Journal of Convex Analysis 26 (2019), No. 4, 1125--1144
Copyright Heldermann Verlag 2019
An Abstract Variational Theorem
Warren B. Moors
Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand
w.moors@auckland.ac.nz
Neset Oezkan Tan
Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand
neset.tan@auckland.ac.nz
[Abstract-pdf]
Let $(X, \|\cdot\|)$ be a Banach space and $f\colon X \to \mathbb{R} \cup \{\infty\}$ be a proper function. Then the {\it Fenchel conjugate of} $f$ is the function $f^*\colon X^* \to \mathbb{R} \cup \{\infty\}$ defined by, $$ f^*(x^*) := \sup\{(x^*-f)(x):x \in X\}. $$ In this article we will prove a theorem more general than the following. \par\medskip {\bf Theorem:} Let $f\colon X \to \mathbb{R} \cup \{\infty\}$ be a proper function on a Banach space $(X,\|\cdot\|)$. If there is a nonempty open subset $A$ of $\mathrm{Dom}(f^*)$ such that $\mathrm{argmax}(x^*-f) \not= \varnothing$ for each $x^* \in A$, then there is a dense and $G_\delta$ subset $R$ of $A$ such that $(x^*-f) \colon X \to \mathbb{R} \cup \{-\infty\}$ has a strong maximum for each $x^* \in R$. In addition, if $0 \in A$ and $0<\varepsilon$ then there is an $x^* \in X^*$ with $\|x^*\| < \varepsilon$ such that $(x^* -f) \colon X \to \mathbb{R} \cup \{-\infty\}$ has a strong maximum.
Keywords: Variational theorem, James' weak compactness theorem, convex analysis.
MSC: 46B20; 46B10, 46B50
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