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非线性Lipschitz算子半群的渐近性质及其应用
引用本文:彭济根,徐宗本.非线性Lipschitz算子半群的渐近性质及其应用[J].数学学报,2002,45(6):1099-110.
作者姓名:彭济根  徐宗本
作者单位:西安交通大学应用数学研究中心及信息与系统科学研究所,陕西,西安,710049
基金项目:国家自然科学基金资助项目(10101019),西安大通大学理科基金资助项目
摘    要:本文对一类非线性算子半群————Lipschitz算子半群的渐近性质进行研究,刻划了非线性Lipschitz算子半群所具有的基本渐近性质(这些性质与线性算子半群所具有的基本渐近性质相一致),证明了作为线性算子对数范数的非线性推广,Dahlquist数能用于刻划非线性Lipschitz算子半群的渐近性质.为克服Dahlquist数只对Lips-chitz算子有定义的缺点,本文引入一个全新的特征数:广义 Dahlquist数,并证明广义Dahlquist数比Dahlquist数能更为精确地刻划Lipschitz算子半群的渐近性质.作为应用,得到关于 Hopfield型神经网络全局指数稳定性的一个新结果.

关 键 词:非线性  Lipschitz  算子半群  生成元  渐近性质  全局指数稳定性
文章编号:0583-1431(2002)06-1099-08
修稿时间:2001年3月21日

On Asymptotic Behaviours of Nonlinear Semigroup of Lipschitz Operators with Applications
Ji Gen PENG,Zong Ben XU.On Asymptotic Behaviours of Nonlinear Semigroup of Lipschitz Operators with Applications[J].Acta Mathematica Sinica,2002,45(6):1099-110.
Authors:Ji Gen PENG  Zong Ben XU
Institution:Ji Gen PENG Zong Ben XU(Research Center for Applied Mathematics, and Institute for Information & System Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China)
Abstract:Abstract This paper is concerned with the asymptotic behaviors of nonlinear semi-group of Lipschitz operators. A series of basic properties similar to those of linear semigroups are characterized. It is shown that the Dahlquist constant, which is a non-linear extension of logarithmic norm of a bounded linear operator, can be applied to characterize the asymptotic behaviours of nonlinear semigroup of Lipschitz operators. Moreover, in order to characterize the asymptotic behaviours of nonlinear semigroups much better, a novel quantity of nonlinear operator, named the generalized Dahlquist constant, is defined. Unlike Dahlqust constant only defined for Lipschitz operator, this new quantity can be defined well for nonlinear operator. As an application, a new result on global exponential stability of Hopfield-type neural networks is proved.
Keywords:Nonlinear semigroup of Lipschitz operators  Generators  Asymptotic be-haviours  Global exponential stability
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