Oscillation of Second Order Nonlinear Differential Equations with a Damping Term
Xue Mi^{1,}^{ *}, Ying Huang^{1,}^{ 2}, Desheng Li^{1}
^{1}School of Mathematics and System Sciences, Shenyang Normal University, Shenyang, Liaoning, P. R. China
^{2}School of Mathematics, Jilin University, Changchun, Jilin, P. R. China
Email address:
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.
References