International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 1

ISSN 2229-5518

B. Selvaraj1, I. Mohammed Ali Jaffer2

1The Dean of Science and Humanities, Nehru Institute of Engineering and Technology

2Research Scholar, Department of Mathematics, Karunya University, Coimbatore, Tamil Nadu, India

Key words: Linear, Nonlinear, Difference equations, Oscillations and Non-oscillation.

—————————— • ——————————

1 Introduction

We are concerned with the oscillatory properties of all solutions of third order nonlinear difference equations of the form

2 qn x

+ c x

+ p x

+ q f ( x

) = 0; *n *= 0,1, 2,...

(1.1)

\ an

n n n cr

)

n n n n +1

2 qn ( x ) x

+ c x

+ p x

+ q f ( x

) = 0; *n *= 0,1, 2,...

(1.2)

\ an

n n n n cr

)

n n n n +1

2 qn x

+ c x

+ q f ( x

) = 0; *n *= 0,1, 2,...

(1.3)

2 qn ( x ) x

\ an

+ c x

n n n cr

)

+ p x

n n +1

+ q f ( x

) = 0; *n *= 0,1, 2,...

(1.4)

\ an

n n n n cr

)

n n n n +1

Where the following conditions are assumed to hold.

(H1) {*a*n },{ *p*n },{*q*n } and {*c*n } are real positive sequence and *q*n 0

for infinitely many values of n .

(H2)

f : R � R is continues and

*xf *( *x*) > 0 for all *x * 0 .

(H3) there exists a real valued function g such that

f (un ) f (vn ) = g (un , vn )[(un + cnun cr ) (vn + cn vn cr )], for all un 0, vn 0, c 0, n > cr > 0 and

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 2

ISSN 2229-5518

g (un , vn )

L > 0 E R.

(H4) : R � R is continues for all x

0, ( *x*n ) > 0 .

(H5)

(*n *+ 1) *p*n .

n= M

(H6)

q 2

n .

n= 0 an

(H7) (*n *+ 1)*q*n = .

n= 0

(H8) n

= .

n= 0 nqn

By a solution of equation (1.1) –(1.4), we mean a real sequence {*x*n } satisfying (1.1)-(1.4) for

*n *= 0,1, 2,... .A solution {*x*n }

is said to be oscillatory if it is neither eventually positive nor eventually

negative. Otherwise, it is called non-oscillatory. The forward difference operator is defined by

xn = xn+1 xn

In recent years, much research is going in the study of oscillatory behavior of solutions of third order difference equations. For more details on oscillatory behavior of difference equations, one may refer [1-22].

In this section, we present some sufficient condition for the oscillation of all the solutions of

(1.1)-(1.4). We begin with the following lemma.

Let

P(n, s, x) be defined on

N x N x R+ , N = {0,1, 2, ...}, R+ = [0, ) such that for fixed n and

s , the function P(n, s, x) is non-decreasing in x .Let {rn } be a given sequence and the sequences

{xn } and {zn } be defined on N satisfying , for all n E N ,

n 1

xn rn + P(n, s, xs ),

s =0

(2.1)

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 3

ISSN 2229-5518

And

zn = rn +

n 1

s=0

P(n, s, zs ),

(2.2)

respectively. Then

zn s; xn

for all n E N .

This proof can be found in [18].

In addition to (H1), (H2) and (H3).assume that (H5), (H6), (H7) and (H8) hold and let

zn = xn + cn xn cr . Then, every solution of (1.1) is oscillatory.

Suppose the contrary. Then we may assume that {*x*n } be a non oscillatory solution of (1.1),

qn

such that xn > 0(orxn

0) for all *n M*

1, *M *> 0 is an integer and let *b*n = .

an

Equation (1.1) implies

bn+1

zn+1

bn zn

+ pn

xn + qn f ( xn +1 ) = 0

(2.3)

Multiplying (2.3) by

n + 1

f ( xn+1 )

and summing from M to (n

1) , we obtain

n 1 s + 1

bs+1

zs +1

n 1 s + 1

bs zs

n 1 s + 1

+ ps

xs +

n 1

(*s *+ 1)*q*s = 0.

(2.4)

s = M

f ( xs+1 )

s = M

f ( xs +1 )

s= M f ( xs+1 )

s = M

But

n 1 s +1

(*n *+1)*b x*

(*M *+1)*b x*

n 1 b z

n 1 (*s *+1)*b g *(*x x*

) *z z*

bs +1 z

s +1

= n +1 n +1

M +1 M +1

s +2 s +2 +

s +2

s +2,

s +1

s +1

s +2

s =M

f (xs +1 )

f (xn +1 )

f (xMs +1M)

= sf M( xs +2 ) =

f ( xs +1 ) f ( xs +2 )

(2.5)

Also,

n 1 s + 1

(*n *+ 1)*b x*

(*M *+ 1)*b x*

n 1 b z

n 1 (*s *+ 1)*b g *( *x x*

)( *z *)2

bs zs

= n n

M M

s +1 s +1 +

s +1

s +2,

s +1

s +1

(2.6)

s =M f ( xs +1 )

f ( xn +1 )

f ( xM +1 )

s M=

f ( xs +2 ) s M=

f ( xs +1 ) f ( xs +2 )

Substituting (2.5) and (2.6) in (2.4), we have

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 4

ISSN 2229-5518

(*n *+ 1)*b z*

(*n *+1)*b z*

n 1 (*s *+1)*b g *( *x*

, *x *) *z z*

(*s *+1)*b g *( *x*

, *x *)( *z *)2

n +1 n +1

\ f ( xn +1 )

n n +

f ( xn+1 ) )

s = M \

s + 2 s + 2 s +1 s +1 s + 2

f ( xs +1 ) f ( xs + 2 )

s +1 s + 2 s +1 s +1

f ( xs +1 ) f ( xs + 2 ) )

n 1 b z b z

n 1 s +1 n 1

(*M *+1)*b z*

(*M *+ 1)*b z*

s+2 s +2

s +1 s +1 +

p x +

(*s *+1)*q *=

M +1 M +1

M M

(2.7)

s =M \

f ( xs +2 )

f (xs+2 ) )

s =M f ( xs +1 )

s s s

s=M \

f ( xM +1 )

f ( xM +1 ) )

Using Schwarz’s inequality, we have

1

2 2

bs + 2 zs + 2 s;

n 1

(*b*s+ 2 )

2 *n *1

zs + 2

(2.8)

s = M \

f ( xs+ 2 ) )

\ s= M

) \ s = M \ f ( xs + 2 ) ) )

1

2 2

bs +1 zs +1 s;

n 1

(*b*s +1 )

2 *n *1

zs +1

(2.9)

s = M \

f ( xs+ 2 ) )

\ s = M

) \ s = M \ f ( xs+ 2 ) ) )

1

2 2

n 1 (*s *+ 1)*b g *( *x*

, *x *) *z z *n 1

2 *n *1

(*s *+1) *g *( *x*

, *x *) *z z*

s + 2 s + 2 s +1 s +1 s + 2 s;

(*b *)2

s + 2 s +1 s +1 s + 2

(2.10)

s = M \

f ( x

s +1

) *f *( *x*

s + 2 )

) \ s = M

s + 2

) \ s = M \

f ( xs +1

) *f *( *x*

s + 2 ) ) )

1

2 2

n 1 (*s *+ 1)*b g *( *x*

, *x *)( *z *)2 n 1

2 *n *1

(*s *+ 1) *g *( *x*

, *x *)( *z *)4

s +1 s + 2 s +1 s +1 s;

(*b *)2

s + 2 s +1 s +1

(2.11)

s = M \

f ( xs +1

) *f *( *x*s + 2 )

) \ s = M

s +1

) \ s = M \

f ( x

s +1

) *f *( *x*

s + 2 ) ) )

And

1

2 2

(s + 1) ps xs s;

n 1

(*s *+ 1)( *p *)2

2 *n *1

(*s *+ 1)

xs

(2.12)

s = M \

f ( xs +1 )

) \ s =M

) \ s =M

\ f ( xs +1 ) ) )

In view of (2.8), (2.9), (2.10),(2.11) and (2.12), the summation in (2.7) is bounded , we have

(n + 1)bn +1 zn +1

(n +1)bn zn

1

n 1 2

(*b*s + 2 )

1

n 1 2 2

s + 2 +

1

n 1 2

(*b*s +1 )

1

n 1 2 2

s +1

\ f ( xn +1 )

f ( xn+1 )

) \ s = M

) \ s = M \ f ( xs + 2 ) ) )

\ s = M

) \ s = M \ f ( xs + 2 ) ) )

1

n 1 2

n 1 (*s *+ 1) *g *( *x*

1

2 2

, *x *) *z z *n 1

1

2 *n *1

(*s *+ 1) *g *( *x*

1

2 2

, *x *)( *z *)

+ (*b*s +2 )

\ s =M )

\ s= M \

s +2 s +1 s+1 s + 2

f ( xs +1 ) f ( xs + 2 ) ) )

(*b*s +1 )

\ s= M )

\ s =M \

s +2 s +1 s +1

f ( xs +1 ) f ( xs +2 ) ) )

1

2 2

n 1

+ (*s *+ 1)( *p *)2

2 *n *1

(*s *+ 1)

xs

s; (M + 1)bM +1 zM +1

(M + 1)bM zM

n 1

(*s *+ 1)*q*

(2.13)

s s

\ s = M

) \ s = M

\ f ( xs +1 ) ) ) \

f ( xM +1 )

fs (MxM +1 ) ) =

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 5

ISSN 2229-5518

(*n *+ 1) (*b z *)

In view of (H5), (H6) and (H7), we get from (2.13) that

n n � as n � .

f ( xn+1 )

Hence there exists M1

M such that (bn

*z*n ) 0 for *n M *, which implies (*b*n

zn )

k , k > 0

Summing the last inequality from m to (n

1) , we obtain

n 1

(*b*s

n 1

zs ) ( k )

s =m s =m

That is bn zn

k (n m) + bm zm

Therefore bn

zn � as n � . Hence there exists M 2

M1 such that zn

0 for *n M *2

(2.14)

Rewriting (2.7), we have

(*n *+ 1)*b z*

n 1 (*s *+1)*b g *(*x*

, *x *) *z z*

(*n *+1)*b z*

(*M *+1)*b z*

(*M *+ 1)*b z*

n +1 n +1 +

s + 2 s +2 s +1 s +1 s +2 = n n + M +1 M +1

M M

f (xn +1 )

s =M 2

f ( xs +1 ) f (xs +2 )

f ( xn +1 )

f (xM +1 )

f ( xM +1 )

n 1 n 1

(*s *+ 1)*b g *( *x*

, *x *)( *z *)2

M 2 1 (*s *+ 1)*b g *( *x*

, *x *) *z z*

M 2 1

s + 1

(*s *+ 1)*q *+

s +1 s + 2 s +1 s +1

s + 2 s + 2 s +1 s +1 s +2

p x

s s s

s = M s= M 2

f ( xs +1 ) f ( xs + 2 )

s= M

f ( xs+1 ) f ( xs +2 )

s= M f ( xs+1 )

M 2 1 (*s *+ 1)*b g *( *x*

, *x *)( *z *)2

M 2 1

b z b z

n 1 b z b z

n 1 s + 1

+ s +1 s +2 s +1 s +1 +

s +2 s +2

s +1 s +1 +

s +2 s +2

* s *+1 *s *+1

s s

s =M

f ( xs +1 ) f ( xs +2 )

s =M \

f ( xs +2 )

f ( xs +2 ) )

s =M 2 \

f ( xs +2 )

f ( xs +2 ) )

s =M 2 f ( xs +1 )

(2.15)

From (H1), (H7), (2.14) and (2.15), there exists an integer M 3

M 2 , such that

(*n *+ 1)*b z*

n 1 (*s *+ 1)*b g *( *x*

, *x *) *z z*

n+1 n +1 +

f (xn +1 )

s = M 2

s + 2 s + 2 s +1 s +1 s + 2 s;

f ( xs +1 ) f ( xs +2 )

l, l M 3

where l is a positive integer.

(*n *+ 1)*b z*

n 1 (*s *+1)*b g *(*x*

, *x *) *z z*

(2.16)

n +1

n +1

s + 2

s +2

s +1

s +1

s +2 l

f ( xn +1 )

s =M 2

f ( xs +1 ) f (xs +2 )

Let

un+1 =

(*n *+ 1)

zn +1`, (2.16) becomes

u b n 1 (s + 1)b g ( x

, *x *) *z z*

n +1 n +1

f (xn +1 )

l +

s =M 3

s +2 s + 2 s +1 s +1 s + 2 ; n M

f (xs +1 ) f ( xs +2 )

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 6

ISSN 2229-5518

f (x )

n 1 b f (x

)*g *(*x*

, *x *)(

z )u

(i.e) n +1

n +1

s +2 n +1 s + 2 s +1 s + 2 s +1

(2.17)

bn +1

s =M 3

bn +1 f (xs +1 ) f ( xs +2 )

f (x )

n 1 b f (x

) *g *( *x*

, *x *)(

z )v

Also, Let

vn +1

= l n +1 +

bn +1

s =M 3

s +2 n +1 s +2 s +1 s +2 s +1

bn +1 f (xs +1 ) f ( xs +2 )

(2.18)

Using lemma 1, we have, from (2.17) and (2.18)

un +1

vn+1

(2.19)

f ( x )

n 1 b g (x

, *x *)(

z )v

(2.18) implies

vn +1

= n +1 l +

bn +1 \

s = M 3

s +2 s +2 s +1 s +2 s +1

f ( xs +1 ) f ( xs + 2 ) )

This implies that

lf ( x

v 3

)

; *n M*

(2.20)

n+1 3

n +1

From (2.19) and (2.20), we have

(*n *+ 1)

zn+1

lf ( x )

3

bn +1

lf ( xM )

zn+1 s; + (2.21)

n b

( 1)

n+1

n 1 n 1 1

Summing (2.21) from M 3 to (n

1) , we have

zn +1 s;

lf ( x

3

)

(*n *+ 1)*b*

s = M 3

s = M 3

n+1

n 1 1

That is

zn 1

zM 1 s;

lf ( xM

)

( 1)

+ 3 + 3

s = M 3

n + b 1

n +

n 1 1

zn+1 s; zM +1

lf ( xM ) (n + 1)b

s= M 3

n+1

zn = ( xn + cn xn cr ) s; 0 For sufficiently large n ,

Which is a contradiction to the fact that

xn is eventually positive. The proof is similar for the case when

xn is eventually negative. Hence the theorem is completely proved.

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 7

ISSN 2229-5518

Examples

Example 1

Consider the difference equation

2 n x

+ nx

+ 9n

+ 18n + 5

x + xn +1 = 0

(E1)

\ n + 1

n n 3

2*n*2 (*n *+1)(*n *+ 2)

n n(n +1)

All the conditions of Theorem 1 are satisfied. Hence every solution of equation (E1) is oscillatory.

Consider the difference equation

2 n + 1

x + nx

+ 1 n x +

( *x*n+1 ) = 0

(E2)

\ n + 2

*n n *5 n3

n + 1 n

(*n *+ 1)(*n *+ 2)

All the conditions of Theorem 1 are satisfied. Hence every solution of equation (E2) is oscillatory

In addition to (H1), (H2) ,(H3)and (H4).assume that (H5), (H6), (H7) and (H8) hold and let

zn = xn + cn xn cr . Then, every solution of (1.2) is oscillatory.

In addition to (H1), (H2) and (H3).assume that (H6), (H7) and (H8) hold and let

Then, every solution of (1.3) is oscillatory.

zn = xn + cn xn cr .

In addition to (H1), (H2), (H3) and (H4).assume that (H6), (H7) and (H8) hold and let

zn = xn + cn xn cr . Then, every solution of (1.4) is oscillatory.

Proofs of Theorem 2, Theorem 3 and Theorem 4 are similar to the proof of Theorem 1 and hence the details are omitted.

Reference

[1] R.P.Agarwal: *Difference equation and inequalities- theory, methods *and Applications- 2nd edition

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

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 8

ISSN 2229-5518

[3] R.P.Agarwal,Mustafa F.Aktas and A.Tiryaki: *On oscillation criteria for third order nonlinear delay differential equations*-Archivum Mathematicum(BANO)- Tomus 45 (2009),1-18.

[4] W.T.Li, R.P.Agarwal : *Interval oscilation critical for second order non linear differential equations with damping*-Comp.Math.Appl.40: 217-230(2000).

[5] John R.Greaf and E.Thandapani : *Oscillatory and asymptotic behavior of solutions of third order delay difference equations*-Funkcialaj Ekvacioj, 42(1999),355-369.

[6] Said. R.Grace, Ravi P.Agarwal and John R. Greaf : *Oscillation criteria for certain third order nonlinear difference equations *- Appl.Anal. Discrete. Math,3(2009),27-28

[7] Sh.Salem, K.R.Raslam : *Oscillation of some second order damped difference equations*- IJNS. vol.5(2008),No:3 , pp : 246-254

[8] B.Selvaraj and I.Mohammed ali jaffer : Oscillation Behavior of Certain Third order Linear

Difference Equations-Far East Journal of Mathematical Sciences,Volume 40, Number 2,

2010,pp 169-178.

[9] B.Selvaraj and I.Mohammed ali jaffer :*Oscillatory Properties of Fourth Order Neutral Delay Difference Equations*-Journal of Computer and Mathematical Sciences-An Iternational Research Journal,Vol. 1(3), 364-373(2010).

[10] B.Selvaraj and I.Mohammed ali jaffer *:Oscillation Behavior of Certain Third order Non- linear Difference Equations*-International Journal of Nonlinear Science(Accepted on September 6, 2010).

[11] B.Selvaraj and I.Mohammed ali jaffer : *Oscillation Theorems of Solutions For Certain Third Order Functional Difference Equations With Delay*-Bulletin of Pure and Applied Sciences(Accepted on October 20,2010)

[12] B.Selvaraj and I.Mohammed ali jaffer: *On The Oscillation of the Solution to Third Order*

Difference Equations(Journal of Computer and Mathematical Sciences-An International

Research Journal- Accepted).

IJSER © 2011 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 9

ISSN 2229-5518

[13] B.Selvaraj and J.Daphy Louis Lovenia : *Oscillation behavoir of fourth order neutral difference equations with variable coefficients*- Far East Journal of Mathemati cal Sciences,Vol 35,Issue 2,2009,pp 225-231.

[14] E.Thandapani and B.Selvaraj: E*xistence and Asymptotic Behavior of Non oscillatory Solutions of Certain Non-linear Difference equation -Far East Journal of Mathematical Sciences *14(1)(2004), pp: 9-25.

[15] E.Thandapani and B.Selvaraj*: Oscillatory and Non-oscillatory Behavior of Fourth order*

Quasi-linear Difference equation -Far East Journal of Mathematical Sciences

17(3)(2004)287-307.

[16] E.Thandapani and B.Selvaraj: *Oscillation of Fourth order Quasi-linear Difference equation-Fasci culi Mathematici *Nr, 37(2007),109-119.

[17] E.Thandapani and B.S.Lalli : *Oscillations criteria for a second order damped difference equations*- Appl.math.Lett.vol.8;No:1 ; PP 1-6, 1995

[18] E.Thandapani, I.Gyori and B.S.Lalli : *An application of discrete inequality to second order non-linear oscillation.*- J.Math.Anal.Appl.186(1994),200-208

[19] E.Thandapani and S.Pandian : *On the oscillatory behavior of solutions of second order non- linear difference equations*- ZZA 13, 347-358(1994)

[20] E.Thandapani and S.Pandian : *Oscillation theorem for non- linear second order difference equations with a non linear damping term*- Tamkang J.Math.26(1995),pp 49-58

[21] E.Thandapani : *Asymptopic and oscillatory behavior of solutions of non linear second order difference equations*- Indian J.Pure and Appl.Math, 24(6), 365-372(1993)

[22] E.Thandepani ,K.Ravi : *Oscillation of second order half linear difference equations*-

Appl.Math.Lett.13: 43-49(2000)

IJSER © 2011 http://www.ijser.org