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)
Fixed point theorems for multivalued mappings on complete metric spaces. (English) Zbl 0688.54028

The authors give the following “multi-version” of Caristi’s fixed point theorem [J. Caristi, Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)]. Let (X,d) be a complete metric space, ψ : X(-,+] be a proper, bounded below and lower semicontinuous function and multimap T: XP(X) is such that for every xX, there exists yTx satisfying


Then T has a fixed point.

It is shown that this result is equivalent to the ϵ-variational principle of Ekeland. Then it is used to generalize Nadler’s fixed point theorem and to obtain a common fixed point theorem for a single-valued map and a multimap. Next, some generalizations of Reich’s fixed point theorems for multimaps of contractive type are considered.

{ Reviewer’s remark: Another generalization of the Caristi’s theorem on multifunctions was given in the work of J. Madhusudana Rao [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Ser. 29(77), No.1, 79-80 (1985; Zbl 0561.54041)]}.

Reviewer: V.V.Obukhovskij

54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)