V -Prox-Regular Functions in Smooth Banach Spaces

Messaoud Bounkhel,Mostafa Bachar

doi:10.1155/2020/9465492

Abstract

In this paper, we continue the study of the V -prox-regularity that we have started recently for sets. We define an appropriate concept of the V -prox-regularity for functions in reflexive smooth Banach spaces by adapting the one given in Hilbert spaces. Our main goal is to study the relationship between the V -prox-regularity of a given l.s.c. f and the V -prox-regularity of its epigraph.

Highlights

Introduction and PreliminariesThroughout this work, X will denote a reflexive smooth Banach space unless otherwise specified
J is the normalised duality mapping on X, and V is the functional defined from X∗ × X to 1⁄20, ∞Þ by
The V-proximal normal cone of a nonempty closed subset S in X at x ∈ S is defined as the V-proximal subdifferential of the indicator function of S, that is, Another proximal subdifferential

Summary

Introduction and Preliminaries

Throughout this work, X will denote a reflexive smooth Banach space unless otherwise specified. The V-proximal normal cone of a nonempty closed subset S in X at x ∈ S is defined as the V-proximal subdifferential of the indicator function of S, that is, Another proximal subdifferential. N∂πGπfðSðx;ÞxÞis=d∂eπfiψnSeðdxÞ(.see [5]) geometrically via the V-proximal normal cone of the epigraph as follows: It has been proved in [2] that in general, we have the inclusion ∂π f ðxÞ ⊂ ∂πG f ðxÞ. We recall (see for instance [2]) the definition of the Fréchet subdifferential and Fréchet normal cone as follows: x∗ ∈ ∂F f ðxÞ if and only if for all ε > 0, there exists δ > 0 such that hx∗ ; x − xi ≤ f ðxÞ − f ðxÞ + ε∥x − x∥,∀x ∈ x + δB: ð4Þ.

Generalized V-Prox-Regular Sets

V-Prox-Regular Functions