zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Equivalents of Ekeland’s principle. (English) Zbl 0793.54025

Let (V,d) be a complete metric space; f:V×V(-,+] a function which is lower semicontinuous in the second argument and such that: (1) f(v,v)0; (2) f(u,v)f(u,w)+f(w,v) for u,v,wV; (3) there exists v 0 V such that inf{f(v 0 ,v):vV}>-.

The authors prove the following theorem and demonstrate its equivalence to the Ekeland variational principle, the Caristi-Kirk fixed point theorem and the Takahashi minimization principle. Let S 0 ={vV:f(v 0 ,v)+d(v 0 ,v)0} and ΨV be such that for every v ¯S 0 Ψ there exists vV such that vv ¯ and f(v ¯,v)+d(v ¯,v)0. Then there exists v * S 0 Ψ.

54E50Complete metric spaces
49J40Variational methods including variational inequalities
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49J27Optimal control problems in abstract spaces (existence)
49J45Optimal control problems involving semicontinuity and convergence; relaxation
54C60Set-valued maps (general topology)