In this paper, we are concerned with a class of second-order nonlinear differential equations with damping term. By using the generalized Riccati technique and the integral averaging technique of Philos-type, two new oscillation criteria are obtained for every solution of the equations to be oscillatory, which extend and improve some known results in the literature recently.
Zhang and Yan discussed respectively the solutions’ oscillation of the second order nonlinear differential equation with damping in [
and obtained some useful results. On this basis, the paper continues this discussion of Equation (1). For Equation (1), assume that
(A1)
(A2)
(A3)
In this paper, we assume that each solution
Many exceptions of Equation (1) have emerged in the literature, for example, the paper [
and the associated equations have been studied by many authors with a number of important results of oscillation. We recommend References [
Using Philos-type integral average conditions, the new oscillatory results of Equation (1) is given as below. Function classes P is introduced, we define that
1)
2) H exists non-positive and continuously partial derivatives for the second variable in
Theorem 1. Assume that (A1) - (A3) hold, and
then Equation (1) is oscillatory.
Proof. Suppose that
From Equation (1), we get
So when
By the division integral formula and applying Equation (4), we have
So when
By (9), when
The two sides of (10) are divided by
Corollary 1. In Theorem 1, if the condition (5) is replaced by the following conditions:
1)
2)
then Equation (1) is oscillatory.
Remark 1. In Theorem 1, if we select different functions
of Equation (1) can be obtained. For example, you can select
If the condition (5) is not satisfied, we can apply the following guidelines for determining oscillation of Equation (1).
Theorem 2. Assume that (A1) - (A3) hold, and
If there is a continuously differentiable function
and continuously function
hold when
where
Proof. Suppose that
We can get (10) as the proof of Theorem 1, i.e.
The two sides of the above result are divided by
Define
By (9), we have
and by (14), we get
and
By (13) and (22) there is a sequence
such that
When
So for sufficiently large n, there is
Because
is increasing, we get
From (26) and (28),
On the other hand, by the Schwarz inequality, we get
So
From (24), we have
There is an contradiction with (28) and (29). If
we obtain a contradiction to (15). This completes the proof. □
Remark 2. The theorems of this paper improve or extend the results in [
Finally, we give two examples.
Example 1. Consider the second-order differential equation with damping
where
Now let
Example 2. Consider the second-order differential equation with damping
Here
Now let
fied. By Theorem 2, Equation (35) is oscillatory on
In this paper, the two well-known results of Philos on the second order linear differential equation are extended to the second order nonlinear differential equations with damping term. As we all know, the motions under ideal conditions and vacuum are rare, but the motions with damping and disturbances are widespread. The discussion on the oscillation of the differential equation with damping term in our paper is of more practical significance. Moreover, the previous study on oscillation of the equation always assumed that
It is a deficiency of this paper that there is no discussion on delay. So in the follow-up study we will discuss the oscillation of the second order delay differential equations with damping, second order neutral delay differential equations and higher order delay differential equations with damping.
We thank the Editor and the referee for their comments. Research of Q. Zhang is funded by the Natural Science Foundation of Shandong Province of China grant ZR2013AM003. This support is greatly appreciated.
Quanxin Zhang,Xia Song,Shouhua Liu, (2016) New Oscillation Criteria for the Second Order Nonlinear Differential Equations with Damping. Journal of Applied Mathematics and Physics,04,1179-1185. doi: 10.4236/jamp.2016.47122