IMPACT: International Journal of Research in Applied, Natural and Social Sciences (IMPACT: IJRANSS) ISSN(E): 2321-8851; ISSN(P): 2347-4580 Vol. 3, Issue 3, Mar 2015, 79-86 © Impact Journals
NEW CRITERIA FOR OSCILLATION OF SECOND ORDER NONLINEAR DYNAMIC EQUATIONS WITH DAMPING ON TIME SCALES M. M. A. EL-SHEIKH1, R. SALLAM2 & NAHED A. MOHAMADY3 1,2 3
Department of Mathematics, Faculty of Science, Menofia University, Egypt
Department of Mathematics, Faculty of Information Technology, University of Pannonia, Hungary
ABSTRACT ( )
∆
The oscillation of solutions of the second order nonlinear damped dynamic equation ( ( ) ( ( ))
( ) + ( , ( ( ))) = 0 on an arbitrary time scale
∆
( ))∆ +
is investigated. A generalized Riccati transformation is
applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic equation. Several new sufficient conditions for the oscillation of solutions are obtained to extend some known results in the literature.
KEYWORDS: Damped Delay Dynamic Equations, Oscillation Criteria, Time Scales INTRODUCTION Much recent attention has been given to dynamic equations on time scales, we refer the reader to the landmark paper of S. Hilger [1]. Since then, several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal, Bohner, O'Regan and Peterson [2]. A time scale numbers
. Thus,
; ; ;
is an arbitrary nonempty closed subset of the real
, i.e., the real numbers, the integers, the natural numbers, and the nonnegative integers are
examples of time scales. On any time scale , we define the forward and backward jump operators by ( ) = inf ∈ , > , A point t
( ) = sup ∈ , < .
∈ , > #$ is said to be left dense if ( ) = , right dense if < % and
( ) = , left
scattered if ( ) < , and right scattered if ( ) > . A function ∶ → is called rd-continuous provided that it is
continuous at right dense points of T, and its left-sided limits exist (finite) at left-dense points of . The set of rd, continuous functions is denoted by )*+ ( , ). By )*+ ( , ), we mean the set of functions whose delta derivative belongs to
)*+ ( , ). In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions
of various equations on time scales (see [5],[6],[7]). However, there are few results dealing the oscillation of solutions of delay dynamic equations on time scales [8-13]. Following this trend, we are concerned in this paper with oscillation for the second-order nonlinear delay dynamic equations of the type ( ( ) ( ( ))
∆
( ))∆ + ( )
∆
( ) + ( , ( ( ))) = 0 (1.1)
We assume that
(./ ) , and are real-valued positive rd-continuous functions defined on
,
0 ≤ ( ) ≤ 1 and there are two positive constants 1, , 12 such that 1, ≤ ( ( )) ≤ 12 . Impact Factor(JCC): 1.8207 - This article can be downloaded from www.impactjournals.us
80
M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady
(.3 ) ∶ → is strictly increasing, and ( ) ≤ and
→ ∞ as → ∞
(.4 ) ( , %) ∈ )*+ ( × , ) satisfies % ( , %) > 0, for % ≠ 0 and there exists a positive rd-continuous function 7 defined on
such that 8
9(:,;) ;
8 ≥ 7( ) for % ≠ 0.
2. MAIN RESULTS We need the following lemma for the proof of main results. Lemma 2.1. Assume that ( ) is a positive solution of Eq. (1.1) on [ , ∞) > . If ?
@ AB(C) (:, :G ) ∞ DE F(C) ∆ *(:)
= ∞
(2.1)
( )7( ) ∆ = ∞ ,
(2.2)
and ?
∞
then there exists (I)
(II)
∆(
,
∈ [ , ∞) > , such that
) > 0, ( ( ) ( ( )) is decreasing.
J(:) :
∆
( ))∆ < 0 for
,
∈ [ , ∞) >
Proof. Assume that ( ) is a positive solution of Eq. (1.1) on [ , ∞) > . Pick ( ( )) > 0 on [ 2 , ∞) > . Then without loss of generality we can take
we have
∆(
( ( ) K ( )L
))∆ + ( )
Putting O( ) = − ( ) K ( )L
P −O ∆ ( ) − Thus O∆ ( ) > −
( )
( ) K ( )L
∆(
∆(
) = − K ,
N(
∆(
2
∈ [ , ∞) > , such that
) < 0 for all ≥
2
)L < 0.
), then we can write (2.3) in the form
QO( ) < 0
( ) QO( ) 1, ( )
Therefore
O( ) > O( 2 )R ST(:) ( , 2 ) UE *(:)
i.e. − ( ) K ( )L
∆(
) > − ( 2 ) K ( 2 )L
∆(
2 )R ST(:) ( UE *(:)
, 2 )
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( ) > 0 and
≥ , . Now from (1.1)
(2.3)
81
New Criteria for Oscillation of Second Order Nonlinear Dynamic Equations with Damping on Time Scales
∆(
( 2 ) K ( 2 )L
) <
K ( )L
∆(
R ST(:) ( 2 ) UE *(:)
( )
, 2)
Then ( 2 ) K ( 2 )L 1,
( ) < ( V) +
∆(
By (2.1) we can see that ( ) → −∞ as
( ( ) ( ( ))
∆
2)
W
:
:Y
R ST(X) ( , UE *(X)
( )
2)
∆
→ ∞ which contradicts the assumption ( ) > 0 so
( )) < 0 for all large . Now to prove (II), we define Z ( ) = ( ) − ∆
∆
[ 2 , ∞) > such that Z ( ) > 0. Suppose this is false, then Z ( ) < 0 on [ V , ∞) > , thus (
( )
∆(
) ∆ =
) − P ( ) −Z ( ) = > 0. ( ) ( )
is strictly increasing on [ V , ∞) > .Pick
J(:) :
Hence
( ( )) ( ( [ )) ≥ ≔ ] > 0 ( ) ( [ )
[
∆(
) > 0 and hence
( ), if there is a
∈ [ V , ∞) > , so that ( ) ≥ ( [ ) for all ≥
[.
V
∈
Then
K ( )L ≥ ] ( ).
(2.4)
But from (1.1) we have ( ( ) K ( )L
∆(
))∆ + ( )
∆(
∆(
))∆ ≤ − K ,
) = − K ,
N(
)L < 0.
i.e. ( ( ) K ( )L ( ) K ( )L
∆(
N(
)L ≤ −7( )
) − ( [ ) K ( [ )L
∆(
N(
) ≤ −Q]7( ) ( ) :
[ ) ≤ −Q] W 7( ) ( ) ∆ :^
i.e. ( [ ) K ( [ )L
∆(
[)
:
≥ ] W 7( ) ( ) ∆ :^
This contradicts (2.2). Therefore Z ( ) > 0 for all ∈ [ V , ∞) > , and hence ( ) > (
( )
So
) ∆ =
J(:) :
∆(
) − P ( ) −Z ( ) = < 0. ( ) ( )
∆(
). Also we see that
is strictly decreasing on [ V , ∞) > .
Theorem 2.2. Suppose that (2.1) holds. Furthermore, suppose that there exists a positive ∆ −differentiable function _( )
such that, for all sufficiently large :
lim % W b :→∞
:E
,
∈ [ , ∞) > , one has
_( )7( ) ( ) 12 ( )c2 ( ) − e ∆ = ∞ (2.5) ( ) 4_( ) ( )
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82
M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady
where c( ): = _ ∆ ( ) −
i(:)T(:) Uj *(:)
. Then Eq. (1.1) is oscillatory.
Proof. Suppose to the contrary that ( ) is a nonoscillatory solution of (1.1). We may assume that ( ) > 0,
K ( )L >
0 ∀ ∈ [ , , ∞) > . The proof when ( ) is eventually negative is similar. From Lemma 2.1 and Eq. (1.1) it follows that ∆(
K ( )L > 0,
) > 0, ( ( ) ( ( ))
l( ) = _( )
( ) K ( )L ( )
∆
∆(
)
( ))∆ < 0 ∀ ≥
, .
Define
Then l
∆(
=
) = _ m
∆ o
∆ ∆
∆
_∆ o ( l + _o
o
n +_m
)
−_
From (1.1) and (.4 ) we have
l∆ ( ) ≤
_∆ o _ l − _o
l∆ ( ) ≤
_∆ o _ ( l − _o 12
o
∆ ∆
(
∆ 2
n
o
)
− _7
∆
−_ o
Since > , then in view of (./ ), and (
Since
J :
l∆ ( ) ≤
=
r
iq
o
)
∆ o
N
− _7
is strictly decreasing and
N
o
o
∆
∆
) < 0, we have
∆ ∆
−
_ ( 12
o
) (
∆ o
< ; < , then
_ _7 _ _∆ o (lo )2 l − lo − − _o 12 _ o 12 (_ o )2
lo −
io
Uj *:(i q )j
(lo )2 −
Jp
Jq
o
)
∆ o
≥ , N
o
o Jq J
≥
o :
, thus
isN o
(2.6)
i.e. l
∆(
2
1 _ c 12 12 c2 _7 o u )≤t ou l − v + − _ 12 _ o 2 _ 4_
therefore l∆ ( ) ≤
12 c2 _7 − 4_
Integrating this inequality from :
W b :j
2
to
, we get
_( )7( ) ( ) 12 ( )c2 ( ) − e ∆ ≤ l( 2 ) − l( ) ≤ l( 2 ) ( ) 4_( ) ( )
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83
New Criteria for Oscillation of Second Order Nonlinear Dynamic Equations with Damping on Time Scales
This contradicts with (2.5), hence the proof is completed. By choosing _( ) = 1, ≥
in Theorem 2.2 we
have the following oscillation result.
Corollary 2.3. Assume that the assumptions of Theorem 2.2 hold and for all sufficiently large 2( ) : 7( ) ( ) − e∆ = ∞ lim % W b :→∞ ( ) 412 ( ) ( ) :E
,,
Then every solution of Eq. (1.1) is oscillatory on [ , ∞) > . By choosing _( ) = , ≥
in Theorem 2.2 we
have the following oscillation result.
Corollary 2.4. Assume that the assumptions of Theorem 2.2 hold. Then every solution of Eq. (1.1) is oscillatory on
[ , ∞) > provided that :
lim % W b
:→∞
:E
7( ) ( ) 12 ( )c2 ( ) − e∆ = ∞ ( ) 4 ( )
where c( ) = 1 −
:T(:)
Uj *(:)
.
Now, we define the function space ℜ as follows: x ∈ ℜ provided x is defined for
[ , ∞) > , x( , ) = 0, x( ; ) ≥ 0 and x has a nonpositive continuous ∆ −partial derivative x
≤ ≤ , , ∈
∆y (
, ) ≥ 0 with
respect to the second variable and satisfies for some ℎ ∈ ℜ. The following theorem extends Theorem 2.2 of [5]. x ∆y ( , ) + x( , )
c( ) ℎ( , ) {x( , ) =− o o _ ( ) _ ( )
Theorem 2.5. Suppose that the assumptions of Theorem 2.2 hold. If there exists a positive functions x, ℎ ∈ ℜ such that for all sufficiently large lim sup
:→∞
,
∈ [ , ∞) > , one has then every solution of Eq. (1.1) is oscillatory on [ , ∞) > .
: 1 _( )7( ) ( ) 12 ( )ℎ2 ( , ) W b x( , ) − e ∆ = ∞, (2.7) ( ) 4_( ) ( ) x( , ) :G
Proof. Suppose to the contrary that ( ) is a nonoscillatory solution of (1.1). Then as in Theorem 2.2 we have K ( )L >
0,
have
∆(
) > 0, ( ( ) ( ( ))
∆
( ))∆ < 0 ∀ ≥
,
∈ [ , ∞) > . We define l( ) as in Theorem 2.2, then from (2.6) we
_7 c _ (lo )2 (2.8) ≤ −l∆ + o lo − _ 12 (_ o )2 Multiplying (2.8) by x( , ) and integrating from
,
to , we get
: : _( )7( ) ( ) c( ) o x( , ) ∆ ≤ W l∆ ( )x( , ) ∆ + W o l ( )x( , ) ∆ ( ) :E :E :E _ ( )
W
:
− W
:
:E 12
_( ) ( )
( ) K_ o ( )L
2
:
(lo )2 ( )x( , ) ∆
c( ) o l ( )x( , ) ∆ o( ) _ :E
= l( , )x( , , ) + W x ∆y ( , ) lo ( ) ∆ + W :E
:
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84
M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady
− W
_( ) ( )
:
( )
:E 12
K_ o (
)L
2
(lo )2 ( )x( , ) ∆
Thus
_( )7( ) ( ) x( , ) ∆ ≤ l( , )x( , , ) ( ) :E
W
:
2
: _( ) ( )x( , ) o ℎ( , ) 12 ( ) 1 12 ( ) ℎ2 ( , ) u u −W ~ o l ( )+ • ∆ + W ∆ 4_( ) ( ) 12 ( ) 2 _( ) ( ) :E _ ( ) :E :
12 ( ) ℎ2 ( , ) ∆ 4_( ) ( ) :E
≤ l( , )x( , , ) + W
:
i.e.
: 1 _( )7( ) ( ) 12 ( )ℎ2 ( , ) W b x( , ) − e ∆ ≤ l( , ) ( ) x( , , ) :G 4_( ) ( )
which contradicts (2.7) and hence the proof is completed. Theorem 2.6. Suppose that (2.1) holds. Furthermore, suppose that there exists a function €( ) such that ( )€( ) is a
∆ − P differentiable function and there exists a positive real rd-function •( ) such that, for all sufficiently large [ , ∞) > , one has
:
lim % W ‚( ) − :→∞
:E
where „( ): =
:*(:)† q (:)‡j (:) Uj o(:)
−
:T(:)† q (:) Uj o(:)
,
∈
ƒ2 ( ) ∆ = ∞, (2.9) 4„( ) † q (:):
Uj *(:)o(:)† j (:)
, ƒ( ): =
† ∆ (:) †(:)
−
:† q (:)T(:)
Uj *(:)o(:)†(:)
− • o ( )[ ( )€( )]∆ . Then Eq. (1.1) is oscillatory.
+
2:† q (:)‡(:) Uj o(:)†(:)
, and
‚( ): =
† q (:)s(:)N(:) o(:)
+
Proof. Suppose to the contrary that ( ) is a nonoscillatory solution of (1.1). We may assume that ( ) > 0, K ( )L >
0 ∀ ∈ [ , , ∞) > . The proof when ( ) is eventually negative is similar. From Lemma 2.1 and Eq. (1.1) it follows that K ( )L > 0,
∆(
) > 0, ( ( ) ( ( ))
%( ) = •( )[
( ) K ( )L ( )
∆(
)
∆
( ))∆ < 0 ∀ ≥
+ ( )€( )]
,
∈ [ , ∞) > . Define
Then %∆ ( ) = • ∆ m
•∆ ( = % + •o •
∆
o
+ €n + • o m )
∆ ∆
− •o
From (1.1) and (.4 ) we have
(
)
∆
∆ 2
o
+ €n
∆
+• o [ €]∆
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85
New Criteria for Oscillation of Second Order Nonlinear Dynamic Equations with Damping on Time Scales
%∆ ( ) ≤
•∆ •o %− •
%∆ ( ) ≤
•∆ •o %− • 12
∆
o
− •o 7
Using Lemma 2.1 and (./ ), we get
N
o
−
•o
o
% ( − €)2 +• o [ €]∆ •
% •o 7 • o %2 2• o % • o €2 ‰ − €Š − − + − + • o [ €]∆ • 12 • 2 12 • 12
i.e. %∆ ( ) ≤
•∆ •o •o € •o 7 •o 2• o % • o €2 2 %− %+ − − % + − + • o [ €]∆ • 12 • 12 12 • 2 12 • 12
= −„%2 + ƒ% − ‚ = −[√„ % − i.e. %∆ ( ) ≤
ƒ
2√„
]2 +
ƒ2 −‚ 4„
ƒ2 ( ) − ‚( ) 4„( )
By integrating the above inequality, we obtain :
W ‚( ) − :j
ƒ2 ( ) ∆ ≤ %( 2 ) − %( ) ≤ %( 2 ) 4„( )
This contradicts (2.9), and the proof is completed. We can get the following result by choosing •( ) = 1 in
Theorem 2.6.
Corollary 2.7. Assume that the assumptions of Theorem 2.5 hold and for some :
lim % W ‚( ) − :→∞
:E
where „( ): =
ƒ2 ( ) ∆ = ∞, 4„( )
:
Uj *(:)o(:)
Then Eq. (1.1) is oscillatory.
, ƒ( ): = −
:T(:)
Uj *(:)o(:)
+
2:‡(:)
, and‚( ): =
Uj o(:)
,
sufficiently large . We have
s(:)N(:) o(:)
+
:*(:)‡j (:) Uj o(:)
−
:T(:)‡(:) Uj o(:)
− [ ( )€( )]∆ .
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M. M. A. El-Sheikh, R. Sallam & Nahed A. Mohamady
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