Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 873459, 22 pages
doi:10.1155/2010/873459
Research Article
Nonoscillation of First-Order Dynamic Equations
with Several Delays
Elena Braverman
1
and Bas¸ak Karpuz
2
1
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N. W., Calgary,
AB, Canada T2N 1N4
2
Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University,
03200 Afyonkarahisar, Turkey
Correspondence should be addressed to Elena Braverman, [email protected]
Received 18 February 2010; Accepted 21 July 2010
Academic Editor: John Graef
Copyright q 2010 E. Braverman and B. Karpuz. 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.
For dynamic equations on time scales with positive variable coefficients and several delays, we
prove that nonoscillation is equivalent to the existence of a positive solution for the generalized
characteristic inequality and to the positivity of the fundamental function. Based on this result,
comparison tests are developed. The nonoscillation criterion is illustrated by examples which are
neither delay-differential nor classical difference equations.
1. Introduction
Oscillation of first-order delay-difference and differential equations has been extensively
studied in the last two decades. As is well known, most results for delay differential equations
0
,t
0
1,
}
,
1.1
where Δ is the forward difference operator defined by Δxt : xt 1 − xt, and the delay
2 Advances in Difference Equations
differential equation
x
t
n
i1
A
i
t
x
α
i
t
t
i∈
1,n
N
A
i
t
x
α
i
t
0fort ∈
t
0
, ∞
T
,
1.3
0
, ∞
T
. Let us denote
α
min
t
: min
i∈
1,n
N
{
α
i
t
}
for t ∈
t
0
, ∞
T
,t
t
0
, ∞
T
, R and 1.3 is satisfied on t
0
, ∞
T
identically. For a given function ϕ ∈ C
rd
t
−1
,t
0
T
, R, 1.3 admits a unique solution satisfying
x ϕ on t
−1
,t
0
T
see 5, Theorem 3.1. As usual, a solution of 1.3 is called eventually
positive if there exists s ∈ t
0
, ∞
T
such that x>0ons, ∞
T
,R
λ∈R
e
−λA
t, α
t
λ
> 1,
1.5
then every solution of the equation
x
Δ
t
A
t
x
α
σt
α
t
A
η
Δη>1. 1.7
Then every solution of 1.6 is oscillatory.
The present paper is mainly concerned with the existence of nonoscillatory solutions.
So far, only few sufficient nonoscillation conditions have been known for dynamic equations
on time scales. In particular, the following theorem, which is a sufficient condition for the
existence of a nonoscillatory solution of 1.3, was proven in 7.
Theorem C see 7, Theorem 2. Suppose that A ∈ C
rd
t
0
, ∞
T
, R
0
and there exist a constant
λ ∈ R
and a point t
1
where t
2
∈ t
1
, ∞
T
satisfies αt ≥ t
1
for all t ∈ t
2
, ∞
T
. Then, 1.6 has a nonoscillatory solution.
In 10, Theorem 3.1, and Corollary 3.3, Agwo extended Theorem C to 1.3.
Theorem D see 10, Corollary 3.3. Suppose that A
i
∈ C
rd
t
0
, ∞
T
, R
0
for all i ∈ 1,n
N
and there exist a constant λ ∈ R
and t
t
e
−λA
t, α
i
t
,
1.9
where A :
i∈1,n
N
A
i
on t
0
, ∞
T
. Then, 1.3 has a nonoscillatory solution.
As was mentioned above, there are presently only few results on nonoscillation of
1.3; the aim of the present paper is to partially fill up this gap. To this end, we present a
nonoscillation criterion; based on it, comparison theorems on oscillation and nonoscillation
of solutions to 1.3 are obtained. Thus, solutions of two different equations and/or two
different solutions of the same equation are compared, which allows to deduce oscillation
and nonoscillation results.
t
f
t
for t ∈
t
0
, ∞
T
x
t
0
x
0
,x
t
ϕ
t
for t ∈
∈ C
rd
t
0
, ∞
T
, R is the coefficient
corresponding to the delay function α
i
for all i ∈ 1,n
N
. We assume that for all i ∈ 1,n
N
,
A
i
∈ C
rd
t
0
, ∞
T
, R, α
i
is a delay function satisfying α
i
∈ C
rd
t
0
t∞ for all i ∈ 1,n
N
.
For convenience in the notation and simplicity in the proofs, we suppose that functions
vanish out of their specified domains, that is, let f : D → R be defined for some D ⊂ R, then
it is always understood that ftχ
D
tft for t ∈ R, where χ
D
is the characteristic function
of D defined by χ
D
t ≡ 1fort ∈ D and χ
D
t ≡ 0fort
/
∈D.
Definition 2.1. Let s ∈ T,ands
−1
: inf
t∈s,∞
T
{α
min
t}.ThesolutionX X·,s : s
−1
, ∞
T
→
R of the initial value problem
t
χ
{s}
t
for t ∈ s
−1
,s
T
,
2.2
which satisfies X·,s ∈ C
1
rd
s, ∞
T
, R, is called the fundamental solution of 2.1.
The following lemma see 5, Lemma 3.1 is extensively used in the sequel; it gives a
solution representation formula for 2.1 in terms of the fundamental solution.
Lemma 2.2. Let x be a solution of 2.1,thenx can b e written in the following form:
x
t
x
0
X
0
X
t, σ
η
A
i
η
ϕ
α
i
η
Δη for t ∈
t
0
, ∞
T
.
2.3
As functions are assumed to vanish out of their domains, ϕα
i
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
0
X
t, t
0
t
t
0
X
t, σ
η
f
η
Δη, t ∈
t
0
, ∞
T
,
x
0
,t t
0
,
ϕ
t
,t∈
t
−1
,t
0
T
2.4
defined by the right hand side in 2.3 solves 2.1. For t ∈ t
0
, ∞
T
X
Δ
t, t
0
t
t
0
X
Δ
t, σ
η
f
η
Δη X
σ
t
,σ
α
i
η
Δη −X
σ
t
,σ
t
i∈
1,n
N
A
i
t
ϕ
α
0
X
α
j
t
,σ
η
f
η
Δη
−
t
t
0
X
α
j
t
,σ
t
ϕ
α
j
t
f
t
2.5
for all t ∈ t
0
, ∞
T
. After making some arrangements, we get
y
Δ
t
−
j∈It
A
j
,σ
η
f
η
Δη
−
α
j
t
t
0
X
α
j
t
,σ
η
i∈
t
f
t
−
j∈I
t
A
j
t
y
α
j
t
−
j∈J
, ∞
T
. The proof is therefore completed.
6 Advances in Difference Equations
Example 2.3. Consider the following first-order dynamic equation:
x
Δ
t
A
t
x
t
0fort ∈
t
0
, ∞
T
,
2.7
then the fundamental solution of 2.7 can be easily computed as Xt, se
−A
t, s for s, t ∈
0
, ∞
T
x
t
0
x
0
2.8
can be written in the form
x
t
x
0
e
−A
t, t
0
t
t
0
t
x
α
t
≤ 0 for t ∈
t
0
, ∞
T
,
2.10
where A ∈ C
rd
t
0
, ∞
T
, R
0
and α is a delay function, has a solution x which satisfies xt > 0 for
all t ∈ t
1
, ∞
∈ T, and assume that α
i
,β
i
∈ C
rd
t
0
, ∞
T
, T, α
i
t,β
i
t ≤ t for
all t ∈ t
0
, ∞
T
, K
i
∈ C
rd
T × T, R
0
for all i ∈ 1,n
N
, and two functions f,g ∈ C
rd
β
i
η
Δη g
t
∀t ∈
t
0
, ∞
T
.
2.11
Then, nonnegativity of g on t
0
, ∞
T
implies the same for f.
Proof. Assume for the sake of contradiction that g is nonnegative but f becomes negative at
some points in t
0
, ∞
T
.Set
1
>t
1
, then we must have ft ≥ 0 for all t ∈ t
0
,t
1
T
and f
σ
t
1
< 0; otherwise,
Advances in Difference Equations 7
this contradicts the fact that t
1
is maximal. It follows from 2.11 that after we have applied
the formula for Δ-integrals, we have
f
σ
t
1
i∈
Δη
i∈
1,n
N
μ
t
1
K
i
σ
t
1
,t
1
f
β
i
3
∈ t
1
, ∞
T
, then for each i ∈ 0,n
N
,we
may find M
i
∈ R
such that K
i
t, s ≤ M
i
for all t ∈ t
1
,t
3
T
and all s ∈ α
i
t,t
T
.SetM :
i∈1,n
N
fη ≥ 2ft
2
and ft
2
< 0. Note that inf
η∈t
0
,t
2
T
fηinf
η∈t
1
,t
2
T
fη since f ≥ 0ont
0
,t
1
T
. Then, we get
f
t
2
≥
⎛
⎝
i∈
1,n
N
t
2
t
1
M
i
Δη
⎞
⎠
inf
η∈
t
0
,t
2
T
f
,
2.14
which yields the contradiction 1 ≤ 2/3 by canceling the negative terms ft
2
on both sides of
the inequality. This completes the proof.
The following lemma will be applied in the sequel.
Lemma 2.6 see 6, Lemma 2. Assume that A ∈ C
rd
T, R
0
satisfies −A ∈R
T, R, then one
has
1 −
t
s
A
η
Δη ≤ e
−A
t, s
t
x
α
i
t
0fort ∈
t
0
, ∞
T
3.1
8 Advances in Difference Equations
and the corresponding inequalities
x
Δ
t
i∈
1,n
1,n
N
A
i
t
x
α
i
t
≥ 0fort ∈
t
0
, ∞
T
3.3
under the same assumptions which were formulated for 2.1. We now prove the following
result, which plays a major role throughout the paper.
Theorem 3.1. Suppose that for all i ∈ 1,n
N
, α
i
1
, ∞
T
, R
0
such that −Λ ∈
R
t
1
, ∞
T
, R and for all t ∈ t
1
, ∞
T
Λ
t
≥
i∈
1,n
N
A
i
T
for some fixed t
1
∈ t
0
, ∞
T
,thenX·,s > 0 holds on s, ∞
T
for
any s ∈ t
1
, ∞
T
.
Proof. Let us prove the implications as follows: i⇒ii⇒iii⇒iv⇒i.
i⇒ii This part is trivial, since any eventually positive solution of 3.1 satisfies 3.2
too, which indicates that its negative satisfies 3.3.
ii⇒iii Let x be an eventually positive solution of 3.2, the case where x is an
eventually negative solution to 3.3 is equivalent, and thus we omit it. Let us assume that
there exists t
1
∈ t
0
, ∞
T
such that xt > 0andxα
i
t > 0 for all t ∈ t
1
for t ∈
t
1
, ∞
T
.
3.5
Evidently Λ ∈ C
rd
t
1
, ∞
T
, R
0
.From3.5,weseethatΛ satisfies the ordinary dynamic
equation
x
Δ
t
Λ
t
t
x
t
1
e
−Λ
t, t
1
∀t ∈
t
1
, ∞
T
. 3.7
Hence, using 3.7 in 3.2, for all t ∈ t
1
, ∞
T
,weobtain
−Λ
t
α
i
t
,t
1
≤ 0.
3.8
Since xt
1
> 0, then by 4, Theorem 2.36 we have
Λ
t
≥
i∈
1,n
N
A
i
t
−Λ
t, α
i
t
.
3.9
⇒iv Let Λ ∈ C
rd
t
0
, ∞
T
, R
0
satisfy −Λ ∈R
t
1
, ∞
T
, R and 3.4 on t
1
, ∞
T
, where
t
t
x
α
i
t
f
t
for t ∈
t
1
, ∞
T
x
t
≡ 0fort ∈
t
0
,t
1
t
1
, ∞
T
x
t
1
0,
3.11
which has the unique solution
x
t
t
t
1
e
−Λ
t, σ
η
−Λ
t, σ
η
g
η
Δη e
−Λ
σ
t
,σ
t
g
t
i∈1,n
N
A
3.13
10 Advances in Difference Equations
which can be rewritten as
f
t
−Λ
t
t
t
1
e
−Λ
t, σ
η
g
η
Δη g
t
g
η
Δη
−
i∈
1,n
N
A
i
t
e
−Λ
t, α
i
t
t
α
i
t
t
α
i
t
e
−Λ
t, σ
η
g
η
Δη f
t
3.15
for all t ∈ t
1
, ∞
T
1
, ∞
T
,i∈
1,n
N
Υ
0
t
:Λ
t
−
i∈
1,n
N
A
i
t
implies the same for x
on t
1
, ∞
T
by 3.12. On the other hand, by Lemma 2.2, x has the following representation:
x
t
t
t
1
X
t, σ
η
f
η
Δη for t ∈
t
1
, ∞
, ∞
T
, we are led to the contradiction xt
2
< 0,
where x is defined by 3.17. To prove eventual positivity of X,set
x
t
:
⎧
⎨
⎩
X
t, s
− e
−Λ
t, s
for t ∈
t
1
, ∞
T
iv⇒i Clearly, X·,t
0
is an eventually positive solution of 3.1.
The proof is therefore completed.
Remark 3.2. Note that Theorem 3.1 for 1.6 includes Theorem C, by letting Λt : λAt
for t ∈ t
1
, ∞
T
, where λ ∈ R
satisfies −λA ∈R
t
1
, ∞
T
, R. And Theorem 3.1 reduces
to Theorem D, by letting Λt : λ
i∈1,n
T
A
i
t for t ∈ t
1
, ∞
T
, where λ ∈ R
0
, ∞
T
and
x
0
≥ 0,then
x
t
:
⎧
⎨
⎩
x
0
e
−Λ
t, t
0
for t ∈
t
0
, ∞
T
i∈
1,n
N
A
i
t
e
λt−α
i
t
∀t ∈
t
1
, ∞
.
3.20
Then, the delay-differential equation 1.2 has an eventually positive solution, and the
fundamental solution X satisfies X·,s > 0ons, ∞
T
for any s ∈ t
1
, ∞
T
because we may
1
, ∞
hZ
.
3.21
Then, the following delay h-difference equation:
Δ
h
x
t
i∈
1,n
N
A
i
t
x
α
i
for t ∈
t
0
, ∞
hZ
,
3.23
12 Advances in Difference Equations
has an eventually positive solution, and the fundamental solution X satisfies X·,s > 0on
s, ∞
hZ
⊂ t
1
, ∞
hZ
because we may let Λt :≡ 1 − λ/h for t ∈ t
0
, ∞
hZ
. Notice that if for
all t ∈ t
1
, ∞
hZ
and all i ∈ 1,n
N
, A
i
N
A
i
t
λ
−log
q
t/α
i
t
∀t ∈
t
1
, ∞
q
Z
.
3.24
Then, the following delay q-difference equation:
D
q
x
t
is defined by
D
q
x
t
:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
x
qt
− x
t
q − 1
q
Z
.Noticethat
if for all t ∈ t
1
, ∞
hZ
and all i ∈ 1,n
N
, tA
i
t and t/α
i
t are constants, then 3.24 becomes
an algebraic inequality.
4. Comparison Theorems
In this section, we state comparison results on oscillation and nonoscillation of delay dynamic
equations. To this end, consider 3.1 together with the following equation:
x
Δ
t
i∈
1,n
N
, R and β
i
∈ C
rd
t
0
, ∞
T
, T is a delay function for all i ∈
1,n
N
.LetY be the fundamental solution of 4.1.
Theorem 4.1. Suppose that B
i
∈ C
rd
t
0
, ∞
T
, R
0
, A
i
≥ B
i
and α
i
≤ β
0
with −Λ ∈R
t
1
, ∞
T
, R such that 3.4 holds on t
1
, ∞
T
.NotethatΛ ∈ C
rd
t
1
, ∞
T
, R
0
Advances in Difference Equations 13
and −Λ ∈R
t
1
, ∞
T
, R imply that e
1,n
N
A
i
t
e
−Λ
t, α
i
t
≥
i∈
1,n
N
B
i
t
e
−Λ
Δ
t
i∈
1,n
N
B
i
t
x
α
i
t
0fort ∈
t
0
, ∞
T
i
on t
1
, ∞
T
for all i ∈ 1,n
N
and some
fixed t
1
∈ t
0
, ∞
T
, and that X·,s > 0 on s, ∞
T
for any s ∈ t
1
, ∞
T
. Then, Y·,s ≥X·,s holds
on s, ∞
T
for any s ∈ t
1
, ∞
T
.
Proof. From 4.3, any fixed s ∈ t
1
i∈
1,n
N
A
i
t
− B
i
t
Y
α
i
t
,s
.
4.4
− B
i
η
Y
α
i
η
,s
Δη
4.5
for all t ∈ s, ∞
T
. Lemma 2.5 implies nonnegativity of Y·,s since X·,s > 0on
s, ∞
T
⊂ t
1
, ∞
T
and the kernels of the integrals in 4.5 are nonnegative. Then dropping the
nonnegative integrals on the right-hand side of 4.5,wegetYt, s ≥Xt, s for all t ∈ s, ∞
T
.
The proof is hence completed.
0
, ∞
T
,
4.6
where A
i
t : max{A
i
t, 0} for t ∈ t
0
, ∞
T
and A
i
,α
i
aresameasin3.1 for all i ∈ 1,n
N
, has
an eventually positive solution, then so does 3.1.
Proof. By Theorem 3.1, we know that the fundamental solution of the corresponding
differential equation
x
Δ
t
i
≥ A
i
holds on t
0
, ∞
T
for all i ∈ 1,n
N
. The proof is hence
completed.
We now compare two solutions of 2.1 and the following initial value problem:
x
Δ
t
i∈
1,n
N
B
i
t
t
for t ∈
t
−1
,t
0
T
,
4.8
where n ∈ N, x
0
,ϕand α
i
for all i ∈ 1,n
N
are the same as in 2.1 and B
i
,g ∈ C
rd
t
0
, ∞
T
, R
for all i ∈ 1,n
N
for any s ∈ t
0
, ∞
T
.
Rearranging 2.1, we have
x
Δ
t
i∈
1,n
N
B
i
t
x
α
i
t
for all t ∈ t
0
, ∞
T
. In view of the solution representation formula 2.3, for all t ∈ t
0
, ∞
T
,
Advances in Difference Equations 15
we have
x
t
x
0
Y
t, t
0
t
t
0
Y
t, σ
,∞
T
α
i
η
x
α
i
η
⎤
⎦
Δη
−
i∈
1,n
N
t
t
0
Y
Y
t, σ
η
g
η
Δη −
i∈
1,n
N
t
t
0
Y
t, σ
η
B
i
Δ
t
2
t
3
x
3
t
3
− 2
3
2t
3
for t ∈
3
√
3, ∞
3
√
N
x
− x
t
3
√
t
3
1 −t
for t ∈
3
√
N
4.12
and
y
Δ
t
1
t
3
y
3
t
3
.
4.13
Denoting by x and y the solutions of 4.11 and 4.13, respectively. Then, y ≥ x on
3
√
3, ∞
3
√
N
by Theorem 4.5. For the graph of 30 iterates, see Figure 1.
Corollary 4.7. Suppose that A
i
∈ C
rd
t
0
, ∞
T
, R
0
for all i ∈ 1,n
N
and X·,s > 0 on s, ∞
T
for
any s ∈ t
0
, ∞
T
N
.
Corollary 4.8. Let x be a solution of 3.1, and Y·,s > 0 on s, ∞
T
for any s ∈ t
0
, ∞
T
be the
fundamental solution of
x
Δ
t
i∈
1,n
N
A
i
t
x
Theorem 4.9. Suppose that there exist t
1
∈ t
0
, ∞
T
and Λ ∈ C
rd
t
1
, ∞
T
, R
0
such that −Λ ∈
R
t
1
, ∞
T
, R and for all t ∈ t
1
, ∞
T
i∈
1,n
Proof. By Corollary 4.4,itsuffices to prove that 4.6 has an eventually positive solution. For
this purpose, by Theorem 3.1, it is enough to demonstrate that Λ satisfies
Λ
t
≥
i∈
1,n
N
A
i
t
e
−Λ
t, α
i
t
∀t ∈
t
Lemma 2.6, for all t ∈ t
1
, ∞
T
, we have
Λ
t
≥
i∈
1,n
N
A
i
t
1 −
t
α
min
t
Λ
i
t
e
−Λ
t, α
min
t
≥
i∈
1,n
N
A
i
t
e
−Λ
t, α
i
t
α
min
t
i∈
1,n
N
A
i
η
Δη ≤ M ∀t ∈
t
2
, ∞
T
,
4.18
where t
2
N
A
i
∈R
t
1
, ∞
T
, R and
t
α
min
t
i∈
1,n
N
A
i
η
Example 4.12. Let a
i
∈ R
, p
i
∈ N for i ∈ 1,n
N
and q ∈ 1, ∞. We consider the following
q-difference equation
D
q
x
t
i∈
1,n
N
a
i
t
x
t
q
N
a
i
> 0.
4.22
Then, we have
t
t/q
p
i∈
1,n
N
a
i
η
Δη a
t
t/q
p
1
η
1 −
a
q − 1
2ap
q − 1
2p − 1
2p
> 0 ∀t ∈
1, ∞
q
Z
,
4.24
which implies that the regressivity condition in Corollary 4.10 holds. So that 4.21 has an
eventually positive solution if
λ
1 − Mλ
1
4aα
0
T
, then for the solution x of
x
Δ
t
i∈1,n
N
A
i
t
x
α
i
t
0 for t ∈
t
0
, ∞
T
.
Proof. As in the proof of Theorem 4.9, we deduce that there exists Λ satisfying 3.4. Hence,
X·,s > 0ons, ∞
T
for any s ∈ t
0
, ∞
T
. By the solution representation formula 2.3,we
get
x
t
x
0
X
t, t
0
−
i∈
1,n
N
t
:
⎧
⎨
⎩
x
0
e
−Λ
t, t
0
for t ∈
t
0
, ∞
T
,
x
0
for t ∈
t
−1
,t
0
A
i
t
x
α
i
t
g
t
for t ∈
t
0
, ∞
T
x
t
≡ x
0
for t ∈
η
g
η
Δ − x
0
i∈
1,n
N
t
t
0
X
t, σ
η
A
i
η
χ
T
, which completes the proof.
Theorem 4.14. Suppose that A
i
∈ C
rd
t
0
, ∞
T
, R
0
for all i ∈ 1,n
N
, X·,s > 0 on s, ∞
T
for
any s ∈ t
0
, ∞
T
, and the solution y of the initial value problem
y
Δ
t
−1
,t
0
T
4.31
is positive. If x
0
≥ y
0
> 0 and y
0
≥ ϕ ≥ 0 on t
−1
,t
0
T
, then the solution x of 4.26 is positive on
t
0
, ∞
T
.
Proof. Solution representation formula 2.3 implies for a solution of 4.31 that
y
t
y
χ
t
−1
,t
0
T
α
i
η
Δη
⎤
⎦
≤ x
0
X
t, t
0
−
t
t
0
X
T
since x
0
≥ y
0
and x
0
≥ ϕ ≥ 0ont
−1
,t
0
T
. Hence, x ≥ y>0 holds on
t
0
, ∞
T
. Thus, the proof is completed.
20 Advances in Difference Equations
Theorem 4.15. Suppose that A
i
∈ C
rd
t
0
, ∞
T
, R
1,n
N
A
i
t
y
α
i
t
0 for t ∈
t
0
, ∞
T
y
t
0
y
0
,x
T
.
Proof. The proof is similar to that of Theorem 4.13.
We give the following example as an application of Theorem 4.15.
Example 4.16. Let T N
3
: {n
3
: n ∈ N}, and consider the following initial value problems:
x
Δ
t
1
t
x
3
√
t − 3
3
0fort ∈
64, ∞
t 1
3
− x
t
3
√
t 1
3
− t
for t ∈ N
3
4.35
and
y
Δ
t
1
t
y
of 7 iterates, see Figure 2, where x>yby Theorem 4.15.
5. Discussion
In this paper, we have extended to equations on time scales most results obtained in
2, 3: nonoscillation criteria, comparison theorems, and efficient nonoscillation conditions.
However, there are some relevant problems that have not been considered.
Advances in Difference Equations 21
12001000800600400200
0.2
0.4
0.6
0.8
1
x
y
Figure 2: The graph of 7 iterates for the solutions of 4.34 and 4.36 illustrates the result of Theorem 4.15,
here xt >yt for all t ∈ 64, ∞
N
3
.
P1 In 2, it was demonstrated that equations with positive coefficients has slowly
oscillating solutions only if it is oscillatory. The notion of slowly oscillating solutions can be
easily extended to equations on time scales in such a way that it generalizes the one discussed
in 2.
Definition 5.1. A solution x of 3.1 is said to be slowly oscillating if it is oscillating and for
every t
1
∈ t
0
, ∞
T
∈ t
3
, ∞
T
.
Is the following proposition valid?
Proposition 5.2. Suppose that for all i ∈ 1,n
N
, α
i
∈ C
rd
t
0
, ∞
T
, T is a delay function and A
i
∈
C
rd
t
0
, ∞
T
, R
.If 3.1 is nonoscillatory, then the equation has no slowly oscillating solutions.
P2 In Section 4, oscillation properties of equations with different coefficients, delays
and initial functions were compared, as well as two solutions of equations with the same
∈ t
2
, ∞
T
with
α
min
t ≥ t
2
for all t ∈ t
3
, ∞
T
such that x>0ont
2
,t
3
T
and x
/
≥0ont
3
, ∞
T
. Therefore, we
have
A
i
,t
3
T
α
i
t
x
α
i
t
/
≡0 5.1
22 Advances in Difference Equations
for all t ∈ t
3
, ∞
T
and all i ∈ 1,n
N
. It follows from Lemma 2.2 that
x
t
η
χ
t
2
,t
3
T
α
i
η
x
α
i
η
Δη
≤−
t
t
3
X
α
i
η
Δη
5.2
for all t ∈ t
3
, ∞
T
. Since the integrand is nonnegative and not identically zero by 5.1,we
learn that the right-hand side of 5.2 is negative on t
3
, ∞
T
;thatis,x<0ont
3
, ∞
T
. Hence,
x is nonoscillatory, which is the contradiction justifying the proposition.
Thus, under the assumptions of Proposition 5.2 existence of a slowly oscillating
solution of 3.1 implies oscillation of all solutions.
Acknowledgment
E. Braverman was partially supported by NSERC research grant.
References
1 S. Hilger, Ein Maßkettenkalk
¨
10 H. A. Agwo, “On the oscillation of first order delay dynamic equations with variable coefficients,”
The Rocky Mountain Journal of Mathematics, vol. 38, no. 1, pp. 1–18, 2008.
11 Y. S¸ahiner and I. P. Stavroulakis, “Oscillations of first order delay dynamic equations,” Dynamic
Systems and Applications, vol. 15, no. 3-4, pp. 645–655, 2006.
12 B. G. Zhang and C. J. Tian, “Nonexistence and existence of positive solutions for difference equations
with unbounded delay,” Computers & Mathematics with Applications, vol. 36, no. 1, pp. 1–8, 1998.
Copyright: Tài liệu đại học ©