Received 28 May 2016; accepted 27 June 2016; published 30 June 2016
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) is continuously differentiable;
(A2) are continuous functions, and for arbitrarily large t, , can change sign;
(A3) is continuously differentiable and when, , ,.
In this paper, we assume that each solution of Equation (1) can be extended to. A solution is said to be regular if there exists t on arbitrary interval, such that. A regular solution is said to be oscillatory, if it has arbitrarily large zeros; otherwise it is said to be nonoscillatory. Equation (1) is called oscillatory if all its regular solutions are oscillatory.
Many exceptions of Equation (1) have emerged in the literature, for example, the paper  discussed the oscillation of the second order linear differential equation with damping
2. Main Results
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
is called function belong to the class P, if there is satisfying
2) H exists non-positive and continuously partial derivatives for the second variable in, and satisfies the equation
Theorem 1. Assume that (A1) - (A3) hold, and,. The function belongs to the class of functions P and (4) holds. If there is an continuously differentiable function
then Equation (1) is oscillatory.
Proof. Suppose that is a nonoscillatory solution of Equation (1). We may assume without loss of gene- rality that with. we consider the function
From Equation (1), we get
So when we have
By the division integral formula and applying Equation (4), we have
So when, it follows
By (9), when, we get
The two sides of (10) are divided by, and we calculate the limit of the two sides of (10) when. So we have a contradiction to the condition (5). This completes the proof.
Corollary 1. In Theorem 1, if the condition (5) is replaced by the following conditions:
then Equation (1) is oscillatory.
Remark 1. In Theorem 1, if we select different functions and the different oscillation criteria
of Equation (1) can be obtained. For example, you can select or.
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. belongs to the class of functions P and (4) holds. Besides,
If there is a continuously differentiable function to make
and continuously function to make
hold when. Besides,
where. Then Equation (1) is oscillatory.
Proof. Suppose that is a nonoscillatory solutions of (1). And when, Define
We can get (10) as the proof of Theorem 1, i.e.
The two sides of the above result are divided by, then we calculate the limit of the two sides when. By (14), we get So
By (9), we have
and by (14), we get
By (13) and (22) there is a sequence
When we calculate the supper limit of (20) and apply (21), it follows
So for sufficiently large n, there is
is increasing, we get Where or is a positive constant. Assume that, then
and by (26), we have
From (26) and (28), where is a constant.That is for sufficiently large
On the other hand, by the Schwarz inequality, we get
From (24), we have
There is an contradiction with (28) and (29). If with (18) we get
we obtain a contradiction to (15). This completes the proof. □
Remark 2. The theorems of this paper improve or extend the results in  -  . For Equation (1), Theorem 1 and 2 are new.
Finally, we give two examples.
Example 1. Consider the second-order differential equation with damping
Now let, , , It is easy to verify that Equation (34) satisfies all the conditions of Theorem 1, so by Theorem 1, Equation (34) is oscillatory.
Example 2. Consider the second-order differential equation with damping
Now let, , , , so all the conditions of Theorem 2 are satis-
fied. By Theorem 2, Equation (35) is oscillatory on. But the other known results cannot be applied in Equation (35).
3. Conclusions and Outlook
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, but the sign of and in our paper may change. Therefore, in this paper we extend and improve some of the results that are known in the previous study.
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.
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