Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 43, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
INTERVAL OSCILLATION CRITERIA FOR SECOND-ORDER
FORCED DELAY DIFFERENTIAL EQUATIONS UNDER
IMPULSE EFFECTS
QIAOLUAN LI, WING-SUM CHEUNG
Abstract. We establish some oscillation criteria for a forced second-order
differential equation with impulses. These results extend some well-known
results for forced second-order impulsive differential equations with delay.
1. Introduction
In this article, we consider the second-order impulsive delay differential equation
n
X
0
0
qi (t)Φαi (x(t − τ )) = e(t), t ≥ t0 , t 6= tk ,
(p(t)x (t)) + q(t)x(t − τ ) +
(1.1)
i=1
x(t+
k ) = ak x(tk ),
0
x0 (t+
k ) = bk x (tk ),
k = 1, 2, . . . ,
where
x(t−
k ) := lim− x(t),
x(t+
k ) := lim+ x(t),
t→tk
t→tk
x(tk + h) − x(tk )
x(tk + h) − x(tk )
, x0 (t+
,
k ) := lim+
h
h
h→0
Φα (s) := |s|α−1 s, 0 ≤ t0 < t1 < · · · < tk < . . . , limk→∞ tk = ∞, the exponents of
the nonlinearities satisfy
x0 (t−
k ) := lim−
h→0
α1 > α2 > · · · > αm > 1 > αm+1 > · · · > αn > 0,
the functions p, q, qi , e are piecewise left continuous at each tk , more precisely, they
belong to the set
P LC[t0 , ∞) := h : [t0 , ∞) → R|, h is continuous on each interval (tk , tk+1 ),
−
h(t±
k ) exists, and h(tk ) = h(tk ) for all k ∈ N ,
and p > 0 is a nondecreasing function.
A function x ∈ P LC[t0 , ∞) is said to be a solution of (1.1) if x(t) satisfies (1.1),
and x(t) and x0 (t) are left continuous at every tk , k ∈ N.
In the past few decades, there has been a great deal of work on the oscillatory
behavior of the solutions of second order differential equations, see [2, 13] and the
2000 Mathematics Subject Classification. 34K11, 34C15.
Key words and phrases. Oscillation; impulse; delay differential equation.
c
2013
Texas State University - San Marcos.
Submitted July 11, 2012. Published February 8, 2013.
1
2
QIAOLUAN LI, WING-SUM CHEUNG
EJDE-2013/43
references cited therein. Impulsive differential equations are an effective tool for
the simulation of processes and phenomena observed in control theory, population
dynamics, economics, etc. Research in this direction was initiated by Gopalsamy
and Zhang in [6]. Since then there has been an increasing interest in finding the
oscillation criteria for such equations, see [1, 3, 4, 8, 9, 10, 11] their references.
Liu and Xu [8], obtained several oscillation theorems for the equation
(r(t)x0 (t))0 + p(t)|x(t)|α−1 x(t) = q(t),
x(t+
k)
= ak x(tk ),
0
x
(t+
k)
0
= bk x (tk ),
x(t+
0)
t ≥ t0 , t 6= tk ,
= x0 ,
0
x0 (t+
0 ) = x0 ,
(1.2)
which is a special case of (1.1).
More recently, Guvenilir [7] established interval criteria for the oscillation of
second-order functional differential equations with oscillatory potentials for the
equation
(k(t)x0 (t))0 + p(t)x(g(t)) + q(t)|x(g(t))|γ−1 x(g(t)) = e(t),
t ≥ 0.
(1.3)
We note that when g(t) = t − τ , this equation is included in (1.1).
In this article, some new sufficient conditions for the oscillation of solutions of
(1.1) are presented, and illustrated by an example. It should be noted that the
derivation in this work adopts new estimates which are not a routine extension of
the existing techniques used for the non-delay case.
As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily
large zeros; otherwise the solution is said to be non-oscillatory.
2. Main Results
We will assume the following three conditions throughout this article.
(H1) τ ≥ 0, bk , ak > 0, tk+1 − tk > τ , k = 1, 2, . . . ; α1 > α2 > · · · > αm > 1 >
αm+1 > · · · > αn > 0, (n > m ≥ 1).
(H2) p, q, qi , e ∈ P LC[t0 , ∞).
(H3) For any T ≥ 0, there exist intervals [c1 , d1 ] and [c2 , d2 ] contained in [T, ∞)
such that c1 < d1 ≤ d1 + τ ≤ c2 < d2 , cj , dj 6∈ {tk }, j = 1, 2, k = 1, 2, . . .
and
q(t) ≥ 0,
qi (t) ≥ 0
for t ∈ [c1 − τ, d1 ] ∪ [c2 − τ, d2 ],
e(t) ≤ 0
for t ∈ [c1 − τ, d1 ];
e(t) ≥ 0
for t ∈ [c2 − τ, d2 ].
i = 1, 2, . . . , n;
Denote
I(s) := max{i : t0 < ti < s};
βj := max{p(t) : t ∈ [cj , dj ]},
j = 1, 2;
Ω := {ω ∈ C 1 [cj , dj ] : ω(t) 6≡ 0, ω(cj ) = ω(dj ) = 0, j = 1, 2};
Γ := G ∈ C 1 [cj , dj ] : G ≥ 0, G 6≡ 0, G(cj ) = G(dj ) = 0,
p
G0 (t) = 2g(t) G(t), j = 1, 2 .
Before giving the main results, we introduce the following Lemma.
Lemma 2.1 ([12]). For any n-tuple (α1 , . . . , αn ) satisfying α1 > · · · > αm > 1 >
αm+1 > · · · > αn > 0, there exists an n-tuple (η1 , . . . , ηn ) satisfying either (a)
n
n
X
X
αi ηi = 1,
ηi < 1, 0 < ηi < 1,
i=1
i=1
EJDE-2013/43
INTERVAL OSCILLATION CRITERIA
3
or (b)
n
X
αi ηi = 1,
i=1
n
X
ηi = 1,
0 < ηi < 1.
i=1
Theorem 2.2. Assume that conditions (H1)–(H3) hold, and that there exists ω ∈ Ω
such that
Z dj
Z tI(c )+1
j
p(t)ω 02 (t)dt −
Q(t)QjI(cj ) (t)ω 2 (t)dt
cj
cj
I(dj )
−
Z
X
k=I(cj )+2
tk
tk−1
Q(t)Qjk (t)ω 2 (t)dt
Z
dj
−
tI(dj )
Q(t)QjI(dj ) (t)ω 2 (t)dt
(2.1)
≤ L(ω, cj , dj ),
where L(ω, cj , dj ) := 0 for I(cj ) = I(dj ), and
L(ω, cj , dj )
aI(cj )+1 − bI(cj )+1
:= βj ω 2 (tI(cj )+1 )
+
aI(cj )+1 (tI(cj )+1 − cj )
I(dj )
X
ω 2 (tk )
k=I(cj )+2
ak − bk ak (tk − tk−1 )
for I(cj ) < I(dj ), j = 1, 2,
Q(t) := q(t) + η0−η0
n
Y
ηi−ηi qiηi (t)|e(t)|η0 ,
i=1
(
Qjk (t) :=
t−tk
ak τ +bk (t−tk ) ,
t−tk −τ
t−tk ,
t ∈ (tk , tk + τ ),
t ∈ [tk + τ, tk+1 ],
k = I(cj ), I(cj ) + 1, . . . , I(dj ),
and η1 , η2 , . . . , ηn are positive constants satisfying part (a) of Lemma 2.1. Then all
solutions of (1.1) are oscillatory.
Proof. Suppose that x(t) is a non-oscillatory solution of (1.1). By re-defining t0
if necessary, without loss of generality, we may assume that x(t − τ ) > 0 for all
t ≥ t0 > 0. Define the Riccati transformation
v(t) :=
p(t)x0 (t)
.
x(t)
It follows from (1.1) that v(t) satisfies
n
v 0 (t) = −q(t)
x(t − τ ) X
x(t − τ )
e(t)
v 2 (t)
−
qi (t)|x(t − τ )|αi −1
+
−
,
x(t)
x(t)
x(t)
p(t)
i=1
(2.2)
bk
for all t 6= tk , t ≥ t0 , and v(t+
k ) = ak v(tk ) for all k ∈ N.
From the assumptions, we can choose c1 , d1 ≥ t0 such that q(t) ≥ 0 and qi (t) ≥ 0
for t ∈ [c1 − τ, d1 ], i = 1, 2, . . . , n, and e(t)P
≤ 0 for t ∈ [c1 − τ,P
d1 ]. By Lemma 2.1,
n
n
there existPηi > 0, i = 1, . . . , n, such that i=1 αi ηi = 1 and i=1 ηi < 1. Define
n
η0 := 1 − i=1 ηi and let
e(t)x(t − τ ) −1
u0 := η0−1 x (t − τ ),
x(t)
4
QIAOLUAN LI, WING-SUM CHEUNG
ui := ηi−1 qi (t)
x(t − τ ) αi −1
x
(t − τ ),
x(t)
EJDE-2013/43
i = 1, 2, . . . , n .
Then by the arithmetic-geometric mean inequality (see [5]), we have
n
X
ηi ui ≥
i=0
n
Y
uηi i
i=0
and so
v 0 (t) ≤ −η0−η0
n
Y
ηi−ηi qiηi (t)
i=1
xηi (t − τ ) (αi −1)ηi
(t − τ )|e(t)|η0
x
xηi (t)
x(t − τ ) v 2 (t)
xη0 (t − τ ) −η0
(t
−
τ
)
−
q(t)
×
x
−
,
xη0 (t)
x(t)
p(t)
(2.3)
t 6= tk .
Since
n
Y
xηi (t − τ )
i=0
xηi (t)
n
Y
=
xη0 +η1 +···+ηn (t − τ )
x(t − τ )
=
,
η
+η
+···+η
0
1
n
x
(t)
x(t)
x(αi −1)ηi (t − τ )x−η0 (t − τ ) = 1 ,
i=1
we obtain
v 0 (t) ≤ −q(t)
n
Y
x(t − τ )
v 2 (t)
x(t − τ )
ηi−ηi qiηi (t)
− η0−η0
|e(t)|η0 −
x(t)
x(t)
p(t)
i=1
x(t − τ ) v 2 (t)
−
,
= −Q(t)
x(t)
p(t)
(2.4)
t 6= tk .
Multiply both sides of (2.4) by ω 2 (t), with w as prescribed in the hypothesis of the
theorem. Then integrate from c1 to d1 ; using integration by parts on the left side,
we have
I(d1 )
X
ω 2 (tk )[v(tk ) − v(t+
k )]
k=I(c1 )+1
Z
d1
≤2
c1
d1
Z
=2
ω(t)ω 0 (t)v(t)dt −
Z
ω(t)ω 0 (t)v(t)dt −
Z
c1
d1
c1
tI(c1 )+1
c1
I(d1 )−1
ω 2 (t)Q(t)
x(t − τ )
dt −
x(t)
Z
d1
c1
x(t − τ )
ω 2 (t)Q(t)
dt
x(t)
v 2 (t)ω 2 (t)
dt
p(t)
(2.5)
tk+1
x(t − τ )
dt
x(t)
k=I(c1 )+1 tk
Z d1
Z d1 2
x(t − τ )
v (t)ω 2 (t)
−
ω 2 (t)Q(t)
dt −
dt.
x(t)
p(t)
tI(d )
c1
−
X
Z
ω 2 (t)Q(t)
1
)
To estimate x(t−τ
x(t) , we first consider the situation where I(c1 ) < I(d1 ). In this
case, all the impulsive moments in [c1 , d1 ] are tI(c1 )+1 , tI(c1 )+2 , . . . , tI(d1 ) .
Case (1). t ∈ (tk , tk+1 ] ⊂ [c1 , d1 ].
EJDE-2013/43
INTERVAL OSCILLATION CRITERIA
5
(i) If t ∈ [tk + τ, tk+1 ], then t − τ ∈ [tk , tk+1 − τ ]. Since tk+1 − tk > τ , there are
no impulsive moments in (t − τ, t). As in the proof of [1, Lemma 2.4], we have
0
x(t) > x(t) − x(t+
k ) = x (ξ)(t − tk ),
ξ ∈ (tk , t).
Since the function p(t)x0 (t) is nonincreasing,
x(t) > x0 (ξ)(t − tk ) >
p(t)x0 (t)
(t − tk ).
p(ξ)
From the fact that p(t) is nondecreasing, we have
p(t)x0 (t)
p(ξ)
p(t)
<
<
.
x(t)
t − tk
t − tk
0
(t)
)
t−tk −τ
1
We obtain xx(t)
< t−t
. Upon integrating from t−τ to t, we obtain x(t−τ
x(t) > t−tk .
k
(ii) If t ∈ (tk , tk + τ ), then t − τ ∈ (tk − τ, tk ), and there is an impulsive moment
0
(s)
< s−t1k +τ for s ∈ (tk − τ, tk ]. Upon
tk in (t − τ, t). Similar to (i), we obtain xx(s)
integrating from t−τ to tk , we obtain
tk ), we have
x(t−τ )
x(tk )
>
t−tk
τ .
0 +
Since x(t)−x(t+
k ) < x (tk )(t−
x0 (t+
x(t)
bk x0 (tk )
k)
<
1
+
(t
−
t
)
=
1
+
(t − tk ) .
k
ak x(tk )
x(t+
x(t+
k)
k)
Using
x0 (tk )
x(tk )
<
1
τ
and x(t+
k ) = ak x(tk ), this implies
x(tk )
τ
>
.
x(t)
ak τ + bk (t − tk )
Therefore,
t − tk
x(t − τ )
>
.
x(t)
ak τ + bk (t − tk )
Case (2). t ∈ [c1 , tI(c1 )+1 ]. We consider three sub-cases.
(i) If tI(c1 ) > c1 − τ , t ∈ [tI(c1 ) + τ, tI(c1 )+1 ], then there are no impulsive moments
t−τ −tI(c1 )
)
in (t − τ, t). Making a similar analysis of case 1(i), we obtain x(t−τ
x(t) > t−tI(c1 ) .
(ii) If tI(c1 ) > c1 − τ , t ∈ [c1 , tI(c1 ) + τ ), then t − τ ∈ [c1 − τ, tI(c1 ) ) and there is
an impulsive moment tI(c1 ) in (t − τ, t). Similar to case 1(ii), we have
t − tI(c1 )
x(t − τ )
>
.
x(t)
aI(c1 ) τ + bI(c1 ) (t − tI(c1 ) )
(iii) If tI(c1 ) < c1 − τ , then there are no impulsive moments in (t − τ, t). So
t − τ − tI(c1 )
x(t − τ )
>
.
x(t)
t − tI(c1 )
Case (3). t ∈ (tI(d1 ) , d1 ]. There are three sub-cases to consider:
(i) If tI(d1 ) + τ < d1 , t ∈ [tI(d1 ) + τ, d1 ], then there are no impulsive moments in
(t − τ, t). Similar to case 2(i), we have
t − τ − tI(d1 )
x(t − τ )
>
.
x(t)
t − tI(d1 )
6
QIAOLUAN LI, WING-SUM CHEUNG
EJDE-2013/43
(ii) If tI(d1 ) + τ < d1 , t ∈ [tI(d1 ) , tI(d1 ) + τ ), then there is an impulsive moment
tI(d1 ) . Similar to case 2(ii), we obtain
t − tI(d1 )
x(t − τ )
>
.
x(t)
aI(d1 ) τ + bI(d1 ) (t − tI(d1 ) )
(iii) If tI(d1 ) + τ ≥ d1 , then there is an impulsive moment tI(d1 ) in (t − τ, t).
Similar to case 3(ii), we obtain
t − tI(d1 )
x(t − τ )
>
.
x(t)
aI(d1 ) τ + bI(d1 ) (t − tI(d1 ) )
Combining all these cases, we have

Q1 (t), for t ∈ [c1 , tI(c1 )+1 ],
x(t − τ )  1I(c1 )
> Qk (t),
for t ∈ (tk , tk+1 ], k = I(c1 ) + 1, . . . , I(d1 ) − 1,

x(t)
 1
QI(d1 ) (t), for t ∈ (tI(d1 ) , d1 ].
Hence by (2.5), we have
I(d1 )
X
ω 2 (tk )[v(tk ) − v(t+
k )]
k=I(c1 )+1
d1
Z
≤2
Z
0
ω(t)ω (t)v(t)dt −
c1
c1
I(d1 )−1
X
−
k=I(c1 )+1
d1
Z
−
c1
tI(c1 )+1
c1
I(d1 )−1
X
−
k=I(c1 )+1
Z
d1
tI (d1 )
tI(c1 )+1
Z
+
c1
I(d1 )−1
X
k=I(c1 )+1
Z
tk+1
ω 2 (t)Q(t)Q1k (t)dt −
tk
Z
d1
tI(d1 )
ω 2 (t)Q(t)Q1I(d1 ) (t)dt
d1
+
tI(d1 )
1
[p(t)ω 0 (t) − v(t)ω(t)]2 dt
p(t)
Z tk+1
1
[p(t)ω 0 (t) − v(t)ω(t)]2 dt
p(t)
tk
1
[p(t)ω 0 (t) − v(t)ω(t)]2 dt
p(t)
−
+
Z
ω 2 (t)Q(t)Q1I(c1 ) (t)dt
v 2 (t)ω 2 (t)
dt
p(t)
Z
=−
tI(c1 )+1
[p(t)ω 02 (t) − Q(t)Q1I(c1 ) (t)ω 2 (t)]dt
Z
tk+1
[p(t)ω 02 (t) − Q(t)Q1k (t)ω 2 (t)]dt
tk
[p(t)ω 02 (t) − Q(t)Q1I(d1 ) (t)ω 2 (t)]dt .
EJDE-2013/43
INTERVAL OSCILLATION CRITERIA
7
Hence we have
I(d1 )
X
ω 2 (tk )[v(tk ) − v(t+
k )]
k=I(c1 )+1
Z
tI(c1 )+1
<
c1
[p(t)ω 02 (t) − Q(t)Q1I(c1 ) (t)ω 2 (t)]dt
I(d1 )−1
X
+
k=I(c1 )+1
Z
d1
+
tI(d1 )
Z
tk+1
(2.6)
[p(t)ω 02 (t) − Q(t)Q1k (t)ω 2 (t)]dt
tk
[p(t)ω 02 (t) − Q(t)Q1I(d1 ) (t)ω 2 (t)]dt ,
for if not, we must have p(t)ω 0 (t) = v(t)ω(t) or x(t)ω 0 (t) = x0 (t)ω(t) on [c1 , d1 ].
Upon integrating, x(t) will be a multiple of ω(t), which contradicts the facts that
ω vanishes at c1 and d1 while x(t) does not.
0
On the other hand, since p(t)x0 (t) < 0 for all t ∈ (c1 , tI(c1 )+1 ], p(t)x0 (t) is
nonincreasing in (c1 , tI(c1 )+1 ]. Thus
x(t) > x(t) − x(c1 ) = x0 (ξ)(t − c1 ) ≥
p(t)x0 (t)
x(t)
and hence
<
p(ξ)
t−c1 .
p(t)x0 (t)
(t − c1 ),
p(ξ)
for some ξ ∈ (c1 , t) ,
Letting t → t−
I(c1 )+1 , we have
v(tI(c1 )+1 ) ≤
β1
.
tI(c1 )+1 − c1
(2.7)
Making a similar analysis on (tk−1 , tk ], k = I(c1 ) + 2, . . . , I(d1 ), it is not difficult
to see that
β1
v(tk ) ≤
.
(2.8)
tk − tk−1
Here we must point out that (2.7) and (2.8) play a key role in our method for
estimating v(tj ), which is different from the usual techniques for the case without
impulses. From (2.7) and (2.8), and noting that ak ≤ bk , we have
I(d1 )
X
k=I(c1 )+1
≥ β1
h
ak − bk 2
ω (tk )v(tk )
ak
I(d1 )
X
k=I(c1 )+2
i
aI(c1 )+1 − bI(c1 )+1
a k − bk
ω 2 (tk ) +
ω 2 (tI(c1 )+1 )
ak (tk − tk−1 )
aI(c1 )+1 (tI(c1 )+1 − c1 )
= L(ω, c1 , d1 ) .
Since
I(d1 )
X
I(d1 )
ω 2 (tk )[v(tk ) − v(t+
k )] =
k=I(c1 )+1
X
k=I(c1 )+1
ak − bk 2
ω (tk )v(tk ) ,
ak
by (2.6), we have
Z
tI(c1 )+1
L(ω, c1 , d1 ) <
c1
[p(t)ω 02 (t) − Q(t)Q1I(c1 ) (t)ω 2 (t)]dt
8
QIAOLUAN LI, WING-SUM CHEUNG
I(d1 )−1
X
+
k=I(c1 )+1
Z
d1
+
tI(d1 )
Z
tk+1
EJDE-2013/43
[p(t)ω 02 (t) − Q(t)Q1k (t)ω 2 (t)]dt
tk
[p(t)ω 02 (t) − Q(t)Q1I(d1 ) (t)ω 2 (t)]dt ,
which contradicts (2.1).
If I(c1 ) = I(d1 ), then L(ω, c1 , d1 ) = 0, and there are no impulsive moments in
[c1 , d1 ]. Similar to the proof of (2.6), we obtain
Z d1
[p(t)ω 02 (t) − Q(t)QI(c1 ) (t)ω 2 (t)]dt > 0 .
(2.9)
c1
This again contradicts our assumption. Finally, if x(t) is eventually negative, we
can consider [c2 , d2 ] and reach a similar contradiction. The proof of Theorem 2.2
is complete.
Theorem 2.3. Assume conditions (H1)–(H3) hold, ak ≤ bk and there exists a
G ∈ Γ such that
Z dj
Z tI(c )+1
j
p(t)g 2 (t)dt −
Q(t)G(t)QjI(cj ) (t)dt
cj
cj
I(dj )−1
X
−
Z
k=I(cj )+1
tk+1
tk
Q(t)G(t)Qjk (t)dt −
Z
dj
tI (dj )
Q(t)G(t)QjI(dj ) (t)dt
(2.10)
≤ R(G, cj , dj ) ,
where R(G, cj , dj ) := 0 for I(cj ) = I(dj ), j = 1, 2, and
R(G, cj , dj )
:=
aI(cj )+1 − bI(cj )+1
G(tI(cj )+1 )βj +
aI(cj )+1 (tI(cj )+1 − c1 )
I(dj )
X
k=I(cj )+2
ak − bk
βj
G(tk )
ak tk − tk−1
for I(cj ) < I(dj ), then all solutions of (1.1) are oscillatory.
Proof. Similar to the proof of Theorem 2.2, suppose x(t − τ ) > 0 for t ≥ t0 . If
I(c1 ) < I(d1 ), multiplying G(t) throughout (2.4) and integrating over [c1 , d1 ], we
obtain
I(d1 )
X
G(tk )
k=I(c1 )+1
ak − bk
v(tk )
ak
d1
Z d1 2
Z d1
p
x(t − τ )
v (t)G(t)
Q(t)G(t)
dt −
dt + 2
v(t)g(t) G(t)dt
x(t)
p(t)
c1
c1
c1
Z d1 s
Z
d1
2
p
G(t)
<−
v(t) − p(t)g(t) dt +
p(t)g 2 (t)dt
p(t)
c1
c1
Z tI(c )+1
I(d1 )−1 Z tk+1
X
1
−
Q(t)G(t)Q1I(c1 ) (t)dt −
Q(t)G(t)Q1k (t)dt
Z
≤−
c1
k=I(c1 )+1
tk
EJDE-2013/43
INTERVAL OSCILLATION CRITERIA
d1
Z
−
tI(d1 )
Z
9
Q(t)G(t)Q1I(d1 ) (t)dt
d1
Z
p(t)g 2 (t)dt −
≤
c1
tI(c1 )+1
c1
I(d1 )−1
−
tk+1
Z
X
Q(t)G(t)Q1I(c1 ) (t)dt
Q(t)G(t)Q1k (t)dt
Q(t)G(t)Q1I(d1 ) (t)dt .
−
tk
k=I(c1 )+1
d1
Z
tI(d1 )
On the other hand, from the proof of Theorem 2.2, we have
v(tI(c1 )+1 ) ≤
β1
,
tI(c1 )+1 − c1
v(tk ) ≤
β1
,
tk − tk−1
(2.11)
for k = I(c1 ) + 2, . . . , I(d1 ). So
I(d1 )
X
k=I(c1 )+1
≥
ak − bk
G(tk )v(tk )
ak
aI(c1 )+1 − bI(c1 )+1
G(tI(c1 )+1 )β1 +
aI(c1 )+1 (tI(c1 )+1 − c1 )
I(d1 )
X
k=I(c1 )+2
ak − bk
β1
G(tk )
ak tk − tk−1
= R(G, c1 , d1 ) .
This contradicts (2.10). If I(c1 ) = I(d1 ), the proof is similar to that of Theorem
2.2, and so it is omitted here. The proof of Theorem 2.3 is complete.
Next, let D = {(t, s) : t0 ≤ s ≤ t}. A function H ∈ C(D, R) is said to belong to
the class H if
(A1) H(t, t) = 0, H(t, s) > 0 for t > s; and
∂H
(A2) H has partial derivatives ∂H
∂t and ∂s on D such that
p
∂H
= 2h1 (t, s) H(t, s),
∂t
p
∂H
= −2h2 (t, s) H(t, s).
∂s
Similar to [8, Theorem 2.3], we have the following Theorem.
Theorem 2.4. Assume the conditions (H1)–(H3) hold. Suppose that there are
δj ∈ (cj , dj ), j = 1, 2, and H ∈ H such that
h
1
H(dj , δj )
+
Z
1
H(δj , cj )
dj
Z
dj
Q(s)Qj (s)H(dj , s)ds −
δj
i
p(s)h22 (dj , s)ds
δj
hZ
δj
cj
> P (H, cj , dj ) ,
Z
δj
Q(s)Qj (s)H(s, cj )ds −
cj
i
p(s)h21 (s, cj )ds
(2.12)
10
QIAOLUAN LI, WING-SUM CHEUNG
EJDE-2013/43
where P (H, cj , dj ) := 0 for I(cj ) = I(dj ), and
bI(δj )+1 − aI(δj )+1
βj
P (H, cj , dj ) :=
H(dj , tI(δj )+1 )
H(dj , δj )
aI(δj )+1 (tI(δj )+1 − δj )
I(dj )
X
+
bi − ai ai (ti − ti−1 )
H(dj , ti )
i=I(δj )+2
+
bI(cj )+1 − aI(cj )+1
βj
H(tI(cj )+1 , cj )
H(δj , cj )
aI(cj )+1 (tI(cj )+1 − cj )
I(δj )
X
+
H(ti , cj )
i=I(cj )+2
(2.13)
bi − ai ai (ti − ti−1 )
for I(cj ) < I(dj ), j = 1, 2. Then all solutions of (1.1) are oscillatory.
Example 2.5. Consider the impulsive differential equation
π
π 3
π
x00 (t) + m cos(t/2)x(t − ) + 8 cos(t/2)|x(t − )| 2 x(t − )
8
8
8
π −1
t
π
π
3 t
+ cos |x(t − )| 2 x(t − ) = sin , t 6= 2kπ ± ,
2
8
8
2
4
π
+
0 +
0
x(tk ) = ak x(tk ), x (tk ) = ak x (tk ), tk = 2kπ ± .
4
(2.14)
In this equation, τ = π/8, tk+1 − tk ≥ π/2 > π/8, α1 = 5/2, α2 = 1/2, and
m is a positive constant. For any T > 0, we can choose k large enough such that
T < c1 = 4kπ − π2 < d1 = 4kπ and c2 = 4kπ + π8 < d2 = 4kπ + π2 satisfy (H3), then
there is an impulsive moment tk = 4kπ − π4 in [c1 , d1 ] and an impulsive moment
tk+1 = 4kπ + π4 in [c2 , d2 ]. Let ω(t) = sin(8t) ∈ Ωω (cj , dj ), j = 1, 2, we have
Z d1
Z d1
(ω 0 (t))2 dt = 32
(cos 16t + 1)dt = 16π,
(2.15)
c1
c1
tI(c1 ) = 4kπ − 47 π, tI(d1 ) = 4kπ − π4 . Choose η0 = η1 = η2 = 1/3. Then
1
3
1/3
t 1/3
Q(t) = m cos(t/2) + ( )−1/3 8 cos(t/2)
cos(t/2) sin 3
2
t
= cos( )(m − 3 sin1/3 t)
2
≥ m cos(t/2).
(2.16)
Hence
Z
4kπ− π
4
Q(t)
4kπ− π
2
Z
t − π8 − tI(c1 )
sin2 (8t)dt
t − tI(c1 )
4kπ− π
8
+
Q(t)
4kπ− π
4
Z
4kπ
+
4kπ− π
8
>
9
m
10
Z
t − tI(d1 )
sin2 (8t)dt
aI(d1 ) (t + π8 − tI(d1 ) )
t − π8 − tI(d1 )
Q(t)
sin2 (8t)dt
t − tI(d1 )
4kπ− π
4
4kπ− π
2
cos(t/2) sin2 (8t)dt > 16π
(2.17)
EJDE-2013/43
INTERVAL OSCILLATION CRITERIA
11
for m large enough. On the other hand, note that ak = bk > 0, so that L(ω, cj , dj ) =
0. It follows from Theorem 2.2 that all the solutions of (2.14) are oscillatory.
Acknowledgments. The authors are very grateful to the anonymous referee for
his/her careful reading of the original manuscript, and for the helpful suggestions.
W.-S. Cheung was partially supported by grant HKU7016/07P from the Research Grants Council of the Hong Kong SAR, China. Q. Li was partially supported
by grants 11071054 from the NNSF of China, A2011205012 from the Natural Science Foundation of Hebei Province, and L2009Z02 from the Main Foundation of
Hebei Normal University.
References
[1] R. P. Agarwal, D. R. Anderson, A. Zafer; Interval oscillation criteria for second-order forced
delay dynamic equations with mixed nonlinearities, Comput. Math. Appl. 59 (2010), 977-993.
[2] D. R. Anderson; Interval criteria for oscillation of nonlinear second order dynamic equations
on time scales, Nonlinear Anal. 69 (2008), 4614-4623.
[3] D. D. Bainov, M. B. Dimitrova, A. B. Dishliev; Oscillation of the bounded solutions of
impulsive differential-difference equations of second order, Appl. Math. Comput. 114 (2000),
61-68.
[4] D. D. Bainov, P. S. Simeonov; Periodic solutions of linear impulsive differential equations
with delay, Commun. Appl. Anal. 7 (2003), 7-29.
[5] E. F. Beckenbach, R. Bellman; Inequalities, Springer, Berlin, 1961.
[6] K. Gopalsamy, B. G. Zhang; On delay differential equations with impulses, J. Math. Anal.
Appl. 139 (1989), 110-122.
[7] A. F. Guvenilir; Interval oscillation of second-order functional differential equations with
oscillatory potentials, Nonlinear Anal. 71 (2009), e2849-e2854.
[8] X. X. Liu, Z. T. Xu; Oscillation of a forced super-linear second order differential equation
with impulses, Comput. Math. Appl. 53 (2007), 1740-1749.
[9] Z. Liu, Y. G. Sun; Interval criteria for oscillation of a forced impulsive differential equation
with Riemann-Stieltjes integral, Comput. Math. Appl. 63 (2012), 1577-1586.
[10] A. Özbekler, A. Zafer; Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects, Math. Comput. Modelling. 50 (2009), 59-65.
[11] A. Özbekler, A. Zafer; Oscillation of solutions of second order mixed nonlinear differential
equations under impulsive perturbations, Comput. Math. Appl. 61 (2011), 933-940.
[12] Y. G. Sun, J. S. W. Wong; Oscillation criteria for second order forced ordinary differential
equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007), 549-560.
[13] Z. W. Zheng, X. Wang, H. M. Han; Oscillation criteria for forced second order differential
equations with mixed nonlinearities, Appl. Math. Lett. 22 (2009), 1096-1101.
Qiaoluan Li
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang
050016, China
E-mail address: qll71125@163.com
Wing-Sum Cheung
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong
Kong, China
E-mail address: wscheung@hkucc.hku.hk