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Smooth variational principles, Asplund spaces, weak Asplund spaces

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Part of the Lecture Notes in Mathematics book series (LNM, volume 1364)

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

Keywords

Banach Space Maximal Monotone Operator Winning Strategy Nonempty Open Subset Bump Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

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