一类差分方程的全局渐近稳定性

Global Asymptotic Stability of a Class of Difference Equations

本文主要研究差分方程x_(n+1)=∑ti=1a_ix_(n-m_i)/(q+∑t i=1c_ix_(n-m_i)+∑s k=1 b_kx_(n-n_k)),n=0,1,…的全局性质,记A=∑t i=1 a_i,B=∑s k=1 b_k,C=∑t i=1 c_i和l=max{m_t,n_s},其中a_i0,c_i>0 for all 1≤i≤t,b>0 for all 1≤k≤s,q<A,B≠C,0≤m_1<m_2<…<m_t,0≤n_1<n_2<…>n_s,and{m_1,m_2,…,m_t}∩{n_1,n_2,…,n_s}=?,初始值为正实数。通过构造恰当的方程组和二元函数,证明该方程的唯一平衡解是局部稳定的并且是全局吸引子,得到其平衡解是全局渐近稳定的结论。

In this paper, the global characteristics of the following difference equations are investigated.xn＋1=∑i=1^taixn-mi/q＋∑i=1^tcixn-mi＋∑k=1^sbkxn-nkLet A =∑i=1^tai,B=∑k=1^tbk,C=∑i=1^tciand l =max{mr ,n,}, where ai ai〉0,ci〉0（i=1,2,…,t）,bk〉0（k=1,2,…,s）,q〈A,B≠C,0≤m1〈m2〈…〈mt,0≤n1,〈n2〈…〈ns,and{m1,m2,…,mt}∩{n1,n2,…,ns}=Ф, and the initial values are positive. By constructing a suitable system of equations and binary functions, it is proved that the unique equilibrium solution of the equation is locally stable and a global attractor. In other words, the solution is globally asymptotically stable.