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The equi-boundedness of solutions and the stability of the zero of nonlinear difference equation with bounded multiple delay $$x_{n+1} = \lambda_nx_n+ \sum_{i = 1}^r\alpha^i_nF(x_{n-m_n}), \quad n = 0, 1, \cdots$$ are investigated. Keywork: stability, fixed point theorem, contraction mapping, nonlinear difference equation, equi-boundedness. | VNU Journal of Science Mathematics - Physics 25 2009 91-98 Boundedness and Stability for a nonlinear difference equation with multiple delay Dinh Cong Huong Ngo Thi Hong Dept of Math Quy Nhon University 170 An Duong Vuong Quy Nhon Binh Dinh Vietnam Received 24 February 2009 received in revised form 11 July 2009 Abstract. The equi-boundedness of solutions and the stability of the zero of nonlinear difference equation with bounded multiple delay C j nF xn mn n 0 1 i l are investigated. Keywork stability fixed point theorem contraction mapping nonlinear difference equation equi-boundedness. 1. Introduction Let R denote the set of real numbers z the set of integers and z the set of positive integers numbers. In this paper we study the equi-boundedness of solutions and the stability of the zero of nonlinear difference equation with bounded multiple delay C nXn H 2 f l C j Tl 0 1 1.1 2 1 where cL for i 1 2 r and À are functions mapping z to R F maps R to R m maps z to z . The properties of solutions of delay nonlinear difference equations has been studied extensively in recent years see for example the work in 1-6 and the references cited therein. In 1 2 and 3 the authors studied the oscillation and the asymptotic behaviour of solutions of the following nonlinear difference equations xn a n xn-m 0 n 0 1 2 xr - xn oii n xn_mi 0 n 0 1 2 2 1 i- i - xn a n f xn m 0 n 0 1 2 and 21 1 I 11 11 X zzZ22j i l Corresponding author. Tel. 0984769741 E-mail dconghuong@yahoo.com 91 92 D.c. Huong N.T. Hong VNU Journal of Science Mathematics - Physics 25 2009 91-98 It is clear that these equations are particular cases of 1.1 . We are particularly motivated by the work of the authors 1-6 on the stability boundedness and convergence of solutions of difference equations. Throughout this paper we assume that there is a K 0 so that if I a I K then F x Kịxị. If m is bounded and the maximum of m is k then for any integer no f 0 we define Zo to be the set of integers in no k no . If m is .
