Approximate Solutions of Variational Inequalities and the Ekeland Principle - Raffaele Chiappinelli; David E. Edmunds (Mathematics)
Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual <semantics>X*</semantics>. We study the so-called variational inequality problem: given <semantics>y∈X*,,</semantics> does there exist <semantics>x0∈K</semantics> such that (in standard notation) <semantics><mfenced separators="" open="⟨" close="⟩">F(x0)−y,<mspace width="3.33333pt"></mspace>x−x0</mfenced>≥0</semantics> for all <semantics>x∈K?</semantics> 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.