一类高阶非线性时滞差分方程的全局吸引性

Global attractivity for a class of higher-order nonlinear difference equations

考虑高阶非线性差分方程xn+1=f(xn,xn-1,…,xn-k),n=0,1,…,其中f∈C[(0,∞)k+1,(0,∞)],f(u0,u1,…,uk)关于ui(i=0,1,…,k)均为严格单调递减的,且初值x-k,…,x0均为正.利用分析理论中的极限方法和迭代方法以及不等式技巧,分别给出了该方程的正平衡解是全局吸引的若干充分条件.将所得结论应用于非线性差分方程xn+1=∑ki=0Aixnpi-i,n=0,1,…,其中Ai,pi>0,i=0,1,…,k,且初值x-k,…,x0均为正,得到了该方程的正平衡解是方程的所有正解的全局吸引子的一个充分条件,部分地回答了Ladas和Kocic提出的一个公开问题.

Consider the higher order nonlinear difference equation x（n＋1）=f（xn,x（n-1）,…,x（n-k））,n=0,1,…,where f∈C[（0,∞）^k＋1,（0,∞）],f（u0,u1,…,uk） is strictly non -increasing in ui （i = O, 1 ,＇.＂ ,k ） , and the initial conditions x -k,… ,X0 are positive. By using the limit method and iteration method of the analysis theory and inequality technique, the sufficient conditions are obtained for global attractivity of the positive equilibrium of the nonlinear difference equation. Appling the obtained results to the following nonlinear difference equation x（n＋1）=^k ∑i=0 Ai/x^Pi （n-i）,n=0,1,…, where Ai,pi〉0,i=0,1,…,k, and the initial conditions x-k,…,x0 are positive, a sufficient condition is obtained that all positive equilibriums of the equation are global attractors, which answers partially the open problem raised by Kocic and Ladas.