Oscillation of Second Order Nonlinear Neutral Differential Equations with Mixed Neutral Term ()
1. Introduction
In this paper, we are concerned with the oscillatory behavior of solutions of the second order nonlinear neutral differential equation of the form
(1)
where
, subject to the following conditions:
(C1)
, and
for all
;
(C2)
, and
;
(C3)
are nonnegative constants,
,
, and
for any
;
(C4)
for
, k is a constant.
By a solution of Equation (1), we mean a continuous function x defined on an interval
such that
is continuously differentiable and x satisfies Equation (1) for all
. We consider only solu-
tions satisfying condition
, and tacitly assume that Equation (1) possess such solu-
tions. As usual, a solution of Equation (1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise we call it nonosicllatory.
From the literature, it is known that second order neutral functional differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems. For further applications and questions regarding existence and uniqueness of solutions of neutral functional differential equations, see [1] -[3] .
In recent years, there has been an increasing interest in establishing conditions for the oscillation or nonoscilla- tion of solution of neutral functional differential equations, see [4] -[20] for example, and the references cited therein.
In [21] , Xu and Meng obtained some sufficient conditions which guarantees that every solution x of equation (1) when
, oscillates or
.
Ye and Xu [22] studied equation when
, and established some new oscillation criteria for Equation (1).
In [23] , Han et al. considered Equation (1) with
and
, and obtained some sufficient conditions which ensure that every solution of Equation (1) is oscillatory.
In [24] , the present authors established some sufficient conditions for the oscillation of all solutions of
Equation (1) when
. Therefore in this paper we try to obtain some new oscillation criteria for
Equation (1). In Section 2, we use Riccati transformation technique to obtain some sufficient conditions for the oscillation of all solutions of Equation (1). Examples are provided in Section 3 to illustrate the main results.
2. Oscillation Results
In this section, we obtain some new oscillation criteria for the Equation (1). We begin with the following theorem.
Theorem 2.1 If
(2)
and
(3)
where
,
, and
then every solution of Equation (1)
is oscillatory.
Proof. Suppose that
is a nonsocillatory solution of Equation (1). Without loss of generality, we may assume that there exists
such that.
and
for all
. From the definition of
, we have
, and from Equation (1),
is nonincreasing eventually. Hence, it is easy to conclude that there exist two possible cases of the sign of
, that is,
or
for all
.
First assume that
for all
. From the Equation (1), we have
![]()
or
(4)
Integrating (4) from
to
and using the fact
for
, we obtain
![]()
a contradiction to (2.1).
If
, then we define the function
by
(5)
Clearly
. Nothing that
is nonincreasing, we obtain
![]()
Dividing the last inequality by
and integrating it from
to
, we obtain
![]()
Letting
in the last inequality, we see that
![]()
Therefore,
(6)
From (5), we have
(7)
Next, we introduce another function
by
(8)
Clearly
. Noting that
is nonincreasing, we have
. Then,
. From (6), we obtain
(9)
Similarly, we introduce another function
by
(10)
Clearly
. Since
is nonincreasing, we have
![]()
Dividing the last inequality by
and integrating it from
to
, we obtain
![]()
Letting
, we see that
(11)
Differentiating (5), we obtain
(12)
Differentiating (8), we have
(13)
Differentiating (10), we have
(14)
Inview of (12), (13) and (14), we can obtain
(15)
From (4) and (15), we obtain
(16)
Multiplying (16) by
and integrating from
to
, we have
![]()
From the above inequality, we obtain
![]()
Thus, it follows that
![]()
By (7), (9) and (11), we obtain that
![]()
which contradicts (3). The proof is now complete. ![]()
Corollary 2.1. Assume that
with
for
. Further assume that (2.1) and (3) hold. Then every solution of Equation (1) is oscillatory.
Proof. The proof follows from Theorem 2.1. ![]()
Theorem 2.2. Assume that
for
. If condition (2.1) holds and
(17)
then every solution of Equation (1) is oscillatory.
Proof. Let
be a nonsocillatory solution of Equation (1). Without loss of generality, we may assume that there exists
such that
and
for all
. By equation (1),
is nonincreasing eventually. Hence, it is easy to conclude that there exist two possible cases of the sign of
, that is,
or
for all
. If
, then we are back to the case of Theorem 2.1, and we can obtain a contradiction to (2.1). If
, then we define
and
as in Theorem 2.1. Then proceed as in the proof of Theorem 2.1, we obtain (7), (9), (11) and (16) for
. Multiplying (16) by
and integrating from
to
yields
(18)
It follows from (C2) and (7) that
![]()
![]()
Inview of (9), we have
![]()
![]()
From (11), we obtain
![]()
![]()
Therefore from (18), we obtain
![]()
which is a contradiction with (17). The proof is now complete. ![]()
Corollary 2.2. Assume that
for
. In condition (2.1) and (17) hold, then every solution of Equation (1) is oscillatory.
Proof. The proof follows from Theorem 2.2. ![]()
To prove our next theorem, we need a class of function
and the operator T defined as follows:
Following [16] , we say that a function
belongs to the function class
, denoted by
if
, where
, which satisfies
and
for
, and has the partial derivative
on
such that
is locally integrable with
respect to
in
.
Define the operator
by
(19)
for
and
. The function
is defined by
(20)
then, it is easy to see that
is a linear operator and
(21)
Theorem 2.3. Assume that
, and there exist functions
and
such that
(22)
and
(23)
where
is defined as in Theorem 2.1, the operator
defined by (19), and
is defined by (20). Then every solution of Equation (1) is oscillatory.
Proof. Let
be a nonoscillatory solution of Equation (1). Then there exists a
such that
for all
. Without loss of generality, we may assume that
and
for all
. Then proceeding as in the proof of Theorem 2.1 we have
or
and
for all
.
First assume that
and
for all
. Define
(24)
Then
, and
(25)
Since
is nonincreasing and
is increasing. Next, define
(26)
Then
, and
(27)
Since
is nonincreasing,
is increasing and
. Again, define
(28)
Then
, and
(29)
Since
is nonincreasing,
is increasing and
. Combining (25) and (29), and then using (4), we obtain
(30)
Now applying the operator
to (30) and then using (21), we have
![]()
From the last inequality, we obtain
![]()
or
![]()
Taking the sup limit in the last inequality, we obtain a contradiction with (22).
Next consider the case
and
for all
. From the proof of Theorem 2.1, we have the inequality (16). Now apply the operator
to (16) and then using (21), we have
![]()
From the last inequality, we obtain
![]()
or
![]()
Taking the sup limit in the last inequality, we obtain a contradiction with (23). The proof is now completed. ![]()
Remark 2.1. With different choices of functions
and
, Theorem 2.3 can be stated with different con- ditions for oscillations of Equation (1).
For example, if we take
for
, then
![]()
From Theorem 2.3, we obtain the following oscillation criteria for Equation (1).
Corollary 2.3. Assume that
, and there exists a function
such that
![]()
and
![]()
where
and
. Then every solution of Equation (1) is oscillatory.
3. Examples
In this section, we provide three examples to illustrate the main results.
Example 3.1. Consider the neutral differential equation
(31)
Here
, and
. By taking
and
, it
is easy to see that all conditions of Theorem 2.1 are satisfied and hence every solution of Equation (31) is oscillatory.
Example 3.2. Consider the neutral differential equation
(32)
Here
, and
. By taking
and
, it is easy to see that all conditions of Corollary 2.3 are satisfied and hence every solution of Equation (32) is oscillatory.
We conclude this paper with the following remark.
Remark 3.1. The results presented in [24] are not applicable to Equations (31) and (32) since in these
equations
and the neutral term contains advanced arguments. Therefore, our results com-
plement and generalize some of the known results in the literature.
NOTES
*Corresponding author.