Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Messaoud Bounkhel,Mostafa Bachar

doi:10.3390/math8112066

Abstract

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

Highlights

Introduction and PreliminariesIn all the paper, X will denote a reflexive smooth Banach space, unless otherwise specified.We quote from [1] the definition of the V-proximalsubdifferential.Definition 1 ([1])
We recall that the V-proximal subdifferential of f at x is defined as x ∗ ∈ ∂π f ( x ) if and only if there exists σ > 0 such that: h x ∗ ; x 0 − x i ≤ f ( x 0 ) − f ( x ) + σV ( J ( x ), x 0 )), ∀ x 0 near x
The V-proximal normal cone of a non-empty closed subset S in X at x ∈ S is defined as the V-proximal subdifferential of the indicator function of S, that is N π (S; x ) = ∂π ψS ( x )

Summary

Introduction and Preliminaries

X will denote a reflexive smooth Banach space, unless otherwise specified. The V-proximal normal cone of a non-empty closed subset S in X at x ∈ S is defined as the V-proximal subdifferential of the indicator function of S, that is N π (S; x ) = ∂π ψS ( x ). G f ( x ) is defined geometrically, in [1], via the V-proximal normal cone of the epigraph as follows:. In [3], the first author extended the concept to nonconvex closed sets It was proven in [1] that the V-proximal normal cone is characterised in terms of the generalised projection as follows:. The Fréchet normal cone N F (S; x ) of a non-empty closed subset S in X at x ∈ S is defined as.

V-Proximal Trustworthy Spaces

On Generalised V-Prox-Regular Sets

V-Primal Lower Nice Functions