 Open Access
 Total Downloads : 321
 Authors : P. Mohankumar, A. Ramesh
 Paper ID : IJERTV2IS70591
 Volume & Issue : Volume 02, Issue 07 (July 2013)
 Published (First Online): 17072013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Oscillatory Behaviour Of The Solution Of The Third Order Nonlinear Neutral Delay Difference Equation
P. Mohankumar (1) and A. Ramesh (2)

Prof of Mathematics, Department of Mathematics, Aarupadiveedu Institute of Technology, Vinayaka Mission University, Kancheepuram, Tamilnadu, India603 104

Senior Lecturer and Head of the Department of Mathematics, District Institute of Education and Training, Uthamacholapuram, Salem636 010
Abstract
In this paper we study oscillatory behaviour of the solution of the third order nonlinear neutral delay difference equation of the form
2 a x p x f n, n 0, n N n
oscillatory. The forward difference operator
xn=xn+1 – xn
2. Main Result
In this section we state and prove some lemmas which are useful in establish main result for the sake of convenience we will use of following
n n n nk
0 notations.
t
t
n1 s1
R(n)
Key words: Oscillation, third order, Nonlinear Neutral Delay difference equations
and
sn0 t n0 at
t 1
t 1
n1 s1
R(n, N )
1. Introduction
sN sN at
We are concerned with the oscillatory behaviour of the solution of the third order nonlinear neutral
Let
x
x
n nn0
be a real sequences we will also
delay difference equations of the form
associated sequences zn
2 a x p x
f n, n 0, n N n
zn xn pnk n N n0 (2.1)
n n n nk
0
(1.1)
Where pn and k have been defined above
First we give some relation between the sequence
xn and zn
Where the following conditions are assumed to
hold.
Let
x
x
n nn0
be positive sequence, zn be
(H1) an is a positive sequence of real numbers
sequence by (1.2)
n (i) lim xn then lim zn
for nN(n0) such that
a
x
x
nn0 n
(ii) If zn converges to zer then so does
(H2) pn is a real sequence such that
xn
0 < < 1 for all nN(n0)
(H3) k is a non negative integer and { } is a sequence of positive integer with lim (n)
x
(H4) : 0 Ã— is continuous and f(n,u) is nondecreasing in u with u f(n,u)> 0 for all u 0 and all nN(n0) and f(n,u) 0 eventually.
By a solution of equation (1.1) we mean real sequence xn satisfying (1.1)
Proof: The proof can be found in [9]
xn
xn
Lemma 2.2
Let is an eventually positive solution of
nn0
equation (1.1) then there only the following two cases for n large enough

xn 0, zn 0, zn o, an zn 0, an zn o
n={n0,n0+1,n0+2,…….} a solution xn is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called non

xn 0, zn 0, zn o, an zn 0, an zn o
Lemma 2.3
We have
z 1 a z R
for
If N n0
then
lim
R(n, N) 1
n N
(n) 2 n n (n)
Lemma 2.4
x
R(n)
The proof is complete
x
x
Let
n nn0
is an eventually positive solution of
Lemma 2.7
If x
is an eventually positive solution of
equation (1.1) then there exists an integer
n nn0
N N n0 and a constant k1 0 such that
equation (1.1) then there exist an integer

n N n0 such that (1 pn )zn xn zn
for
a z R(n) z k R(n), n N

n n n 1
Lemma 2.5
n N
x
x
Proof: If
n nn0
is an eventually positive
nn
nn
Let xn is an eventually positive solution of
0
solution of equation (1.1) for n N . Then from
equation (1.1) then there exist an integer
the definition of zn we have zn xn
for
n1 N n0 such that for any integer N n1 we
n1
n N from lemma 2.2 we have
zn 0 and
have
zn R(s, N ) f (s, (n)), n N
s N
zn o for n N
The proof of lemmas can be found [7] and [8]
z x

px
x z

p x
x
x
Lemma 2.6
If
n nn0
is an eventually positive solution of
n n nk
xn zn pn znk
n n n nk
equation (1.1) then there exist an integer
n N n0 such that
1 pn zn
for n N
z 1 (a z
)R (n)
for n N
also if
This completes the proof.
n 2 n n
(n) n , then
z 1 (a z )R
(n) 2 n n (n)
for n N
(2.2)
Theorem 2.8
Assume that there exists real sequences qn such
f (n,u)
Proof: From Lemma 2.2 we have for
n n1 N (n0 )
that
(2.4)
Mq 0
u n
for all u 0, n n0
zn 0
zn
n1
o and
n1 1
2 a z
0
and (n) n l
such that
n n
n n
n
where l is a sequence pn
( )2
z z
a z
limsup [(1 p )q
s ]
n s z s
x
s z l s
2M R(s l) 2
sn1
sn1 az
sn0 s
n1 s1
n1 s1
a
a
t t
t t
1 a z
sn1 s t n1
(2.5)
Then all solutions of equation (1.1) are oscillatory.
n1 s n
Proof: Let xn be a nonoscillatory solutions of
(an zn ) 1
(1.1) and assume without loss of generality the
sn1 as
x is eventually positive. From Lemmas 2.2 and
(a z )R(n, n ) (2.3) n
n n 1
2.7 we have z
o, z 0, z
o and
From lemma 2.3 we conclude that there exist an
1
1
integer n N such that R(n, n ) 1 R(n)
n nl n
anzn 0 for n N and
2 x (1 p )z
for n N
n n
n n
Since 2 a z
0
and (n) n
nl n nl
Define
REFERENCES
n
n
n an zn ,
znl
n N

R. P. Agarwal: Difference equation andinequalities theory, methods and Applications 2nd edition.
Then in view of Lemma 2.6, (2.4) and (2.5) we have
2 a z a z a z
z

R.P.Agarwal, Martin Bohner, Said R.Grace, Donal O'Regan: Discreteoscillation theoryCMIA Book Series, Volume 1, ISBN : 9775945194.
n n n n n n
n n n nl

B.Selvaraj and I.Mohammed Ali Jaffer
z z
z z
n 2
nl nl
n nl n n
n nl n n
Mq 1 p n
n
Mq (1 p ) n 1 2R n l
:Oscillation Behavior of Certain Thirdorder Linear Difference EquationsFarEast Journal of Mathematical Sciences,
Volume 40, Number 2, pp 169178(2010).

B.Selvaraj and I.Mohammed Ali Jaffer:Oscillatory Properties of Fourth OrderNeutral Delay Difference EquationsJournal
n nl n n
n 2n
of Computer and Mathematical Sciences
Mq (1 p
2
n
n
)
AnIternational ResearchJournal,Vol. 1(3) 364373
(2010).
n
n
n nl n 2 R(n l) Summing the last inequality from N to n N , we obtain

B. Selvaraj and I. Mohammed AliJaffer: Oscillation Behavior of CertainThird order Non linear Difference EquationsInternational Journal ofNonlinear Science (Accepted on September 6,
n
n
sn0
s [(1

pzl )qs
(s )
2
2
s
s
2M R(s l) 2 ]
N
M
(2010).


B.Selvaraj and I.Mohammed Ali JafferOscillation Theorems of Solutions ForCertain
and this contradicts (2.5). Thus the proof is complete.
For the linear equation
3 x p x q x 0 (2.6)
Third Order FunctionalDifference Equations With DelayBulletin of Pure and Applied Sciences(Accepted on October 20, (2010).

E.Thandapani and B.Selvaraj: Existenceand
n n n
n n
Asymptotic Behavior of Nonoscillatory Solutions
Where and are nonnegative integers less than n we obtain from Theorem 2.8 the following corollary
Corollary 2.7
of Certain Nonlinear Difference equation Far East Journal of Mathematical Sciences 14(1),pp: 925 (2004).

E.Thandapani and B.Selvaraj:Oscillatory and NonoscillatoryBehavior of Fourth order Quasi
Suppose qn 0 for all n n0 and there exists positive sequences n such that
n ( )2
limsup [(1 p )q ]
linearDifference equation Far East Journalof Mathematical Sciences 17(3), 287307 (2004).

E.Thandapani and B. Selvaraj:Oscillation of Fourth order QuasilinearDifference equationFasci culi Mathematici Nr, 37, 109119 (2007).
s
s
x
sn0
s z l s
2M R(s l) 2
s
s
then all solutions of equation 2.5 are oscillatory. The proof is complete
Example : Consider the difference equations
2
2
n(n
n(n
1)
1)
x x nx3
x x nx3
0; n
0; n
3
3
1 1
n
n 1
n1
n1
(2.7)
it is easy to see all solutions of the equations(2.7) are oscillatory