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席鸿建,孙太祥,赵金凤.两类非线性差分方程的全局渐近稳定性[J].广西科学,2006,13(2):93-95. [点击复制]
- XI Hong-jian,SUN Tai-xiang,ZHAO Jin-feng.Global Asymptotic Stability of Two Families of Nonlinear Difference Equations[J].Guangxi Sciences,2006,13(2):93-95. [点击复制]
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| 两类非线性差分方程的全局渐近稳定性 |
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席鸿建1, 孙太祥2, 赵金凤2
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| (1.广西财经学院数学系, 广西南宁 530003;2.广西大学数学与信息科学学院, 广西南宁 530004) |
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| 摘要: |
| 利用泛函分析方法证明差分方程xn+1=(∑i∈Zk-{j,s,t}xn-i+xn-tr+xn-jxn-sm+A)/(∑i∈Zk-{j,s,t}xn-i+xn-sm+xn-jxn-tr+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. |
| 关键词: 差分方程 平衡点 全局渐近稳定性 |
| DOI： |
| 投稿时间：2005-12-21 |
| 基金项目:Supported by NSF of China (10361001,10461001) and NSF of Guangxi(0447004) |
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| Global Asymptotic Stability of Two Families of Nonlinear Difference Equations |
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XI Hong-jian1, SUN Tai-xiang2, ZHAO Jin-feng2
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| (1.Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi, 530003, China;2.College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, China) |
| Abstract: |
| Two families of difference equations are discussed.They are the form xn+1= (∑i∈Zk-{j, s, t}xn-i+xn-tr+xn-jxn-sm+A)/ (∑i∈Zk-{j, s, t}xn-i+xn-sm+xn-jxn-tr+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 x=1 is globally asymptotically stable, which includes the corresponding results of the references[3~5, 7]. |
| Key words: difference equation equilibrium global asymptotic stability |
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