A coercive James's weak compactness theorem and nonlinear variational problems
type: Item
instance of: scholarly article
title: A coercive James's weak compactness theorem and nonlinear variational problems
zbMATH Open document ID: 1248.46015
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-12-08T00:00:00Z
review text: Let \(E\) be a Banach space, \(E^{\ast }\) its dual and \(\partial f:E\rightrightarrows E^{\ast }\) denote the Fenchel-Moreau subdifferential of the function \(f:E\rightarrow \mathbb{R}\cup \{+\infty \}\), that is, \(x^{\ast }\in \partial f(x_{0}),\) if and only if \(x_{0}\) is a (global) maximizer of the function \(x^{\ast }-f\) on \(E\). Provided \(\partial f(E)=E^{\ast }\) (equivalently, for all \(x^{\ast }\in E^{\ast }\), the function \(x^{\ast }-f\) attains its supremum in \(E\)), the authors show that the function \(f\) is \(1\)-coercive (that is, \(f(x)/\|x\|\) tends to \(+\infty \) as \(\|x\|\) goes to \(+\infty \)) if and only if the subdifferential \(\partial f\) maps bounded subsets of \(E\) to bounded subsets of \(E^{\ast }\), and under this assumption, the sublevel sets \([f\leq c],\) \(c\in \mathbb{R}\), of \(f\) are relatively weakly compact. This result yields the following generalization of the classical James theorem: if \(A\) is a weakly closed subset of \(E\), \(\psi :A\rightarrow \mathbb{R}\) is a bounded function and for all \(x^{\ast }\in E^{\ast }\) the function \(x^{\ast }-\psi \), defined in \(A\), attains its supremum in \(A,\) then \(A\) is weakly compact. (Taking in particular \(A\) to be the unit ball of \(E\) and \(\psi \) any constant function there, we obtain the classical characterization of reflexivity of \(E\) via norm attaining functionals.) In the last part of the paper, the authors revisit some results from mathematical finance as well as surjectivity results for monotone coercive functionals, and come up with the assertion that reflexivity is the unique possible framework for the development of existence results for a wide class of nonlinear variational problems.
reviewed by: A. Daniilidis
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 91B30
zbMATH DE Number: 5987793
zbMATH Keywords: James's theorem
zbMATH Keywords: weak compactness
zbMATH Keywords: coercive function
zbMATH Keywords: subdifferential
zbMATH Keywords: nonlinear variational problems
zbMATH Keywords: mathematical finance
author: José Orihuela
author: M. Ruiz Galán
MaRDI profile type: Publication
full work available at URL: j.na.2011.08.062
OpenAlex ID: W2036971432
cites work: In a nonreflexive space the subdifferential is not onto
cites work: Reflexivity and the sup of linear functionals
cites work: A new minimax theorem and a perturbed James's theorem
cites work: Law invariant risk measures have the Fatou property
cites work: Differentiability Properties of Utility Functions
cites work: Convex functions, monotone operators and differentiability.
cites work: Techniques of variational analysis
cites work: Convex numerical radius
cites work: ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES
cites work: New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals
cites work: A one perturbation variational principle and applications
cites work: Lebesgue property for convex risk measures on Orlicz spaces
cites work: On Sequences without Weak ∗ Convergent Convex Block Subsequences
cites work: Weakly Compact Sets
cites work: A counterexample for a sup theorem in normed spaces
cites work: Stochastic finance. An introduction in discrete time
cites work: Methods in Nonlinear Analysis
cites work: The extension of bilinear functionals
cites work: Q3372673
cites work: Numerical treatment of partial differential equations. Revised translation of the 3rd German edition of `Numerische Behandlung partieller Differentialgleichungen' by Martin Stynes.
cites work: Q5852076
cites work: Characterization of the subdifferentials of convex functions
DOI: 10.1016/J.NA.2011.08.062
description: scientific article; zbMATH DE number 5987793