zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Coincidence theorems for set-valued mappings and Ekeland’s variational principle in fuzzy metric spaces. (English) Zbl 0867.54018
Summary: We establish a coincidence theorem for set-valued mappings in fuzzy metric spaces with a view to generalizing Downing-Kirk’s fixed point theorem [{\it D. Downing} and {\it W. A. Kirk}, Pac. J. Math. 69, 339-346 (1977; Zbl 0357.47036)] in metric spaces. As consequences, we obtain {\it J. Caristi}’s coincidence theorem [Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)] for set-valued mappings and a more general type of {\it I. Ekeland}’s variational principle [J. Math. Anal. Appl. 47, 324-353 (1974; Zbl 0286.49015)] in fuzzy metric spaces. Further, we also give a direct simple proof of the equivalence between these two theorems in fuzzy metric spaces. Some applications of these results to probabilistic metric spaces are presented.

MSC:
54A40Fuzzy topology
54H25Fixed-point and coincidence theorems in topological spaces
WorldCat.org
Full Text: DOI
References:
[1] Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. amer. Math. soc. 215, 241-251 (1976) · Zbl 0305.47029
[2] Chang, S. S.; Luo, Q.: Set-valued caristi’s fixed point theorem and Ekeland’s variational principle. Appl. math. Mech. 10, 119-121 (1989) · Zbl 0738.49009
[3] Downing, D.; Kirk, W. A.: A generalization of caristi’s theorem with applications to nonlinear mapping theory. Pacific J. Math. 69, 339-346 (1977) · Zbl 0357.47036
[4] Ekeland, I.: On the variational principle. J. math. Anal. appl. 47, 324-353 (1974) · Zbl 0286.49015
[5] Ekeland, I.: Nonconvex minimization problems. Bull. amer. Math. soc. (New series) 1, 443-474 (1979) · Zbl 0441.49011
[6] Fang, J. X.: A note on fixed point theorems of hadžíc. Fuzzy sets and systems 48, 391-395 (1991)
[7] Hadžíc, O.: Fixed point theorems for multi-valued mappings in some classes of fuzzy metric spaces. Fuzzy sets and systems 29, 115-125 (1989) · Zbl 0681.54023
[8] He, P. J.: The variational principle in fuzzy metric spaces and its applications. Fuzzy sets and systems 45, 389-394 (1992) · Zbl 0754.54005
[9] Jung, J. S.; Cho, Y. J.; Kim, J. K.: Minimization theorems for fixed point theorems in fuzzy metric spaces and applications. Fuzzy sets and systems 61, 199-207 (1994) · Zbl 0845.54004
[10] Kaleva, O.; Seikkala, S.: On fuzzy metric spaces. Fuzzy sets and systems 12, 215-229 (1984) · Zbl 0558.54003
[11] Mizoguchi, N.; Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. math. Anal. appl. 141, 177-188 (1989) · Zbl 0688.54028
[12] Park, S.: On extensions of the caristi-kirk fixed point theorem. J. korean math. Soc. 19, 143-151 (1983) · Zbl 0526.54032
[13] Schweizer, B.; Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313-334 (1960) · Zbl 0091.29801