非线性Lipschitz算子的一个特征不变量及其应用

A Novel Characteristic Constant of Nonlinear Lipschitz Operator and its Applications

本文通过推广有界线性算子对偶到非线性Lipschitz算子的方法,将谱半径的概念推广到非线性情形,从而得到一个有关非线性Lipschitz算子的特征数．作为应用,本文在一定条件下证明:非线性离散系统的收敛性可由算子T在各点处的．Jacobi矩阵谱半径确定,从而部分地证明LaSalle提出的一个公开性猜想．

In this paper, the dual notion of linear bounded operator is generalized to the nonlinear case, and then a novel quantity characterizing the convergence of nonlinear discrete system xn＋1 = Txn is developed. This new quantity is independent of the choice of equivalent metrics of the concerned space. As an application example, it is proved that under some common conditions the convergence of nonlinear discrete system xn＋1= Txn can really be characterized with the spectral radius of the Jacobi matrices of T＇（x）, and therefore, one of LaSalle＇s conjectures is partially proved.