关键词: 差分方程, 正解, 全局渐近稳定

Abstract: In this paper, the global characteristics of the following difference equations are investigated:$x_{\text{n+1}}=\frac{{\sum^t_{i=1}} a_i x_{n-m_i}}{q+{\sum^t_{i=1}} c_i x_{n-m_i}+{\sum^t_{k=1}} b_k x_{n-n_k}}$.Let $A=\sum^t_{i=1} a_i$,$B=\sum^s_{k=1} b_k$,$B=\sum^s_{k=1} c_i$and $l=\max\left \{m_t,n_s\right \}$, where a_i>0 ,c_i>0 for all 1≤i≤t, b_k>0 for all 1≤k≤s,q12<…t, 0≤n₁2<…s and {m₁,m₂,…,m_t}∩{n₁,n₂,…,n_s}=?, 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.

Key words: difference equations, positive solution, global asymptotically stable