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On the metrizability of cone metric spaces. (English) Zbl 1206.54026
Summary: We show in this paper that a (complete) cone metric space (X,E,P,d) is indeed (completely) metrizable for a suitable metric D. Moreover, given any finite number of contractions f 1 ,,f n on the cone metric space (X,E,P,d),D can be defined in such a way that these functions become also contractions on (X,D).
MSC:
54E35Metric spaces, metrizability
References:
[1]Amini-Harandi, A.; Fakhar, M.: Fixed point theory in cone metric spaces obtained via the scalarization method, Computers and mathematics with applications 59, No. 11, 3529-3534 (2010) · Zbl 1197.54055 · doi:10.1016/j.camwa.2010.03.046
[2]Z. Kadelburg, S. Radenovic, V. Rakocevic, A note on equivalence of some metric and cone metric fixed point results, Applied Mathematics Letters, doi:10.1016/j.aml.2010.10.030, in press.
[3]Huang, L. -G.; Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings, Journal of mathematical analysis and applications 332, 1468-1476 (2007) · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087
[4]Raja, P.; Vaezpour, S. M.: Some extensions of Banach’s fixed point theorem in complete cone metric spaces, Journal of fixed point theory and applications, 1-11 (2008) · Zbl 1148.54339 · doi:10.1155/2008/768294
[5]Rezapour, Sh.; Hamlbarani, R.: Some notes on the paper ”cone metric spaces and fixed point theorems of contractive mappings”, Journal of mathematical analysis and applications 345, No. 2, 719-724 (2008) · Zbl 1145.54045 · doi:10.1016/j.jmaa.2008.04.049
[6]Du, W. S.: A note on cone metric fixed point theory and its equivalence, Nonlinear analysis TMA 72, 2259-2261 (2010) · Zbl 1205.54040 · doi:10.1016/j.na.2009.10.026