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

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 oscillatory if all of it’s nontrivial solutions are oscillatory. 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  , where     0 , : D t s t s t    , which satisfies: (H1)   , 0 H t t  and   , 0 H t s  for


Introduction
Oscillation of some even order differential equations have been studied by many authors.For instance, see [1][2][3][4][5][6][7] and the references therein.We deal with the oscillatory behavior of the even order neutral differential equations with deviating arguments of the form where 2 n  is even, throughout this paper, it is assumed that: (A 1 ) and there exists a func- . By a solution of Equation ( 1) we mean a function

 
x t which has the property that   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 Theorem A ([6, Theorem 2.1]).Assume that (A 1 ) -(A 5 ) hold, let the functions , , H h k satisfy (H 1 ) and (H 2 ), suppose holds for every and If there exists a function and where , then every solution of Equation ( 1) is oscillatory.
In Theorem A and B, function , so each of the condition (2), ( 4), ( 5) and ( 6) has as many as l conditions.Meanwhile, the Riccati func- tion 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 obtained for Equation (1) by Meng and Xu [6].In our paper, we redefine the functions  and provide some new oscillation criteria for oscillation of Equation (1).

Main Results
In the sequel, we need the following lemmas: Let   x t be a n times differentiable function on There exists a for all large t .

Lemma 2.3([7]).
Suppose that   x t is an eventually positive solution of Equation ( 1) Theorem 2.1 Assume that (A 1 ) -(A 5 ) hold, let the functions , , H h k satisfy (H 1 ) and (H 2 ), suppose holds for every 0 r t  and for some 1   , where    , 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 even-tually positive (when   x t is eventually negative, the proof is similar).
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 which contradicts (7).This completes the proof of the Theorem.
The assumption (7) in Theorem 2.1 can fail, consequently, Theorem 2.1 does not apply.The following results provide some essentially new oscillation criteria for Equation (1).
Theorem 2.2 Assume that (A 1 )-(A 5 ) hold, the functions , , , H h k F and G be the same as in Theorem 2.1, suppose that If there exists a function for all 0 t T t   and for some 1 and . Then every solution of Equation ( 1) is oscillatory.
Proof.Assume to the contrary that ( 1) is non-oscillatory.Following the proof of Theorem 2.1, without loss of generality, assume for all 0 t T t   and for some 1   , we obtain For all 0 T t  and for any 1   , by (9) we have and especially Now, we claim that Let  be any arbitrary positive number, from (14) there exists a 1 0 t t  such that, ,  Obviously our results are superior to the results obtained before.

Acknowledegments
, so condition (4) in Theorem B is not satisfied, these show that Theorem B cannot be applied to Equation (17).