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Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. (English) Zbl 1145.54045
Summary: {\it L. G. Huang} and {\it X. Zhang} [J. Math. Anal. Appl. 332, No. 2, 1468--1476 (2007; Zbl 1118.54022)] reviewed cone metric spaces. We prove that there are no normal cones with normal constant $M<1$ and for each $k>1$ there are cones with normal constant $M>k$. Also, by providing non-normal cones and omitting the assumption of normality in some results of [loc. cit.], we obtain generalizations of the results.

54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
[1] Long-Guang, Huang; Xian, Zhang: 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
[2] Mohebi, H.: Topical functions and their properties in a class of ordered Banach spaces, , 343-361 (2005) · Zbl 1124.90048
[3] Mohebi, H.; Sadeghi, H.; Rubinov, A. M.: Best approximation in a class of normed spaces with star-shaped cone, Numer. funct. Anal. optim. 27, No. 3 -- 4, 411-436 (2006) · Zbl 1098.41036 · doi:10.1080/01630560600686231