Two necessary and sufficient conditions for the oscillation of the bounded solutions of the secondorder nonlinear delay differential equation
are obtained by constructing the sequence of functions and using inequality technique.
AMS Mathematics Subject Classification: 34C10.
1. Introduction
Consider the secondorder nonlinear delay differential equation
The paper assumes the following conditions hold:

(H1)a(t) ∈C1([t0, ∞), (0, ∞)),q(t) ∈C([t0, ∞), [0, ∞)), and witht→ ∞,

(H2)ƒ(x) ∈ C(R,R) is nondecreasing function , andwithx≠ 0;

(H3)τ(t) ∈ C([t0, ∞),R),τ(t) ≤t, and
We call that
x
(
t
) ∈ C
^{1}
([
T_{x}
, ∞), R) (
T_{x}
≥
t
_{0}
) is the solution of equation (1.1) if
a
(
t
)
x
′(
t
) ∈ C
^{1}
([
T_{x}
, ∞). R) and
x
(
t
) satisfy (1.1) for
t
∈ [
T_{x}
, ∞). We suppose that every solution of (1.1) can be extended in [
t
_{0}
, +∞). In any infinite interval [
T
, +∞), we call
x
(
t
) is a regular solution of (1.1) if
x
(
t
) is not the eventually identically zero. The regular solution of (1.1) is said to be oscillatory in case it has arbitrarily large zero point; otherwise, the solution is said to be nonoscillatory.
For the equation (1.1), if
a
(
t
) ≡ 1, the equation (1.1) becomes
For the equation (1.2), if
ƒ
(
x
) =
x
,
τ
(
t
) =
t
, and
q
(
t
) =
c
(
t
), the equation (1.2) is simplified to be the secondorder linear differential equation
There are some oscillation criteria for the equation (1.3), and one of the most important criteria is given by Wintner
[1]
as follows: If
the equation (1.3) is oscillatory. In 1978, Kamenev
[2]
improved the result of Wintner. He proved that if
where λ is a constant and λ > 1, the equation (1.3) is oscillatory.
In recent years, the oscillation theory and its application of differential equations have been greatly concerned. For example, you can see the recent monographs
[3

5]
. In particular, the result on oscillation criteria of secondorder differential equation is very rich (see
[6

16]
), but the most results obtained establish the sufficient condition of the oscillation for differential equations. Generally, the necessary and sufficient condition is difficult to obtain. This article discusses the oscillation for the bounded solution of the equation (1.1), and establishes two necessary and sufficient conditions of oscillation for the bounded solution of (1.1) by constructing a sequence of functions and using inequality technique.
2. Main results
Theorem 2.1.
Suppose that
(
H
_{1}
)−(
H
_{3}
)
hold. Then the necessary and sufficient condition of the oscillation for every bounded solution of the equation (1.1) is
Proof
. Sufficiency. Suppose that there is nonoscillatory bounded solution
x
(
t
) of the equation (1.1). Without loss of generality we assume that
x
(
t
) is eventually positive, then there exists
t
_{1}
(
t
_{1}
≥
t
_{0}
) such that as
t
≥
t
_{1}
,
From the equation (1.1), we get
Thus, we can determine that
Actually, if there exists
t
_{2}
(
t
_{2}
≥
t
_{1}
) such that
noticing
a
(
t
)
x
′(
t
) is nonincreasing, we can obtain
as
t
≥
t
_{2}
, i.e.
Integrating the above formula from
t
_{2}
to
t
, by (
H
_{1}
), we get that
This contradicts with that
x
(
t
) is eventually positive, so (2.2) holds. Thus, for
t
≥
t
_{1}
, we have
Then there exists
t
_{3}
(
t
_{3}
≥
t
_{2}
) and
l
(
l
> 0) such that when
t
≥
t
_{3}
,
Substituting (2.3) into equation (1.1), we get
Multiplying the both ends of the above formula by
A
(
t
) , we can get
Because
we can obtain
Substituting the above into (2.5), we get
Integrating the above formula from
t
_{3}
to
t
, we obtain
Because
x
(
t
) is increasing and bounded, there exists
C
(
C
> 0) such that
This contradicts with the condition (2.1). The proof of sufficient section is completed.
Necessity. If
then there is
T
≥
t
_{0}
such that for
t
≥
T
, we have
Constructing the sequence of functions such that
for
k
= 1, 2, · · · , Then
Suppose that
Noticing
ƒ
(
u
) is nondecreasing, then
Thus, by mathematical induction, for any positive integer
k
, we get
Therefore, the limit of the sequence of functions {
x_{k}
(
t
)} exists, i.e.
and 1 ≤
x
(
t
) ≤ 2,
t
≥
τ
(
T
): Applying Lebesgue control convergence theorem to (2.6), we can get
Derivation of the both sides of the above formula and multiplying them by
a
(
t
), we get that for
t
>
T
Continually after derivation of the both sides of the above formula, we get
It is easy to see that
x
(
t
) is an eventually positive bounded solution of (1.1), which is contradictory to that every bounded solution of the equation (1.1) is oscillatory. The proof is completed.
Remark 2.1.
For linear case of (1.1) (i.e.
ƒ
(
x
) =
x
), oscillation criterion relative to Theorem 2.1 has been obtained in Corollary 2 in
[17]
.
By Theorem 2.1, we can obtain for equation (1.2)
Corollary 2.2.
Suppose that
(
H
_{1}
)−(
H
_{3}
)
hold. Then every bounded solution of equation (1.2) is oscillatory if and only if
Corollary 2.3.
Assume that
(
H
_{1}
)−(
H
_{3}
)
hold. Then every bounded solution of (1.1) is oscillatory if and only if
Proof
. Sufficiency. Suppose that there is nonoscillatory bounded solution
x
(
t
) of the equation (1.1). Without loss of generality, we assume that
x
(
t
) is eventually positive, then there exists
t
_{1}
(
t
_{1}
≥
t
_{0}
) such that
for
t
≥
t
_{1}
. Using arguments similar to ones in the proof of Theorem 2.1, we can get (2.4), i.e.
Integrating the above from
t
to
t
+
v
, we get
Noticing (2.2) and letting
v
→ ∞, we get that
Integrating the above from
T
(
T
≥
t
_{2}
) to
t
(
t
≥
T
), it follows
Let
t
→ ∞ to acquire the limits of both sides of the above. Because
x
(
t
) is bounded and increasing, it is easy to get
This is contradictory to the condition (2.8). The proof of sufficiency is completed.
Necessity. Suppose that
and there exists
T
(
T
≥
t
_{0}
) such that
for
t
≥
T
. Construct the sequence of functions and let
k
= 1, 2, · · · . Similarly to the proof of Theorem 2.1, by the mathematical induction for any positive integer
k
, we have
So the limit of {
x_{k}
(
t
)} exists, i.e.
and 1 ≤
x
(
t
) ≤ 2,
t
≥
τ
(
T
): By Lebesgue control convergence theorem to (2.9), it follows that
Derivation of the both sides of the above and multiplying them by
a
(
t
), we get that for
t
>
T
Continuing to take the derivatives of the both sides of the above, we can get
Thus
x
(
t
) is an eventually positive bounded solution of (1.1), which is contradictory to that every bounded solution of the equation (1.1) is oscillatory. The proof is completed.
By Theorem 2.3, we can get for equation (1.2).
Corollary 2.4.
Suppose that
(
H
_{1}
)−(<
H
_{3}
)
hold. Then every bounded solution of (1.2) is oscillatory if and only if
Example 2.5.
Consider secondorder linear differential equation
Here
The conditions (H
_{1}
)(H
_{3}
), (2.1) and (2.8) are clearly satisfied. Altogether, by Theorems 2.1 and 2.3, every bounded solution of the equation (2.11) is oscillatory. In fact,
x
(
t
) = cos ln
t
is a bounded oscillatory solution of the equation (2.11).
BIO
Xia Song received M.Sc. from Shandong University, China. She is now a lecturer of Department of Mathematics in Binzhou University, Shandong, China. Her current research interests include differential equations and dynamical systems.
Department of Mathematics, Binzhou University, Shandong 256603, P.R. China.
email: songxia119@163.com
Quanxin Zhang was made professor in 1997. He is the incumbent Dean of the Department of Mathematics in Binzhou University, he was selected as the “ExcellentWellknown Teacher of Shandong Province”. His current research interests include differential equations and dynamical systems.
Department of Mathematics, Binzhou University, Shandong 256603, P.R. China.
email: 3314744@163.com
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