Some Oscillation Results for Second
Order Linear Delay Dynamic Equations
Lynn Erbe and Allan Peterson
Abstract. We obtain some oscillation theorems for linear delay
dynamic equations on a time scale. We illustrate the results by a
number of examples.
1. Preliminary Results
Consider the second order linear delay dynamic equation
n
X
∆
∆
(1.1)
L[x](t) := (r(t)x (t)) +
qi (t)x(τi (t)) = 0.
i=1
We will be interested in obtaining oscillation theorems for (1.1) by
comparing the solutions to a related equation without delay of the
form
n
X
(1.2)
(r(t)x∆ )∆ +
Qi (t)xσ = 0,
i=1
for which many oscillation results are known. We recall that a solution
of (1.1) or (1.2) is nonoscillatory if it is eventually of one sign. If a
solution changes sign infinitely often it is said to be oscillatory.
Let T be a time scale (nonempty closed subset of the reals R) which
is unbounded above. We assume that the coefficient functions qi (t) ≥ 0,
i = 1, 2, · · · , n, and r(t) > 0 are rd-continuous on the time scale interval
[a, ∞)T := [a,P
∞) ∩ T, (i.e., r, qi ∈ Crd ([a, ∞)T ). Furthermore, we will
assume that ni=1 qi (t) 6≡ 0 (for all large t). We will also assume that
the delay functions τi : [a, ∞)T → T are rd-continuous, τi (t) ≤ t, and
τi (t) → ∞ as t → ∞, i = 1, 2, · · · , n. For details concerning calculus
2000 Mathematics Subject Classification. 39A10.
Key words and phrases. comparison theorems, linear oscillation, delay equations, time scale.
2
LYNN ERBE AND ALLAN PETERSON
on time scales and other pertinent definitions, we refer to the books [3],
[4], and [11]. Stability and oscillation questions for certain first order
delay dynamic equations have been considered in [1] for example.
We start with several auxiliary lemmas which are crucial in the
proof of the main results. The first lemma is usually referred to as the
Riccati technique. Denote
S[z] =
z2
.
r(t) + µ(t)z
Lemma 1.1 ([12], [6]). The equation
(1.3)
Lrq [x] := (r(t)x∆ )∆ + q(t)xσ = 0
is nonoscillatory if and only if there is a function z satisfying the Riccati
dynamic inequality
(1.4)
z ∆ (t) + q(t) + S[z](t) ≤ 0
with r(t) + µ(t)z(t) > 0 for large t.
That is, if x(t) is a solution of (1.3) that is of one sign for all large
∆ (t)
t ∈ [a, ∞)T , then z(t) := r(t)x
satisfies (1.4) with r(t) + µ(t)z(t) > 0
x(t)
for large t. Conversely, if z(t) solves (1.4) with r(t) + µ(t)z(t) > 0 for
large t, then (1.3) has a solution x(t) which is of one sign for all large
t.
We will use this lemma to show that a nonoscillatory solution of
(1.1) leads to a solution of the Riccati dynamic inequality (1.4). In order to do this, we introduce the auxiliary functions H(t, t1 ) and ηi (t, t1 )
defined by
Z t
1
H(τi (t), t1 )
∆s, and ηi (t, t1 ) :=
, l ≤ i ≤ n.
H(t, t1 ) :=
H(σ(t), t1 )
t1 r(s)
We may then establish the following result.
Lemma 1.2. Let x(t) be a solution of (1.1) which satisfies
x(t) > 0,
x∆ (t) > 0,
(r(t)x∆ (t))∆ ≤ 0
for all t ≥ τi (t) ≥ T ≥ a. Then for each 1 ≤ i ≤ n we have
x(τi (t)) > ηi (t, T )xσ (t),
t ≥ τi (t) > T.
DELAY DYNAMIC EQUATIONS
3
Proof. For t ≥ τi (t) > T ≥ a we have
Z σ(t)
x(σ(t)) − x(τi (t)) =
x∆ (s)∆s
τi (t)
Z
σ(t)
1
r(s)x∆ (s)∆s
τi (t) r(s)
Z σ(t)
∆s
∆
≤ r(τi (t))x (τi (t))
τi (t) r(s)
=
which yields
xσ (t) ≤ x(τi (t)) + r(τi (t))x∆ (τi (t))H(σ(t), τi (t)).
Dividing both sides of this inequalilty by x(τi (t)) we get
(1.5)
xσ (t)
r(τi (t))x∆ (τi (t))
≤1+
H(σ(t), τi (t)).
x(τi (t))
x(τi (t))
Also, we have
Z
τi (t)
x(τi (t)) − x(T ) =
x∆ (s)∆s
T
∆
Z
τi (t)
≥ r(τi (t))x (τi (t))
T
∆s
r(s)
and so
x(τi (t)) ≥ x(T ) + r(τi (t))x∆ (τi (t))H(τi (t), T )
> r(τi (t))x∆ (τi (t))H(τi (t), T ).
Therefore, we have
(1.6)
1
r(τi (t))x∆ (τi (t))
<
.
x(τi (t))
H(τi (t), T )
Hence, from (1.5) and (1.6) we have
xσ (t)
H(σ(t), τi (t))
< 1+
x(τi (t))
H(τi (t), T )
H(σ(t), T )
1
=
=
.
H(τi (t), T )
ηi (t, T )
This gives us the desired result
x(τi (t)) > xσ (t)ηi (t, T ).
4
LYNN ERBE AND ALLAN PETERSON
P
Lemma 1.3. Assume qi (t) ≥ 0, 1 ≤ i ≤ n, and ni=1 qi (t) 6≡ 0 for
large t. Let x be a solution of (1.1) with x(t) > 0, t ∈ [t0 , ∞)T and
assume further that
Z ∞
∆t
= ∞.
t0 r(t)
Then there exists a T ∈ [t0 , ∞)T such that
x∆ (t) > 0,
x(t) > 0,
(r(t)x∆ (t))∆ ≤ 0
and
for t ∈ [T, ∞)T .
Proof. We can suppose that t1 ≥ t0 is such that x(t) > 0, x(τi (t)) >
0, t ≥ t1 , for all 1 ≤ i ≤ n. Then we have
∆
∆
(r(t)x (t)) = −
n
X
qi (t)x(τi (t)) ≤ 0,
t ∈ [t1 , ∞)T ,
i=1
and so r(t)x∆ (t) is decreasing for t ∈ [t1 , ∞)T . Therefore, if x∆ (t2 ) ≤ 0
for some t2 ∈ [t1 , ∞)T , then it follows that
r(t)x∆ (t) ≤ 0,
t ∈ [t2 , ∞)T .
If x∆ (t3 ) < 0 for some t3 ≥ t2 , then an integration gives
Z t
x∆ (s)∆s = x(t) − x(t3 )
t3
∆
Z
t
∆s
t3 r(s)
as t → ∞,
≤ r(t3 )x (t3 )
→ −∞,
which gives us a contradiction. Hence, x∆ (t) ≡ 0 for t ∈ [t2 , ∞)T and
this means x(t) ≡ constant for t ∈ [t2 , ∞)T . But, then
∆
∆
(r(t)x (t)) ≡ 0 ≡ −
n
X
qi (t)x(τi (t)) 6≡ 0,
i=1
which is a contradiction. Hence, it follows that
x∆ (t) > 0,
t ∈ [t1 , ∞)T ,
and (r(t)x∆ (t))∆ ≤ 0,
t ∈ [t1 , ∞)T .
2. Main Results
We may now apply the previous lemmas to obtain our first oscillation result.
DELAY DYNAMIC EQUATIONS
5
R∞
Theorem 2.1. Assume r(t) > 0P
with a 1/r(t) ∆t = ∞ and assume that qi (t) ≥ 0, 1 ≤ i ≤ n, and ni=1 qi (t) 6≡ 0, for all sufficiently
large t. If
(2.1)
(r(t)x∆ )∆ + Q(t, T )xσ = 0,
where, for t ∈ (T, ∞)T ,
Q(t, T ) :=
n
X
ηi (t, T )qi (t),
i=1
is oscillatory on (T, ∞)T for all sufficiently large T , then all solutions
of (1.1) are oscillatory.
Proof. If not, assume that x(t) is a solution of (1.1) of one sign for
t ≥ t1 ≥ a and without loss of generality let us suppose that x(t) > 0,
t ∈ [t1 , ∞)T . Then by Lemma 1.2 and Lemma 1.3, there exists a
T ∈ [t1 , ∞)T , sufficiently large, such that
x(t) > 0,
x∆ (t) > 0,
t ∈ [T, ∞)T ,
σ
x(τi (t)) ≥ ηi (t, T )x (t), τi (t) ∈ (T, ∞)T , for all 1 ≤ i ≤ n,
and (2.1) is oscillatory on [a, ∞)T . Consequently, we have that x(t) > 0
satisfies x∆ (t) > 0 and
(r(t)x∆ )∆ + Q(t, T )xσ ≤ 0, t ∈ (T, ∞)T .
∆
(t)
, then z(t) > 0 and
If we set z(t) := r(t)x
x(t)
x(t)(r(t)x∆ (t))∆ − r(t)(x∆ (t))2
x(t)xσ (t)
1 2 x(t)
≤ −Q(t, T ) −
z (t) σ
r(t)
x (t)
2
z (t)
x(t)
= −Q(t, T ) −
r(t) x(t) + µ(t)x∆ (t)
z 2 (t)
.
= −Q(t, T ) −
r(t) + µ(t)z(t)
z ∆ (t) =
Therefore, since r(t) + µ(t)z(t) > 0 and z is a solution of
z ∆ + Q(t, T ) + S[z](t) ≤ 0
for large t, we get by Lemma 1.1 that the linear equation
(2.2)
(r(t)x∆ )∆ + Q(t, T )xσ = 0
is nonoscillatory on (T, ∞)T . This contradiction proves the result. We may establish a number of corollaries by using Theorem 2.1 and
known criteria for linear second order dynamic equations (cf. [2], [5-8],
[10] and [12-15]). For example, we have the following result.
6
LYNN ERBE AND ALLAN PETERSON
Corollary 2.2. Assume
Z ∞
Z ∞
∆s
=∞=
Q(s, T )∆s.
r(s)
T
T
Then all solutions of (1.1) are oscillatory.
Proof. Corollary 2.2 follows from the Fite–Wintner–Leighton criterion which says that all solutions of (1.3) are oscillatory (cf. [2])
if
Z ∞
Z ∞
1
q(t)∆t.
∆t = ∞ =
r(t)
a
a
For the case r(t) ≡ 1 and a single delay, (1.1) becomes
x∆∆ + q(t)x(τ (t)) = 0.
(2.3)
In this case, η(t, T ) =
Q(t, T ) =
τ (t)−T
,
σ(t)−T
so that
τ (t) − T
τ (t)
q(t) ∼
q(t) as t → ∞.
σ(t) − T
σ(t)
Therefore, if
∞
Z
a
τ (t)
q(t)∆t = ∞,
σ(t)
then all solutions of (2.3) are oscillatory.
We next consider the dynamic equation
(2.4)
x∆∆ +
γ
x(τ (t)) = 0.
tτ (t)
We have the following.
Corollary 2.3. All solutions of (2.4) are oscillatory if γ >
limt→∞ µ(t)
= 0.
t
1
4
and
Proof. We use the fact that all solutions of
γ σ
(2.5)
x∆∆ +
x =0
tσ(t)
are oscillatory if γ >
Theorem 2.1.
1
4
and limt→∞
µ(t)
t
= 0 (cf. [12]) along with
DELAY DYNAMIC EQUATIONS
7
3. Examples
In this section we give examples of our main results.
= 0 (e.g.,
Example 3.1. If T is any time scale with limt→∞ µ(t)
t
T = R or T = Z), then all solutions of (2.5), or more generally
x
∆∆
n
X
γi
+
x(τi (t)) = 0
tτi (t)
i=1
are oscillatory provided
γ̄ :=
n
X
i=1
1
γi > .
4
To see this, we observe that in this case
Q(t, T ) =
n
X
ηi (t, T )qi (t)
i=1
n
X
γi
τi (t) − T
=
tτi (t) σ(t) − T
i=1
and
n
X
γi
τi (t) − T
γ̄
∼
tτi (t) σ(t) − T
tσ(t)
i=1
as t → ∞.
Therefore, since (2.5) with γ replaced by γ̄ is oscillatory, the result now
follows from Theorem 2.1.
Example 3.2. If T = q N0 , q > 1, then the q-difference equation
γ σ
x∆∆ +
x =0
tσ(t)
1
is oscillatory iff γ > (√q+1)
[5], [12]). Therefore, the delay q2 (cf.
difference equation (where τi : T = q N0 → T)
x∆∆ +
n
X
γi
x(τi (t)) = 0
tτ
(t)
i
i=1
is oscillatory provided
γ̄ =
n
X
i=1
1
γi > √
.
( q + 1)2
Additional examples may be readily given. We leave this to the
interested reader.
8
LYNN ERBE AND ALLAN PETERSON
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Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130 USA
E-mail address: [email protected], [email protected]