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Oscillation Criteria of second Order Non-Linear Differential
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.
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
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, ,,, ,,
 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)
0 for at least one t
Definition2:A nontrivial solution x(t) of the differential
equation (2) is said to be oscillatory if it has arbitrarily large
zeros on [t
), otherwise it said to be " non oscillatory ".
Definition3:We say that the differential equation (1)
oscillatory if an equivalent differential equation (2) is
2. Main Results
In  the author considered a class of systems equivalent
to the second order non-linear differential equation (1). The
standard equivalent system
while he worked on a wider class of systems of the form
If ĮW!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
System (5) cab be transformed into
, 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  any solution x(t) of (3) is a
solution is a solution of (6).
Theorem1: Let h(x) be continuous and continuously
differentiable on (-
then any solution of the differential equation (2) is either
oscillatory or tends monotonically to zero as t
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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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
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
Theorem2: If In addition to hypotheses (7) we assume that
for some x(t)>0
Then every solution x(t) of the differential equation(2) is
Proof: As in theorem 1,we want to show that x(t) doesn't tend
monotonically to zero as t
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
.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
Integrating (10) from Į to t we get
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,) where x(t)>0 and then
x’(t)>0 for large t.
This completes the proof of the theorem.
Consider the second order nonlinear order differential
for this differential equation we have
. Then the equivalent second order
differential equation to (12) is
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
To show the applicability of Theorem 2 it is clear that the
hypothesis is satisfied hence
Hence Theorem 2 is applicable.
To show the applicability of Theorem 3 the hypothesis is
satisfied as follows
Hence Theorem 3 is applicable.
I would like to extend my thanks to the University of
Sharjah for its support.
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