# 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)
Fixed point theorem of generalized quasi-contractive mapping in cone metric space. (English) Zbl 1231.65101
Summary: We introduce the generalized quasi-contractive mapping $f$ in a cone metric space $\left(X,d\right)$. $f$ is called a generalized quasi-contractive if there is a real $\lambda \in \left[0,1\right)$ such that for all $x,y\in X$, $d\left(fx,fy\right)\le \lambda s$ for some $s\in co\left\{0,d\left(fx,fy\right),d\left(x,y\right),d\left(x,fx\right),d\left(y,fy\right),d\left(x,fy\right),d\left(y,fx\right)\right\}$. It is proved that if $X$ is a complete cone metric space with normal cone then $f$ has a unique fixed point. A example is given, which shows that our result is a genuine generalization of quasi-contractive mapping.
##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties
##### References:
 [1] Ćirić, Lj.B.: A generalization of banachs contraction principle, Proc. amer. Math. soc. 45, 267-273 (1974) · Zbl 0291.54056 · doi:10.2307/2040075 [2] Gajić, Lj.; Rakocević, V.: Pair of non-self-mappings and common fixed points, Appl. math. Comput. 187, 999-1006 (2007) · Zbl 1118.54304 · doi:10.1016/j.amc.2006.09.143 [3] Rakocević, V.: Functional analysis, (1994) [4] Guang, H. L.; Xian, Z.: Cone metric spaces and fixed point theorems of contractive mappings, J. math. Anal. appl. 332, 1468-1476 (2007) · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087 [5] Ilić, Dejan; Rakocević, Vladimir: Quasi-contraction on a cone metric space, Appl. math. Lett. 22, 728-731 (2009) · Zbl 1179.54060 · doi:10.1016/j.aml.2008.08.011