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

ISSN 2229-5518

Oscillation Properties of Solutions for Certain Nonlinear

Difference Equations of Third Order

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

Abstract : In this paper some sufficient conditions for the oscillation of all solutions of certain difference equations are obtained. Examples are given to illustrate the results.

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

AMS Subject Classification: 39 A 11.

—————————— • ——————————

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) {an },{ pn },{qn } and {cn } are real positive sequence and qn 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

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g (un , vn )

L > 0 E R.

(H4) : R R is continues for all x

0, ( xn ) > 0 .

(H5)

(n + 1) pn .

n= M

(H6)

q 2

n .

n= 0 an

(H7) (n + 1)qn = .

n= 0


(H8) n

= .

n= 0 nqn

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

n = 0,1, 2,... .A solution {xn }

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].

2 Main Results

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.

Lemma 1

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)

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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].

Theorem 1

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.

Proof:

Suppose the contrary. Then we may assume that {xn } 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 bn = .

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)qs = 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

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(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

(bs+ 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

(bs +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 ( xs + 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

(bs + 2 )

1

n 1 2 2

s + 2 +

1

n 1 2

(bs +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 )

+ (bs +2 )

\ s =M )

\ s= M \

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

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

(bs +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 ) ) =

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(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

zn ) 0 for n M , which implies (bn

zn )

k , k > 0

Summing the last inequality from m to (n

1) , we obtain

n 1

(bs

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 )

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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.

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Examples

Example 1

Consider the difference equation

2 n x

+ nx

+ 9n

+ 18n + 5

x + xn +1 = 0

(E1)

\ n + 1

n n 3

2n2 (n +1)(n + 2)

n n(n +1)

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

Example 2

Consider the difference equation

2 n + 1

x + nx

+ 1 n x +

( xn+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

Theorem 2

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.

Theorem 3

In addition to (H1), (H2) and (H3).assume that (H6), (H7) and (H8) hold and let
Then, every solution of (1.3) is oscillatory.

Theorem 4

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

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[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.

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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.

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