局部Lipschitz连续函数差的刻画

Characterizations of the Difference of Local Lipschitz Continuous Functions

在Banach空间中利用广义方向导数和Clarke次微分的定义,指出两个局部Lipschitz连续函数差与Clarke次微分之间的关系。在此基础上,指出如果两个局部Lipschitz连续函数f,g:X→R是Clarke正则的,那么结果退化到经典意义下ε次微分与局部Lipschitz连续函数差的关系,并指出了当函数h是可微偶凸函数时,在定理1的条件下两个局部Lipschitz连续函数的Clarke次微分之间的关系,最后指出当两个局部Lipschitz连续函数差为常数时,两个函数的Clarke次微分之间的关系。

The definition of generalized directional derivative and Clarke subdifferential are used in Banach spaces, and the relation between the difference of two local Lipschitz continuous functions and Clarke subdifferential is pointed out. Based on this, if the two local Lipschitz continuous functions fand g： X → R are Clarke regular, the result is degenerated to the relationship between the classical ε-subdifferential and the difference of two local Lipschitz continuous functions. Then when h is a function which is differentiable, even and convex, under conditions of the theorem 1, the relationship of Clarke differ- entials of two local Lipschitz continuous functions is pointed out. Finally, when the difference of two local Lipschitz continu- ous functions is a constant, the relationship of Clarke differentials of the two functions is pointed out.