非线性Lipschitz算子半群的渐近性质及其应用

On Asymptotic Behaviours of Nonlinear Semigroup of Lipschitz Operators with Applications

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.

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.