −σ 1 λ
q
q
q
q1 2
q1 i0,tk −τ
λtk −τ
c1 ≥ 0.
λ N ϕtk − σθtk − σ − θtk − pϕtk − σe
−
1 λ
q
q
3.62
Therefore, the proof of Lemma 3.4 is completed.
Now, considering the sequence of functions
zn t − zn−1 t pzn−1 t − τ λn
t−σ
wsds t0
q1
cn λn
λn
,
wtk q t0 ≤tk <t−σ
q
3.63
n 1, 2, . . . ,
where z0 t is the function defined by 3.45, λ0 is the number defined in Lemma 3.3a,
m
N n
n > 0,
2 e−γ 1 q1 /q γ 2 − pe−γτ 1 q1 /q γ 1
1/2
,
λn λ2n−1 N
3.64
cn − zn t0 σ pzn t0 σ − τ .
The repeated applications of Lemma 3.4 lead to
λn ∈ Λzn−1 for n 1, 2, . . . .
3.65
Clearly
lim λn ∞,
n→∞
3.66
18
Journal of Inequalities and Applications
which contradicts with the fact proved in Lemma 3.3b that
λn ≤ γ
for n 1, 2, . . . .
3.67
Therefore, the proof of Theorem 3.1 is completed.
4. Oscillation of the Bounded Solutions
In Section 2, we complete the case of W − ∅. Now we consider the conditions ensuring
∅, that is, considering the conditions ensuring oscillation of the
oscillation of the case of W − /
bounded solutions of 1.1. Then in view of Lemma 2.1c, p < −1.
We are looking for a positive solution of 1.1 with the form
i0,t
,
yt e−λt 1 − μ
4.1
where λ, μ > 0 are constants.
Substituting 4.1 in 1.1, just like in Section 2, we can obtain the characteristic
equation of 1.1 as follows:
Fλ ≡ λ pλ e
2
2 λτ
n1
q1
1− λ
q
qe
λσ
n2
q1
1− λ
q
0,
4.2
where
it, t − τ n1 ,
it, t − σ n2 .
4.3
Theorem 4.1. If the condition (H) holds, then the following assertions are equivalent.
a Each bounded regular solution of the equation 1.5 is oscillatory.
b The characteristic equation 4.2 has no real roots λ ∈ 0, q/q1 .
The proof of a ⇒ b is obvious. In fact, if the characteristic equation 4.2 has a real
root λ ∈ 0, q/q1 , then μ q1 /qλ ≥ 0 and 0 < 1 − μ ≤ 1, therefore the function
i0,t
yt e−λt 1 − μ
4.4
is a bounded positive solution of 1.1.
The proof of b ⇒ a is quite complicated and will be accomplished by establishing
a series of lemmas.
Journal of Inequalities and Applications
19
As in Section 3 we assume, without further mention, that 1.2 holds. We also assume
that 4.2 has no real roots λ ∈ 0, q/q1 . For the sake of contradiction, we assume that 1.1
has a bounded finally positive solution yt.
Also like the case in Section 3, for each function w ∈ W − , define the set
⎧
⎪
⎨
−w t λ2 wt ≤ 0
Λw λ ≥ 0
q
−Δw tk 1 λ2 wtk ≤ 0
⎪
⎩
q
⎫
finally ⎪
⎬
.
finally ⎪
⎭
4.5
Clearly, 0 ∈ Λw and if λ ∈ Λw, then 0, λ ⊆ Λw. That is, Λw is a nonempty
subinterval of R .
Lemma 4.2. There exists a positive constant m such that
q1 n1
q1 n2
λσ
qe
≥m
1− λ
1− λ
Fλ ≡ λ pλ e
q
q
2
2 λτ
q
∀λ ∈ 0,
.
q1
4.6
Proof. We clearly see that
F0 q > 0,
q−
F
q1
q
q1
2
>0
4.7
and so there exists a positive constant m such that
Fλ ≥ m
q
.
∀λ ∈ 0,
q1
4.8
Lemma 4.3. (a) If w ∈ W − and l ∈ N with −lτ > τ − σ, then
λ1 ≡
q
k1
−p
1/2
∈ Λw.
4.9
(b) If w ∈ W − and λ ∈ Λw, then
w t λwt ≤ 0,
Δwtk t/
tk ,
q1
λwtk ≤ 0.
q
(c) Λw is bounded above by a positive constant λ2 for any w ∈ W − .
4.10
20
Journal of Inequalities and Applications
Proof. a Let l ∈ N with −lτ > τ − σ.
For w ∈ W − we have
− wt pwt − τ > 0
4.11
l
l
wt < −pwt − τ < −p wt − lτ < −p wt τ − σ.
4.12
wt
wt τ − σ > l ,
−p
4.13
and so
That is,
as
w t q
wt τ − σ > 0,
p
t/
tk ,
q1
Δw tk wtk τ − σ > 0.
p
4.14
So
q
/ tk ,
k1 wt > 0, t −p
q
q1
wtk > 0.
Δw tk −
q −pk1
w t − 4.15
That is,
q
k1
−p
1/2
∈ Λw.
4.16
Journal of Inequalities and Applications
21
b Let
ψt w t λwt,
ψtk t/
tk ,
4.17
q1
Δwtk λwtk ,
q
then
ψ t − λψt w t − λ2 wt ≥ 0,
Δψtk −
t/
tk ,
q1
q1
λψtk Δw tk − λ2 wtk ≥ 0.
q
q
4.18
Let
ut ψte
−λt
q1 i0,t
,
1− λ
q
4.19
then
q1 i0,t u t e−λt 1 − λ
ψ t − λψt ≥ 0, t / tk ,
q
q1 i0,tk q1
Δutk e−λtk 1 − λ
Δψt − λψtk ≥ 0.
q
q
4.20
Therefore, ut is a nondecreasing function.
Note that
lim ut 0.
t→∞
4.21
We know that ut ≤ 0 and so
ψt ≤ 0,
ψtk ≤ 0.
4.22
22
Journal of Inequalities and Applications
c Otherwise,
λ2 ≡
1 ln −p ∈ Λw for some w ∈ W − .
−τ
4.23
Then from part b
w t λ2 wt ≤ 0,
4.24
we know that the function wteλ2 t is nonincreasing and
wteλ2 t ≥ wt − τeλ2 t−τ
4.25
wt ≥ wt − τe−λ2 τ −pwt − τ.
4.26
or
This contradicts with3.54; therefore, The proof of Lemma 4.3 is completed.
Lemma 4.4. Suppose that m is a constant defined in Lemma 4.2 and T > σ. If w ∈ W − , λ sup Λw, and
N
e
m
−it−T,t −it−T,t
n > 0,
q/λ2 eλT 1 − q1 /q λ
− peλT 1 − q1 /q γ 1
1 − q1 /q λ
4.27
−λT
then
λ2 N
1/2
4.28
∈ Λz,
where
zt g t pg t − τ q
t−σ
gξdξ q1
t−T ≤tk <t−σ
t−T
λ
2
∞
t−T
2 q1
gξdξ λ
q
gtk 4.29
gtk tk ≥t−T
and
gt w t − λw t.
4.30
Journal of Inequalities and Applications
23
Proof. Clearly, zt is an element of W − . If
φt e
λt
−i0,t
q1
1− λ
q
−g t > 0,
4.31
then
q1 −i0,t 4
−w t λ2 w t < 0,
φ t eλt 1 − λ
q
q1 i0,t
−λt
−g t e
φt, t /
tk ,
1− λ
q
q1 i0,tk q1
q1
φtk .
−Δgtk − g tk e−λtk 1 − λ
q
q
q
4.32
Let
θt e
−λt
q1 i0,t
,
1− λ
q
4.33
then
θ t −λθt,
Δθtk −
t/
tk ,
4.34
q1
λθtk .
q
By integrating 4.34 from t to t1 , we get
θt1 − θt −λ
t1
t
θξdξ −
t≤tk <t1
Δθtk .
4.35
24
Journal of Inequalities and Applications
Let t1 → ∞; then
−θt −λ
∞
θξdξ −
t
q1 λ θtk .
q tk ≥t
4.36
Integrating 4.32 from t to ∞, we get
gt ∞
e
−λξ
t
∞
t
φξdξ −
Δgtk tk ≥t
q1 −λtk
q1 i0,ξ
q1 i0,tk e−λξ 1 − λ
φξdξ e
φtk 1− λ
q
q tk ≥t
q
∞
≤ φt
t
i0,ξ
q1
1− λ
q
q1 θξdξ θtk q tk ≥t
4.37
q1 i0,t
1
1
1
φtθt φte−λt 1 − λ
− g t.
λ
λ
q
λ
That is,
g t λgt ≤ 0.
4.38
By integrating again from t to ∞, we find
0 ≥ −gt λ
∞
gξdξ t
1
g t λ
λ
∞
q1 q1 1
λ gtk ≥ g t λ
gξdξ λ gtk q tk ≥t
λ
q tk ≥t
t
∞
t
q1 q1 gξdξ λ
gtk − λ
gξdξ − λ
gtk .
q tk ≥t−T
q t−T ≤tk <t
t−T
t−T
4.39
Then from 4.35 we have
g t λ
2
∞
t
q1 2 q1
2
gξdξ λ
gtk ≤ λ
gξdξ λ2
gtk q tk ≥t−T
q t−T ≤tk <t
t−T
t−T
≤ λ2
t
t−T
1
q1
1
θξφξdξ λ2
θtk φtk λ
q t−T ≤tk <t λ
≤ φt − T θt − T − θt
4.40
Journal of Inequalities and Applications
25
and
t−σ
q
gξdξ q1
t−T
t−T ≤tk <t−σ
gtk ≤
q
φt − T θt − T − θt − σ.
λ2
4.41
Using 4.29 and 4.32 together with 4.38–4.41, we see that
z t −q − λ2 g t − T q1 i0,t−T λ2 q e−λt−T 1 − λ
φt − T ,
q
t−σ
gξdξ q1
zt g t pg t − τ q
t−T
≤ φt − T θt − T − θt t−T ≤tk <t−σ
gtk λ2
∞
gξdξ λ2
t−T
q
φt − T θt − T − θt − σ pg t − τ.
λ2
q1 gtk q tk ≥t−T
4.42
Therefore,
z t − λ2 N zt
q1 i0,t−T ≥ λ2 q e−λt−T 1 − λ
φt − T q
q
2
− λ N φt − T θt − T − θt 2 φt − T θt − T − θt − σ pg t − τ
λ
q1 i0,t−T φt − T λ2 q e−λt−T 1 − λ
q
q1 i0,t−T q1 i0,t
2
−λt−T −λt
− λ N φt − T e
−e
1− λ
1− λ
q
q
q1 i0,t−T q1 i0,t−σ
q
−λt−T −λt−σ
−e
1− λ
1− λ
φt − T 2 e
q
q
λ
q1 i0,t−τ
−λt−τ
.
1− λ
−pφt − τe
q
4.43
26
Journal of Inequalities and Applications
For T > 0 and τ < 0, we have t − T < t − τ. From the fact that φt is nonincreasing, we clearly
see that φt − T > φt − τ. Therefore,
z t − λ2 N zt
q1 i0,t
≥ φt − T e−λt 1 − λ
q
!
q1 −it−T,t 2
2
λT
1− λ
×
λ q e
− λ N
q
q
q1 −it−T,t
q1 −it−T,t
q1 n2
λT
λσ
−1 2 e
−e
1− λ
1− λ
1− λ
× e
q
q
q
λ
λT
−pe
λτ
"
q1 n1
1− λ
q
q1 i0,t
φt − T e−λt 1 − λ
q
!
q1 n1
q1 n2
× λ2 pλ2 eλτ 1 − λ
qeλσ 1 − λ
q
q
−N e
λT
q1 −it−T,t
q1 −it−T,t
q1 n2
q
λT
λσ
1− λ
1− λ
1− λ
−1 2 e
−e
q
q
q
λ
−pe
≥ φt − T e
−λt
λτ
n1 "
q1
1− λ
q
q1 i0,t
1− λ
q
!
q1 n1
q1 n2
× λ2 pλ2 eλτ 1 − λ
qeλσ 1 − λ
−N
q
q
"
q1 −it−T,t q λT
q1 −it−T,t
q1 n1
λτ
2e
− pe
1− λ
1− λ
1− λ
× e
q
q
q
λ
λT
≥ φt − T e
−λt
q1 i0,t
1− λ
m − m 0.
q
4.44
Journal of Inequalities and Applications
27
Analogously, we have
Δz tk z tk − z t−k
g tk pg tk − τ q g tk − σ − g tk − T − λ2 g tk − T
− g t−k pg t−k − τ q g t−k − σ − g t−k − T
− λ2 g t−k − T
Δg tk pΔg tk − τ q Δgtk − σ − Δgtk − T − λ2 Δgtk − T 4.45
−q1 g tk − σ q Δgtk − σ − Δgtk − T − λ2 Δgtk − T −qΔgtk − T − λ2 Δgtk − T q1 2
q1 i0,tk −T λ q e−λtk −T 1 − λ
φtk − T .
q
q
So
Δz tk −
q1 2
λ N zt
q
!
q1 2
q1 i0,tk −T ≥
λ q e−λtk −T 1 − λ
φtk − T − λ2 N
q
q
"
q
× φtk − T θtk − T − θtk 2 φtk − T θtk − T − θtk − σ pg tk − τ
λ
≥ 0.
4.46
Therefore, the proof of Lemma 4.4 is completed.
Now, we consider the sequence of functions
zn t zn−1 t pzn−1 t − τ q
λ2
∞
t−σ
zn−1 ξdξ q1
t−T
q1 zn−1 ξdξ λ2
zn−1 tk ,
q tk ≥t−T
t−T
zn−1 tk t−T ≤tk <t−σ
4.47
n 1, 2, . . . ,
28
Journal of Inequalities and Applications
where, z0 t is the function zt defined in 4.29,
λ0 sup Λw,
1/2
λn λ2n−1 N
,
N
e
m
−it−T,t q λT −it−T,t
n > 0.
2 e 1 − q1 /q λ
− peλT 1 − q1 /q γ 1
1 − q1 /q λ
λ
4.48
−λT
The repeated applications of Lemma 4.4 lead to
λn ∈ Λzn−1 for n 1, 2, . . . .
4.49
Clearly,
lim λn ∞,
n→∞
4.50
which contradicts with the fact proved in Lemma 4.3c that Λw is bounded above for any
w ∈ W − . Therefore, the proof of Theorem 4.1 is completed.
Remark 4.5. Let n1 n2 0, q1 0, then 1.1 reduces to 1.5, and 3.3 and 4.2 reduce to
Fλ ≡ λ2 pλ2 eλτ qeλσ 0.
4.51
From our Theorems 3.1 and 4.1, we get the following well-known results:
Corollary 4.6. The following assertions are equivalent.
a Each unbounded regular solution of the equation 1.5 is oscillatory.
b The characteristic equation 4.51 has no positive real roots.
Corollary 4.7. The following assertions are equivalent.
a Each bounded regular solution of the equation 1.5 is oscillatory.
b The characteristic equation 4.51 has no real root λ ∈ 0, ∞.
Acknowledgment
The authors wish to thank the anonymous referee for the very careful reading of the
manuscript and fruitful comments and suggestions.
Journal of Inequalities and Applications
29
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