V-Proximal Trustworthy Banach Spaces

Messaoud Bounkhel,Mostafa Bachar

doi:10.1155/2020/4274160

Abstract

In a recent work (2016), the first author proved the fuzzy sum rule for the V-proximal subdifferential under some natural assumptions on an equivalent norm of the Banach spaces. In the present paper, we are going to prove that the class of Banach spaces satisfying the fuzzy sum rule is very large and contains all Lp spaces 1<p<∞ as well as the sequence spaces lp1<p<∞, the Sobolev spaces Wp,n1<p<∞, and the Schatten trace ideals Cp1<p<∞.

Highlights

The Fréchet and proximal normal cones are defined as N FðS ; xÞ = ∂FψSðxÞ and NPðS ; xÞ = ∂PψSðxÞÞ
Using the same terminology used in Ioffe [6], we will say that X is a V-proximal trustworthy space provided that for any ε > 0, any two functions f1, f2 : X ⟶ R ∪ f∞g and any u ∈ X such that f1 is lower semicontinuous and f2 is Lipschitz around u, the following fuzzy sum rule holds: Journal of Function Spaces n o
U f i ðu, εÞ ≔ fx ∈ u + εB such that j f iðxÞ − f iðuÞj < εg and B∗ denotes the closed unit ball in X∗. It has been proved in Theorem 2.3 in [1] that if X is a Banach space with an equivalent norm k⋅k such that k·ks is C2-differentiable on X \{0} and let V be the functional associated to that norm k⋅k, X is a V -proximal trustworthy space

Summary

Introduction

Let X be a Banach space with dual X∗, f : X ⟶ R ∪ f∞g a function, x ∈ dom f : fx ∈ X : f ðxÞ 0 and q ∈ ð1, 2, VðJðxÞ, xÞ ≤ σkx − xkq, for all x ∈ x + δB: ð15Þ

Results

Conclusion