两类非线性差分方程的全局渐近稳定性(英文)

Global Asymptotic Stability of Two Families of Nonlinear Difference Equations

利用泛函分析方法证明差分方程xn＋1=∑i∈Zk-{j,s,t}^xn-i＋xn-t^r＋xn-jxn-s^m＋A/∑i∈Zk-{j,s,t}^xn-i＋xn-s^m＋xn-jxn-t^r＋A,n=0,1,…,其中k∈{2,3,…},j,s,t∈Zk≡{0,1,…,k}（s≠t,j￠{s,t}）,A,r,m∈[0,＋∞]且初始条件x-k,x-k＋1,…,x0∈（0,＋∞），和差分方程xn＋1=∑i∈Zk-{j0,j1,…,js}^xn-i＋xn-j0＋xn-j1…xn-js＋1/∑i∈Zk-{j0,j1,…js-1}^xn-i＋xn-j0xn-j1…xn-js-1,n=0,1,…,其中k∈{1，2,3,…},1≤s≤k,{j0,…,js}包函Zk（ji≠jl对i≠l）且初始条件x-k,x-k＋1,…,x0∈（0,＋∞）的唯一平衡点^-x=1是全局渐近稳定的， 该结果推广了文献[3～5，7]中相应的结果．

Two families of difference equations are discussed. They are the form xn＋1=∑i∈Zk-{j,s,t}^xn-i＋xn-t^r＋xn-jxn-s^m＋A/∑i∈Zk-{j,s,t}^xn-i＋xn-s^m＋xn-jxn-t^r＋A,n=0,1,… where k∈{2,3,…},j,s,t∈Zk≡{0,1,…,k} with s≠t and j￠{s,t},A,r,m∈[0,＋∞] and the initial values x-k,x-k＋1,…,x0∈（0,＋∞） ,and the form xn＋1=∑i∈Zk-{j0,j1,…,js}^xn-i＋xn-j0＋xn-j1…xn-js＋1/∑i∈Zk-{j0,j1,…js-1}^xn-i＋xn-j0xn-j1…xn-js-1,n=0,1,… wherek k∈{1,2,3,…},1≤s≤k,{j0,…,js}belong to Zk with ji≠jl for i≠l and the initial values x-k,x-k＋1,…,x0∈（0,＋∞）For these difference equations,it is proved that the unique equilibrium ^-x ： 1 is globally asymptotically stable,which includes the corresponding results of the references [3-5,7].