Research Journal of Applied Sciences, Engineering and Technology 7(16): 3342-3347, 2014
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2014
Submitted: September 28, 2013
Accepted: November 19, 2013
Published: April 25, 2014
Oscillation Criteria for Ordinary Differential Equations of Second Order Nonlinear with
Alternating 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,
43600 UKM Bangi, Selangor, Malaysia
Abstract: Several new oscillation criteria are established under quite general assumptions. Our results generalize
and extend some earlier results of the literature. Examples are given to illustrate the results. We employ the
averaging technique to obtain sufficient conditions for oscillation in the solutions of our equation.
Keywords: Averaging technique, nonlinear differential equations, oscillation criteria, Riccati substitution, second
order
INTRODUCTION
The study of the oscillation of second order
nonlinear ordinary differential equations with
alternating coefficients is of special interest because of
the fact that many physical systems are modeled by
second-order nonlinear ordinary differential equations.
For example, the so called Emden-Fowler equation
arises in the study of gas dynamics and fluid mechanics.
This equation appears also in the study of relativistic
mechanics, nuclear physics and in the study of
chemically reacting systems. The averaging techniques
are used also in study of the nonlinear oscillations.
This study is concerned with the oscillation of
solutions of the second order nonlinear differential
equation:
oscillatory if it has a sequence of zero clustering at ∞
and non-oscillatory otherwise. Thus a non-oscillatory is
either eventually positive or eventually negative.
Equation (1) is called oscillatory if all its solutions are
oscillatory and otherwise it is called non-oscillatory.
Both oscillatory and non-oscillatory behavior of
solutions for various classes of second order differential
equations have been widely discussed in the literature,
see for example; Elabbasy and Elsharabasy (1997),
Grace and Lalli (1980, 1992), Grace (1990), Kiran and
Rogovchenko (2001), Philos (1975), Wong (1986),
Wong and Yeh (1992), Wintner (1949), Yan (1986) and
Yeh (1982). There are a great number of papers
addressing particular cases of Eq. (1), such as the
following equations (Li, 1998; Manojlovic, 1999, 1991;
Tiryakiand Ayanlar, 2004):
(r (t )ψ ( x(t )) f ( x′(t )))′ + q(t ) g ( x(t )) =
H (t , x′(t ), x(t )), (1)
where,
q and r : Continuous functions on the interval (t 0 , ∞), t 0
≥0
r(t)
: A positive function
ψ and f : Continuous functions on the real line ℝ, with
ψ (x) >0, ∀ x ∈ ℝ and yf (y) >0 for y ≠ 0
g
: Continuously differentiable function on the
real line, ℝ, except possible at 0 with xg (x) >0
and g' (x) ≥k>0 for all x ≠ 0 and k is a constant
H
: A continuous function on [t 0 , ∞) × ℝ2 , with
H (t , y, x)
≤ p (t ) , ∀ 𝑡𝑡 ∈ [t 0 , ∞), y ∈ ℝ and x ≠ 0
g ( x)
By a solution of (1), we mean a nontrivial function
x (t) satisfying (1). A solution x (t) is said to be
H (t ),
( r (t ) x′(t )α )′ + q(t ) g ( x(t )) =
(2)
H (t ),
( r (t )ψ ( x) x′(t ) )′ + q (t ) g ( x) =
(3)
H (t , x(t ), x′(t )),
( r (t ) x′(t ) )′ + Q(t , x) =
(4)
(5)
H (t ).
( r (t )ψ ( x(t )) f ( x′(t )) )′ + q(t ) g ( x(t )) =
An important tool in the study of oscillatory
behaviour of solutions of these equations is the
averaging technique which goes back as far as the
classical result of Wintner (1949) which states that (2)
with r (t) = 1, α = 1, g (x) = x and H (t) = 0 is oscillatory
if:
Corresponding Author: Ambarka A. Salhin, School of Mathematical Sciences, Faculty of Science and Technology, Universiti
Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
3342
Res. J. App. Sci. Eng. Technol., 7(16): 3342-3347, 2014
′
t s
1
q (u ) du ds = ∞.
t →∞ t ∫ ∫
t0 t0
lim
w′(t )
=
In this study, the comparison between our results
and the previously known results are presented and
some examples of the main results are illustrated.
Furthermore, we expand some of the previous
equations, (Baculikova, 2006; Tiryaki and Ayanlar,
2004) as well as the expansion and development of
some of the previous conditions (Elabbasy et al., 2005;
Graef et al., 1978; Remili, 2008).
ρ (t ) [ r (t )ψ ( x(t )) f ( x′(t ))]
ρ ′(t )r (t )ψ ( x(t )) f ( x′(t ))
+
g ( x(t ))
g ( x(t ))
ρ (t )r (t )ψ ( x(t )) f ( x′(t )) g ′( x(t )) x′(t ))
,
−
g 2 ( x(t ))
ρ (t ) H (t , x′(t ), x(t ))
=
w′(t )
− ρ (t )q (t ) +
ρ ′(t )
w(t )
ρ (t )
g ( x(t ))
1
−
x′(t ) g ′( x(t )) w2 (t ).
ρ (t )r (t )ψ ( x(t )) f ( x′(t ))
From conditions O 1 and O 3 we obtain:
MAIN RESULTS
w′(t ) ≤ ρ (t ) ( p (t ) − q (t ) ) +
In this section, we use the Riccati technique to
establish sufficient conditions for Eq. (1) to be
oscillatory.
ρ ′(t )
K
w(t ) −
w2 (t ) ,
ρ (t )
ρ (t )r (t )k2
M
ρ ′(t )
w(t ) −
w2 (t ) − w′(t ).
ρ (t )
ρ (t )r (t )
ρ (t ) ( q(t ) − p (t ) ) ≤
Theorem 1: Suppose that:
Integrating from t 0 to t gives following result:
O 1 : g ′( x) ≥ K > 0 for a constant K and x ≠ 0,
ψ ( x)
±∞
O 2 : ∫ ψ ( y ) dy < ∞ for all ε > 0 ,
±ε g ( y )
O 3 : 0 < k ≤ f ( y) ≤ k
1
2
y
for all constants k1 , k2 , y = x′(t ) ≠ 0 and lim
t →0
f ( y)
y
t
t
M
2
ρ ′( s )

0
t0
t
∫ ρ ( s ) ( q( s ) − p ( s ) ) −
t0
2
t

r ( s ) ρ ( s ) ρ ′( s ) 
r ( s ) ρ ( s )  ρ ′( s ) 
M
ds ≤ w(t0 ) − w(t ) − ∫ 
w( s ) −
 ds,
4 M  ρ ( s ) 
r( s)ρ ( s)
2 M ρ ( s) 
t0 
2
≤ w(t0 ) − w(t ),
or,
exist.
t

t0

∫  ρ ( s) ( q( s) − p( s) ) −
There also exists a positive function ρ ∈ C 1 [t 0 , ∞)
such that ( ρ (t )r (t ) )′ ≤ 0 for all t ≥ t0 , and:
2
r ( s ) ρ ( s )  ρ ′( s )  
ρ (t )r (t )ψ ( x (t )) f ( x′(t ))
.
 ds ≤ w(t0 ) −


4 M  ρ ( s)  
g ( x (t ))

Using condition O 3 :
O4:
t
ρ (t )r (t )ψ ( x(t )) x′(t )
r ( s ) ( ρ ′( s )) 2 
ds ≤ w(t0 ) − k1
.
ρ ( s) 
g ( x(t ))

∫  ρ (s) ( q(s) − p(s) ) − 4M
t s

1 
( ρ ′(u )) 2
lim sup ∫ ∫  ρ (u )[ q (u ) − p (u )] −
r (u )  du ds = ∞,
t →∞
t t0 t0 
4 M ρ (u )

t0
Taking a second integration from t 0 to t we have:
where,
t s
M =

∫ ρ ( s) ( q( s) − p( s) ) ds ≤ w(t ) − w(t ) − ∫  r( s)ρ ( s) w ( s) − ρ ( s) w( s)  ds,
t0
K
k2
.
t0 t0
t
(7)
Then Eq. (1) is oscillatory.
Proof: On the contrary we assume that Eq. (1) has a
non-oscillatory solution, x(t ) . We suppose without loss
of generality that x (t ) > 0 for all t ∈ [t0 , ∞) . We
Since r (t ) ρ (t ) is non-increasing, then by the
Bonnet’s Theorem there exists an η ∈ [t 0 , t ] such that:
t
− k1 ∫ ρ ( s )r ( s )
t0
define the function w(t ) as:
w(t ) =
r (u ) ( ρ ′(u )) 2 
ρ ( s)r ( s)ψ ( x( s)) x′( s )
duds ≤ w(t0 )(t − t0 ) − k1 ∫
ds .
ρ (u ) 
g ( x( s ))
t0

∫ ∫  ρ (u) ( q(u) − p(u) ) − 4M
ρ (t )r (t )ψ ( x(t )) f ( x′(t ))
g ( x(t ))
η
ψ ( x( s )) x′( s )
ψ ( x( s )) x′( s )
ds =
ds
−k1r (t0 ) ρ (t0 ) ∫
g ( x( s ))
g ( x( s ))
t
0
= k1r (t0 ) ρ (t0 )
x ( t0 )
∫
x (η )
for all t ≥ t0 .
ψ ( y)
dy
g ( y)
0,

∞
< 
ψ ( y)
k1r (t0 ) ρ (t0 ) ∫ g ( y ) dy
ε

(6)
Differentiating Eq. (6) and substituting Eq. (1)
imply:
hence,
3343
if x(t0 ) < x(η ),
if x(t0 ) > x(η ),
Res. J. App. Sci. Eng. Technol., 7(16): 3342-3347, 2014
t
− ∞ < − k1 ∫ r ( s ) ρ ( s )
ψ ( x) x′( s )
g ( x)
t0
1

t
s
1  1
1
u
= lim sup ∫  ∫ 1  + sin u − 3  −
× 0 du
t →∞
t t0  t0  2
u  4M

= ∞.
ds < L1 ,
where,
∞
L1 = k1r (t0 ) ρ (t0 ) ∫
ε
ψ ( y)
g ( y)
By Theorem 1, this equation is oscillatory.
dy .
Remark 1: Theorem 1 improves and expands to the
Theorem 4 of Graef et al. (1978) and to the Theorem 1
of Elabbasy et al. (2005).
Equation (7) becomes:
t s
r (u ) ( ρ ′(u )) 2 
duds ≤ w(t0 )(t − t0 ) + L1.
ρ (u ) 

∫ ∫  ρ (u ) ( q(u ) − p(u ) ) − 4M
t0 t0
Theorem 2: Assume that O 2 holds and:
(8)
Dividing (8) by t and taking the upper limit as 𝑡𝑡 → ∞:
lim sup
t →∞
t s
1 
r (u ) ( ρ ′(u )) 2 
1
sup [ w(t0 )(t − t0 ) + L1 ]
 ρ (u ) ( q (u ) − p (u ) ) −
 du ds ≤ lim
t →∞
t t∫0 t∫0 
4 M ρ (u ) 
t
< ∞.
This contradicts the assumption O 4 , which
completes the proof.
O 5 : f ( y ) > L > 0; for a constant L and y ≠ 0,
y
there exists a positive continuously differentiable
∞
function ρ defined as in Theorem 1 and ∫𝑡𝑡 𝜌𝜌(𝑠𝑠)𝑑𝑑𝑑𝑑 =
0
∞ Then Eq. (1) is oscillatory if:
O6:
t

lim sup  ∫ ρ ( s )ds 
t →∞
 t0

Example 1: Consider the following equation:
′
1 
( x′(t ))3    1
x 7 (t ) cos t sin x′(t )
π

, t≥ .
=
+  + sin t  x3 (t )
  3x′(t ) +


2
( x′(t )) + 1    2
2
x 4 (t ) + 1 t 3

t 
(
)
Notice that:
−1
t
s




t0
 t0

∞.
∫ ρ (s) ∫ [ q(u ) − p(u )] du ds =
Proof: On the contrary we assume that Eq. (1) has a
non-oscillatory solution x (t). We suppose without loss
of generality that x (t) >0 for all t ∈ (t 0 , ∞). We define
the function w (t) as:
t
w(t ) = ρ (t ) ∫
H (t , x′(t ), x(t )) x 7 cos t sin x′ 1 1
=
× 3 ≤=
p (t ) , ∀x′ ∈  , x ∈  and t ≥ t0 ,
g ( x(t ))
( x 4 + 1)t 3
x
t3
t0
r ( s )ψ ( x( s )) f ( x′( s )) for all t ≥ t .
0
ds
g ( x( s ))
Thus, for every t ≥ t0 we obtain:
and
r ( s )ψ ( x( s )) f ( x′( s ))
ρ (t )r (t )ψ ( x(t )) f ( x′(t )) (9)
.
ds +
g ( x( s ))
g ( x(t ))
t0
g ′( x) 3 x 2
=
≥ 3= k ,
ψ ( x) 1
±∞
∫
ψ ( y)
±ε
g ( y)
3 < 3+
dy = −
1
2 y2
t
=
w′(t ) ρ ′(t ) ∫
±∞
=
±ε
1
< ∞,
2ε 2
From Eq. (1) and definition function H, we have:
′
f ( x′(t )) ]
[ r (t )ψ ( x(t )) =
g ( x(t ))
( x′(t )) 2
f ( y)
3.
< 4 , ∀y ≠ 0, lim
=
y →0
( x′(t )) 2 + 1
y
1
− 2 < 0,
1 ρ ′(t ) =∈
0 C1 [t0 , ∞ ) , ( ρ (t )r (t ) )′ =
ρ (t ) =⇒
t
O4:
lim sup
H (t , x, x′)
− q (t )
g ( x(t ))
′
[ r (t )ψ ( x(t )) f ( x′(t ))]
Let,
t →∞


 ds


g ( x(t ))
≤ p (t ) − q (t ).
Integrate the above inequality from t 0 to t and
integrate the left hand side by parts we obtain:
t s
1 
r (u ) ( ρ ′(u )) 2 
 ρ (u ) ( q (u ) − p (u ) ) −
 duds
∫
∫
t t0 t0 
4 M ρ (u ) 
[ r (t )ψ ( x(t )) f ( x′(t ))] − B + t r (s)ψ ( x(s)) f ( x′(s)) g ′( x(s)) x′(s) ds ≤ t
g ( x(t ))
3344
∫
t0
g 2 ( x( s ))
∫ p(s) − q(s)ds,
t0
Res. J. App. Sci. Eng. Technol., 7(16): 3342-3347, 2014
B=
if x(t0 ) < x(η ),
0

∞
<
ψ ( y)
L
r
(
t
)
ρ
(
t
)
dy
0
0 ∫

ε g ( y)

where,
r (t0 )ψ ( x(t0 )) f ( x′(t0 ))
.
g ( x(t0 ))
Now, multiply the last inequality by ρ (t) we obtain:
if x(t0 ) > x(η ),
hence,
ρ (t )r (t )ψ ( x (t )) f ( x′(t ))
(10)
≤ Bρ (t ) − ρ (t ) ∫ q( s ) − p( s )ds.
g ( x (t ))
t
t
t
−∞ < − ∫
0
Substituting (10) in (9) we have:
ρ ( s )r ( s )ψ ( x( s )) x′( s )
g ( x( s ))
t0
ds ≤ B1 ,
where,
r ( s )ψ ( x( s )) f ( x′( s ))
ds ≤ ρ (t ) B − ρ (t ) ∫ (q ( s ) − p ( s ))ds.
g ( x( s ))
t0
t0
t
t
w′(t ) − ρ ′(t ) ∫
B1 = ρ (t0 )r (t0 )
x ( t0 )
∫η
ψ ( y)
g ( y)
x( )
dy.
Integrate from t 0 to t we have:
Consequently:
t
s
t
t
s
t0
t0
t0
t0
t0
∫ ρ (s)∫ (q(u ) − p(u ))du ds ≤ w(t0 ) − w(t ) + B ∫ ρ (s)ds + ∫ ρ ′(s)∫
r (u )ψ ( x(u )) f ( x′(u ))
du ds
g ( x(u ))
t


s


t
t0

t0

t0
∫  ρ (s)∫ (q(u ) − p(u ))du  ds ≤ M + B ∫ ρ (s)ds,
(11)
Now evaluate using integration by parts:
where,
r (u )ψ ( x(u )) f ( x′(u ))
r (u )ψ ( x(u )) f ( x′(u ))
du ds = ρ ( s ) ∫
du
∫ ρ ′(s)t∫
g
(
x
(
u
))
g ( x(u ))
t0
t
0
0
t
s
t
−∫
s
ρ ( s )r ( s )ψ ( x( s )) f ( x′( s ))
g ( x( s ))
t0
0
t
M = w (t 0 ) + LB 1
t0
Then,
ds
t

 ∫ ρ ( s )ds 
 t0

r ( s )ψ ( x( s )) f ( x′( s ))
ρ ( s )r ( s )ψ ( x( s )) f ( x′( s ))
ρ (t ) ∫
ds − ∫
ds
g
(
x
(
s
))
g ( x( s ))
t0
t0
t
t
t
= w(t ) − ∫
ρ ( s )r ( s )ψ ( x( s )) f ( x′( s ))
g ( x( s ))
t0
Substituting (12) in (11) we obtain:
t
t0

s


t0

t
t
t0
t0
∫  ρ (s)∫ (q(u ) − p(u ))du  ds ≤ w(t ) − w(t ) + B ∫ ρ (s)ds + w(t ) − ∫
ρ ( s )r ( s )ψ ( x( s )) f ( x′( s ))
0
t
t
≤ w(t0 ) + B ∫ ρ ( s )ds − ∫
t0
t0
g ( x( s ))
ρ ( s )r ( s )ψ ( x( s )) f ( x′( s ))
g ( x( s ))




s


t0
t
t
t
t0
t0
t0
(
g ( x( s ))
+ B.
∫ ρ (s)ds
)
(
x 3 (t )
( x′(t ))2
=
sin t
, t ≥ t0 1 .
2
′(t ))2 + 1
x
(
x (t ) + 1
2
)
H (t , x′(t ), x(t ))
( x′(t )) 2
1
= x 3 (t )sin t
×
g ( x(t ))
( x′(t )) 2 + 1 x(t ) 1 + x 2 (t )
ds.
(
±∞
ψ ( y)
±∞
±
±
η
= ρ (t0 )r (t0 )
y
dy ∫ =
dy
∫ε=
g ( y)
ε (1 + y )
2 2
ψ ( x( s )) x′( s )
ds =
ds
− ρ (t0 )r (t0 ) ∫
g ( x( s ))
t0
x ( t0 )
∫η
x( )
ψ ( y)
g ( y)
)
2
≤ sin
=
=
t p (t ), ∀x′ ∈  , x ∈  and t ≥
t 0 1,
( ρ (t )r (t ) )′ ≤ 0 then by Bonnet’s Theorem
g ( x( s ))

t0
′
2
1 2 ′  3
=
+ t x(t ) 1 + x 2 (t )
x
t
x
t
(
)
(
)
 t

ds.
ρ ( s )r ( s )ψ ( x( s )) x′( s )
0
ρ ( s )r ( s )ψ ( x( s )) x′( s )

ds
there exist a η ∈ [t0 , t ] such that:
−∫
t0
M
t
Taking the upper limit as 𝑡𝑡 → ∞ which contradicts
the assumption O 6 . This completes the proof.
(13)
Since

s
Notice that:
∫  ρ (s)∫ (q(u ) − p(u ))du  ds ≤ w(t ) + B ∫ ρ (s)ds − L ∫
t0

Example 2: Consider the nonlinear differential
equation:
From condition O 5 we get:
t
t
∫  ρ (s)∫ (q(u ) − p(u ))du  ds ≤
t0
(12)
ds.
−1
Let,
r (t )
=
dy
3345
1
, ρ (t ) 1
=
t
1
(
4 1+ ε 2
)
2
< ∞.
Res. J. App. Sci. Eng. Technol., 7(16): 3342-3347, 2014
We have:
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( ρ (t )r (t ) )′ =
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<0
t2
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∞
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∫t ρ (s)ds = ∫1 ds = ∞,
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equations with damping term. Electron. J. Differ.
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


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



differential equations with damping. J. Aust. Math.

1  t5
1
1
1
Soc. A, 49: 43-54.
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+ sin t − t − t cos1 −
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
t →∞
t − 1  20
4
20
4

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117-192.
′
′
𝑥𝑥 , 𝑔𝑔 �𝑥𝑥 (𝑡𝑡)� = 𝑥𝑥 and 𝐻𝐻 �𝑡𝑡, 𝑥𝑥 (𝑡𝑡), 𝑥𝑥 (𝑡𝑡)� = 0 then
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ACKNOWLEDGMENT
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This research has been completed with the support
order nonlinear differential equations. Proc. Amer.
of these grants: UKM-DLP-2011-049 and FRGS/
1/2012/SG04/UKM/01/1.
Math. Soc., 98: 109-112.
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