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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

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On the theory of subdifferentials

Alexander D. Ioffe
Published Online: 2012-01-24 | DOI: https://doi.org/10.1515/ana-2011-0003


The theory presented in the paper consists of two parts. The first is devoted to basic concepts and principles such as the very concept of a subdifferential, trustworthiness and its characterizations, geometric consistence, fuzzy principles and calculus rules, methods of creation of new subdifferentials etc. In the second part we study certain specific subdifferentials, namely, subdifferentials associated with bornologies, their limiting versions and metric modifications. For each subdifferential we verify that the basic properties discussed in the first part are satisfied and prove calculus rules for two main operations: summation and partial minimization. Separate sections are devoted to the Fréchet and limiting Fréchet subdifferentials and to the approximate G-subdifferential. For the last two new definitions are given which lead to a certain unification and simplification of analysis. The sum rule for these two subdifferentials is proved with the so-called “linear metric qualification condition”, so far the most general. In the last section we briefly discuss how other operations reduce to the two mentioned basic operations and give the corresponding calculus rules (with suitable versions of the metric qualification conditions).

Keywords.: Banach space; subdifferential calculus; limiting subdifferentials; metric modification; G$G$-subdifferential; normal cone; coderivative; trustworthiness; separable space; separable reduction

About the article

Received: 2011-05-04

Accepted: 2011-11-21

Published Online: 2012-01-24

Published in Print: 2012-02-01

Citation Information: Advances in Nonlinear Analysis, Volume 1, Issue 1, Pages 47–120, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/ana-2011-0003.

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