# 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)
Distance in cone metric spaces and common fixed point theorems. (English) Zbl 1230.54048
Let $(X,d)$ be a cone metric space over a normal cone $P$ in a real Banach space $E$ in the sense of {\it L.-G. Huang} and {\it X. Zhang} [J. Math. Anal. Appl. 332, No. 2, 1468--1476 (2007; Zbl 1118.54022)]. Let $q:X\times X\to E$ satisfy the following properties: (q1) $\theta\preceq q(x,y)$ for all $x,y\in X$; (q2) $q(x,z)\preceq q(x,y)+q(y,z)$ for all $x,y,z\in X$; (q3) if $\{y_n\}$ is a sequence in $(X,d)$ converging to $y\in X$ and for some $x\in X$ and $u=u_x\in P$, $q(x,y_n)\preceq u$ for each $n\geq1$, then $q(x,y)\preceq u$; (q4) for each $c\in E$ with $\theta\ll c$, there exists $e\in E$ with $\theta\ll e$, such that $q(z,x)\ll e$ and $q(z,y)\ll e$ imply $d(x,y)\ll c$. The function $q$ is called a $c$-distance in $(X,d)$ (this notion is a cone metric version of the notion of $\omega$-distance of {\it O. Kada, T. Suzuki} and {\it W. Takahashi} [Math. Japon. 44, No. 2, 381--391 (1996; Zbl 0897.54029)]). The authors prove the following common fixed point result in terms of a $c$-distance. Let $a_i\in(0,1)$, $i=1,2,3,4$ be constants with $a_1+2a_2+a_3+a_4<1$ and let $f,g:X\to X$ be two mappings satisfying the condition $q(fx,fy)\preceq a_1q(gx,gy)+a_2q(gx,fy)+a_3q(gx,fx)+a_4q(gy,fy)$ for all $x,y\in X$. Suppose that $gX\subset fX$ and $gX$ is a complete subset of $X$. If $f$ and $g$ satisfy $\inf\{\|q(fx,y)\|+\|q(gx,fy)\|+\|q(gx,fx)\|:x\in X\}>0$ for all $y\in X$ with $y\ne fy$ or $y\ne gy$, then $f$ and $g$ have a common fixed point in $X$. No compatibility assumptions have to be used.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces
Full Text: