下半连续函数的Proximal-次微分与广义中值定理

数学物理学报 ›› 1998, Vol. 18 ›› Issue (3): 324-329.

下半连续函数的Proximal-次微分与广义中值定理

郭兴明

上海大学上海市应用数学和力学研究所上海 200072

收稿日期:1996-11-25 出版日期:1998-09-26 发布日期:1998-09-26

Proximal-subdifferential of lower semicontinuous functions and generalized mean value theorems

Guo Xingming

Shanghai University;Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072

Received:1996-11-25 Online:1998-09-26 Published:1998-09-26

摘要/Abstract

摘要： 该文对定义在Hilbert空间E上的一般下半连续函数证明了如[9]中形式的逼近中值定理在Proximal-次微分意义下也成立.若E=[a,b]⊂R,则得到了不等式形式的中值定理.作为应用给出了函数凸性、Lipschitz性质及常数性质的Proximal-次微分刻划.

关键词: Proximal-次微分, 广义中值定理, 非光滑分析

Abstract: This present paper show that, for general lower semicontinuous functions defined on Hilbert space E,the approximate mean value theorem such as that in[9] is also valid for Proximal-subdiferential. If E=[a,b]⊂R, the mean value theorem in inequality form is obtained. By use of the approximate mean value theorem, the Proximal-subdifferential descriptions of convexity, Lipschitz and constant properties of functions is given.

Key words: Proximal-subdifferential, Generalized mean value theorems, Nonsmooth analysis

郭兴明. 下半连续函数的Proximal-次微分与广义中值定理[J]. 数学物理学报, 1998, 18(3): 324-329.

Guo Xingming. Proximal-subdifferential of lower semicontinuous functions and generalized mean value theorems[J]. Acta mathematica scientia,Series A, 1998, 18(3): 324-329.