“Ekeland's variational principle”的意思、由来-开放百科全书

In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland,^[1]^[2]^[3] is a theorem that asserts that there exists nearly optimal solutions to some optimization problems.

Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem cannot be applied. Ekeland's principle relies on the completeness of the metric space.^[4]

Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.^[4]^[5]

Ekeland's principle has been shown by F. Sullivan to be equivalent to completeness of metric spaces.

Ekeland was associated with the Paris Dauphine University when he proposed this theorem.^[1]

Statement of the theorem

Let (X, d) be a complete metric space, and let F: X → R ∪ {+∞} be a lower semicontinuous functional on X that is bounded below and not identically equal to +∞. Fix ε > 0 and a point u ∈ X such that

References

1. ^¹{{cite journal| doi = 10.1016/0022-247X(74)90025-0| last = Ekeland| first = Ivar| title = On the variational principle| journal = J. Math. Anal. Appl.| volume = 47| year = 1974| pages = 324–353| issn = 0022-247X}}
2. ^{{cite journal|last=Ekeland|first=Ivar||title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|ref=harv}}
3. ^{{cite book|last1=Ekeland|first1=Ivar|last2=Temam|first2=Roger|authorlink=Roger Temam|title=Convex analysis and variational problems|edition=Corrected reprinting of the (1976) North-Holland|series=Classics in applied mathematics|volume=28 |publisher=Society for Industrial and Applied Mathematics (SIAM)|location=Philadelphia, PA|year=1999|pages=357–373|isbn=0-89871-450-8|mr=1727362|ref=harv}}
4. ^¹{{cite book |author1=Kirk, William A. |author2=Goebel, Kazimierz | title = Topics in Metric Fixed Point Theory | year = 1990 | publisher = Cambridge University Press | isbn = 0-521-38289-0}}
5. ^{{cite book|last=Ok|first=Efe|title=Real Analysis with Economic Applications|publisher=Princeton University Press|year=2007|pages=664|chapter=D: Continuity I|isbn=978-0-691-11768-3|url=http://homepages.nyu.edu/~eo1/Book-PDF/Ekeland.pdf|accessdate=January 31, 2009}}

Statement of the theorem

References

Further reading