Approximate Solutions of Variational Inequalities and the Ekeland Principle

Raffaele Chiappinelli () and David E. Edmunds
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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

Mathematics, 2025, vol. 13, issue 6, 1-15

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.

Keywords: coercive operators and functionals; strongly monotone operator; minimization on convex sets; pseudo-monotone operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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