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Subdifferential determination of a primal lower regular function on a Banach space

M. Ivanov, M. Konstantinov, N. Zlateva

TL;DR

This work extends the Thibault–Zagrodny subdifferential determination from finite dimensions and Hilbert spaces to general Banach spaces by employing a slope-based framework. It develops a robust slope toolkit, relates slopes to the Clarke–Rockafellar subdifferential via $F$-regularity, and proves that if two proper l.s.c. $plr$ functions have identical subdifferentials locally, they differ by a constant on a neighborhood. The main result provides a Banach-space analogue of local constancy under subdifferential equality, with the neighborhood radius explicitly given as $\hat{\delta}=\min\{\delta/2, 1/(18c)\}$. Overall, the paper enhances non-convex variational calculus on Banach spaces and offers practical slope-based methods for local comparison and stability analyses.

Abstract

We generalize to Banach space Thibault-Zagrodny Theorem that if $f$ and $g$ are primal lower regular functions and $\partial f = \partial g$ locally, then $f$ and $g$ locally differ by a constant.

Subdifferential determination of a primal lower regular function on a Banach space

TL;DR

This work extends the Thibault–Zagrodny subdifferential determination from finite dimensions and Hilbert spaces to general Banach spaces by employing a slope-based framework. It develops a robust slope toolkit, relates slopes to the Clarke–Rockafellar subdifferential via -regularity, and proves that if two proper l.s.c. functions have identical subdifferentials locally, they differ by a constant on a neighborhood. The main result provides a Banach-space analogue of local constancy under subdifferential equality, with the neighborhood radius explicitly given as . Overall, the paper enhances non-convex variational calculus on Banach spaces and offers practical slope-based methods for local comparison and stability analyses.

Abstract

We generalize to Banach space Thibault-Zagrodny Theorem that if and are primal lower regular functions and locally, then and locally differ by a constant.

Paper Structure

This paper contains 4 sections, 20 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Slope regularity
  3. Clarke-Rockafellar subdifferential
  4. Primal lower regular functions

Key Result

Theorem 2

Let $(E,\|\cdot\|)$ be a Banach space. Let the functions $f,g:E\to\mathbb{R}\cup \{+\infty\}$ be proper and lower semicontinuous. Let for some $c,\delta > 0$. Let Then there is a constant $a\in\mathbb{R}$ such that where

Theorems & Definitions (42)

  • Definition 1: cf. thibault-book, Definition 11.1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6: Ekeland
  • Lemma 7
  • ...and 32 more