Oscillatory Behavior of Second Order Neutral Delay Difference Equations

Oscillatory Behavior of Second Order Neutral Delay Difference Equations

B. SELVARAJ ¹ and G. GOMATHI JAWAHAR ²

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

Coimbatore-641105, India.

2 Assistant Professor, Department of Mathematics, Karunya University, Coimbatore-641114, India

(Received on : March 23, 2012) ABSTRACT

In this paper, we obtain some sufficient conditions for the oscillation of second order neutral delay difference equation of the form

) ( ,

0 ) ( )

( ₀

2 y _n + p _n y _{n k} + q _n f y _{n l} = n ∈ N n

∆ _{− − ,}

where k,l >0.

Keywords: Neutral Delay Difference Equation, Oscillatory solution.

1.1 INTRODUCTION

Difference equations with discrete and continuous arguments are playing a fundamental role in nonlinear phenomena and in process occurring in various drastically different systems. In the past few decades the study of difference equations has already drawn a great deal of attention, not only among mathematicians themselves, but from various other disciplines as well.

Many statements concerning the theory of linear differential equations are also valid for the corresponding difference equations. To a certain extent, the growing interest in

difference equations may be also attributed to their simplicity.

With the use of a computer one can easily discover that difference equations posses fascinating properties with a great deal of structure and regularity. Ofcourse all computer observations and predictions must also be proven analytically. Difference equations are often rearranged as a recursive formula so that a system output can be computed from the input signal and past outputs.

In this paper, we obtain some

sufficient condition for the oscillation of

second order neutral delay difference equation of the form

) ( , 0

) ( )

(

0 2

n N n

y f q y

p

y _{n n n k n n l}

∈

=

+ +

∆ _{− −}

(1.1.1) Where k,l >0.

Here we assume the following conditions.

H ₁ : { p _n } is an increasing sequence.

H 2 : { q _n } is an positive sequence.

H 3 : f is a continuous function such that f ( u ) ≥ u and f ( uv ) ≥ f ( u ). f ( v ) 1.2 EXISTENCE OF OSCILLATORY SOLUTIONS

In this section, we study the structure of the oscillatory solutions of the equation 1.1.1.

Theorem 1. 2.1

Suppose Q _n = q _n f ( 1 − p _{n − l} ) and

∞

→ m lim ,

sup

0

∞

∑ =

= m

m n

Q n

Then every solution of the equation (1.1.1) oscillates.

Proof

Suppose { y _n } is a nonoscillatory solution, without loss of generality we may assume that { y _n } is eventually positive.

Let z _n = y _n + p _n y _{n − k}

From the equation (1.1.1),

. 0 )

2 = − ( <

∆ z _n q _n f y _{n −l} Hence ∆ z _n is a decreasing function.

Also, ∆ z n = z n +1 − z n .

Hence

0 )

1 (

1

1 + − + >

=

∆ z _n y _{n +} p _{n +} y _{n − k +} y _n p _n y _{n − k}

Since { z _n } is eventually positive, ∆ z n > 0 Also, y n = z n − p n y n ₋ k

) ( _{n k n k n k}

n n

n z p z p y

y = − ₋ − _{− −}

k n n n

n z p z

y > − _{− .}

Hence y n > ( 1 − p n ) z n .

Hence for some l > 0 , y n ₋ l > ( 1 − p n ₋ l ) z n ₋ l

Using H 3, f ( y _{n − l} ) ≥ f ( 1 − p _{n − l} ) f ( z _{n − l} ) . From the equation(1.1.1),

)

2 (

l n n

n q f y

z = − ₋

∆ ≤ − q _n f ( 1 − p _{n − l} ) f ( z _{n − l} ) Hence ∆ ² z _n ≤ − Q _n f ( z _{n − l} ) _.

Using H 3, ∆ ² z n ≤ − Q n z n ₋ l ^.

Therefore ∆ ² z n + Q n z n _−l ≤ 0 . (1.2.1) Define

l n

n n

n z

w z

−

= ρ ∆

where ρ _n = max( l k , ) _.

Then w n > 0 Equation (1.2.1) becomes, ∆ ⁽

n l n n z w

ρ ^{− ) +} .

≤ 0

−l n n z

Q

Hence

n n

n l n n l n n

n w z w z

ρ ρ

ρ ρ

1

) (

+

−

− − ∆

∆ ≤ − Q n z n ₋ l

1

) (

+

∆ − n

l n n z w

ρ _{n n}

n l n n z w

ρ ρ

ρ

+ 1

− ∆

− ≤ − Q n z n ₋ l ^.

1 1

+

−

−

+ ∆ + ∆

n

n l n l n

n z z w

w

ρ ⁿ _n ^{n l} _n ⁿ z w

ρ

ρ ρ

+ 1

− ∆

− ≤

l n n z Q ₋

− .

w n

∆ ≤

n n

n

w n

ρ ρ

ρ

+ 1

∆

Q n

−

l n n

l n n

z z w

− +

− + ∆

−

1 1

ρ (1.2.2) Taking summation from m ₀ to m , for some m > m ₀ ,

n

n w

w +1 − ≤ ∑

= m

m

n

₀

n n

n n

w n

ρ ρ ρ ρ

1

1 )

(

+ + −

−

− ∑

= m

m n

Q

n 0

∑

= m

m

n

₀

n n l

l n l n n

z z z

w

− +

− +

−

+ −

1 1

1 ( )

ρ

1 m

0

m w

w ₊ − ≤ ∑

= m

m

n

₀

n n

n n

w n

ρ ρ

ρ ρ

1

1 )

(

+ + −

- ∑

= m

m n

₀

( Q _n +

l n n

l n n

z z w

− +

− + ∆

1

1 )

ρ ⁾

+1 − w m ∑

= m

m

n

₀

n n

n n

w n

ρ ρ

ρ ρ

1

1 )

(

+ + −

+ ∑

= m

m

n

₀

n n l

l n n

z z w

− +

− + ∆

1 1

ρ ≤ w m

₀

− ∑

= m

m n

₀

Q n

when m → ∞ , w _m → −∞ .

which is a contradiction to the assumption . Hence every solution of the equation (1.1.1) is an oscillatory solution.

Corollary 1.2.1 If

equation (1.1.1) becomes

) ( ,

0 ₀

2 y _n + q _n y _{n l} = n ∈ N n

∆ _{− .} (1.2.3)

If y _n is an eventually positive solution of the equation (1.2.3), then there exists

)

( n ₀ N n ∈ Such that

Proof

Let y _n be an eventually positive solution of the equation (1.2.3). Hence there exists

) ( ₀

1 N n

n ∈ such that y _n >0 and y _{n − l} >0 for n ≥ n _{1 .} .

From the equation (1.2.3),

) ( ,

0 ₀

2 y _n = − q _n y _{n l} < n ∈ N n

∆ ₋ , (1.2.4)

Hence ∆ y _{n +1} < ∆ y _n . So ∆ y _n is eventually decreasing. Since q _n is a positive function, the decreasing function ∆ y _n is either eventually positive or eventually negative.

Suppose there exists n ₂ ≥ n ₁ such that

<

∆ y n

₂

0.

∑

= m

m

n

₀

, 0 , )

( y _{n − l} = y _{n − l} p _n = f

, 0 ,

0 ∆ ≥

>

≥ _{n − k n}

n y y

y _∆ ² _{y n < 0}

Taking summation from n _{2 to} s the equation (1.2.4) becomes,

∑

∑ = ∆ + ≤ = ^s ∆

n n

n s

n n

n y

y

2 2

1

Hence we have, y _{s + 2} − y _{s + 1} ≤

2

2

1 n

n y

y ₊ −

+ 2

y s ≤

n

2

∆ y ^+y s+1

Hence as n → ∞ , y _n → −∞ .

Which is a contradiction.

Hence for all n∈N(n 0 ).

Corollary 1.2.2 If

then every solution of the equation (1.2.3) oscillates.

Proof

Suppose to the contrary that y n is a nonoscillatory solution of the solution of the equation (1.2.3).

Define x _n = ∆ y _n

Hence y _{n − l} − y _{n − l − 1} = x _{n − l − 1}

Therefore y _{n − l} ≥ x _{n − l − 1} (1.2.5) From the equations (1.2.5) and (1.2.4), we get

1

2 = − − ≤ − − −

∆

=

∆ x _n y _n q _n y _{n l} q _n x _{n l}

Clearly y _n is an eventually positive solution of

.

1 ≤ 0 +

∆ x _n q _n x _{n −l −}

Taking summation from n − l − 1

to s , we have

0≥ {( ) ₁ }

1

1 − −

−

− + − +

∑ n n n l s

l n

n x q x

x

≥ x _{s + 1} − x _{n − l − 1} + 1 1

−

− −

−

∑ n n l s

l n

x q

= x _{s + 1} + ( ₁

1

) 1 _{− −}

−

−

∑ n − n l s

l n

x q

Hence,

∞

→ n

lim

sup ( x _{s + 1} + ( 1

1

) 1 _{− −}

−

−

∑ n − n l s

l n

x

q ) ≤ 0.

Therefore

∞

→

n lim sup n s

l n

∑ q

−

− 1

≤ 0.

This is a contradiction to our assumption.

Hence every solution of the equation (1.2.4) has oscillatory solution.

1.3 EXAMPLES

Example 1.3.1

Consider the second order difference equation

0 , 0

4 ₃

2 + = >

∆ y _n y _{n −} n

,

≥ 0

∆ y n

1 sup

lim

1

>

∑ −

∞ −

→

s l

n n

n q

Here

3 ,

3 =

= q n

l

Also, lim sup

∞

→

n 1

1

∑ >

−

− s

l n

q n

Hence all the conditions of the corollary 1.2.2 are satisfied.

Hence all its solutions are oscillatory. One such solution is (-1) ⁿ .

Example 1.3.2

Consider the second order difference equation

3 , 3 0

) 1 ( 4

3

2 = >

− + +

∆ y ₋ n

n y _n n _n

Here

3 3 ,

) 1 ( , 4

3 >

−

= +

= n

n q n

l _n

Also, lim sup

∞

→

n 1

1

∑ >

−

− s

l n

q n

Hence all the conditions of the corollary 1.2.2 are satisfied.

Hence all its solutions are oscillatory. One such solution is n(-1) ⁿ .

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