两类非线性差分方程的全局渐近稳定性

利用泛函分析方法证明差分方程xn 1=∑i∈Zk-{j,s,t}xn-i xrn-t xn-jxmn-s A∑i∈Zk-{j,s,t}xn-i xnm-s xn-jxnr-t 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-j0xn-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}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.

Global Asymptotic Stability of Two Families of Nonlinear Difference Equations

Two families of difference equations are discussed. They are the form xn+1=∑i∈Zk-{j,s,t}xn-i+xrn-t+xn-jxmn-s+A/∑i∈Zk-{j,s,t}xn-i+xmn-s+xn-jxrn-t+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-j0xn-j1...xn-js+1/∑i∈Zk-{j0,j1,...,js-1}xn-i+xn-j0xn-j1...xn-js-1,n=0,1,...,where k∈{1,2,3,...},1≤s≤k,{j0...,js}( )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 =1 is globally asymptotically stable,which includes the corresponding results of the references 3～5,7].