Electronic Journal of Differential Equations, Vol. 2009(2009), No. 122, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATION OF SOLUTIONS TO IMPULSIVE DYNAMIC
EQUATIONS ON TIME SCALES
QIAOLUAN LI, FANG GUO
Abstract. In this article, we study the oscillation of second order impulsive
dynamic equations on time scales. The effect of the moments of impulse are
fixed. Using Riccati transformation techniques, we obtain some conditions for
the oscillation of all solutions
1. Introduction
This paper concerns the oscillation of second-order impulsive dynamic equations
on time scales. We consider the system
(a(t)x∆ (t))∆ + p(t)x(σ(t)) = 0,
x(t+
k ) = bk x(tk ),
x(t+
0)
t ∈ JT := [t0 , ∞) ∩ T, t 6= tk , k = 1, 2, . . . ,
∆
x∆ (t+
k ) = ck x (tk ),
= x(t0 ),
∆
x
(t+
0)
k = 1, 2, . . . ,
∆
= x (t0 ),
(1.1)
where T is a time scales, unbounded-above, with tk ∈ T, 0 ≤ t0 < t1 < t2 < · · · <
∆ +
∆
tk < . . . , limk→∞ tk = ∞ and y(t+
k ) = limh→0+ y(tk +h), y (tk ) = limh→0+ y (tk +
∆
h), which represent right limits of y(t), y (t) at t = tk in the sense of time scales.
∆ −
We can define y(t−
k ), y (tk ) similarly.
In this paper, we assume that a(t) ∈ Crd (T, R+ ), p(t) ∈ Crd (T, R+ ), bk > 0,
ck > 0, dk = cbkk , tk are right dense, where Crd denotes the set of rd-continuous
functions, σ(t) := inf{s ∈ T : s > t}, R+ = {x : x > 0}.
Definition 1.1. A function x is a solution of (1.1), if it satisfies (a(t)x∆ (t))∆ +
p(t)x(σ(t)) = 0 a.e. on JT \{tk }, k = 1, 2, . . . , and for each k = 1, 2, . . . , x satis∆ +
∆
fies the impulsive condition x(t+
k ) = bk x(tk ), x (tk ) = ck x (tk ) and the initial
+
∆ +
∆
condition x(t0 ) = x(t0 ), x (t0 ) = x (t0 ).
Definition 1.2. A solution x of (1.1) is oscillatory if it is neither eventually positive
nor eventually negative; otherwise it is called non-oscillatory. Equation (1.1) is
called oscillatory if all solutions are oscillatory.
2000 Mathematics Subject Classification. 34A60, 39A12, 34K25.
Key words and phrases. Oscillation; time scales; impulsive dynamic equations.
c
2009
Texas State University - San Marcos.
Submitted February 26, 2009. Published September 29, 2009.
Supported by grant 07M004 from the Natural Science Foundation of Hebei Province,
and by the Main Foundation of Hebei Normal University.
1
2
Q. LI, F. GUO
EJDE-2009/122
In recently years, there has been an increasing interest in studying the oscillation
and non-oscillation of solutions of various equations on time scales, and we refer
the reader to papers [5, 10, 11, 12] and references cited therein. The time scales
calculus has a tremendous potential for applications in mathematical models of real
processes. Impulsive dynamic equations on time scales have been investigated by
Agarwal [1], Benchohra [2] and so forth. Benchohra [2] considered the existence of
extremal solutions for a class of second order impulsive dynamic equations on time
sales.
The oscillation of impulsive differential equations and difference equations have
been investigated by many authors and numerous papers have been published on
this class of equations and good results were obtained (see [7, 9] and the references
therein). But fewer papers are on the oscillation of impulsive dynamic equations
on time scales.
For example, Huang [8] considered the equation
y ∆∆ (t) + f (t, y σ (t)) = 0,
y(t+
k)
= gk (y(tk )),
y(t+
0)
t ∈ JT := [0, ∞) ∩ T,
y
∆
(t+
k)
= y(t0 ),
∆
= hk (y (tk )),
y ∆ (t+
0)
t 6= tk , k = 1, 2, . . . ,
k = 1, 2, . . . ,
= y ∆ (t0 ).
Using Riccati transformation techniques, they obtain sufficient conditions for oscillations of all solutions.
2. Results
In the following, we assume the solutions of (1.1) exist in JT . To the best of our
knowledge, the question of the oscillation for second order self-conjugate impulsive
dynamic equations has not been yet considered.
Lemma 2.1. Suppose that x(t) > 0, t ≥ t00 ≥ t0 is a solution of (1.1). If
Z t2
Z t3
Z tn+1
Z t1
∆s
∆s
∆s
∆s
+ d1
+ d1 d2
+ · · · + d1 d2 . . . d n
+ · · · = ∞,
a(s)
t1 a(s)
t2 a(s)
tn
t0 a(s)
(2.1)
∆
0
then x∆ (t+
)
≥
0
and
x
(t)
≥
0
for
t
∈
(t
,
t
]
,
where
t
≥
t
k k+1 T
k
0
k
The proof is similar to that in [8, Lemma 2.1]; so we omit it. We remark that
when a(t) ≡ 1, Lemma 2.1 reduces to [8, Lemma 2.1].
Lemma 2.2. Assume that q(t) ∈ Crd (T, R+ ), if
ω ∆∆ (t) + q(t)ω ∆ (t) ≤ 0
has a positive solution, then
ω ∆∆ (t) + q(t)ω ∆ (t) = 0
has a positive solution.
The proof is similar to that in [6, Lemma 4.1.2]; so we omit it.
Theorem 2.3. Assume that a(t) ≡ 1 and (2.1) holds.
Y
∆
Y
∆
d−1
+
d−1
k ω
k p(t)ω(t) = 0,
T ≤tk <t
T ≤tk <t
is oscillatory, then (1.1) is oscillatory.
a.e. t > T ≥ t0
(2.2)
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OSCILLATION OF SOLUTIONS
3
Proof. Suppose to the contrary that (1.1) has a non-oscillatory solution x(t), we
may assume that x(t) is eventually positive solution of (1.1); i.e., x(t) > 0, t ≥
T ≥ t0 . From Lemma 2.1, we have x∆ (t) ≥ 0, x∆ (t+
k ) ≥ 0, t ≥ T , t ∈ T. Let
z(t) =
x∆ (t)
x(t)
≥ 0. For t 6= tk , we get
z ∆ (t) =
x∆∆ (t)x(t) − (x∆ (t))2
z 2 (t)
= −p(t) −
,
x(t)x(σ(t))
1 + µ(t)z(t)
z(t+
k)=
x∆ (t+
ck x∆ (tk )
k)
= dk z(tk ).
=
+
bk x(tk )
x(tk )
Thus we arrive at
z 2 (t)
+ p(t) = 0,
1 + µ(t)z(t)
z ∆ (t) +
t ∈ [T, ∞) ∩ T, t 6= tk ,
z(t+
k ) = dk z(tk ).
Q
Now we define v(t) = ( T ≤tk <t d−1
k )z(t), t > T , t ∈ T. Then for tn > T ,
Y
Y
+
v(t+
d−1
d−1
n) = (
k )z(tn ) = (
k )dn z(tn ) = v(tn ),
T ≤tk ≤tn
T ≤tk ≤tn
which implies that v(t) is rd-continuous on (T, ∞) ∩ T. For t 6= tn , we have
Y
∆
v ∆ (t) =
d−1
k z (t)
T ≤tk <t
z 2 (t)
]
1 + µ(t)z(t)
T ≤tk <t
Q
Y
( T ≤tk <t dk )2 v 2 (t)
Q
=
d−1
[−p(t)
−
]
k
1 + T ≤tk <t dk µ(t)v(t)
Y
=
d−1
k [−p(t) −
T ≤tk <t
=−
Y
dk
T ≤tk <t
1+
Q
v 2 (t)
−
T ≤tk <t dk µ(t)v(t)
Y
d−1
k p(t).
T ≤tk <t
For t = tn , the left-hand derivative of v(t) at t = tn is given by
Y
∆ −
v ∆ (t−
d−1
n) =
k z (tn )
T ≤tk <tn
Y
=
d−1
lim− [−p(t) −
k
t→tn
T ≤tk <tn
z 2 (t)
]
1 + µ(t)z(t)
z 2 (tn )
]
1 + µ(tn )z(tn )
T ≤tk <tn
Q
2 2
Y
T ≤tk <tn dk v (tn )
−1
Q
=
dk [−p(tn ) −
]
1 + µ(tn ) T ≤tk <tn dk v(tn )
Y
=
d−1
k [−p(tn ) −
T ≤tk <tn
=−
Y
T ≤tk <tn
d−1
k p(tn ) −
Y
T ≤tk <tn
dk
1 + µ(tn )
v 2 (t )
Q n
T ≤tk <tn
dk v(tn )
].
4
Q. LI, F. GUO
EJDE-2009/122
Similarly, we obtain
Y
v ∆ (t+
n) = −
dk
T ≤tk ≤tn
1+
Q
v 2 (tn )
−
T ≤tk ≤tn dk µ(tn )v(tn )
Y
d−1
k p(tn ).
T ≤tk ≤tn
So for t > T ,
Y
v ∆ (t) +
dk
T ≤tk <t
Now we define q(t) =
1+
Q
Q
v 2 (t)
+
T ≤tk <t dk µ(t)v(t)
Y
d−1
k p(t) = 0,
a.e.
(2.3)
T ≤tk <t
dk v(t), w(t) = eq (t, t0 ) > 0, t > T . Then
Y
w∆ (t) = q(t)w(t) =
dk v(t)w(t),
T ≤tk <t
T ≤tk <t
Y
∆
d−1
k w
∆
= (v(t)w(t))∆ = w∆ (t)v(t) + w(σ(t))v ∆ (t)
T ≤tk <t
Y
=
dk v 2 (t)w(t) + eq (σ(t), t0 )v ∆ (t).
T ≤tk <t
Since
Y
eq (σ(t), t0 ) = (1 + µ(t)q(t))eq (t, t0 ) = (1 + µ(t)v(t)
dk )w,
T ≤tk <t
by (2.3) we obtain
Y
∆
Y
Y
∆
d−1
w
= w[
dk v 2 (t) + (1 + µ(t)v(t)
dk )v ∆ (t)]
k
T ≤tk <t
T ≤tk <t
T ≤tk <t
Y
= −w[1 + µ(t)v(t)
T ≤tk <t
Y
≤ −w(t)
d−1
k p(t),
Y
dk ]
d−1
k p(t)
T ≤tk <t
a.e.
T ≤tk <t
This implies
Y
∆
d−1
k w
∆
Y
d−1
k p(t)w ≤ 0,
a.e.
has a positive solution. By Lemma 2.2, we obtain
Y
∆
Y
∆
d−1
+
d−1
k ω
k p(t)ω = 0,
a.e.
T ≤tk <t
T ≤tk <t
+
T ≤tk <t
T ≤tk <t
has a positive solution, a contradiction, and so, the proof is complete.
Theorem 2.4. Assume that bk = ck , tk are right dense for all k = 1, 2, . . . . Then
the oscillation of all solutions of (1.1) is equivalent to the oscillation of all solutions
of the equation
(a(t)y ∆ (t))∆ + p(t)y(σ(t)) = 0.
(2.4)
Q
Proof. Let y(t) be any solution of (2.4). Set x(t) = y(t) t0 ≤tk <t bk for t > t0 , then
Y
+
x(t+
bk = bn x(tn ).
n ) = y(tn )
t0 ≤tk ≤tn
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OSCILLATION OF SOLUTIONS
5
Furthermore, for t 6= tn , we have
Y
x∆ (t) = (
bk )y ∆ (t),
t0 ≤tk <t
∆
x
(t+
n)
Y
=(
∆
∆
∆
bk )y (t+
n ) = bn x (tn ) = cn x (tn ).
t0 ≤tk ≤tn
For t 6= tn ,
Y
(a(t)x∆ (t))∆ = [a(t)(
bk )y ∆ (t)]∆ =
t0 ≤tk <t
Y
= −p(t)
Y
bk (a(t)y ∆ (t))∆
t0 ≤tk <t
Y
bk y(σ(t)) = −p(t)
t0 ≤tk <t
bk y(σ(t))
t0 ≤tk <σ(t)
= −p(t)x(σ(t)).
Thus x(t) is the solution of (1.1).
Q
Conversely, if x(t) is the solution of (1.1). Set y(t) = x(t) t0 ≤tk <t b−1
k . Thus
we have
Y
+
y(t+
b−1
n ) = x(tn )
k = y(tn ).
t0 ≤tk ≤tn
Furthermore for t 6= tn , n = 1, 2, . . . , we have
Y
y ∆ (t) = x∆ (t)
b−1
k ,
t0 ≤tk <t
y
∆
(t−
n)
∆
=x
Y
(t−
n)
b−1
k
t0 ≤tk <tn
Y
∆ +
y ∆ (t+
n ) = x (tn )
Y
= x∆ (tn )
b−1
k ,
t0 ≤tk <tn
b−1
k
Y
= x∆ (tn )
t0 ≤tk ≤tn
b−1
k .
t0 ≤tk <tn
For t 6= tn , n = 1, 2, . . . ,
Y
(a(t)y ∆ (t))∆ = [a(t)(
∆
∆
b−1
k )x (t)] =
t0 ≤tk <t
Y
= −p(t)
Y
∆
∆
b−1
k (a(t)x (t))
t0 ≤tk <t
b−1
k x(σ(t))
t0 ≤tk <t
Y
= −p(t)
b−1
k x(σ(t))
t0 ≤tk <σ(t)
= −p(t)y(σ(t)).
For t = tn ,
Y
∆ − ∆
−
(a(t−
n )y (tn )) = (a(tn )
∆ − ∆
b−1
k x (tn ))
t0 ≤tk <tn
Y
=−
b−1
k p(tn )x(σ(tn )) = −p(tn )y(σ(tn )),
t0 ≤tk <tn
∆ + ∆
(a(t+
n )y (tn ))
∆
= (a(tn )y ∆ (t+
n )) = −p(tn )y(σ(tn )).
Thus we arrive at
(a(t)y ∆ (t))∆ + p(t)y(σ(t)) = 0.
This shows that y(t) is a solution of (2.4). The proof is complete.
6
Q. LI, F. GUO
EJDE-2009/122
Example 2.5. Consider the equation
1
t ∈ T = P 12 , 12 , t 6= k + , k = 0, 1, 2, . . . ,
5
1 +
1
1 +
1
∆
∆
x((k + ) ) = bk x(k + ), x ((k + ) ) = bk x (k + ), bk > 0, k = 1, 2, . . . ,
5
5
5
5
1 +
1
∆ 1 +
∆ 1
x(( ) ) = x( ), x (( ) ) = x ( ),
5
5
5
5
(2.5)
S
∞
where p(t) ∈ Crd (T, R+ ), P 12 , 21 = k=0 [k, k + 21 ]. Assume that for each t0 ≥ 0 there
exists k0 ∈ N and l0 ∈ N such that k0 ≥ t0 and
l0 Z k0 +j+ 1
l0 −1
X
2
1X
1
p(t)dt +
p(k0 + j + ) ≥ 4.
2
2
j=1 k0 +j
j=0
x∆∆ + p(t)x(σ(t)) = 0,
From [3, Theorem 4.46], we know that
x∆∆ + p(t)x(σ(t)) = 0
is oscillatory on T. By Theorem 2.4, (2.5) is oscillatory.
Example 2.6. Consider the equation
σ(t) ∆ ∆
x ) + tx(σ(t)) = 0, t ≥ 1, t 6= k, k = 1, 2, . . . ,
t
∆
x(t+
x∆ (t+
bk > 0, k = 1, 2, . . .
k ) = bk x(tk ),
k ) = bk x (tk ),
(
x(1+ ) = x(1),
(2.6)
x∆ (1+ ) = x∆ (1),
where µ(t) = σ(t) − t ≤ ct, c is a positive constant. Since µ(t) ≤ ct, we get
t
t
1
=
≥
.
σ(t)
t + µ(t)
1+c
It is easy to see that
Z
∞
1
Z ∞
t
1
∆t ≥
∆t = ∞,
σ(t)
1+c 1
Z ∞
t∆t = ∞.
1
By [4, Theorem 3.2], we see that
σ(t) ∆ ∆
x ) + tx(σ(t)) = 0,
t
is oscillatory. So (2.6) is oscillatory.
(
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Qiaoluan Li
College of Mathematics and Information Science, Hebei Normal University,
Shijiazhuang, 050016, China
E-mail address: qll71125@163.com
Fang Guo
Department of Mathematics and Computer, Bao Ding University, 071000, China
E-mail address: gf825@126.com