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)
An implicit function implies several contraction conditions. (English) Zbl 1180.54052
The article deals with new generalization of coincidence point theorem. It is assumed that $A, B, C, D$ are self-mappings of a metric space $(X,d)$ satisfying, for all $x, y \in X$, the inequality $$F(d(Ax,By),d(Sx,Ty),d(Ax,Sx),d(By,Ty),d(Sx,By),d(Ty,Ax)) \le 0,$$ where $F: {\Bbb R}_+^6 \to {\Bbb R}$ is a function with properties: (F$_1$) $F(t,0,t,0,0,t) > 0$ for all $t > 0$; (F$_2$) $F(t,0,0,t,t,0) > 0$ for all $t > 0$; (F$_3$) $F(t,t,0,0,t,t) > 0$ for all $t > 0$. Furthermore, it is assumed that there exist two sequences $\{x_n\}, \{y_n\}$ such that $$\lim_{n \to \infty} Ax_n = \lim_{n \to \infty} Sx_n = \lim_{n \to \infty} By_n = \lim_{n \to \infty} Ty_n = t \quad \text{for some} \ t \in X,$$ and that $S(X)$ and $T(X)$ are closed. Under these conditions, it is stated that both pairs $(A,S)$ and $(B,T)$ have a coincidence point and, moreover, $A, B, C, T$ have a unique common fixed point, provided that both pairs $(A,S)$ and $(B,T)$ are weakly compatible ($(A,S)$ is called weakly compatible if $Ax = Sx$ for some $X$ implies $ASx = SAX$, and, similarly, for $(B,T)$). Some modifications of this statement are also given. A number of examples of functions $F$ are presented. Comparisons between old and new results are gathered at the end of the article.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces