tailieunhanh - Báo cáo "On the Oscillation, the Convergence, and the Boundedness of Solutions for a Neutral Difference Equation "

On the Oscillation, the Convergence, and the Boundedness of Solutions for a Neutral Difference Equation In this paper, the oscillation, convergence and boundedness for neutral difference equations $$\Delta(x_n + \delta_nx_{n-\tau}) + \sum_{i = 1}^r\alpha_i(n)F(x_{n-m_i})=0, \quad n = 0, 1, \cdots$$ are investigated. Keywork: Neutral difference equation, oscillation, nonoscillation, convergence, boundedness. | VNU Journal of Science Mathematics - Physics 26 2010 155-162 On the Oscillation the Convergence and the Boundedness of Solutions for a Neutral Difference Equation Dinh Cong Huong Dept of Math Quy Nhon University 170 An Duong Vuong Quynhon Binhdinh Vietnam Received 14 April 2009 Abstract. In this paper the oscillation convergence and boundedness for neutral difference equations A z ỗnxn_T y oti n F xn_mi 0 n 0 1 i l are investigated. Keywork Neutral difference equation oscillation nonoscillation convergence boundedness. 1. Introduction Recently there has been a considerable interest in the oscillation of the solutions of difference equations of the form A rrra ỏxn_T a rì xn_ơ 0 where n G N the operator A is defined as Axn Xn-Ị-1 xn the function a n is defined on N Ỗ is a constant T is a positive integer and Ơ is a nonnegative integer see for example the work in 1-7 and the references cited therein . In 2 the author obtained some sufficient criterions for the oscillation and convergence of solutions of the difference equation A rrra ổwIIT y ai n F xn_mi 0 i l for n G N n f a for some a G N the operator A is defined as Axn Xn-Ị-1 xn J is a constant T r TO1 TO2 mr are fixed positive integers and the functions otfn are defined on N and the function F is defined on R. Motivated by the work above in this paper we aim to study the oscillation and asymptotic behavior for neutral difference equation A irra Ỗnxn T afri F xn_mịf 0 1 i l where ỏn n G N is not zero for infinitely many values of n and F R R is continuous. Corresponding author. Tel. 0984769741 E-mail dinhconghuong@ 155 156 . Huong VNU Journal of Science Mathematics - Physics 26 2010 155-162 Put A max r TO1 mr . Then by a solution of 1 we mean a function which is defined for n f A and sastisfies the equation 1 for n G N. Clearly if J an n A A 1 0 are given then 1 has a unique solution and it can be constructed recursively. A nontrivial solution x no of 1 is called oscillatory if for any ni f no there .