非线性Lipschitz-α算子的若干性质

Some Properties of Nonlinear Lipschitz-α Operators

本文引入并研究Ｌｉｐｓｃｈｉｔｚ－α算子（简称Ｌｉｐ－α算子）我们首先给出这类算子的定义及其基本性质；然后，讨论Ｌｉｐ－α算子的可逆性并引入它的α－阶条件数，并给出其在研究非线性算子方程扰动问题中的一个应用；其次，还研究了Ｌｉｐ－α算子列的收敛性，引入并研究了Ｌｉｐ－α极限与Ｌｉｐ－α Ｃａｕｃｈｙ列，证明过零Ｌｉｐ－α算子空间是一个Ｂａｎａｃｈ空间．

In this paper, we introduce and discuss nonlinear Lipschitz-α operators (shortly, Lip-α operators). At first, we define Lip-αoperators and obtain some basic properties of them. Secondly, the invertibility of these op erators is discussed and the conditional number of order a of a Lip-αoperator is defined and is used for dealing with 'the Nonlinear Perturbation-Problem' of a nonlinear operator equation. Thirdly, the convergence of a sequence of Lip-αoperators is studied, and the Lip-αlimit and Lip-αCauchy sequence are introduced and discussed. We also prove that the space of all pass-zero Lip-αoperators from a set into a Banach space is a Banach space.