Smooth variational principles, Asplund spaces, weak Asplund spaces

Abstract

It is clear that Ekeland’s variational principle (Lemma 3.13) is an extremely useful form of the “maximality points lemma” (3.12); it was a key step in a sequence of fundamental results. As shown in Ekeland’s survey article [Ek], it has found application in such diverse areas as fixed-point theorems, nonlinear semigroups, optimization, mathematical programming, control theory and global analysis. Recall the statement: If ƒ is lower semicontinuous on E, ε τ 0 and x ₀ is such that ƒ(x₀) ≤ inf _E ƒ + ε, then for any λ τ 0 there exists v ∈ E such that

$$ \lambda ||x_0 - v|| \le f\left( {x_0 } \right) - f\left( v \right) \le \in and f\left( x \right) + \lambda ||x - v|| > f\left( v \right) whenever x \ne v. $$