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Group invariant variational principles

Javier Falcó, Daniel Isert

Abstract

In this paper we introduce a group invariant version of the wellknown Ekeland variational principle. To achieve this, we defne the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.

Group invariant variational principles

Abstract

In this paper we introduce a group invariant version of the wellknown Ekeland variational principle. To achieve this, we defne the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.
Paper Structure (10 sections, 20 theorems, 143 equations)

This paper contains 10 sections, 20 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group invariant Hahn-Banach separating theorems
  4. Ekeland's variational principle for $G$-invariant functions
  5. Consequences of the Ekeland variational principle
  6. Palais-Smale minimizing sequences
  7. Description of $X^{*}_{G}$
  8. A characterization of the completeness of $X_{G}$.
  9. A proof of the Bishop-Phelps theorem by using Ekeland's variational principle
  10. Brø nsted-Rockafellar theorem

Key Result

Lemma 5

Let $X$ be a Banach space, $G \subseteq \mathcal{L}(X)$ be a compact topological group of isometries acting on $X$, and $C \subseteq X$ a $G$-invariant open convex set. If there exists a $G$-invariant point $x_{0} \in X\backslash C$, then we can find $f \in X^{*}_{G}$ such that $f(x) < f(x_{0})$ for

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • Theorem 8
  • proof
  • ...and 32 more