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Oscillation Conditions for a Class of Lienard Equation
Hishyar Kh. Abdullah
Dept. of Mathematics
University of Sharjah,
P.O. Box 27272, Sharjah, U.A.E.
hishyar@sharjah.ac.ae http://www.sharjah.ac.ae
Abstract: - The aim of this paper is to investigate oscillatory properties of a class of a generalized Lienard
equation .Several oscillation conditions are presented that improve the results obtained in the literature. The
results obtained here are new and further improve and complement some known results in the literature. We
extend and improve the oscillation criteria of several authors. Moreover, two examples are presented to
demonstrate the main results.
Key-Words: - Oscillatory, Lienard equation, Second order differential equations, Non-Linear.
1 Introduction
In this paper we are concerned with the
oscillation of a class of Lienard equation of the
form
()+()(())2+() ()= 0, (1.1)
where () and () are continuously
differentiable functions on R .
Many criteria have been found which involves the
behavior of the integral of a combination of the
coefficients of second order nonlinear differential.
This approach has been motivated by many authors
(for example see [1],[2],[3],[4],[5],[6],[7] ,[8],[9]
,[10],[11],[13],[14]and [15] and the authors
therein).Which often studied by reducing the
problem to the estimate of suitable first integral Our
attention is concentrated to such solutions () of
(1.1) which exists on the interval [β,∞) for β > α.
Where α is non negative real number.
Definition 1: A nontrivial solution () of
differential equation (1.1) is said to be oscillatory if
it has arbitrarily large zeroes on [β,∞), for β > α
otherwise it said to be " non oscillatory.
It is well known (see [12] Reid) that either all
solutions of (1.1) are nonoscillatory, or all the
solutions are oscillatory . In the former case, we call
the differential (1.1) nonoscillatory and in the later
case is oscillatory.
2 Main Results
We prove the following theorem
Theorem 1: If
lim
(())2+()
=, (2.1)
and
lim
((())2
4((())2+)
+()
)=. (2.2)
Then the differential equation (1.1) is oscillatory.
Proof: Let () be a nonoscillatory solution of (1.1)
on the interval [α,∞), , without loss of generality its
solution can be supposed such that () > 0 on
[α,∞).
We define ()=()((())1.
Then () is well defined and satisfies the equation
()=()2
+()
()2
() (), (2.3)
rewriting equation (2.3) we have
()=
()2
+()
1
2()
+()
2()2
+()
1
2
2
+()2
4()2
+()
.
Integrating both sides of the above equation from α
to t we get
H. Kh. Abdullah
International Journal of Mathematical and Computational Methods
http://www.iaras.org/iaras/journals/ijmcm
ISSN: 2367-895X
325
Volume 1, 2016
()
=()
()2
+()
1
2()
+()
2()2
+()
1
2
2
+()2
4()2
+()
. (2.4)
Using the hypotheses (2.1) of the theorem there
exist β >α we get
()()2
+()
1
2()2
.
Define
()=()2
+()
1
2()2
. (2.5)
Thus () ().
Now differentiating equation (2.5) with respect to t
we get
()()2
+()
()2.
Therefor
()2
+()
()
()2 .
Integrating both sides of this inequality with respect
to (with replaced by s) from β to for > β we
get ()2
+()
1
()
.
Which contradicts the hypothesis of the theorem.
Hence the differential equation (1.1) is oscillatory.
This completes the proof.
Theorem 2: If
lim
((()))2
()()+(())
=, (2.6)
and
lim
()()+(())
=. (2.7)
Then the differential equation (1.1) is oscillatory.
Proof: Let () be a nonoscillatory solution of
(1.1) on the interval [α,∞), , without loss of
generality its solution can be supposed such that
()> 0 on [α,∞).
We define ()=()()1
.
Then () is well defined and satisfies the
equation
()=()()
+()
()2
()(). (2.8)
Rewriting equation (2.8) we have
()
=
()()+()
1
2()
+()
2()()+()
1
2
2
+()2
4()()+()
.
Integrating both sides of this equation from α to
t we get
H. Kh. Abdullah
International Journal of Mathematical and Computational Methods
http://www.iaras.org/iaras/journals/ijmcm
ISSN: 2367-895X
326
Volume 1, 2016
()
=()
()()+()
1
2()
+()
2()()+()
1
2
2
+()2
4()()+()
. (2.9)
Using the hypothesis (2.6) of the theorem 2
there exist β > α such that
()
()()
+()
1
2()
+()
2()()+()
1
2
2.
Define
()
()()+()
1
2()
+()
2()()+()
1
2
2. (2.10)
Thus ()().
Now differentiating equation (2.10) with
respect to t we get
() ()()
+()
()2.
Therefor
()()+()
()
()2
Integrating both sides of this inequality with
respect to t (with t replaced by s ) from β to t for
t > β we get
()()+()
1
()1
(),
we conclude that
lim
()()
+()
1
() .
Which contradicts the hypothesis of the
theorem. Hence the differential equation (1.1) is
oscillatory.
This completes the proof.
3 Examples
The following examples illustrate the applicability
of the theorems.
Example 1: The applicability of theorem 1.
Consider the second nonlinear order differential
equation
()+() (())2+()()= 0, (3.1)
for this differential equation we have ()=
() and ()=().
To show the applicability the hypothesis (2.1) of
theorem 1
H. Kh. Abdullah
International Journal of Mathematical and Computational Methods
http://www.iaras.org/iaras/journals/ijmcm
ISSN: 2367-895X
327
Volume 1, 2016
lim
(())2+()
=lim
(22)
= lim
= lim
]
= .
To show the applicability of the hypothesis (2.2)
of theorem 1
lim
((())2
4((())2+)
+()
)
= lim
2
4(22)
=lim
()
8
= .
Therefore the theorem implies that the
differential equation is oscillatory.
Example 2:
The applicability of theorem 2.
Consider the second nonlinear order differential
equation
()+2()1
1 + (())2 (())2(1 + (())2)()
= 0, (3.2)
for this differential equation we have ()=
2()1
1+(())2 and ()=(1 + (())2).
To show the applicability the hypothesis (2.6) of
theorem 2
lim
((()))2
()()+(())
= lim
(1 + (())2)2
2()1
1 + (())2(1(())2)+ 2()
= lim
(1 + (())2)2=.
And the applicability of the hypothesis (2.7) of
theorem 2
lim
()()+(())
= lim
2()1
(1 + (())2)(1
(())2)+ 2()
=lim
(1)
= .
Hence the theorem is applicable.
4 Conclusion
In this paper, we have discussed some conditions for
oscillation of class of lienard equation (1,1), where
() and () are continuously
differentiable functions on R . Under certain
assumptions, we have derived a complete
characterization of an eventually positive solution
() of (1.1). By using generalized Riccati
techniques, we have proved that under a number of
conditions that every solution () of (1.1) is
oscillatory. Also, we have given two examples to
illustrate the obtained results.
5 Acknowledgement
I would like to extend my thanks to the University
of Sharjah for its support.
References:
[1] H. Kh. Abdullah, On the Oscillation of
Second Order Nonlinear Differential
Equations, International Journal of Applied
Mathematical Research, Vol.3, No.1, 2013,
pp. 1-6.
[2] H. Kh. Abdullah, Oscillation Criteria of
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[3] H. Kh. Abdullah, The Oscillation of the
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International Journal of Pure and Applied
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[4] H. Kh. Abdullah, Sufficient Conditions for
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Differential Equations, International
H. Kh. Abdullah
International Journal of Mathematical and Computational Methods
http://www.iaras.org/iaras/journals/ijmcm
ISSN: 2367-895X
328
Volume 1, 2016
Journal of Differential Equations and
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[5] H. Kh. Abdullah, Oscillation Conditions of
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H. Kh. Abdullah
International Journal of Mathematical and Computational Methods
http://www.iaras.org/iaras/journals/ijmcm
ISSN: 2367-895X
329
Volume 1, 2016
