Vol 9, No 2 (2018)

(1)

Available online throug

ISSN 2229 – 5046

OSCILLATION OF SECOND ORDER

NONLINEAR NEUTRAL ADVANCED FUNCTIONAL DIFFERENCE EQUATIONS

YADAIAH ARUPULA

a,

**_{*, V. DHARMAIAH}**

b

_{AND MAHESH CHALPURI}

c

a,b,c

_{Department of Mathematics Osmania University, Hyderabad -07, India.}

(Received On: 28-12-17; Revised & Accepted On: 07-02-18)

ABSTRACT

I

n this paper we establish some sufficient conditions for the oscillations of all solutions of the second order nonlinear neutral advanced functional difference Equation

∆

[

r

(

n

)

∆

(

x

(

n

)

+

p

(

n

)

x

(

n

+

τ

))

]

+

q

(

n

)

f

(

n

+

σ

))

=

0 :

n

≥

n

0

,

_(*)

where

( )

0

1

n n

r n

∞

=

≤ ∞

∑

and

0 ≤

p n

( )

≤

p

₀

< ∞

,

Τ

is an integer, and

σ

is a positive integer. The result proved here improve some known results in the literature.

Keywords: Neutral functional difference equation, oscillation and nonoscillation solutions.

1. INTRODUCTION

In this paper we consider the following second order nonlinear neutral advanced functional difference equations of the form:

( ) ( )

(

( ) (

)

[

r

n

∆

x

n

+

p

n

x

n

+

] ( ) (

+

q

n

f

(

x

n

+

)

=

0 :

n

≥

n

0

,

∆

τ

σ

(1)

where

∆

is the forward difference operator defined by

∆

x

( ) (

n

=

x

n

+

1 ) ( )

−

x

n

,

n

is a positive integer,

n

₀ is a

fixed positive integer,

x

is a delay function.

The following conditions are assumed to hold through the paper;

(a)

{ }

( )

∞₌ 0

n n

n

r

is a sequence of positive real numbers;

(b)

{ }

( )

∞₌ 0

n n

n

p

is a sequence of nonnegative real numbers with the property that

0 <

p

( )

n

≤

p

0

<

∞

;

(c)

{ }

( )

∞₌ 0

n n

n

q

is a sequence of nonnegative real numbers and

q

( )

n

is not identically zero for large values of

n

;

(d)

( )

0 >

≥

k

u

f

for

u

≠

0 ,

k

is constant;

\and

(e)

τ

is an integer and

σ

is a positive integer.

We shall consider the following two cases, separately

( )

∑

∞

=

∞

=

0

1

n

r

n

(2)

and

( )

1 .

0

∞

<

∑

∞ =n

n

r

n

(3)

Corresponding Author: Yadaiah Arupula

a,

**_*,**

(2)

We note that the second order neutral functional difference equations have applications in problems dealing with vibrating masses attached to elastic ball and in some variational problems. In recent years there has been an increasing interest in obtaining sufficient conditions for the oscillation or nonoscillation of solutions of different classes of difference equations. We refer [1, 2, 6, 7, 13]. Also the oscillatory behaviour of neutral functional difference equations has been the subject of intensive study, see for example, [3-5, 8-12, 14].

In [12], Zhang et al. established that every solution of the equation

( ) ( )

(

)

[

∆

−

]

+

(

,

(

( )

)

=

0 ;

∆

a

n

x

n

px

x

τ

f

n

x

σ

n

≥

n

₀ (4)

is either oscillatory or eventually

where

p

,

q

,

σ

( )

n

are non-negative constants,

f

is a bounded set,

a

( )

n

is a positive constant.

( )

n

≤

p

x

(

n

−

τ

)

x

if

( )

∑ ∑

∞

= =

∞

=

n

s

m

i i

p

s

q

,

0

where

( ) ( )

,

≥

q

n

>

0 u

u

n

f

for

u

≠

0 .

In [3], Budincevic established that every solution of equation

( ) ( )

(

)

[

∆

+

−

0

] ( ) (

+

(

−

0

)

=

0 ∆

a

n

x

n

px

n

q

n

f

x

n

m

(5) is either oscillatory or else

x

( )

n

→

0

as

n

→

∞

.

In [7], Murugesan et al. established that every solution of equation (1) is oscillatory if

τ

>

0 ,

0 ≤

p

( )

n

≤

p

₀

<

∞

and at least one of the first order advanced difference inequalities

where

Q

₁

( )

n

,

Q

₂

( )

n

are non-negative functions

( )

( ) (

)

0

1

1 0

≥

+

−

∆

Q

n

w

n

σ

p

n

w

(6)

( )

( ) (

)

0

1

2 0

≥

+

−

∆

Q

n

w

n

σ

p

n

w

(7)

has no positive solution .

In this paper our aim is to obtain sufficient conditions for oscillation of all solution of equation (1) under the condition

( )

∑

∞

=

∞

=

0

1

n

r

n

or

( )

.

1

0

∑

∞ =

∞

<

n

r

n

Let

n

₀ be a fixed nonnegative integer. By a solution of (1) we mean a nontrival real sequence

{ }

x

( )

n

which is defined for

n

≥

n

₀ and satisfies equation (1) for

n

≥

n

₀. A solution

{ }

x

( )

n

of (1) is said to be oscillatory if for every positive integer integer

N

>

0 ,

there exists an

n

≥

N

such that

x

( ) (

n

x

n

+

1 )

≤

0

otherwise

{ }

x

( )

n

is said to be

nonoscillatory. Equation (1) is said to be oscillatory if all it solutions are oscillatory. In the sequel for the sake of

convenience, where we write a functional inequality without specifying its domain of validity we assume that it hold for all sufficiently large

n

2. MAIN RESULTS

In this section, we establish some new oscillation criteria for oscillatory of equation

( )

1 .

For the sake of convenience, we define the following notations

( )

n

:

=

min

{

q

( ) (

n

,

q

n

+

τ

)

}

,

Q

( )

(

∆

ρ

n

)

₊

=

max

{

0 ,

∆

ρ

( )

n

}

,

(3)

( )

_∑

−

_{( )}

=

1

0

,

1

n

s

r

s

n

R

and

( )

∑

∞

=

n

s

r

s

n

1 .

δ

Theorem 2.1: Assume that (2) holds and

τ

>

0 .

Moreover suppose that there exist a positive real valued sequence

( )

{ }

∞

=n0

n

p

such that

( ) ( ) (

(

) ( ) ( )

_{( )}

(

)

.

4

1 sup

lim

0

2

0

=

∞













+

∆

−

∑

=

+

∞ →

n

n s

n

s

r

p

s

Q

s

k

ρ

Then every solution of equation (1) is oscillatory.

Proof: Assume the contrary. without loss of generality we assume that

{ }

x

( )

n

is an eventually positive solution of (1) .Then there exists integer

n

₁

≥

n

₀ such that

x

( )

n

>

0

for all

n

≥

n

1

.

Define

( )

(

)

(

).

)

(

n

=

x

n

+

p

n

x

n

+

τ

z

₍₉₎

Then

z

( )

n

>

0

for all

n

≥

n

₁_. From

( )

1

, we have

( ) ( )

(

r

n

∆

z

n

)

≤

−

kq

( ) (

n

x

n

+

)

≤

0 ;

n

≥

n

1

∆

σ

(10) Therefore the sequence

{

r

( ) ( )

n

∆

z

n

}

is non-increasing. We claim that

( )

>

0 ∆

z

n

for

n

≥

n

₁

.

_.₍₁₁₎

If not,there exists an integer

n

₂

≥

n

₁ such that

∆

z

( )

n

₂

<

0

. Then from (10), we obtain

( ) ( ) ( ) ( )

n

z

n

r

n

2

z

n

2

,

r

∆

≤

∆

n

≥

n

2

.

Here

( ) ( ) ( ) ( )

_{( )}

1 .

1

2 2 2

2

∑

− =

∆

+

≤

n

s

r

s

n

z

n

r

n

z

n

z

Letting

n

→

∞

_{we get}

z

( )

n

→

∞

.

This contradiction proves that

∆

z

( )

n

>

0

for

n

≥

n

1

.

On the other hand, using

the definition of

z

( )

n

and applying (1) for all sufficiently large values of

n

,

( ) ( )

[

∆

]

+

( ) (

+

)

+

₀

∆

[

(

+

) (

∆

+

)

]

+

₀

(

+

) (

+

)

≤

0 ∆

r

n

z

n

kq

n

x

n

σ

p

r

n

τ

z

n

τ

p

kq

n

τ

x

n

σ

τ

( ) ( )

[

∆

]

+

0

∆

[

(

+

) (

∆

+

)

]

+

( ) (

+

)

≤

0 .

∆

r

n

z

n

p

r

n

τ

z

n

τ

kQ

n

z

n

σ

Since

∆

z

( )

n

>

0

we have

z

(

n

+

σ

) ( )

≥

z

n

and hence

( ) ( )

[

∆

]

+

0

∆

[

(

+

) (

∆

+

)

]

+

( ) ( )

≤

0 .

∆

r

n

z

n

p

r

n

τ

z

n

τ

kQ

n

z

n

₍₁₂₎

Define

( ) ( ) ( ) ( )

_{( )}

,

n

z

n

z

n

r

n

w

=

ρ

∆

n

≥

n

1₍₁₃₎

Clearly

w

( )

n

>

0 .

From (9)we have

(

n

1 ) (

z

n

1 ) ( ) ( )

r

n

z

n

.

r

+

∆

+

≤

∆

From (13), we have

( ) ( ) ( ) ( )

(

_{( )}

)

( ) (

_{( ) (}

) (

₎

)

( ) (

₍

) (

₎

)

( )

n

z

n

z

n

r

n

z

n

z

n

z

n

z

n

r

n

z

n

z

n

r

n

w

ρ

∆

ρ

+

∆

+

∆

+

∆

+

−

∆

=

∆

1

1 ( ) ( ) ( )

(

_{( )}

)

( ) (

₍

_{) ( )}

)

(

₍

)

_{) ( )}

n

w

n

r

n

w

n

z

n

z

n

r

n

ρ

∆

+

−

∆

≤

1

2 2

( ) ( ) ( )

(

_{( )}

)

( ) (

₍

_{) ( )}

)

(

₍

( )

)

_{) (}

1 )

.

1

2 2

+

∆

+

−

∆

≤

+

n

w

n

r

n

w

n

z

n

z

n

r

n

ρ

(4)

Similarly, we define

( ) ( ) (

) (

_{( )}

)

,

n

z

n

z

n

r

n

v

=

ρ

+

τ

∆

+

τ

n

≥

n

₁

.

₍₁₅₎

From (9) we have

r

(

n

+

τ

+

1 ) (

∆

z

n

+

τ

+

1 ) ( ) ( )

≤

r

n

∆

z

n

.

Using this in (15), we have

( ) ( ) (

(

_{( )}

) (

)

( ) (

₍

_{) ( )}

)

₍

( )

_{) (}

1 )

.

1

2 2

+

∆

+

−

+

∆

+

∆

≤

∆

v

n

r

n

v

n

z

n

z

n

r

n

v

ρ

τ

ρ

(16)

It follows from (14) and (16), that

( )

( ) ( ) ( ) ( )

[

_{( )}

]

( ) (

[

_{( )}

) (

)

]

n

z

n

z

n

r

n

p

n

z

n

z

n

r

n

v

p

n

w

+

∆

≤

ρ

∆

+

ρ

∆

+

τ

∆

+

τ

∆

₀ ₀

( ) (

)

(

) ( )

(

( )

)

) (

)

( ) (

(

n

) ( )

r

n

)

n

v

n

p

n

w

n

r

n

w

n

1

2 2 0 2 2

+

−

+

∆

+

−

+

ρ

( )

(

)

(

) (

1 )

.

1

0

+

∆

+

n

v

n

p

ρ

In view of (12) and the above inequality, we obtain

( )

( ) ( )

( ) (

₍

_{) ( )}

)

(

₍

( )

)

_{) (}

1 )

1

2 2

0

₊

+

∆

+

−

≤

∆

+

∆

+

n

w

n

r

n

w

n

Q

n

k

n

v

p

n

w

ρ

( ) (

)

(

1 ) ( )

(

( )

1 )

)

(

1 )

1

0 2

2

0

₊

+

∆

+

−

+

_v

_n

n

p

n

r

n

v

n

p

ρ

( ) ( ) (

) ( )

(

_{( )}

(

( )

)

.

4

1

₀ 2

n

s

n

r

p

n

Q

n

k

ρ

₊

+

∆

+

−

≤

Summing the above inequality from

n

to

n

−

1

, we have

( )

( ) ( ) (

) ( )

(

_{( )}

(

( )

)

.

4

1

1 2 0 1 0 1 0 1

∑

− = +













+

∆

−

+

≤

+

n n s

s

r

p

s

Q

s

kp

n

v

p

n

w

n

v

p

n

w

ρ

It follows that

( ) ( ) (

) ( )

(

_{( )}

(

( )

)

( )

1 0

( )

1

1 2 0 1

4

1 n

v

p

n

w

s

r

p

s

Q

s

k

n n s

+

≤













₊

_∆

−

∑

− = +

ρ

.

which contradicts (8). This completes the proof. Choosing

ρ

( ) (

n

=

R

n

+

σ

)

.

By Theorem 2.1 we have the following

results.

Corollary 2.2: Assume that (2) holds and

τ

>

0 .

_If

(

) ( )

_{( ) (}

(

)

₎

0

1 lim sup

,

4

n

n _{s n}

p

kR s

Q s

r s R s

σ

→∞ =



+



+

−

= ∞



₊







∑

(17)

then every solution of equation (1) is oscillatory.

Corollary 2.3: Assume that (2) hold and

τ

>

0 .

If

(

)

(

) ( )

,

4

1 ln

1 inf

lim

0 0

k

p

s

Q

s

R

n

R

n n s n

+

>

+

∑

= ∞

→

σ

(18)

(5)

Proof: From (18), there exists an

ε

>

0

such that for all large

n

,

(

σ

)

(

σ

) ( )

+

ε

+

>

+

∑

−

=

1

0

4

1 ln

1

n

s

k

p

s

Q

s

R

n

R

.

It follows that

(

σ

) ( )



(

+

σ

)

≥

ε

(

+

σ

)









 +

−

+

∑

=

n

R

s

R

k

p

s

Q

s

R

n

n s

ln

4

1

0

.

That is

(

) ( )

( ) (

)

(

)

(

)

0 1

0 0

0

1

1 ln

ln

4

n

s n

p

R s

Q s

R n

kr s R s

k

σ

ε

σ

−

=



₊



_

₊

_

+

−

≥

+



₊















∑

(19)

Now, it is obvious that (19) implies (17) and the assertion of corollary 2.3follows from corollary 2.2

τ

>

0 .

If

( ) (

2

) ( )

1

0

lim inf

,

4

n

p

Q n R

n

r n

k

σ

→∞

+



+



>





(20)

then every solution of (1) is oscillatory.

Proof: It is easy to verify that (20) yields the existence

ε

>

0

such that for all large

n

,

( ) (

+

σ

) ( )

≥

+

ε

k

p

n

r

n

R

n

Q

4

1

₀

2

Multiplying by

(

n

) ( )

r

n

R

+

σ

1

on both sides of the above inequality, we have

(

) ( )

_{( ) (}

₎

₍

_{) ( )}

s

r

s

R

s

R

s

kr

p

s

Q

s

R

σ

ε

σ

+

≥

+

−

+

4

1

₀

which implies that (17) holds. Therefore by Corollary 2.2. every so solution of equation (1) is oscillatory.

Next, choosing

ρ

( )

n

=

n

.

By Theorem 2.1 we have the following result.

τ

>

0 .

if

( ) (

_{( )}

) ( )

0

1 lim sup

,

4

n

n _{s n}

p r s

ksQ s

s

→∞ ₌



+



−

= ∞









∑

(21)

Theorem 2.6: Assume that (3) holds,

τ

>

0

and

σ

<

τ

. Suppose also that there exists a positive real valued sequence

( )

{ }

∞

=n0

n

ρ

such that (8) holds, and

(

) (

)

(

_{( ) (}

( )

)

₎

0

0 2

1 lim sup

1

4

1

n

n _{s n}

p

s

k

s

Q s

r s

s

δ

τ

δ

→∞ ₌



+



+

− −

= ∞





+









∑

(22)

Proof: Assume the contrary. without loss of generality; we may suppose that

{ }

x

( )

n

is an eventually positive solution of equation (1)Then there exists an integer

n

₁

≥

n

₀ such that

x

(

n

+

σ

−

τ

)

>

0

for all

n

≥

n

₁. From (1) it follows that

{

r

( ) ( )

n

∆

z

n

}

is non-increasing eventually, where

z

( )

n

is defined by (9). Consequently, it is easy to conclude that there exist two possible cases of the sign of

∆

z

( )

n

, that is,

∆

z

( )

n

>

0

or

∆

z

( )

n

<

0

eventually. If

∆

z

( )

n

>

0

eventually, then we are back to the case of Theorem 2.1 and we can get a contradiction to (8). If

∆

z

( )

n

<

0

,

1

2

n

≥

_, then we define the sequence

{ }

v

( )

n

by

( )

r n

( ) ( )

_{( )}

z n

,

v n

z n

∆

(6)

Clearly

v

( )

n

<

0 .

Noting that

{

r

( ) ( )

n

∆

z

n

}

is

n

non increasing, we get

r

( ) ( ) ( ) ( )

s

∆

z

s

≤

r

n

∆

z

n

,

s

≥

n

≥

n

3

.

Dividing the above by

r

( )

s

and summing it from

n

_to

l

−

1

, we obtain

( ) ( ) ( ) ( )

_{( )}

1 ,.

1

∑

− =

∆

+

≤

l

n

s

r

s

n

z

n

r

n

z

l

z

3

n

l

≥

Letting

l

→

∞

in the above inequality, we see that

( ) ( ) ( ) ( )

,

.

0 ≤

z

n

+

r

n

∆

z

n

δ

n

≥

n

₃_. Therefore,

( ) ( )

( )

n

1 ,

n

3.

n

z

n

z

n

r

∆

_δ

_≥

₋

_≥

.

From (23), we have

( ) ( )

3

1 v n

δ

n

0,

n

− ≤

≤

≥

(24)

Similarly, we introduce the sequence

{

w

( )

n

}

where

( ) (

) (

_{( )}

)

,

n

z

n

z

n

r

n

w

=

−

τ

∆

−

τ

n

≥

n

3₍₂₅₎

Obviously

w

( )

n

<

0 .

Noting that

{

r

( ) ( )

n

∆

z

n

}

is non increasing, we have

r

(

n

−

τ

) (

∆

z

n

−

τ

) ( ) ( )

≥

r

n

∆

z

n

. Then

( ) ( )

n

v

n

w

≥

. From (24), we have

( ) ( )

3.

1 w n

δ

n

0,

n

.

− ≤

≤

≥

(26)

From (24) and (25), we have

( )

(

r n

( ) ( )

_{( )}

z n

)

v

2

_{( )}

( )

n

v n

z n

r n

∆

=

−

(27)

and

( )

(

r n

(

_{( )}

) (

z n

)

w

2

_{( )}

( )

n

.

w n

z n

r n

τ

∆

− ∆

−

∆

=

−

(28)

In view of (27) and (28) we obtain,

( )

n

p

0

( )

n

(

) (

_{( )}

)

0

(

( ) ( )

_{( )}

)

2

_{( )}

( )

2

_{( )}

( )

.

r n

z n

r n

z n

w

n

v

n

w

v

p

z n

r n

τ

∆

− ∆

−

∆

+ ∆

≤

+

−

∆

(29)

On the otherhand, proceeding as in the proof of theorem 2.1, we have

(

) (

)

(

r

n

−

τ

∆

z

n

−

τ

)

+

p

∆

(

r

( ) ( )

n

∆

z

n

)

≤

−

kQ

(

n

−

τ

) ( )

z

n

∆

₀ .

Using the above inequality in (29), we have

( )

0

( )

(

)

2

_{( )}

( )

2

_{( )}

( )

.

w

n

v

n

w n

p

v n

kQ n

r n

τ

∆

+ ∆

≤ −

− −

−

(30)

Multiplying (30) by

δ

(

n

+

1 )

and summing from

n

₃ to

n

−

1 ,

we have

( ) ( ) (

) ( )

_∑

−

_{( )}

( )

_∑

_{( ) (}

( )

)

=

−

=

+

−

3 3 1 1 2

3 3

1

n

n s

n

n s

s

r

s

w

s

r

s

w

n

w

n

w

n

δ

( ) ( )

(

) ( )

_∑

−

( )

_{( )}

=

+

−

+

₀ ₀ ₃ ₃ ₀ 1

3

1

n

s

r

s

v

p

n

v

n

p

n

v

n

p

δ

( )

( ) (

)

(

) (

)

∑

−

∑

=

−

=

≤

+

−

+

0 1 2 1

3 3

.

0

1

n

n s

n

n s

s

Q

k

s

r

s

v

(7)

From the above inequality, we have

( ) ( ) (

) ( )

( ) ( )

(

) ( )

(

) (

)

_{( ) (}

( )

₎

3 3

3 3 0 0 3 3

1 1

0

2

1

0.

4

1

n n

s n s n

n w n

n

w n

p

n v n

p

n

v n

p

s

k

s

Q s

r s

s

δ

τ

δ

− −

= =

−

+

−

+

− −

≤

+

∑

Thus, it follows from the above inequality that

( ) ( )

(

) (

_{( ) (}

) ( )

₎

3 1

0

0 2

1

4

1

n

s n

p

s

n w n

p

n v n

k

s

Q s

r s

s

δ

τ

δ

−

=



+



+

_

+

− −

_

+





∑

≤

δ

(

n

3

+

1 ) ( )

w

n

3

+

p

0

δ

(

n

3

+

1 ) ( )

v

n

3

.

By (24) and (26), we get a contradiction with (22) and this completes the proof.

Corollary 2.7: Assume that (3) holds,

τ

>

0

and

σ

≤

τ

.

_{Further assume that (22) one of the conditions (17), (18),}

(20) and (21) holds. Then every solution of equation (1) is oscillatory.

τ

>

0

and

σ

≤

τ

.

_{Furthermore suppose that there exists a positive real valued} sequence

{ }

( )

∞₌

0

n n

n

ρ

such that (8) holds, and

(

) (

)

0 2

lim sup

1

n

n _{s n}

s

Q s

δ

τ

→∞

∑

₌

+

−

= ∞

(31)

Proof: Assume the contrary. Without loss of generality, we assume that

{ }

x

( )

n

is an eventually positive solution of equation (1). Then there exists an integer

n

₁

≥

n

₀ such that

x

(

n

−

τ

)

>

0

for all

n

0

≥

n

1

.

By

( )

1

,

{

r

( ) ( )

n

∆

z

n

}

is

non-increasing, eventually where

z

( )

n

is defined by (9) . Consequently, it is easy to conclude that there exist two possible cases of the sign of

∆

z

( )

n

that is,

∆

z

( )

n

>

0

or

∆

z

( )

n

<

0

eventually. If

∆

z

( )

n

<

0

,

n

≥

n

₂

≥

n

₁, then

we define

w

( )

n

and

v

( )

n

as in theorem 2.6, Then proceed as in the proof of theorem 2.6, we obtain (24), (26) and

(30) Multiplying (30) by

δ

2

(

n

+

1 )

,

and summing from

n

₃ to

n

−

1

_{, where}

n

₃

≥

n

+

τ

,

we get

( ) ( )

(

) (

)

_∑

−

( ) (

₍

₎

)

( ) ( )

=

+

−

1 2

0 3

3 2 2

3

1

2

1

n

n s

n

v

n

p

s

r

s

w

n

w

n

w

n

δ

(

) ( )

_∑

−

( ) (

₍

₎

)

_∑

( ) (

_{( )}

)

=

−

=

+

−

3 3 1 1 2 2

2 0

3 3

1

2

1

n

n s

n

s

r

s

w

s

r

s

v

n

v

n

p

δ

( )

( ) (

)

(

) (

)

∑

−

∑

=

−

=

≤

+

−

+

1 2 1 2

2

0

3 3

.

0

1

n

n s

n

n s

s

Q

k

s

r

s

v

p

δ

τ

δ

(32)

It follows from (3) and (24) that

( ) (

)

(

)

( ) (

(

)

(

)

,

1

3 3

3

∞

<

+

≤

+

≤

+

_∑

∑

∞

= ∞

=n s n s n

s

r

s

r

s

w

s

r

s

w

δ

and

( ) (

)

( )

1

3

( )

1 .

3

1 2 2

∞

<

≤

+

∑

∞

= −

= s n

n

s

r

s

r

s

w

δ

In view of (26), we have

( ) (

)

(

)

( ) (

(

)

(

)

,

1

3 3

3

∞

<

+

≤

+

≤

+

_∑

∑

∞

= ∞

=n s n s n

s

r

s

r

s

v

s

r

s

v

δ

( ) (

)

( )

1 ( )

1 .

2 2

∞

<

≤

+

∑

∞

= ∞

=n s n

s

r

s

r

s

v

δ

(8)

From (32), we get

(

1 ) (

)

,

sup

lim

0

2

+

−

<

∞

∑

= ∞ →

n

n s n

s

Q

s

τ

δ

which is a contradiction with (31). This completes the proof.

Corollary 2.9: Assume that (3) holds,

τ

>

0

and

σ

≤

τ

Suppose se also that one of conditions (17), (18) and (20) and (21) holds, also condition (31) holds The every solution of equation (1) is oscillatory.

Theorem 2.10: Assume that (2) holds and

τ

<

0 .

_{Also, suppose that there exist a positive real valued sequence}

( )

{ }

∞

=n0

n

ρ

such that

( ) ( )

(

) (

_{( )}

)

(

( )

)

0

2

0

1 lim sup

.

4

n

n _{s n}

p r s

s

k

s Q s

s

τ

ρ

+

→∞ =



₊

_∆





₋



_{= ∞}









∑

(33)

Proof: Assume the contrary. Without loss of generality we may assume that

{ }

x

( )

n

is an eventually positive solution of equation (1). Then there exists an integer

n

₁

≥

n

₀ such that

x

(

n

−

τ

)

>

0

for all

n

≥

n

.

Define

z

( )

n

by (9). Similar to the proof of Theorem 2.1, there exists an integer

n

₂

≥

n

₁ such that

∆

z

(

n

+

τ

)

>

0

for

n

≥

n

₂and

( ) ( )

0

(

) (

)

( ) (

)

0

2

.

r n

z n

p

r n

τ

z n

τ

kQ n z n

τ

fo r n

n

∆

_



∆



_

+



_

+ ∆

+



_

+

≤

≥

₍₃₄₎

Define the sequence

{

w

( )

n

}

by

( ) ( ) ( ) ( )

₍

₎

,

τ

ρ

+

∆

=

n

z

n

z

n

r

n

w

n

≥

n

₂

.

(35)

Then

w

( )

n

>

0

. By

( )

10

we get

(

n

+

1 ) (

∆

z

n

+

1 ) (

≤

r

n

+

τ

) (

∆

z

n

+

τ

)

,

r

and

( ) ( ) ( ) ( )

(

₍

₎

)

₍

( ) (

_{) (}

)

₎

(

₍

)

_{) ( )}

n

w

n

r

n

w

n

z

n

z

n

r

n

w

ρ

τ

ρ

τ

ρ

∆

+

−

+

∆

≤

∆

1

2 2

( ) ( ) ( )

(

₍

₎

)

₍

( ) (

_{) (}

)

₎

(

₍

₊

)

_{) ( )}

(

∆

)

+

−

+

∆

≤

p

n

w

n

r

n

w

n

z

n

z

n

r

n

1

2 2

ρ

τ

ρ

τ

ρ

(36)

Again define the sequence

{ }

v

( )

n

by

( ) ( ) (

₍

) (

₎

)

,

n

₂

.

n

z

n

z

n

r

n

v

≥

+

∆

+

=

τ

ρ

(37)

The rest of the proof is similar to that of Theorem 2.1 and so is omitted. This completes the proof.

Choosing

ρ

( ) (

n

=

R

n

+

τ

)

.

_{By Theorem 2.10 we have the following oscillation criteria.}

τ

<

0

. If

(

) ( )

₍

_{) (}

₎

.

4

1 sup

lim

0

=

∞













+

−

+

∑

= ∞ →

n

n s

n

r

s

R

s

p

s

Q

s

kR

τ

(38)

τ

<

0

. If

(

)

(

) ( )

4 ,

1 ln

1 inf

lim

0

1

0

k

p

s

Q

s

R

n

R

n

n s n

+

>

+

∑

−

= ∞

→

τ

(39)

(9)

τ

<

0

. If

( ) (

) (

)

[

]

,

4

1 inf

lim

2 0

k

p

n

r

n

R

n

Q

n

+

>

+

∞

→

τ

(40) then every solution of equation (1) is oscillatory.

τ

<

0

. If

( ) (

) (

)

,

4

1 sup

lim

0

=

∞









+

−

∑

= ∞ →

n

n s

n

s

r

p

s

ksQ

τ

(41)

Next, we will give the following results under the case when (3) holds and

σ

<

τ

.

τ

<

0

and

σ

≤

−

τ

.

_{Further, suppose that there exists a positive real valued}

sequence

{ }

( )

∞₌ 0

n n

n

ρ

such that (33) holds. Suppose also that one of the following holds:

(

) (

_{( ) (}

) ( )

₎

,

1

4

1

1 sup

lim

0

2

0

=

∞













+

−

+

∑

= ∞ →

n

n s

n

r

s

p

s

Q

s

k

δ

τ

δ

(42)

(

) (

)

0 2

lim sup

1 .

n

n _{s n}

s

Q s

δ

τ

→∞

∑

₌

+

= ∞

(43)

Proof: Assume the contrary. Without loss of generality. we may assume that

{ }

x

( )

n

.

is an eventually positive

solution of equation (1). Then there exists an integer

n

₁

≥

n

₀ such that

x

( )

n

>

0

for all

n

≥

n

₁. From (1)

( ) ( )

{

r

n

∆

z

n

}

is non-increasing eventually where

z

( )

n

is defined by (9). Consequently, it is easy to conclude that there exist two possible cases of sign of

∆

z

( )

n

. That is,

∆

z

( )

n

>

0

or

∆

z

( )

n

<

0

eventually. If

∆

z

( )

n

>

0

eventually,

then we are back of the case of theorem 2.10 and we can get a contradiction to (33). If

∆

z

( )

n

<

0

eventually, then

there exists an integer

n

₂

≥

n

₁ such that

∆

z

( )

n

<

0

for all

n

≥

n

₂

.

Define the sequences

{

w

( )

n

}

and

{ }

v

( )

n

as follows:

( ) (

) (

)

( )

n

z

n

z

n

r

n

w

=

+

τ

∆

+

τ

for

n

≥

n

₃

=

n

₂

+

2 τ

and

( ) (

) (

_{( )}

)

n

z

n

z

n

r

n

v

=

+

2 τ

∆

+

2 τ

for

n

≥

n

₃

=

n

₂

+

2 τ

The rest of the proof can be proceed as in theorem 2.6 or theorem 2.8 we can obtain a contradiction to (42) or (43) respectively. This completes the proof.