Research Article
Oscillation Criteria of Second-Order Nonlinear Differential
Equations with Variable Coefficients
Ambarka A. Salhin, Ummul Khair Salma Din,
Rokiah Rozita Ahmad, and Mohd Salmi Md Noorani
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi,
Selangor, Malaysia
Correspondence should be addressed to Ambarka A. Salhin; amb80ark@yahoo.com
Received 31 August 2013; Revised 27 December 2013; Accepted 31 December 2013; Published 18 February 2014
Academic Editor: M. De la Sen
Copyright © 2014 Ambarka A. Salhin et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Some new oscillation criteria are given for second-order nonlinear diferential equations with variable coefcients. Our results
generalize and extend some of the well-known results in the literatures. Some examples are considered to illustrate the main results.
1. Introduction
Since many problems in physics, chemically reacting sys-
tems, celestial mechanics, and others felds are modeled by
second-order nonlinear diferential equations, the oscillatory
and asymptotic behaviors of solutions of such diferential
equations have been investigated by many authors [1–9].
Investigation of nonlinear diferential equation in this paper
is motivated by [1], where some of the conditions required
in the theorems contain the unknown solution (). It seems
that any verifcation of such conditions is questionable.
In this paper, we consider the oscillation of second-order
nonlinear forced diferential equation
( ()(()) (
()))
+()(())
=(,
(),()), ∈[
0
,∞),
(1)
where , ∈ ([
0
,∞), R) and ,,∈(R, R) and is a
continuous function on [
0
,∞)× R
2
.
Consider the following with respect to (1):
(A
1
) ()>0, ≥0;
(A
2
) () ∈ (R), ()>0 for all ;
(A
3
) () > 0, ∈
1
(R) for ̸ =0;
(A
4
) (,,)/() ≤ () for all ∈ [
0
,∞); , ∈ R
and ̸ =0.
We say that a nonzero function : [
0
,
1
) → (−∞,∞),
1
>
0
is called a nontrivial solution of (1) if () satisfes (1) for all
∈[
0
,
1
). A solution () of (1) is called oscillatory if there
exists a sequence {
}
∞
=1
of points in the interval [
0
,∞), so
that lim
→∞
=∞ and (
)=0, ∈ N; otherwise it is
called nonoscillatory. Equation (1) is called oscillatory if all its
solutions are oscillatory.
We note that if (
()) =
() and (,(),
()) = 0,
then (1) becomes the diferential equation of the form
( ()(())
())
+()(())=0,
∈[
0
,∞),
0
>0.
(2)
Te oscillation of solutions of (2) has been studied by El-
sheikh [1], Manojlovic [10], and Ohriska and Zulova [5] under
diferent conditions. Also, if (()) = 1, (
()) =
()
and ()(()) = (,), then (1) becomes the diferential
equation of the form
( ()
())
+(,)= (,
(),()),
∈[
0
,∞),
0
>0.
(3)
Te oscillation of solutions of (3) has been studied by
Remili [7, 8], where new results with additional suitable
weighted function are found. Recently, Temtek and Tiryaki
Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2014, Article ID 279236, 9 pages
http://dx.doi.org/10.1155/2014/279236