New Results on Oscillation of even Order Neutral Differential Equations with Deviating Arguments

(1)

New Results on Oscillation of even Order Neutral

Differential Equations with Deviating Arguments

Lianzhong Li1, Fanwei Meng2 1

School of Mathematical and System Sciences, Taishan College, Tai’an, China 2

School of Mathematical Sciences, Qufu Normal University, Qufu, China E-mail: llz3497@163.com, [email protected]

Received January 19, 2011; revised March 15, 2011; accepted March 20, 2011

Abstract

In this paper, we point out some small mistakes in [6] and revise them, we obtain some new oscillation re-sults for certain even order neutral differential equations with deviating arguments. Our rere-sults extend and improve many known oscillation criteria because the article just generalizes Meng and Xu’s results.

Keywords:Oscillation, Neutral Differential Equation, Deviating Argument

1. Introduction

Oscillation of some even order differential equations have been studied by many authors. For instance, see [1-7] and the references therein. We deal with the oscil-latory behavior of the even order neutral differential eq-uations with deviating arguments of the form

 



 



 



 



1

0

0,

n m

i i

i l

j j j

j

x t p t x t

q t f x t

t t



 



 _ 

 

 



 



(1)

where n2 is even, throughout this paper, it is as-sumed that:

(A1) p qi, j C t





0,



,R



,fj C R R uf



,



, j

 

u 0



   

for u0 and fj

 

u is non-decreasing on R , 1, 2, , , 1, 2, , ;

i _ m j _ l

(A2)i C t





0,



,R



, t ti

 

t



   and lim _i

 

,

t t  

1, 2, , i _ m;

(A3)







 

1

, , , , lim

j j j

t

C t R t t t

   



    

and  j

 

t 0, j1, 2,, ;l

(A4) There exists a constant M0 such that

 

sgn

 

M

j

f x x  x for x0,j1, 2,,l;

(A5)

 

1

, 0,1

m

i i

p t p p



 



and there exists a func-

tion q t

 

C t





0,



,R





  such that, q t

 

min



qj

 

t :



1, 2, ,

j _ l .

By a solution of Equation (1) we mean a function

 

x t which has the property that

 



 



1

m

i i i

x t p t x  t













, ,



n x

C t R

  for some txt0 and satisfies

Equa-tion (1) on



t_x,



. We restrict our attention to those solutions x t

 

of Equation (1) which exist on some half-line



tx,



with sup



x t

 

:tT



0 for any

x

T t . A nontrivial solution of Equation (1) is called oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. Equation (1) is said to be os-cillatory if all of it’s nontrivial solutions are osos-cillatory.

Recently, Meng and Xu [6] studied Equation (1) and obtained some sufficient conditions for oscillation of the Equation (1), we list the main results of [6] as follows.

Following Philos [5], we say that a function

 

,

HH t s belongs to a function class W , denotes by

HW , if HC D R



, 



, whereD



 

t s, :t s t0



,

which satisfies: (H1) H t t

 

, 0 and H t s

 

, 0 for 0

t    s t ; (H2) H has a continuous non-positive

partial derivative H S 

 satisfying the condition:

 

_{ }

_{   }

 

,

, ,

H t s k s

h t s H t s

S k s

 

 



for some hL_loc



D R,



, kC1





t₀,

 

, 0,



is a non-decreasing function.

Theorem A ([6, Theorem 2.1]).

Assume that (A1) - (A5) hold, let the functions

, ,

H h ksatisfy (H1) and (H2), suppose

 

1

2

1

lim sup , ,

4

t C F t r _C G t r

 

  

 

(2)

holds for every rt C0, 10,C20, where

 

_{ }

   

 

_{ }

   

_{   }

1

2

1

, , d ,

,

, 1

, d

, ,

l t

j r

j

t

n r

j j

F t r H t s k s q s s

H t r

k s h t s

G t r s

H t r H t s  s  s













and  1 p, then every solution of Equation (1) is oscillatory.

Theorem B ([6, Theorem 2.2]).

Assume that (A1)-(A5) hold, and H h k, , are the same

as in Theorem A, suppose that

 

0

,

inf 0

, lim inf

s t t

H t s H t t

 

 

 _{ }

 

 

  (3)

and

 

0

lim sup ,

t G t t   (4) If there exists a function mC t





0,



,R



such that

for all t T t0.

 

1

2

1

, ,

4 lim inf

t C F t T _C G t T m T

 

 

 

  (5)

and

     

 

0

2 2

lim sup d , 1, 2, ,

n

t _j _j

t t

m

s j l

k

s s s

s

  

 



  



 (6)

where m_

 

t max



m t

 

, 0



, then every solution of Equation (1) is oscillatory.

In Theorem A and B, function G t r

 

, should be

 

,

j

G t r , so each of the condition (2), (4), (5) and (6) has as many as l conditions. Meanwhile, the Riccati func-tion 

 

t is not well-defined and there exist some small errors in the proof of the theorems. The purpose of this paper is further to strengthen oscillation results ob-tained for Equation (1) by Meng and Xu [6]. In our paper, we redefine the functions F

     

t r, ,G t,r , t and provide some new oscillation criteria for oscillation of Equation (1).

2. Main Results

In the sequel, we need the following lemmas: Lemma 2.1 ([1]).

Let x t

 

be a

n

times differentiable function on



t0,



of one sign,

 n

_{ }

₀

x t  on



t0,



which satisfies x( )n

   

t x t 0. Then:

(I1) There exists a t1t0 such that

 

_{ }

, 1, 2, , 1

i

x t i _ n are of one sign on



t1,



; (I2) There exists a number h



1, 3, 5,,n1



when

n

is even, or h



2, 4, 6,_,n1



when n is odd,

such that x t x

 

 i

 

t 0 for i0,1,_,h , tt1;

 

1

 

_{ }

1n i  x t xi t 0

  for i h 1,h2,, ,n tt1.

Lemma 2.2 ([1]).

If x t

 

is as in Lemma 2.1 and xn1

 

t x n

 

t 0 for tt0, then for every 



0  1



, there exists a

constant N0, such that



 



n1  n1

 

Nt t

x  t   X  for all large t.

Lemma 2.3([7]).

Suppose that x t

 

is an eventually positive solution of Equation (1), let

 



 



1 m

i i

i

z t x t p t x  t



 



,

then there exists a number t1t0 such that z t

 

0,

 

1

0, n 0

z t  z  t  and  

 

0, 1

n

t t

z  t . Theorem 2.1

Assume that (A1) - (A5) hold, let the functions H h k, ,

satisfy (H1) and (H2), suppose

 

lim sup , ,

4

t MF t r _NG t r

 





 _ _{ }

 

  (7)

holds for every rt0 and for some  1, where

 

_{ }

     

 

_{ }

   

 

   

2

1

, , d

,

, 1

, d

,

t r

t

l r

n

j j

j

F t r H t s k s q s s H t r

k s h t s

G t r s

H t r

H t s   s  s 









and  1 p, then every solution of Equation (1) is oscillatory.

Proof. Suppose to the contrary that x t

 

is a nonoscillatory solution of Equation (1) and that x t

 

is eventually positive (when x t

 

is eventually negative, the proof is similar).

Let z t

 

be defined as in Lemma 2.3, then following the proof of Theorem 2.1 in [6], without loss of generality, assume there exists a t1t0 such that

 



 





 



( 1)

0, 0, 0,

0, 0

n

j j

x t t t

t t

z

z t z

z

z  





  

 





n 2

 

n 1

_{ }

j N j t t

z  t    z  (by lemma 2.2) and

 

_{ }



_{ }



1

l n

j j

j

t t

z M q z  t



 



for all tt1.

Let

 





1

n

l j j

z k

z t

t t

t 

 

 



(not as [6]),

(3)

 

   

_{   }

 

_{   }

   

2

1

2 , ,

l n

j j

j

N k

Mk q t

t t

t

t t t t t

k t k t t

  

  

  

 

    



 (not as [6]).

Multiplying the above equation, with t replaced by s, by H t s

 

, and integrating it from T to t, for all

1

t T t , for some  1, we obtain

     

   

_{ }

   

 

   

 

   

 

   

 

   

2

1 2

2 2

1 2

2

1

2

1

2

1

, , ,

, ,

,

d , d d

, d d

4 1

, 4

l n

j j

t t t _j

T T T

l n

j j

t t _j

l

T T

n

j j

j

l n

j j

j

l n

j j

j

M H k q s H t T T h s N H s

k

NH

k h

H t T T s s

k NH

s s

t s s s t s s t s s

s

t s s s

s t s

s s

t s s s

t s s

NH

k

h s

s

s t

s

t s s s

k

NH

 

    

  

 

 



  

 

 _ _ _

 

  



 



  



  

 



 













 

   

 

   

2

1

d

,

, ,

d 4

t

T

t

l T

n

j j

j

s

k

s

s h

H t t s

t s s

T T s

NH s

 

   



 

 

 

 

 

 







Hence, we have

 

,

 

,

 

4

MF t T G t T T

N 

 



 

for all t T t1, this gives

 

lim sup , ,

4

t MF t r _NG t r

 





 _ _{ }

 

 

which contradicts (7). This completes the proof of the Theorem.

The assumption (7) in Theorem 2.1 can fail, conse-quently, Theorem 2.1 does not apply. The following re-sults provide some essentially new oscillation criteria for Equation (1).

Theorem 2.2

Assume that (A1)-(A5) hold, the functions H h k F, , ,

and G be the same as in Theorem 2.1, suppose that

 

0 ₀

,

inf 0

, lim inf

s t t

H t s H t t

 

 

 _{ }

 

 

  (8)

If there exists a function mC t





0,



,R



such that

for all t T t0 and for some 1,

 

limsup , , 4

t MF t T _NG t T m T 





 _ _

 

  (9)

and

     

 

0

2 2

1

lim sup d

l n

j j

t j t t

m

s s s

s s

k

  

 





 





(10)

where m_

 

t max



m t

 

, 0



. Then every solution of Equation (1) is oscillatory.

Proof. Assume to the contrary that (1) is non-oscil- latory. Following the proof of Theorem 2.1, without loss of generality, assume for all t T t0 and for some

1

  , we obtain

     

   

 

   

 

   

 

2

1

2

1 2

,

, ,

,

d ,

d 4

d 1

t

T

t

l T

n

j j

j l

n

j j

t _j

T

t s s s

s t s

t s s s

s s

M H k q s H t T T

k h

s NH

NH

s k

 



  

 _



 







 









So, we get

 

1

 

, , ,

4

MF t T G t T T NB t T

N

 

  

 



  

where

 

_{ }

 

_{ }

   

 

2

1 2

0

, ,

, 1

d ,

l n

j j

t j

r

H

B s

H k

t s s s

t r s

r t

 



 

 



(4)

then

 

, ,

4 1

lim in m s

, l up

f i

t

t MF t T _NG t T

T N B t T

 

 

 

 



 _ 

 

 



 

For all Tt0 and for any  1, by (9) we have

 

1

 

lim inf ,

t

T m T  N B t T

 

 



 

So

 

T m T

 

  (11)

and especially

 

0

 

0

 

0

lim inf ,

( 1)

t B t t _N t m t

 _

 

       (12)

Now, we claim that

   

 

0 2

1 2

lim sup d

l n

j j

t _j t t

s s

s k

 

 

 



 





(13)

Suppose to the contrary that

   

 

0 2

1 2

lim sup d

l n

j j

t j t t

s s

s k

 

 

 



 





(14)

By (8), there is a positive constant  satisfying

 

0

,

inf lim inf 0 ,

s t t

H t s

H t t 

 

 

 _{  }

 

 

  (15)

Let  be any arbitrary positive number, from (14) there exists a t1t0 such that,

   

 

0 2

1 2

d

l n

j j

t _j t

s s

s

s s

k

 

 

 











for all tt1

Then, for tt1, we have

 

   

 

   

 

1 0

2

1 2

2 0

0

1

1 2

0

,

, ,

1

d

1

, ,

d d

l n

j j

t _{s j}

t t

l n

j j

t _{s j}

t t

t t

v v

t s d v

t t v

v v

t s v

t t v

t B

H v

H k

H v s

t

H k

H t

 



 



 

 

 _ 

 

 _ _

 

 

 _ 

 

 _ _

 

 











By (15), there exists a t2t1 such that, for all

2

tt ,

 

10

, ,

H t t

H t t  , which implies B t t

 

,0  for all

2

tt . Since  is arbitrary, we have

 

0

 

0

lim inf , lim ,

t B t t tB t t  

which contradicts (12), thus (13) holds. Then by (11) and (13) we get

   

 

   

 

0

0 2

1 2

2

1 2

lim sup

d

d ,

l n

j j

t _j t

l n

j j

t

t t

j t

m s

k

s k

s s

 

 







 



 



  









which contradicts (10). This completes the proof.

Remark 1 Let  1 in Theorem 2.1, Theorem 2.1 reduces to Theorem A [6]; we obtain the same result in Theorem 2.2 in which we omit the assumption (4) in Theorem B [6]. Therefore, Theorem 2.1 and 2.2 are gen-eralizations and improvements of the results obtained in [6].

Remark 2 With an appropriate choices of the func-tions H h, and k, one can derive a number of oscilla-tion criteria for Equaoscilla-tion (1) from our theorems.

Let k t( )1, 0 is a constant, H

  

t s,  t s



,

 

1

, ,

t s s

h   t  t s t0, and we have

 

0

_{ }

 

0

, ,

lim lim 1

, ,

t t

H t s t s

H t t t t 



    for any st0.

Consequently, let  2, using Theorem 2.2, we have:

Corollary 2.1 Assume that (A1)-(A5) and (8) hold,

suppose that there exists a function mC t





0,



,R



such that, for some  1,



  

   

 

2 2

0

2

1

lim sup d

,

l t

n

j j

j t

T M t s q s s

t

N s s

m T t T t

 



 

 _



 

 

 _ _ 

 

 

 









(16) and (10) (with k s

 

1) hold. Then every solution of Equation (1) is oscillatory.

Example 1 Let t



4,



, consider the following second order neutral differential equation

 



 



 



 



0

x t p t x  t  q t x  t

    

(5)

where

 

1,

 

max 2 1







sin , 0 ,



 

2

p t  q t  t t f x x,

 

₄

_

_

4

_

d

1 2 sin

t

t s

s s

 

 



, in this case M 1, Let

1, 2

N   , by direct calculation, we get



  

_{ }



 









 

2 2

1

lim sup d

1

lim sup 1 sin 1 2 sin d cos sin cos 1

t T t

M t s q s s

N s

t

t s s s s s s

t

m T T T T T

 

  

 

 

 

 

 

 

 _      _

 



  



It is easy to verify that (10) holds, therefore, Equation (17) is oscillatory by Corollary 2.1. However, we can easily find that

 







0 2 0

1

lim sup , lim sup t 1 2 sin d

t

t G t t t _t



s  s s 

so condition (4) in Theorem B is not satisfied, these show that Theorem B cannot be applied to Equation (17). Obviously our results are superior to the results obtained before.

3. Acknowledegments

The authors are very grateful to the referee for his/her valuable suggestions.

4. References

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[2] R. P. Agarwal and S. R. Grace, “The Oscillation of High-er OrdHigh-er DiffHigh-erential Equations with Deviating Argu-ments,” Computers & Mathematics with Applications, Vol. 38, No. 3-4, 1999, pp. 185-199.

doi:10.1016/S0898-1221(99)00193-5

[3] Y. Bolat and O. Akin, “Oscillatory Behavior of Higher Order Neutral Type Nonlinear Forced Differential Equa-tion with Oscillating Coefficients,” Journal of Mathe-matical Analysis and Applications, Vol. 290, No. 1, 2004, pp. 302-309.doi:10.1016/j.jmaa.2003.09.062

[4] W. N. Li, “Oscillation of Higher Order Delay Differential Equations of Neutral Type,” The Georgian Mathematical Journal, Vol. 7, No. 2, 2000, pp. 347-353.

[5] Ch. G. Philos, “Oscillation Theorems for Linear Differ-ential Equations of Second Order,” Archiv der Mathe-matik,Vol. 53, No. 5, 1989, p. 483.

doi:10.1007/BF01324723

[6] F. Meng and R. Xu, “Kamenev-Type Oscillation Criteria for Even Order Neutral Differential Equations with Devi-ating Arguments,” Applied Mathematics and Computa-tion, Vol. 190, No. 2, 2007, pp. 1402-1408.

doi:10.1016/j.amc.2007.02.017

[7] Yu. V. Rogovchenko and F. Tuncay, “Oscillation Criteria For Second-Order Nonlinear Differential Equations with Damping,” Nonlinear Analysis: Theory, Methods & Ap-plications, Vol. 69, No. 1, 2008, pp. 208-221.