Applied and Computational Mathematics
Volume 5, Issue 2, April 2016, Pages: 46-50

Oscillation of Second Order Nonlinear Differential Equations with a Damping Term

Xue Mi1, *, Ying Huang1, 2, Desheng Li1

1School of Mathematics and System Sciences, Shenyang Normal University, Shenyang, Liaoning, P. R. China

2School of Mathematics, Jilin University, Changchun, Jilin, P. R. China

Email address:

(Xue Mi)
(Ying Huang)
(Desheng Li)

*Corresponding author

To cite this article:

Xue Mi, Ying Huang, Desheng Li. Oscillation of Second Order Nonlinear Differential Equations with a Damping Term. Applied and Computational Mathematics. Vol. 5, No. 2, 2016, pp. 46-50. doi: 10.11648/j.acm.20160502.12

Received: February 5, 2016; Accepted: March 7, 2016; Published: March 25, 2016


Abstract: A class of second-order nonlinear differential equations with a damping term is investigated in this paper. By using the Riccati transformation technique and general weight functions, we obtain some new sufficient conditions for the oscillation of the equation. Our results improve and extend some known results. Two examples are given to illustrate the main results.

Keywords: Oscillation, Second Order Nonlinear Differential Equation, Damping Term, Riccati Transformation Technique, Weight Function


1. Introduction

In this paper we are concerned with the problem of oscillation of the nonlinear second order differential equation with a damping term

(1)

Several assumptions are as follow:

(I) , ;

(II) , and , for some  and for all . , , and they are both quotients of odd positive integers.

Let, . We say the function  belongs to a class  if:

(i)  for all ,  in ;

(ii) has a continuous and non-positive partial derivative in  with respect to the second variable satisfying the condition

,

for some function .

We shall consider the solutions of Equation (1) which are defined for all large. A solution of Equation (1) is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be non-oscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

Recently, there are many authors who have investigated the oscillation for second order differential equations with a damping term, see [3-16] and the references are cited therein.

Wong [10] has studied the equation

.(2)

Rogovchenko and Tuncay [7], M. Kirane and Yu. V. Rogovchenko [8], Yan [12] have obtained oscillation criteria of the following equation:

.(3)

Theorem A [8]. Assume that the functionsatisfies for some constant and for all. Suppose further that the functions , are such that belongs to the class  and

, for all .

Assume that there exists a function  such that

,

where ,

 and ,

Then Eq.(3) is oscillatory.

More recently, Li et al [9] investigated oscillation criteria for the following equation:

,(4)

where  is a quotient of odd positive integers and  for some .

Theorem B [9]. Suppose that there exists a function  such that, for some  and for some ,

where

,

and .

Then Eq. (4) is oscillatory.

Theorem C [9]. Suppose that there exists a function ,

 and  such that, for some  and for all ,

and where  and  are as in theorem B. If

,

where , then Eq.(4) is oscillatory.

It is obvious that (2), (3) and (4) are special cases of Eq. (1).

Motivated by the idea of Li [9], in this paper we obtain, by using a generalized Riccati technique due to Li [9], several new interval criteria for oscillation, that is, criteria given by the behavior of equation (1) on . Our results improve and extend the results of Li [9], Rogovchenko [3, 7, 8], and Grace [16]. Finally, several examples are inserted to illustrate the main results.

2. Lemmas

Lemma 1. Let  be a ratio of two odd numbers. Then,

.(5)

Lemma 2. Let , ,  and , then

.(6)

3. Conclusions

Theorem 1. Suppose that there exists a function  such that, for some  and for some ,

(7)

where and ,

Then, equation (1) is oscillatory.

Proof. To obtain a contradiction, suppose thatis a non-oscillatory solution of Eq. (1) and let  such that  for all . Without loss of generality, we may assume that  for all  since the similar argument holds also for eventually negative. We define a generalized Riccati substitution by

, (8)

Differentiating (8) and using (1), we obtain

,(9)

where  is a continuous function and , so there exist  and  such that , for all . By lemma 1, let , , then

(10)

Thus, (9) and (10) yield

(11)

Multiplying the both sides of (11) by  and integrating the inequality from  to , we obtain, for all ,

(12)

Let , ,  and , by lemma 2, we have

.

Thus,

.

Hence,

,

which contradicts (7). The proof is complete.

Theorem 2. Suppose that there exist functions, , and  such that, for all ,

(13)

and

(14)

where  and  are as in Theorem 1. If

(15)

where , then equation (1) is oscillatory.

Proof. As in Theorem 1, without loss of generality we may assume that there exists a solution  of Eq. (1) such that  on  for some . Defining again the function  by (8), we arrive at (12) which implies, for all ,

.

By (14), we have

.

Thus for all ,

.

Consequently,

,(16)

and

(17)

Assume that

.(18)

By (13), there exists a such that

.(19)

By (18), for any positive constant , there exists a  such that for all ,

.(20)

Then

.

By (19), there exists a  such that, for all ,

,

which implies that

, .

Since  is an arbitrary positive constant,

,

and that contradicts (15).

Thus

,

and by (16),

,

which contradicts (15). This complete the proof.

4. Examples

Example 1. Consider the following equation

, , (21)

where  are both quotients of odd positive integers,  and.

Let . Then ,  and . We have

By theorem 1, Eq. (21) is oscillatory.

Example 2. Consider the following equation

(22)

where  are both quotients of odd positive integers,  and . Let  and . Then ,  and .

We have

.

It is easy to verify that (15) is satisfied. Hence, Eq. (22) is oscillatory by theorem 2.


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