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Zbl 1221.54059
On cone metric spaces: a survey.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 7, 2591-2601 (2011). ISSN 0362-546X

The article is a very useful survey of different results connected with fixed point theorems in cone metric ($K$-metric) spaces $(X,d)$ with cone metric $d(\cdot,\cdot)$ taking values in a cone $P$ in a Banach space $E$. In particular, the authors, in a short and easy way, prove that numerous fixed point results in cone metric spaces obtained under the assumptions that the cone $P$ is normal and solid, can be reduced to the corresponding results in metric spaces; however, the similar reduction, in the general case when $P$ is not normal, is not possible (in particular, some assertions of {\it M. A. Khamsi} [Fixed Point Theory Appl. 2010, Article ID 315398 (2010; Zbl 1194.54065)] are not true).
[Peter Zabreiko (Minsk)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces

Keywords: cone metric spaces; $K$-metric spaces; normal cone; non-normal cone; solid cone; fixed points; common fixed points

Citations: Zbl 1194.54065

Cited in: Zbl 1251.54045

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