Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space

Abstract

It is known (see [J. M. 157-174]) that the subdifferential of a semismooth or essentially smooth locally Lipschitz continuous function f over a Banach space determines this function up to an additive constant in the sense that any other function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c. Unfortunately, those classes of locally Lipschitz continuous functions do not include proper lower semicontinuous convex functions taking the value +∞ at some points. In this paper a new concept of essentially directionally smooth functions is introduced, and it is also shown, by a detailed analysis of enlarged inclusions of their subdifferentials, that these functions are subdifferentially determined up to an additive constant. It is also proved that the class of such functions contains proper lower semicontinuous convex functions and locally Lipschitz continuous functions which are arcwise essentially smooth. Moreover, it is established that the essentially directional smoothness property is preserved under addition. It is also shown that the class of essentially directionally smooth functions includes that of directionally stable functions (studied in [L. Thibault and D. Zagrodny, Canad. Math. Bull., 48 (2005), pp. 283-301]) as well as several other classes of functions.

C 1,ω (·) -regularity and Lipschitz-like properties of subdifferential

“…The case of condition (a) corresponds to[55, Proposition 4.4] and the cases of conditions (b)-(d) are established in Corollaries 4.9, 4.8 and 4.13, respectively, of the same paper[55]. The final case of approximate convex function follows from,[54, Proposition 3.5] and[55, Proposition 4.15].…”

“…Further, the finiteness over U of the locally Lipschitz continuous arc-wise essentially smooth functions makes clear that neither the extended real-valued convex functions is included in that class nor the converse. The class of essentially directionally smooth functions introduced in [55] allowed their authors on the one hand, to unify several of the above results and, on the other hand, to identify many other interesting amenable functions that are subdifferentially determined. As we shall see below, the class even provides stronger results concerning enlarged inclusion (3.3).…”

“…The above integration property can be expanded to the DC function. In view of this, we need the following property, which is an immediate consequence of [55,Corollary 4.10]. As a simple consequence of Theorem 4.1 and Proposition 4.1 we have the following proposition.…”

“…In this section, we introduce the class of essentially directionally smooth functions. We need first to recall some other concepts which motivated the study in [55] of essentially directionally smooth functions and which will be involved in the present paper.…”

“…The class of eds functions is stable under addition according to the following result of Thibault and Zagrodny [55].…”