Oscillation of the systems of impulsive hyperbolic partial differential equations

(1)

Journal of Chemical and Pharmaceutical Research, 2014, 6(7):1370-1377

Research Article

_{CODEN(USA) : JCPRC5}

ISSN : 0975-7384

Oscillation of the systems of impulsive hyperbolic partial differential

equations

**Xiaoqi Ning* and Shujun You**

Department of Mathematics, Huaihua University, Huaihua, China

_____________________________________________________________________________________________

ABSTRACT

The systems of impulsive hyperbolic partial differential equations with Robin boundary value condition are investigated. Several new sufficient conditions of oscillation for such systems are established by employing impulsive differential inequalities and integration.

Key words: Impulsive; Delay; The systems of partial differential equations; Oscillation

_____________________________________________________________________________________________

INTRODUCTION

As an important study field of impulsive partial differential equation, the oscillation theory of impulsive partial differential equation has various applications therefore it has aroused great study interests in recent years, and the study about oscillation theory of impulsive partial differential equations also gradually draws people's attention [1-6]. This paper considers equations

2 2

1

[ ( , )

( ) (

, )]

[ ( , )

( ) (

, )]

( )

( , )

( )

(

( ), )

( , )

(

( ), ), ( , )

,

1, 2,

,

{1, 2,

, },

( , )

( , ),

1, 2,

,

i i i i

s

i i il i l

l m

ij j j

k m

i k i k k i k

u t x

c t u t

x

u t x

c t u t

x

t

a t

u t x

a t

u t

t x

p t x u t

t x

R

t

t k

i

I

m

u t

x

u t

x

u t x

k

τ

ρ

σ

α

=

+ =

+ −

∂

₊

₋

₊

∂

₊

₋

∂

=

∆

+

∆

−

∈ ×Ω

≠

=

∈

=

−

=

∑

L

,

( , )

,

1, 2,

,

m

i k i k i k

k m

i

I

u t

x

u t

x

u t x

k

i

I

t

β

t

+ −









∈



_∂



₋

₌

_∈

∂



L

(1)

with boundary condition

i

( , )

( , ) ( , )

0,

,

i i k m

u t x

g t x u t x

x

t

t i

I

N

∂

₊

₌

_{∈∂Ω ≠}

_∈

∂

(2)

where,

[0,

),

,

( , ),

n

i i

(2)

2 2 1

( , )

, ( , )

,

.

n i

i m

r r

u t x

t x

G i

I

x

=

∂

=

∈

∂

∑

We study the oscillation problem of equation (1) with boundary condition (2), obtain sufficient condition for the oscillation of all solutions and extend the previous results by using existence condition of the eventually positive solution, combined with the impulsive differential inequality. In the following, we will use the conditions as follows:

1 2

( 1)

0

_k

, lim

_k

,

k

H

t

→+∞

< < < < <

L

= +∞

lim (

( ))

lim (

_j

( ))

.

t→+∞

t

−

σ

t

=

t→+∞

t

−

ρ

t

= +∞

1

0 ≤

α

k

≤

β

k

,

k

=

1, 2,

L

. ,

σ ρ

l

, ,

a a

i il

∈

PC

([0,

+∞

),[0,

+∞

)), ( )

c t

∈

C R R

(

+

,

+

);

(

2)

_ij

( , )

( , ),

_ii

( , )

0,

_ii

( )

min

_ii

( , ),

_ij

( )

sup |

_ij

( , ) |,

x G _{x G}

H

p t x

PC G R

p t x

p t

p t x

p t

p t x

∈ _∈

∈

>

=

1

1,

( )

min{

( )

( )}

0,

,

.

m

ii ji m m

i m

j j i

Q t

p t

i

I

j

I

≤ ≤ _{= ≠}

=

−

_∑

≥

∈

here

PC

denotes piecewise smooth continuous function with the following properties: only discontinuous on

t

=

t k

_k

,

=

1, 2,

L

which are discontinuity point of the first kind and left continuous on

t

=

t k

_k

,

=

1, 2,

L

;

(

H

3)

the

N

in (2) is the unit vector normal to

∂Ω

,pointing out of

Ω

,

g t x

_i

( , )

is continuous nonnegative function on

[0,

+∞ ×∂Ω

)

;

(

H

4)

u t x

_i

( , )

is the solution satisfied the boundary conditions of equation (1), and

u t x

i

( , )

t

∂

is the partial derivative of

u t x

_i

( , )

both are piecewise continuous function with discontinuity point of the first kind on

,

1, 2,

k

t

=

t k

=

L

. Suppose that they satisfy the following equation at impulsion:

u t

i

( , )

k

x

u t x

i

( , ),

k

u t

i

( , )

k

x

(1

a u t x

k

) ( , ),

i k

k

1, 2,

,

i

I

m

;

−

₌

+

_{= +}

₌

_∈

L

(3)

( , )

,

(1

)

,

1, 2,

,

.

i k i k i k i k

k m

u t

x

u t x

u t

x

u t x

k

i

I

t

β

t

− +

∂

₌

∂

_{= +}

∂

₌

_∈

∂

L

(4)

1. THE MAIN RESULT

Theorem 1. Suppose condition

( 1) (

H

−

H

4)

satisfied, if second-order impulsive differential inequalities

0

( )

( ) (

( ))[1

(

( ))]

0,

,

1, 2,

,

( )

(1

) ( ),

1, 2,

,

( )

(1

)

( ),

1, 2,

,

k

k k k

w t

Q t w t

t

c t

t

t t

t k

w t

k

w t

k

σ

α

β

+ +

′′

+

′

+

−

≤





_≥

_≠

₌





= +

=



_′

_{≤ +}

_′

₌



L

(5)

doesn’t have eventually positive solutions, then every non zero solution of problems (1)-(2) is oscillation in domain

G

.

Proof. Suppose systems (1)-(2) have a nonzero and non-oscillation solution

T

1 2

( , )

( ( , ),

( , ),

,

_m

( , )) ,

u t x

=

u t x u t x

_L

u t x

let us assume when

t

≥ ≥

t

₀

0

,we have

|

u t x

i

( , ) | 0,

>

i

=

1, 2,

L

,

m

. Set

sgn ( , ),

( , )

( , ),

i

u t x z t x

i i i

u t x

i

(3)

Then

z t x

_i

( , )

>

0, ( , ) [ ,

t x

∈

t

₀

+∞ ×Ω

)

.

From condition

( 1)

H

we can see there is

T

₁

>

t

₀, when

t

≥

T

₁ we have

z t x

_i

( , )

>

0,

z t

_i

(

−

ρ

_l

( ), )

t x

>

0,

z t

_i

(

−

σ

( ), )

t x

>

0,

( , ) [ ,

t x

∈

T

1

+∞ ×Ω =

)

,

i

1, 2,

L

, ,

m l

=

1, 2,

L

, .

s

When

t

≠

t

_k时, we integrate both sides of equation (1) on

Ω

2 2

1

1 1

( , )

( )

(

, )

( , )

( )

(

, )

( )

( , )

( )

(

( ), )

( , )

(

( ), )

,

1, 2,

,

(

i i

)

(

i i

)

s

i i il i l

l m

ij j m

j

d

u t x dx c t

u t

x dx

u t x dx c t

u t

x dx

dt

a t

u t x dx

a t

u t

t x dx

p t x u t

t x dx t

T i

I

k

τ

ρ

σ

Ω Ω Ω Ω

Ω ₌ Ω

Ω =

+

−

+

−

=

∆

+

∆

−

+

−

≥

∈

=

∫

∑

∫

∑∫

L

(6)

then

(

) (

)

2 2

1

1 1

( , )

( )

(

, )

( , )

( )

(

, )

( )

( , )

( )

(

( ), )

( , ) (

( ), )

,

1, 2,

.

i i i i

s

i i il i l

l m

i

ij j m

j j

d

z t x dx

c t

z t

x dx

z t x dx c t

z t

x dx

dt

a t

z t x dx

a t

z t

t x dx

p t x z t

t x dx t

T i

I

k

τ

ρ

δ

_σ

δ

Ω Ω Ω Ω

Ω ₌ Ω

Ω =

+

−

+

−

=

∆

+

∆

−

+

−

≥

∈

=

∫

∑

∫

∑ ∫

L

(7)

It can be derived from Green Theorem and the boundary condition (2) that

( , )

i

( , ) ( , )

0,

i i i

z t x

z t x dx

ds

g t x z t x ds

N

Ω ∂Ω ∂Ω

∂

∆

=

= −

≤

∂

∫

t

≥

T i

₁

,

∈

I

_m

,

k

=

1, 2,

L

,

(8)

1

(

( ), )

(

( ), )

(

( ), ) (

( ), )

0,

,

1, 2,

,

i l i l

m

z t

t x

z t

t x dx

ds

N

g t

t x z t

t x ds

t

T i

I

k

ρ

Ω ∂Ω

∂Ω

∂

−

∆

−

=

∂

= −

−

≤

≥

∈

=

∫

L

(9)

here

ds

denotes the area element on

∂Ω

. Combining (7),(8),(9) and condition

(

H

2)

, we can work out the form

(

) (

)

2 2

1,

1

( , )

( )

(

, )

( , )

( )

(

, )

( )

(

( ), )

( )

(

( ), )

,

1, 2,

.

i i i i

m

ii i ij j

j j i m

d

z t x dx

c t

z t

x dx

z t x dx c t

z t

x dx

dt

p t

z t

t x dx

p t

z t

t x dx

t

T i

I

k

τ

σ

Ω Ω Ω Ω

Ω _{= ≠} Ω

+

−

+

−

≤ −

−

+

−

≥

∈

=

∫

∑

∫

L

(10)

Set

1

( )

( , )

,

i i m

v t

z t x dx

t

T i

I

Ω

=

_∫

≥

∈

By form (10), we have

1,

1

[ ( )

( ) (

)]

[ ( )

( ) (

)]

[

( ) (

( ))

( ) (

( ))]

0,

,

1, 2,

.

i i i i

m

ii i ij j

j j i k

v t

c t v t

v t

c t v t

p t v t

t

p t v t

t

T t

t k

τ

σ

′′ ′

= ≠

+

−

+

−

+

−

≤

≥

≠

=

∑

L

(4)

Set

1 1

( )

( ),

,

m i i

V t

v t

t

T

=

∑

≥

Combining this form and form (11), we have

1 ,

1

[

( ) (

( ))

( ) (

( ))]

[ ( )

( ) (

)]

[ ( )

( ) (

)]

0,

,

1, 2,

m m

ii i ij j

i j j i

V t

c t V t

V t

c t V t

t

T

p t v t

t

p t v t

t

k

σ

τ

′′

τ

′

= = ≠

+

−

+

−

≤

≥

=

+

∑

−

∑

−

L

(12)

and hence

11 1 1

1, 1

22 2 2

1, 2

1,

1

[ ( )

( ) (

)]

[ ( )

( ) (

)]

( ) (

( ))

( ) (

( ))

( ) (

( ))

(

( ))

( )

(

( ))

( ) (

( ))

0,

,

1, 2,

.

{[

]

[

]

[

]}

m

j j j j

m

j j j j

m

mm m mj j

j j m

V t

c t V t

V t

c t V t

p

t v t

t

p

t v t

t

p

t v t

t

p v t

t

p

t v t

t

p

t v t

t

T k

τ

σ

′′ ′

= ≠

+

−

+

−

+

−

+

−

+

+ +

−

≤

≥

=

∑

L

furthermore we have

11 1 1 22 2 2

1, 1 1, 2

1 1,

[ ( )

( ) (

)]

[ ( )

( ) (

)]

( )

(

( ))

( )

(

( ))

( )

(

( ))

0,

,

1, 2,

,

{[

]

[

]

[

]

}

m m

j j

j j j j

m

mm jm m

j j m

V t

c t V t

V t

c t V t

p

t

p

t v t

t

p

t

p

t v t

t

p

t

p

t v t

t

T k

τ

σ

′′ ′

= ≠ = ≠

= ≠

+

−

+

−

+

−

+

−

+

+ +

−

≤

≥

=

+

∑

L

then

1 1

1, 1

[ ( )

( ) (

)]

[ ( )

( ) (

)]

min

{

( )

}

(

( ))

0,

,

1, 2,

,

m m

ii ji i

i m

j j i i

V t

c t V t

V t

c t V t

p t

v t

t

T k

τ

σ

′′ ′

≤ ≤ _{= ≠} ₌

+

−

+

−

+

−

≤

≥

+

=

∑

L

Consequently, we have

1

[ ( )

( ) (

)]

[ ( )

( ) (

)]

( ) (

( ))

0,

,

1, 2,

.

V t

c t V t

V t

c t V t

Q t V t

t

T k

τ

′′

τ

′

σ

+

−

+

−

+

−

≤

≥

=

L

(13)

Set

1

( )

( ) (

)

( )

0,

,

w t

=

V t

+

c t V t

− ≥

τ

V t

>

t

≥

T

then form (13) can be rewrite as

w t

′′

( )

+

w t

′

( )

+

Q t V t

( ) (

−

σ

( ))

t

≤

0,

t

≥

T k

₁

,

=

1, 2,

L

,

(14) Thus, we have

1

[

e w t

t

′

( )]

′

+

e Q t V t

t

( ) (

−

σ

( ))

t

≤

0,

t

≥

T k

,

=

1, 2,

_L

.

From the above discussion , we can see that

( )

0,

(

( ))

0, [

t

( )]

0,

_k

,

1, 2,

,

w t

>

V t

−

σ

t

>

e w t

′

≤

t

≠

t k

=

_L

So

e w t

t

′

( )

is a monotone decreasing function in

( ,

t t

_k _k₊₁

]

,

w t

′

( )

is also a monotone decreasing function in

1

( ,

t t

_k _k₊

]

, it means

w t

′′ ≤

( )

0,

t

≠

t k

_k

,

=

1, 2,

_L

.

(5)

( , )

(1

) ( , ),

i k

(1

)

i k

,

i k k i k k

u t

x

u t x

u t

x

u t x

t

α

+

β

+

_{= +}

∂

_{= +}

∂

Hence

( , )

sgn ( , )

sgn ( , ), sgn

i k

sgn

i k

,

i k i k

u t

x

u t x

u t

x

u t x

t

+

₌



∂



₌



∂





_∂





_∂











then

1

1 1

1

( ( , )

( , ))

sgn( ( , )) ( , ) sgn( ( , )) ( , )

sgn( ( , ))[ ( , )

( , )]

sgn( ( , )

( )

[ ( )

( )

)

(

]

m

i k i k i

m

i k i k i k i k i

m

i k m

k k

i k i k i

m

i k k i k i

i k i k i

z t

x

z t

x dx

u t

x u t

x

u t

x u t

x dx

u t x

u t

x

u t

x d

V t

x

v

u

t

v

t

x

α

+ −

Ω =

+ + − −

+ −

Ω =

+ −

Ω =

+ −

=

−





=

_

−

_

=

−

=

−

=

−

∑

∑∫

∑

∫

∑∫

∫

, )

( ),

k k

x dx

V t

α

=

furthermore ,we have

( )

( ) (

) [ ( )

( ) (

)]

( )

( )[ (

)

(

)]

[ ( )

( ) (

)]

( ),

k k k k k k k k

k k k k k

k k k k k k

w t

V t

c t V t

V t

c t V t

V t

c t

V t

c t V t

t

τ

α

τ

α ω

+ − + + + − − −

+ − + −

−

=

+

− −

+

−

=

−

+

− −

−

=

+

−

=

Thus, we have

w t

( )

k

w t

( )

k

α

k

w t

( ).

k

+

₋

−

₌

_{Note that}

( )

k

( ),

k

w t

−

=

w t

we can also have

w t

( )

_k+

= +

(1

α

_k

) ( ).

w t

_k (15) Similarly, we can prove

( )

_k

( )

_k _k

( ).

_k

w t

′

+

−

w t

′

−

≤

β

w t

′

note that

1

1 1

(

( )

[ ( )

( )]

, )

( , )

sgn(

)

sgn(

)

( , )

(

sgn(

)

m

i k i k i

m

i k i k i

m

k k i k i

k i k i

m

i k k i

k i

i k

i

z t

x

z t

x

dx

t

u t

x

u t

x

u t

x

u t

x

dx

t

u t x

u t

x

u t

t

V t

v t

+ −

Ω =

+ + − −

Ω =

+ − + −

+ −

Ω =

=



_∂



=



_∂

−

_∂





′

−

′

=

′





_∂



=



_∂

−

_∂







∂

=

−

∂

∑∫

∑

1

, )

( , )

sgn(

)

( )

( ),

m

i k i k k i

m

k i k k k i

x

dx

t

u t x

dx

t

v t

V t

β

Ω =

=







_∂







∂

=

∂

′

=

∑∫

(6)

then

( )

[ ( )

( ) (

)]

[ ( )

( ) (

)]

( ) [ ( ) (

)]

( ) [ ( ) (

)]

( )

( ) (

)

( )

(

)

( ) (

)

( )

(

)

[

k k k k k k k k

k k k k k k

k k k k

w t

V t

c t V t

V t

c t V t

V t

c t V t

V t

c t V t

V t

c t V t

τ

+ − + + + ′ − − − ′

+ + + ′ − − − ′

+ − + + + +

− − − −

′

−

′

=

+

−

+

−

′

=

+

−

′

=

−

+

− +

−

′

−

− −

−

′

=

( )

( )]

( )[ (

)

(

)]

( )[

(

)

(

)]

( )

(

)

( )

(

)

[

( )

( ) (

)

( )

(

)]

[ ( )

( ) (

)]

( ),

k k k k k

k k k

k k k k k k k k

k k k k k k

V t

c t

V t

c t

V t

c t

V t

c t

V t

c t V t

V t

c t V t

w t

τ

β

α

τ

β

τ

β

τ

β

τ

β

+ − + −

+ −

′

−

+

− −

−

′

+

− −

−

′

=

+

− +

−

′

≤

+

− +

−

=

+

−

′

=

thus, we have

( )

_k

( )

_k _k

( ).

_k

w t

′

+

−

w t

′

−

≤

β

w t

′

furthermore , because

w t

′

( )

_k−

=

w t

′

( ),

_k we have

w t

′

( )

_k+

≤ +

(1

β

_k

)

w t

′

( ).

_k (16) By using (15),(16),we can work out

1

( )

0,

,

_k

.

w t

′ ≥

t

≥

T t

≠

t

In fact, if the above form is wrong then

∃ ≥

T

₂

T

₁

,

such that

w T

′

( )

₂

<

0,

let us assume

w T

′

( )

₂

= −

c c

, (

>

0).

By

( )

0,

_k

,

1, 2,

,

w t

′′ ≤

t

≠

t k

=

_L

w t

′

( )

_k+

≤ +

(1

β

_k

)

w t

′

( ),

_k

t

=

t k

_k

,

=

1, 2,

L

,

together with lemma 5.1.3 in [6], we have

2 2

( )

(1

)

(1

),

.

k k

T t t T t t

w t

w T

β

c

β

t

T

< < < <

′

≤

′

∏

+

= −

∏

+

≥

Note that

w t

( )

_k+

= +

(1

α

_k

) ( ),

w t

_k then we have

2

2 2

2

2 2

2

( )

(1

)

(1

)

(1

)

1 (1

)

( )

,

1

k k k

k k

t

k _T k k

T t t s t t T t s t

k

k _T

T t t T t s k

w t

w T

c

ds

w T

c

ds

α

β

α

< < < < < <

< < < <

≤

+

−

+



₊



=

+



−



+





∏

∫

∏

∫

∏

note

0 <

α

_k

≤

β

_k in

( 1)

H

, therefor we have

w t

( )

≤

0,

t

≥

T

₂

,

according the above form , this contradicts to

w t

( )

>

0,

t

≥

T

₁. By

w t

( )

=

V t

( )

+

c t V t

( ) (

−

τ

),

t

≥

T t

₁

,

≠

t k

_k

,

=

1, 2,

L

. together with form (14) we have

1

( )

( )[ (

( ))

(

( )) (

( )

)]

0,

,

_k

,

1, 2,

.

w t

Q t w t

t

c t

t V t

t

T t

t k

σ

τ

′′

+

′

+

−

≤

≥

≠

=

L

(17)

Since

( )

( ),

( )

0,

_k

,

1, 2,

,

(7)

1

( )

( ) (

( ))[1

(

( ))]

0,

,

_k

,

1, 2,

.

w t

′′

+

w t

′

+

Q t w t

−

σ

t

−

c t

−

σ

t

≤

t

≥

T t

≠

t k

=

L

(18)

therefor combining form (18) and form (15),(16) we can see that

w t

( )

is a eventually positive solution of differential inequality (5),this contradicts to condition of theorem 1,and we complete the proof of theorem1. Theorem 2. Suppose condition

( 1) (

H

−

H

4)

satisfied, further more suppose

(i)

0 ≤

σ

( )

t

≤

σ

, 0

≤

ρ

_j

( )

t

≤

ρ

,

j

=

1, 2,

L

, ;

m

(ii) if

1 1

,

k k

α

β

∞ ∞

= =

< +∞

∑

and for each are sufficiently large

_T

, we have

{

}

(1

)

(1

)

( )[1

(

( ))]

lim inf

,

(1

)

(1

)

k k

t

k k

T T

t t s t t t

k k

T

T t t t

Q s

c s

s

dsd

d

η

η η

α

β

σ

η

β

α η

< < < < →+∞

< < < <

+

−

= −∞

+

∏

∫

∏

∫

Then every nonzero solution of problems (1)-(2) is oscillation in domain G

Proof. By theorem 1 we only need to prove the nonexistence of the eventually positive solution of the second order

impulsive differential inequality (5). Suppose

w t

( )

is a eventually positive solution of the impulsive differential inequality (5), then there exists a real number

T

≥

max{ , },

σ ρ

and

w T

(

−

σ

)

≥

0,

in form (5), we notice that

( )

0 w t

′ ≥

and

w t

(

−

σ

( ))

t

≥

0,

t

≥

T t

,

≠

t k

_k

,

=

1, 2,

_L

,

hence we have

( )

(

) ( )[1

(

( ))]

0. w t

′′ +

w T

−

σ

Q t

−

c t

−

σ

t

≤

(19) combining (19) and

w t

′

( )

_k+

≤ +

(1

β

_k

)

w t

′

( ),

_k in (5), we can use lemma 5.1.3 in [6] to get

( )

(1

)

(1

)

{

(

) ( )[1

(

( ))]

}

,

k

k T t t t

k T

s t t

w t

w T

Q s

c s

s

ds

β

σ

< <

′

≤

′

+

−

∏

∫

(20)

and together with

w t

( )

_k+

= +

(1

α

_k

) ( )

w t

_k in (5) we have

( )

(1

)

(1

)

( )

(1

)

(1

)

(

) ( )[1

(

( ))]

( )

(1

)

( )

(1

)

(1

)

(

)

(1

)

(1

)

{

[

] }

k k k

k

k k k

k k

t

k _T k k

T t t t t T t

k T

s t

t

k _T k k

T t t t t T t

t

k k

T T

t t s t

w t

w T

Q s

c s

s

ds d

w T

d

w T

η η

η

η η

α

β

σ

η

α

β η

σ

α

β

< < < < < <

< <

< < < < < <

< < < <

′

≤

+

−

′

=

+

−

+

∏

∫

∏

∫

∏

∫

∏

∫

η

[

−

Q s

( )[1

−

c s

(

−

σ

( ))]

s

]

dsd

η

(21)

(8)

1 1 1 1

1

1 1 1 1

1

1 1

1

1 1 1

1 1

(1

)

(1

)

(1

)

(1

)

(1

)

(1

)

(1

)(

)

(1

)

(1

)(

)

(1

)(

)

k k

j m

t

k k

T

T t t t

j

m m m m

t t t

k k k k

T t t

j

k k k j k

j

m m m

k k k j j

j

k k k j

m

k m m k

d

t

T

t

η η

β

α η

β

α η

β η

α

β

α

β

+ +

< < < < −

=

= = = + =

−

+ =

= = = +

+ =

+

≤

+





=

+

− +



+

−







+

−

< +∞

∏

∫

∑

∏

∫

∑

∏

.

(22)

For

t

>

T

, the inequality (21) with both sides divided by the left side of (22), we obtain

(1

) ( )

( )

(1

)

(1

)

(1

)

(1

)

(

)

(1

)

(1

)

( ) 1

(

( ))

,

(1

)

(1

)

{

[

]}

k

k k k k

k k

k T t t

t t

k k k k

T T

T t t t T t t t

t

k k

T T

t t s t t

k k

T

T t t t

T

t

T

d

T

Q s

c s

s

dsd

d

η η η η

η

η η

α ω

ω

_ω

β

α η

β

α η

ω

σ

α

β

σ

η

β

α η

< <

< < < < < < < <

< < < <

+

′

≤

+

−

+

−

+

∏

∫

∏

∫

∏

∫

Notes

(1

)

(1

)

,

k k

t

k k

T

T t t t

d

t

η η

β

α η

< < < <

+

→ +∞

∏

∫

We get

( )

lim inf

.

(1

)

(1

)

k k

t t

k k

T

T t t t

w t

d

η η

β

α η

→+∞

< < < <

= −∞

+

∏

∫

(23)

According to condition (ii) in theorem 2, on the other hand by

w t

( )

>

0,

t

≥

T

,

we have

( )

lim inf

0,

(1

)

(1

)

k k

t t

k k

T

T t t t

w t

d

η η

β

α η

→+∞

< < < <

≥

+

∏

∫

This form contradicts to (23), hence impulsive differential equation (5) doesn’t have eventually positive solution, and we complete the proof of theorem 2.

REFERENCES

[1] L.P.Luo, Z.G.Ouyang. Oscillation of systems of a class of impulsive delay neutral parabolic equations. Acta

Analysis Functionalis Applicata, 8(3): 257-264, 2006.

[2] Q.Zhao,W.A.Liu. Oscillation of solution of a class of impulsive delay neutral hyperbolic equations. Journal of

Mathematics, 26(5): 563-568, 2006.

[3] Zhang Y T, Luo Q. On the forced oscillation of solutions for systems of impulsive neutral parabolic differential equations with several delays. J. of Math(PRC), 26(3): 272-276, 2006.

[4] L.P.Luo,Z.G.Ouyang . Oscillation of the system of impulsive neutral felay parabolic partial differential equations. Acta Mathematicae Applicatae Sinica, 30(5): 822-830, 2007.

[5] L.P.Luo. Oscillation of impulsive delay neutral hyperbolic equations with nonlinear diffusion coefficient.

Progress in Nature Science, 18(3): 341-344, 2008.