The Oscillation of Second-order Impulsive Delay Difference Equations with Continuous Arguments

Ping Yu (Corresponding author) Department of Science, Yanshan University, Qin Huangdao 066004, China E-mail: swift0910@163.com Xiaozhu Zhong, Ning Li, Wenxia Zhang, Shasha Zhang Department of Science, Yanshan University, Qin Huangdao 066004, China Abstract In this paper, we considered the oscillation of second-order impulsive delay difference equation with continuous arguments, and the sufficient conditions are obtained for oscillation of all solutions and some results in the literatures are improved.


Introduction
Recently, with the development of the medicine, biology mathematics and modern physics and so on, there have been many investigations into the study of delay difference equation.In particular, an extensive literature now exists on the oscillation theory for delay difference equation, and various applications have been found.But the study of the oscillation of impulsive delay difference equations with continuous arguments is very little, and this style of equations has very extensive application.Thus, to study the influence of impulsive to the system's stability has very important value applications.We refer to [1][2][3][4][5][6] and the references cited therein for more details． In this paper, we consider the following second-order impulsive delay difference equations with continuous arguments we define the function satisfy initial condition (2)，then we easy to know that 0 ， (1) and (2) have the unique solution.The solution ( ) t x of equation ( 1) is said to be oscillatory if the terms n x are neither eventually positive nor eventually negative.Otherwise, the solution is called non-oscillatory.

The assistant equation
the solution ( )

Main results and proofs
Theorem 1 If ( ) is the solution of equation(1).Since ( ) is the solution of equation (1), then ( ) ( ) ( ) . We easy to proof that it satisfy equation (3).When it equal to point k t , then is the solution of equation (3), and the proof is completed.
, then all solutions of equation (1) oscillate are equals to all solutions of equation (3) oscillate.
Proof: By the results of theorem 1, we can proof the corollary.Then the proof is completed.
, for all of Then every solution of equation (1) oscillates.
Proof: Firstly, the assistant equation(3)of equation ( 1) can derived to be , equation( 6)can derived to be ( ) Where .
Secondly, by the front conclude, if every solution of equation ( 7) oscillates, then every solution of equation ( 3) oscillates.
Next , we proof that every solution of equation ( 7) oscillates.Assume the contrary.Then equation( 7)may have an eventually positive solution ( ) ,for all to t for equation ( 7), by the integral midst value theorem, then ( ) where is continuous and almost everywhere derivate in ( ) is gradually reduce，and then by ( 9), we have by( 8),( 10)and ( 11),we have ( ) by ( 11) and ( 12), we have (13) by ( 11) and ( 13),we have by simple induction, in the end we have ( ) by ( 11) and ( 14) ,we have As we all know，if ，by the define of ε ,we know if . Thus, by the lemma and (15),we have This contradicts (5), then we prove that the positive solution of equation ( 7) doesn't exist.Similarly, we can proof that the negative solution of equation ( 7) doesn't exist.i.e. the every solution of equation ( 7) oscillates.Thus, the every solution of equation (3) oscillates.By the corollary, the every solution of equation (1) oscillates.
Then the proof is completed.

Conclusion
In this paper, by study the method of the oscillation first-order linear impulsive difference equation with continuous arguments, we study the second-order impulsive delay difference equations with continuous arguments.The sufficient conditions are obtained for oscillation of all solutions and some results in the literatures are improved.