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Approximate Solutions of Variational Inequalities and the Ekeland Principle

Author

Listed:
  • Raffaele Chiappinelli

    (Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, I-53100 Siena, Italy)

  • David E. Edmunds

    (Department of Mathematics, University of Sussex, Brighton BN1 9QH, UK)

Abstract

Let K be a closed convex subset of a real Banach space X , and let F be a map from X to its dual X * . We study the so-called variational inequality problem: given y ∈ X * , , does there exist x 0 ∈ K such that (in standard notation) F ( x 0 ) − y , x − x 0 ≥ 0 for all x ∈ K ? After a short exposition of work in this area, we establish conditions on F sufficient to ensure a positive answer to the question of whether F is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle.

Suggested Citation

  • Raffaele Chiappinelli & David E. Edmunds, 2025. "Approximate Solutions of Variational Inequalities and the Ekeland Principle," Mathematics, MDPI, vol. 13(6), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:6:p:1016-:d:1616855
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