Vol 5, No 3 (2014)

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 5(3), March – 2014 71

ON THE OSCILLATORY BEHAVIOR

FOR A CERTAIN CLASS OF SECOND ORDER DELAY DIFFERENCE EQUATIONS

P. Mohankumar

1

_{and A. Ramesh*}

2

1

_{Professor of Mathematics, Aaarupadaiveedu Institute of Techonology,}

Vinayaka Missions University, Kanchipuram, Tamilnadu, India.

2

_{Senior Lecturer in Mathematics, District Institute of Education and Training,}

Uthamacholapuram, Salem-636 010, Tamilnadu India.

(Received on: 14-03-14; Revised & Accepted on: 25-03-14)

ABSTRACT

I

n this paper, we study the oscillatory behavior for a certain class of second order delay difference equation of the form

( )

1

0

n n n

n

u

q u

a

σ





∆

_

∆

_

+

=





(1.1)

Where

{ } { }

a

_n

,

q

_n are real sequence and

{ }

a

_n

>

0.

Examples are inserted to illustrate the results.

Keywords: Oscillatory, Second order, Double Sequence, Delay Difference equations.

AMS Classification: 39A21.

INTRODUCTION

We consider the second order delay difference equation of the form

( )

1

0

n n n

n

u

q u

a

σ





∆

_

∆

_

+

=





(1.1)

Where

∆

is the forward difference operator is defined by

∆ =

u

_n

u

_n+1

−

u

_n and

{ } { }

a

_n

,

q

_n are real sequence. With respected to the difference equations (1.1) throughout we shall assume that the following conditions holds.

(C1):

{ } { }

a

n

,

q

n are real sequence and

{ }

a

n

>

0

(C2):

σ

( )

n

>

0

is an integer such that

lim ( )

n→∞

σ

n

= ∞

(C3):

0

1

n

n s

s n

R

a

−

=

=

∑

→ ∞

as

n

→ ∞

By a solution of equation (1.1) we mean a real sequence

{ }

u

_n satisfying (1.1) for

n

≥

n

₀. A solution

{ }

u

_n is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called non-oscillatory. For more details on oscillatory behavior difference equation we refer [1-23].

Corresponding author: A. Ramesh*

2

2

_{Senior Lecturer in Mathematics, District Institute of Education and Training,}

IJMA- 5(3), March-2014.

MAIN RESULT

In this section, we present some sufficient conditions for the oscillation of all the solutions of equations (1.1)

Theorem: 1 Assume the (C3) hold

∆

σ

( )

n

≥

0

(

)

0

2 1

(s) ( )

1 ( )

lim sup

4

n

n s

n

s n s n

R

R

p

a R

σ σ

σ −

→∞ ₌

+



_∆





₊



_{= ∞}









∑

(1.2)

Then every solution of equations (1.1) is oscillatory.

Proof: Let

{ }

u

_n be non-oscillatory solution of equation (1.1) without loss of generality, we suppose that

( )

0,

0

n n

u

>

u

_σ

>

and for

n

≥

n

₁ from the equation

1

_n

0

n

u

a





∆

_

∆

_

<





for

n

≥

n

1. Since

1

n n

u

a





∆

_

∆

_





is

non-increasing there exist a non-negative constant

k

and

n

₂

≥

n

₁

1

_n

n

u

k

a

∆ ≤ −

for

n

≥

n k

2

,

>

0

n n

u

ka

∆ ≤ −

,

n

≥

n k

₂

,

>

0

Summing the inequality for

n to n

₂

−

1

2 2

1

n

n n s

s n

u

u

k

a

−

=

≤

−

∑

Letting

n

→ ∞

we have

u

_n

→ ∞

, which is contradiction to the fact that

u

_n is positive.

Then

1

_n

0

1

_n

0

n n

u

and

u

a

a





∆

_

∆

_

>

∆ >





Define

( )

( )

n n

n

n n

R

u

a u

σ

σ

ω

=

∆

(A)

( ) 1 ( )

( ) 1 ( )

1

n n n

n n

n n n n

R

_u

R

u

u

a

a

u

σ σ

σ σ

ω

+

+











 ∆

∆

=

∆



∆



+



∆



_



_



















( ) 1 ( ) ( ) ( ) ( )

( ) 1 ( ) ( 1)

1

n n n n n n

n n

n n n n n

R

u

R

u

u

R

u

u

a

a

u

u

σ σ σ σ σ

σ σ σ

ω

+

+ +



∆

− ∆





 ∆

∆

=

∆



∆



+

















( ) ₁ ( ) ₁ ( ) ( )

( ) 1 ( 1) 1 ( ) ( 1)

1

n _n n _n n n

n n

n n n n n n n

R

u

R

u

u

R

u

u

a

a

u

a

u

u

σ σ σ σ

σ σ σ σ

ω

+ +

+ + + +

∆

∆



 ∆

∆

∆

=

∆

_

∆

_

+

−





( ) ( ) 1 ( )

( ) 1

( 1) 1 ( ) ( 1)

n n n n

n n n n

n n n n

R

R

u

u

R

q

R

a u

u

σ σ σ

σ

σ σ σ

ω

ω

+

+

+ + +

∆

∆

∆

∆

= −

+

−

In the view of (C2) and (1.1)

(

)

2

( ) ( ) 1

( ) 1 2

( 1) 1

(

( 1)

)

n n n

n n n n

n n n

R

R

u

R

q

R

a

u

σ σ

σ

σ σ

ω

ω

+

+

+ + +

∆

∆

∆

= −

+

−

That is

(

)

2

( )n n 1 ( )n n 1

R

a

R

R

_σ

q

σ σ

ω

ω

ω

+ +

+

∆

© 2014, IJMA. All Rights Reserved 73

2 2

1 ( )

( ) ( )

( ) 1

1 ( ) ( 1) 1 ( )

(

)

4

2

n n

n n

n n n n

n n n n n

a

R

R

R

R

q

a

R

R

a

R

σ

σ σ

σ

σ σ σ

ω

+

ω

+ + + +





∆

∆





∆

= −

+

−

−









This implies that

2 ( ) ( )

1 ( )

(

)

4

n

n n n

n n

R

R

q

a

R

σ σ σ

ω

+



∆



∆

< −



_

−



_





(1.3)

Summing the inequality for

n to n

₂

−

1

we have

1 2 2 1 ( ) ( )

1 ( )

(

)

4

n

n

n n n n

s n n n

R

R

q

a

R

σ σ σ

ω

ω

−

= +



∆



≤

−



_

−



_





∑

Letting

n

→ ∞

_{, we have, in view of (1.2) that}

ω

_n

→ ∞

as n

→ ∞

,

which contradicts

ω

_n

>

0

and the proof is complete.

Theorem: 2 Let all the assumption of Theorem 1 holds except the condition (1.2) which changed to

0

2 1

( ) ( )

1 ( )

(

)

1

lim sup

(

)

4

n n r n n r n

s n n n

R

n

s

q R

n

a

R

σ σ σ − →∞ ₌ +



∆



−



_

−



_

= ∞





∑

(1.4)

Then every solutions

{ }

u

_n of equation (1.1) is oscillatory.

Proof: Proceeding as in the proof of Theorem 1, we assume that equations (1.1) non-oscillatory solution, say

( )

0,

0

n n

u

>

u

_σ

>

and for

n

≥

n

₁from the equation (1.3) we have

n

≥

n

₁.

(

)

(

)

1 1 2 1 1 (s) (s) 1 (s)

(

)

4

n n r r s s

s n s s n

R

n

s

R

q

n

s

a R

σ σ σ

ω

− − = + =



∆



−



_

−



_

< −

−

∆





∑

∑

(1.5)

Since

(

)

(

)

₁

(

)

1 1

1 1

1

1

n n

r r r

s s n

s n s n

n

s

ω

r

n

s

ω ω

n n

− − ₋ = =

−

∆

=

−

−

−

∑

∑

(1.6) We get

(

)

1

(

)

1 1

1 1

1 1

1

n r n

r r

s n s

r r

s n s n

n n

r

n

s M

n

s

n

ω

n

n

ω

− − − = =

−





−

≤

_

_

−

−





∑

∑

(1.7) Where 2 (s) (s) 1 (s)

(

)

4

s s s

R

M

R

q

a

R

σ σ σ +

∆

=

−

Letting

n

→ ∞

_, we have

(

)

₁

1

1

1

1

n r

r

s n

r s n

n n

n

s M

n

ω

n

− =

−





−

≤

_

_





∑

(1.8) Then

(

)

₁

1 1

1

lim sup

n r s n r n s n

n

s M

n

ω

−

→∞

∑

₌

−

→

Which contradicts the condition (1.4)

IJMA- 5(3), March-2014.

Next, we present some new oscillation results for equation (1.1) we introduce a double sequence

(

)

{

H m n

,

/

m

≥ ≥

n

0

}

Such that

2

( )

( , )

0

( )

( , )

0 for m

n

o and

( )

( , )

h(m, n)

( , );

i

H m n

for m

o

ii

H m n

ii

L H m n

H m n

for m

n

o

=

≥

>

> ≥

−

=

≥ ≥

Theorem: 3 Assume that (C1)-(C3) holds and let

{ }

u

_n be a positive sequence and assume that there exist a double

sequence

{

H m n

(

,

)

/

m

≥ ≥

n

0

}

such that

0

2 1

( ) ( )

1 1 ( )

(

)

1

lim sup

( , )

( , n )

4

n

n

n n

n

s n n n

R

H n s

q R

H n

a

R

σ σ

σ −

→∞ ₌

+



∆



−

= ∞













∑

(1.9)

( )

( 1)

( , )

( , )

( , )

n

n

R

h n s

where M n s

R

H n s

σ

σ +

∆

=

−

(1.10)

Then every solutions

{ }

u

_n of equation (1.1) is oscillatory

Proof: Let

{ }

u

_n be non-oscillatory solution of equation (1.1). Let us first assume the

{ }

u

_n is eventually positive and

that

u

_n

>

0,

u

_{σ( )n}

>

0

and for

n

≥

n

₁

In the view of (A) and (B)

Let us denote ( ) 1 ( )

2

( 1) ( 1)

n s s

s s

n n

R

a R

and

B

R

R

σ σ

σ σ

γ

+

+ +

∆

=

=

(

)

2

( ) 1 ( ) 1

( ) 1 2

( 1)

(R

( 1)

)

n n n n

n n n n

n n

R

a

R

R

q

R

σ σ

σ

σ σ

ω

ω

ω

+ +

+

+ +

∆

∆

= −

+

−

(

)

2

( ) 1 ( ) 1

( ) 1 2

( 1)

(R

( 1)

)

n n n n

n n n n

n n

R

a

R

R

q

R

σ σ

σ

σ σ

ω

ω

ω

+ +

+

+ +

∆

= −∆

+

−

( )

1 1

1 1

2

(s) 1

( , )

(n, s)

n n

s s s s s s

s n s n

H n s R

_σ

q

H

ω γ ω

B

ω

− −

+

= =





≤

_

−∆ +

−

_

∑

∑

( )

{

}

1

1 1

1 1

2

(s) 2 1 1

( , )

[

( , s)

]

(n ,s)

( , )

n n

n

s s s n s s s s s

s n s n

H n s R

_σ

q

H n

ω

L H

ω

H n s

γ ω

B

ω

− −

= + +

= =





≤

−

+

_

−

_

∑

∑

(

)

(

)

1 1

1

2

1 1 1

( , n )

(n ,s)

( , )

(

( , )

( , )

n

n s s s s

s n

H n

ω

H

h n s

H n s

γ ω

H n s B

ω

−

+ +

=





=

−

∑

_

−

+

_

1

1 1

2

2

1 1

1 1

1

( , )

( , )

( , )

( , n )

(n, s)

2

4

n n

n s s

s n s s n s

M n s

M

n s H n s

H n

H

B

B

B

ω

−

ω

₊ −

= =





=

−



+



+









∑

∑

It follows that

1

2 1

( ) ( )

(

)

1

lim sup

( , )

( , n )

4

n

n

n n n

n

R

H n s

q R

H n

a

R

σ σ

σ

ω

−

→∞ ₌

+



∆



−

≤













© 2014, IJMA. All Rights Reserved 75 Which clearly contradicts (1.9). This contradiction completes our proof

Remarks: By choosing various specific double sequences

{

H m n

( , )

}

we can derive several oscillation criteria for (1.1)

Let us consider the double sequence

{

H m n

( , )

}

defined by

( , )

(

) ,

,

H m n

=

m n

−

µ

m

≥ ≥

n

o

Where

µ

≥

1

is a constant.

Then

H m n

( , )

=

0

for m

≥

0,

H m n

( , )

>

0 fo r

m

> ≥

n

0

and L H m n

₂

( , )

≤

0 o r

m

> ≥

n

0

Hence, we have the following corollary

Corollary: 1 If

0

2 1

( ) ( )

1 ( )

(

)

1

lim sup

(

)

1,

4

n

n

n n

n

s n n n

R

m n

q R

for some

m

a

R

σ µ

σ µ

σ

µ

−

→∞ ₌

+



∆



−



_

−



_

= ∞

≥





∑

Then every solutions

{ }

u

n of equation (1.1) is oscillatory

Example: 1 Consider the delay difference equations

1

( )

1

1

0

( 1)

1

(

1)

0

2

_σ

1,

( 1)

.

λ

µ

−





∆

_

∆

_

+

=

−

−





≥

=

=

n n

n

n

u

u

E

n

n

where

R

the equation E

is Oscillatory

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