Oscillation Criteria of second Order Non-Linear Differential
Equations
Hishyar Kh. Abdullah
Dept. of Mathematics, University of Sharjah, Sharjah, U.A.E.
Abstract—In this paper we are concerned with the oscillation criteria of second order non-linear homogeneous differential
equation. Example have been given to illustrate the results.
Keywords-component; Oscillatory, Second order differential equations, Non-Linear.
1. Introduction
The purpose of this paper is to establish a new oscillation
criteria for the second order non-linear differential equation
with variable coefficients of the form (1)
where is a fixed real number and f(x) and g(x) are
continuously differentiable functions on the interval .
The most studied equations are those equivalent to second
order differential equations of the form
, (2)
where h(x)>0 is a continuously differentiable functions on the
interval .Oscillation criteria for the second order
nonlinear differential equations have been extensively
investigated by authors(for example see[2], [3],[4],[5], [6],[8],
[9] and the authors there in). Where the study is done by
reducing the problem to the estimate of suitable first integral.
Definition1:A solution x(t) of the differential equation (2) is
said to be "nontrivial " if x(t)
Į 0 for at least one t ę
ę
Definition2:A nontrivial solution x(t) of the differential
equation (2) is said to be oscillatory if it has arbitrarily large
zeros on [t
΋ , Ğ ), otherwise it said to be " non oscillatory ".
Definition3:We say that the differential equation (1)
oscillatory if an equivalent differential equation (2) is
oscillatory.
2. Main Results
In [7] the author considered a class of systems equivalent
to the second order non-linear differential equation (1). The
standard equivalent system
(3)
while he worked on a wider class of systems of the form
(4)
If ĮW!then (4) is equivalent to a differential equation of the
type (1). This allows to choose a modified system in order to be
able to cope with different problems related to
(2).Ta king
() g
,where
.
One obtains
. (5)
System (5) cab be transformed into
, (6)
where
, which is equivalent to (2) where
sufficient conditions for solutions of differential equation (1)
to oscillate are given.
Remark: Assume that f(x(t)) and
Let us set
Since , for all is invertible
on I, we define the transformation u=׋(x(t)), acting on I.
Accordion to Lemma1 in [7] any solution x(t) of (3) is a
solution is a solution of (6).
Theorem1: Let h(x) be continuous and continuously
differentiable on (-
Ğ ,0) Ĥ (0, Ğ ) with
and let
t
, (7)
then any solution of the differential equation (2) is either
oscillatory or tends monotonically to zero as t
ėĞ .
Proof: Suppose that x(t) is non-oscillatory solution of (2), and
assume x(t)>0 for some.
From (2) we get
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.
Put
Then
,
then
Since h’(x(t))>0 then
By hypothesis (7)we have
this means we obtain for some constant k>0
Integrating from tЅ to t for tЅ>0 we get
(8)
The right hand side is negative, since x(tЅ)>0, x(t) is positive.
From (8) we conclude
Thus x(t) is oscillatory or tends monotonically to zero as t ėĞ
Theorem2: If In addition to hypotheses (7) we assume that
for some x(t)>0
(9)
Then every solution x(t) of the differential equation(2) is
oscillatory.
Proof: As in theorem 1,we want to show that x(t) doesn't tend
monotonically to zero as t
ėĞ .
Assume x(t)>0 for a>0 on .
Since from (8) we have
then there exists a positive real number m such that
,
This means is bounded below by a finite positive
number, then by hypothesis (9), x(t) doesn't tend
monotonically to zero as t
ėĞ .Then x(t) is oscillatory.
Theorem3: Assume that h(x) satisfies
then every solution x(t) of (2) is oscillatory.
Proof: Let x(t) be non-oscillatory solution of (2), which
without loss of generality, may be assumed to be positive for
large t.
Define
then
or
(10)
Integrating (10) from Į to t we get
(11)
Since and from (11) we get
w(t)<0 from which we get x’(t)<0 for large twhich is a
contradiction (by lamma1II.I.8,[1]) where x(t)>0 and then
x’(t)>0 for large t.
This completes the proof of the theorem.
EXAMPLES
Consider the second order nonlinear order differential
(12)
for this differential equation we have
(
and
. Then the equivalent second order
differential equation to (12) is
(13)
where . To show the applicability of
Theorem 1, the hypothesis is satisfied as follows
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Therefore the Theorem implies that the differential equation is
oscillatory.
To show the applicability of Theorem 2 it is clear that the
hypothesis is satisfied hence
and
Hence Theorem 2 is applicable.
To show the applicability of Theorem 3 the hypothesis is
satisfied as follows
And
Hence Theorem 3 is applicable.
3. Acknowledgment
I would like to extend my thanks to the University of
Sharjah for its support.
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