Báo cáo hóa học: " Research Article Iterated Oscillation Criteria for Delay Dynamic Equations of First Order" pdf

Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 458687, 12 pages
doi:10.1155/2008/458687

Research Article
Iterated Oscillation Criteria for Delay Dynamic
Equations of First Order
ă ă
M. Bohner,1 B. Karpuz,2 and O. Ocalan2
1

Department of Economics and Finance, Missouri University of Science and Technology, Rolla,
MO 65409-0020, USA
2
Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University,
ANS Campus, 03200 Afyonkarahisar, Turkey
Correspondence should be addressed to B. Karpuz,
Received 9 June 2008; Accepted 4 December 2008
Recommended by John Graef
We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic
equations on arbitrary time scales, hence combining and extending results for corresponding
differential and difference equations. Examples, some of which coincide with well-known results
on particular time scales, are provided to illustrate the applicability of our results.
Copyright q 2008 M. Bohner 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.

1. Introduction
Oscillation theory on Z and R has drawn extensive attention in recent years. Most of the
results on Z have corresponding results on R and vice versa because there is a very close

relation between Z and R. This relation has been revealed by Hilger in 1 , which unifies
discrete and continuous analysis by a new theory called time scale theory.
As is well known, a first-order delay differential equation of the form
x t
where t ∈ R and τ ∈ R :

p t x t−τ

0,

1.1

1
e

1.2

0, ∞ , is oscillatory if
t

lim inf
t→∞

p η dη >
t−τ


2

Advances in Difference Equations

holds 2, Theorem 2.3.1 . Also the corresponding result for the difference equation
Δx t
where t ∈ Z, Δx t

x t

p t x t−τ

0,

1.3

1 − x t and τ ∈ N, is
t−1

lim inf
t→∞

τ 1

τ

p η >

τ

η t−τ

1.4

1

2, Theorem 7.5.1 . Li 3 and Shen and Tang 4, 5 improved 1.2 for 1.1 to
lim inf pn t >
t→∞

1
,
en

1.5

where
⎧
⎪1,
⎪
⎨
pn t

n

t

⎪
⎪
⎩

0,

p η pn−1 η dη, n ∈ N.

1.6

t−τ

Note that 1.2 is a particular case of 1.5 with n 1. Also a corresponding result of 1.4 for
1.3 has been given in 6, Corollary 1 , which coincides in the discrete case with our main
result as
lim inf pn t >
t→∞

nτ 1

τ
τ

,

1

1.7

where pn is defined by a similar recursion in 6 , as
⎧
⎪1,
⎪
⎨
pn t

⎪
⎪
⎩

n

t−1

0,

p η pn−1 η , n ∈ N.

1.8

η t−τ

Our results improve and extend the known results in 7, 8 to arbitrary time scales. We refer
the readers to 9, 10 for some new results on the oscillation of delay dynamic equations.
Now, we consider the first-order delay dynamic equation
xΔ t

p t x τ t

0,

1.9

where t ∈ T, T is a time scale i.e., any nonempty closed subset of R with sup T
∞,
∞ and τ t ≤ t for all

p ∈ Crd T, R , the delay function τ : T → T satisfies limt → ∞ τ t
t ∈ T. If T R, then xΔ x the usual derivative , while if T Z, then xΔ Δx the usual


M. Bohner et al.

3

forward difference . On a time scale, the forward jump operator and the graininess function are
defined by
μ t : σ t − t,

σ t : inf t, ∞ T ,

1.10

where t, ∞ T : t, ∞ ∩ T and t ∈ T. We refer the readers to 11, 12 for further results on
time scale calculus.
A function f : T → R is called positively regressive if f ∈ Crd T, R and 1 μ t f t > 0
for all t ∈ T, and we write f ∈ R T . It is well known that if f ∈ R t0 , ∞ T , then there
exists a positive function x satisfying the initial value problem
xΔ t

f t x t ,

x t0

1,

1.11

where t0 ∈ T and t ∈ t0 , ∞ T , and it is called the exponential function and denoted by ef ·, t0 .
Some useful properties of the exponential function can be found in 11, Theorem 2.36 .
The setup of this paper is as follows: while we state and prove our main result in
Section 2, we consider special cases of particular time scales in Section 3.
2. Main results
We state the following lemma, which is an extension of 3, Lemma 2 and improvement of
10, Lemma 2 .
Lemma 2.1. Let x be a nonoscillatory solution of 1.9 . If
t

lim sup
t→∞

p η Δη > 0,

2.1

τ t

then
lim inf yx t < ∞,

2.2

t→∞

where
yx t :

x τ t
x t

for t ∈ t0 , ∞ T .

2.3

Proof. Since 1.9 is linear, we may assume that x is an eventually positive solution. Then, x is
eventually nonincreasing. Let x t , x τ t > 0 for all t ∈ t1 , ∞ T , where t1 ∈ t0 , ∞ T . In view
of 2.1 , there exists ε > 0 and an increasing divergent sequence {ξn }n∈N ⊂ t1 , ∞ T such that
σ ξn
τ ξn

p η Δη ≥

ξn
τ ξn

p η Δη ≥ ε

∀ n ∈ N0 .

2.4


4

Advances in Difference Equations

Now, consider the function Γn : τ ξn , σ ξn

t

Γn t :

→ R defined by

T

τ ξn

ε
p η Δη − .
2

2.5

We see that Γn τ ξn < 0 and Γn ξn > 0 for all n ∈ N. Therefore, there exists ζn ∈ τ ξn , ξn T
such that Γn ζn ≤ 0 and Γn σ ζn ≥ 0 for all n ∈ N. Clearly, {ζn }n∈N ⊂ t1 , ∞ T is a
nondecreasing divergent sequence. Then, for all n ∈ N, we have
σ ζn

2.5

p η Δη

τ ξn

ε
2

≥

Γn σ ζn

ε
2

2.6

≥

ε
ε
− Γn ζn ≥ .
2
2

2.7

and
σ ξn

p η Δη

σ ξn

2.5

ζn

ε
2

p η Δη − Γn ζn

τ ξn

Thus, for all n ∈ N, we can calculate
σ ξn

1.9

x ζn ≥ x ζn − x σ ξn

σ ξn

p η x τ η Δη ≥ x τ ξn

ζn

ε
x τ ξn
2

2.7

≥

≥

ε
x τ ζn
2

≥

ε
x τ ξn
2

σ ζn

p η Δη

ζn

τ ξn

σ ζn

p η x τ η Δη

2.8

τ ξn

2

ε
2

2.6

p η Δη ≥

ε
2

1.9

− x σ ζn

x τ ζn

,

and using 2.3 ,
yx ζn ≤

2
ε

2

.

2.9

Letting n tend to infinity, we see that 2.2 holds.
For the statement of our main results, we introduce

⎧
⎪1,
⎪
⎨
αn t :

⎪
⎪
⎩

n
1

inf

λ>0
−λpαn−1 ∈R τ t ,t

for t ∈ s, ∞ T , where τ n s ∈ t0 , ∞ T .

T

λe−λpαn−1 t, τ t

,

0,

n ∈ N,

2.10


M. Bohner et al.

5

Lemma 2.2. Let x be a nonoscillatory solution of 1.9 . If there exists n0 ∈ N such that
lim inf αn0 t > 1,

2.11

t→∞

then
lim yx t

t→∞

∞,

2.12

where yx is defined in 2.3 .
Proof. Since 1.9 is linear, we may assume that x is an eventually positive solution. Then,
x is eventually nonincreasing. There exists t1 ∈ t0 , ∞ T such that x t , x τ t > 0 for all
t ∈ t1 , ∞ T . Thus, yx t ≥ 1 for all t ∈ t1 , ∞ T . We rewrite 1.9 in the form
xΔ t

yx t p t x t

0

2.13

for t ∈ t1 , ∞ T . Integrating 2.13 from t to σ t , where t ∈ t1 , ∞ T , we get
x σ t

−x t

which implies −yx p ∈ R

t1 , ∞

0

μ t yx t p t x t > −x t 1 − μ t yx t p t ,
T

x t

2.14

. From 2.13 , we see that
x t1 e−yx p t, t1

∀t ∈ t1 , ∞

T

,

2.15

and thus
yx t

where τ t2 ∈ t1 , ∞ T . Note R
Now define

zn t :

e−yx p
t1 , ∞

T

1
t, τ t
⊂ R

∀t ∈ t2 , ∞

τ t ,∞

T

,

⊂ R

T

⎧
⎨yx t ,

n

⎩inf z
n−1 η : η ∈ τ t , t

T

2.16

τ t ,t

T

for t ∈ t2 , ∞ T .

0,

2.17

, n ∈ N.

By the definition 2.17 , we have yx η ≥ z1 t for all η ∈ τ t , t T and all t ∈ t2 , ∞ T , which
yields −z1 t p ∈ R τ t , t T for all t ∈ t2 , ∞ T . Then, we see that
yx t

2.16

e−yx p

1
t, τ t

2.17

≥

e−z1

1
t p t, τ t

z1 t
z1 t e−z1

tp

2.10

t, τ t

≥ α1 t z1 t

2.18

6

Advances in Difference Equations

holds for all t ∈ t2 , ∞

T

see also 13, Corollary 2.11 . Therefore, from 2.13 , we have
xΔ t

z1 t p t α1 t x t ≤ 0

2.19

for t ∈ t2 , ∞ T . Integrating 2.19 from t to σ t , where t ∈ t2 , ∞ T , we get
0≥x σ t

μ t z1 t p t α1 t x t > −x t 1 − μ t z1 t p t α1 t ,

−x t

which implies that −z1 pα1 ∈ R t2 , ∞ T . Thus, −z2 t pα1 ∈ R
where τ t3 ∈ t2 , ∞ T , and we see that

yx t

2.16 , 2.17

≥

1
e−z1 pα1 t, τ t

≥

e−z2

t pα1

T

z2 t

1

2.17

τ t ,t

t, τ t

z2 t e−z2

2.20

for all t ∈ t3 , ∞ T ,

2.10

t pα1

t, τ t

≥ α2 t z2 t
2.21

for all t ∈ t3 , ∞ T . By induction, there exists tn0

1

∈ tn0 , ∞

T

with τ tn0

1

∈ tn0 , ∞

T

yx t ≥ zn0 t αn0 t

and
2.22

for all t ∈ tn0 1 , ∞ T . To prove now 2.12 , we assume on the contrary that lim inft → ∞ yx t <

∞. Taking lim inf on both sides of 2.22 , we get
lim inf yx t ≥ lim inf zn0 t αn0 t
t→∞

t→∞

≥ lim inf zn0 t lim inf αn0 t
t→∞

2.17

2.23

t→∞

lim inf yx t lim inf αn0 t ,
t→∞

t→∞

which implies that lim inft → ∞ αn0 t ≤ 1, contradicting 2.11 . Therefore, 2.12 holds.
Theorem 2.3. Assume 2.1 . If there exists n0 ∈ N such that 2.11 holds, then every solution of
1.9 oscillates on t0 , ∞ T .
Proof. The proof is an immediate consequence of Lemmas 2.1 and 2.2.
We need the following lemmas in the sequel.
Lemma 2.4 see 7, Lemma 2 . For nonnegative p with −p ∈ R

1−

t

s

p η Δη ≤ e−p t, s ≤ exp

−

t
s

s, t

T

p η Δη .

, one has

2.24


M. Bohner et al.

7

Now, we introduce
βn t : sup αn−1 η : η ∈ τ t , t

2.25

T

for n ∈ N and t ∈ s, ∞ T , where τ n s ∈ t0 , ∞ T .
Lemma 2.5. If there exists n0 ∈ N such that

lim sup
t→∞

1
βn0 t

1−

1
αn0 t

>0

2.26

holds, then 2.1 is true.
Proof. There exists t1 ∈ t0 , ∞
Then, Lemma 2.4 implies

αn0 t

2.10

≤

1

e−pαn0 −1 t, τ t

T

such that −pαn0 −1 ∈ R

≤

1
1−

t
τ t

t1 , ∞

T

see the proof of Lemma 2.2 .

1

2.25

p η αn0 −1 η Δη

≤

1 − βn0 t

t
τ t

p η Δη

,

2.27

which yields
t

p η Δη ≥

τ t

1
βn0 t

1−

1
αn0 t

∀t ∈ t1 , ∞

T

.

2.28

In view of 2.26 , taking lim sup on both sides of the above inequality, we see that 2.1 holds.
Hence, the proof is done.
Theorem 2.6. Assume that there exists n0 ∈ N such that 2.26 and 2.11 hold. Then, every solution
of 1.9 is oscillatory on t0 , ∞ T .
Proof. The proof follows from Lemmas 2.1, 2.2, and 2.5.
Remark 2.7. We obtain the main results of 7, 8 by letting n0 1 in Theorem 2.6. In this case,
we have β1 t ≡ 1 for all t ∈ t0 , ∞ T . Note that 2.1 and 2.26 , respectively, reduce tos
lim inf α1 t > 1,
t → ∞

which indicates that 2.26 is implied by 2.1 .

lim sup α1 t > 1,
t→∞

2.29


8

Advances in Difference Equations

3. Particular time scales
This section is dedicated to the calculation of αn on some particular time scales. For
convenience, we set
⎧
⎪1,
⎪

⎨
pn t :

Example 3.1. Clearly, if T

⎪
⎪
⎩

R and τ t

n

t
τ t

t − τ, then 3.1 reduces to 1.6 and thus we have

inf

1
λ exp − λp1 t

α2 t

inf

1
λ exp − eλp2 t

λ>0

3.1

pn−1 η p η Δη, n ∈ N.

α1 t

λ>0

0,

ep1 t ,
3.2
e2 p2 t

by evaluating 2.10 . For the general case, it is easy to see that
e n pn t

αn t

3.3

for n ∈ N. Thus if there exists n0 ∈ N such that
lim inf pn0 t >
t→∞

1
,
en0

3.4

then every solution of 1.1 is oscillatory on t0 , ∞ R . Note that 3.4 implies
lim supt → ∞ p1 t ≥ 1/e > 0. Otherwise, we have lim supt → ∞ pn t < 1/en for n 2, 3, . . . , n0 .
This result for the differential equation 1.1 is a special case of Theorem 2.3 given in Section 2,
and it is presented in 3, Theorem 1 , 4, Corollary 1 , and 5, Corollary 1 .
Example 3.2. Let T
we have

Z and τ t

α1 t

t − τ, where τ ∈ N. Then 3.1 reduces to 1.8 . From 2.10 ,

inf

λ>0
1−λp η >0
η∈ t−τ,t−1 Z

≥

inf

λ>0
1−λp η >0
η∈ t−τ,t−1

≥ inf
λ>0

t−1

−1

1
λ

η t−τ

1
λ

1 t−1
1 − λp η
τ η t−τ

1 − λp η

−τ

3.5

Z

λ
1
1 − p1 t

λ
τ

−τ

τ

1
τ

τ 1

p1 t .


M. Bohner et al.

9

In the second line above, the well-known inequality between the arithmetic and the geometric
mean is used. In the next step, we see that

α2 t

inf

λ>0
1−λp η α1 η >0
η∈ t−τ,t−1 Z

≥

≥

1
λ

t−1

λ>0
1−λ τ 1 /τ τ 1 p1 η p η >0
η∈ t−τ,t−1 Z

inf

λ>0
1−λ τ 1 /τ τ 1 p1 η p η >0
η∈ t−τ,t−1 Z

λ>0

1 − λα1 η p η

η t−τ

inf

≥ inf

−1

1
λ τ 1
1−
λ
τ
τ

t−1

1
λ

1−λ

τ

1

p1 η p η

τ

η t−τ

−1

τ 1

3.6

τ 1

1 t−1
τ 1
1−λ
τ η t−τ
τ

1
λ

−τ

τ 1

τ

p2 t

1

−τ

p1 η p η

2τ 1

p2 t .

τ

By induction, we get

αn t ≥

τ

nτ 1

1

pn t

τ

3.7

for n ∈ N. Therefore, every solution of 1.3 is oscillatory on t0 , ∞
n0 ∈ N satisfying

lim inf pn0 t >
t→∞

provided that there exists

n0 τ 1

τ
τ

Z

1

.

3.8

Note that 3.8 implies that lim supt → ∞ p1 t ≥ τ/ τ 1 τ 1 > 0. Otherwise, we would have
lim supt → ∞ pn t < τ/ τ 1 n τ 1 for n 2, 3, . . . , n0 . This result for the difference equation
1.3 is a special case of Theorem 2.3 given in Section 2, and a similar result has been presented
in 6, Corollary 1 .
t/qτ , where q > 1 and τ ∈ N. This time
Example 3.3. Let T qN0 : {qn : n ∈ N0 } and τ t
scale is different than the well-known time scales R and Z since t s/T for t, s ∈ T. In the
∈
present case, 3.1 reduces to
⎧
⎪1,
⎪
⎨
pn t

⎪ q−1
⎪
⎩

n
τ
η

t
t
t
p η pn−1 η ,
qη
q
q
1

0,

n ∈ N,

3.9


10

Advances in Difference Equations

and the exponential function takes the form
τ

e−p t, q−τ t

t
qη

1− q−1 p

η 1

t
.
qη

3.10

Therefore, one can show
λe−λp t, q−τ t

τ

t
qη

1−λ q−1 p

λ
η 1

λ q−1
≤λ 1−
τ

τ
η

t

qη

t
p η
q
1

τ

t
qη

≤

τ 1

τ
τ

1

3.11
1
p1 t

and
τ

α1 t ≥

τ 1

1

3.12

p1 t .

τ

For the general case, for n ∈ N, it is easy to see that
τ

αn t ≥

1

nτ 1

pn t .

τ

3.13

Therefore, if there exists n0 ∈ N such that
lim inf pn0 t >
t→∞

n0 τ 1

τ
τ

,

1

3.14

x qt − x t
,
q−1 t

3.15

then every solution of
xΔ t

p t x

t
qτ

0,

where xΔ t

is oscillatory on t0 , ∞ qN0 . Clearly, 3.14 ensures lim supt → ∞ p1 t ≥ τ/ τ 1 τ 1 > 0. This
result for the q-difference equation 3.15 is a special case of Theorem 2.3 given in Section 2,

and it has not been presented in the literature thus far.
ξm−τ , where {ξm }m∈N is an increasing divergent
Example 3.4. Let T {ξm : m ∈ N} and τ ξm
sequence and τ ∈ N. Then, the exponential function takes the form
m−1

λe−λp ξm , ξm−τ

λ
η m−τ

1 − λ ξη

1

− ξη p ξη .

3.16


M. Bohner et al.

11

One can show that 2.10 satisfies
nτ 1

τ

αn ξm ≥

τ

pn ξm ,

1

3.17

where 3.1 has the form
⎧
⎪1,
⎪
⎨
pn ξm

⎪
⎪
⎩

n

m−1

ξη

1

− ξη p ξη pn−1 ξη ,

0,
3.18

n ∈ N.

η m−τ

Therefore, existence of n0 ∈ N satisfying

lim inf pn0 ξm >
m→∞

n0 τ 1

τ
τ

3.19

1

ensures by Theorem 2.3 that every solution of
xΔ ξm

p ξm x ξm−τ

0,

where xΔ ξm

x ξm 1 − x ξm
,
ξm 1 − ξm

is oscillatory on ξτ , ∞ T . We note again that lim supm → ∞ p1 ξm ≥ τ/ τ
from 3.19 .

1

τ 1

3.20

> 0 follows

References
1 S. Hilger, Ein Maßkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, Universită t
a
ă
Wurzburg, Wurzburg, Germany, 1988.
ă
ă
2 I. Gy ri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford
o
Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991.
3 B. Li, “Multiple integral average conditions for oscillation of delay differential equations,” Journal of
Mathematical Analysis and Applications, vol. 219, no. 1, pp. 165–178, 1998.
4 J. Shen and X. Tang, “New oscillation criteria for linear delay differential equations,” Computers &
Mathematics with Applications, vol. 36, no. 6, pp. 53–61, 1998.
5 X. Tang and J. Shen, “Oscillations of delay differential equations with variable coefficients,” Journal of

Mathematical Analysis and Applications, vol. 217, no. 1, pp. 32–42, 1998.
6 X. H. Tang and J. S. Yu, “Oscillation of delay difference equation,” Computers & Mathematics with
Applications, vol. 37, no. 7, pp. 11–20, 1999.
7 M. Bohner, “Some oscillation criteria for first order delay dynamic equations,” Far East Journal of
Applied Mathematics, vol. 18, no. 3, pp. 289–304, 2005.
8 B. G. Zhang and X. Deng, “Oscillation of delay differential equations on time scales,” Mathematical
and Computer Modelling, vol. 36, no. 11-13, pp. 1307–1318, 2002.
9 R. Agarwal and M. Bohner, “An oscillation criterion for first order dynamic equations,” to appear in
Functional Differential Equations.
10 Y. Sahiner and I. P. Stavroulakis, “Oscillations of first order delay dynamic equations,” Dynamic
¸
Systems and Applications, vol. 15, no. 3-4, pp. 645–655, 2006.
11 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application,
Birkhă user, Boston, Mass, USA, 2001.
a


12

Advances in Difference Equations

12 M. Bohner and A. Peterso, Eds., Advances in Dynamic Equations on Time Scales, Birkhă user, Boston,
a
Mass, USA, 2003.
13 E. Akin-Bohner, M. Bohner, and F. Akın, “Pachpatte inequalities on time scales,” Journal of Inequalities
in Pure and Applied Mathematics, vol. 6, no. 1, article 6, pp. 1–23, 2005.