Oscillatory Behaviour Of The Solution Of The Third Order Nonlinear Neutral Delay Difference Equation

Oscillatory Behaviour Of The Solution Of The Third Order Nonlinear Neutral Delay Difference Equation

P. Mohankumar (1) and A. Ramesh (2)

1. Prof of Mathematics, Department of Mathematics, Aarupadiveedu Institute of Technology, Vinayaka Mission University, Kancheepuram, Tamilnadu, India-603 104

2. Senior Lecturer and Head of the Department of Mathematics, District Institute of Education and Training, Uthamacholapuram, Salem-636 010

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 × 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

1. 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

2. 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

1. n N n0 such that (1 pn )zn xn zn

   for

   a z R(n) z k R(n), n N

2. 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

1. 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

2. R.P.Agarwal, Martin Bohner, Said R.Grace, Donal O'Regan: Discreteoscillation theory-CMIA Book Series, Volume 1, ISBN : 977-5945-19-4.

   n n n n n n

   n n n nl

3. 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 Equations-FarEast Journal of Mathematical Sciences,

   Volume 40, Number 2, pp 169-178(2010).

4. B.Selvaraj and I.Mohammed Ali Jaffer:Oscillatory Properties of Fourth OrderNeutral Delay Difference Equations-Journal

   n nl n n

   n 2n

   of Computer and Mathematical Sciences-

   Mq (1 p

   2

   n

   n

   )

   AnIternational ResearchJournal,Vol. 1(3) 364-373

   (2010).

   n

   n

   n nl n 2 R(n l) Summing the last inequality from N to n N , we obtain

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   n

   n

   sn0

   s [(1

   • pzl )qs

   (s )

   2

   2

   s

   s

   2M R(s l) 2 ]

   N

   M

   (2010).

6. 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 Delay-Bulletin of Pure and Applied Sciences(Accepted on October 20, (2010).

7. 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: 9-25 (2004).

8. E.Thandapani and B.Selvaraj:Oscillatory and Non-oscillatoryBehavior 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), 287-307 (2004).

9. E.Thandapani and B. Selvaraj:Oscillation of Fourth order Quasi-linearDifference equation-Fasci culi Mathematici Nr, 37, 109-119 (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