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Zbl 1231.65101
Zhang, Xian
Fixed point theorem of generalized quasi-contractive mapping in cone metric space. (English)
[J] Comput. Math. Appl. 62, No. 4, 1627-1633 (2011). ISSN 0898-1221

Summary: We introduce the generalized quasi-contractive mapping $f$ in a cone metric space $(X,d)$. $f$ is called a generalized quasi-contractive if there is a real $\lambda \in [0,1)$ such that for all $x,y\in X$, $d(fx,fy)\le \lambda s$ for some $s\in co\{0,d(fx,fy),d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)\}$. 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 2000:: *65J15 Equations with nonlinear operators (numerical methods)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: cone metric space; fixed point; generalized quasi-contractive mapping

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Zentralblatt MATH Berlin [Germany]