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Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. (English) Zbl 1145.54045
Summary: L. G. Huang and 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.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
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[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
[2] Mohebi, H., Topical functions and their properties in a class of ordered Banach spaces, (), 343-361 · 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, 3-4, 411-436, (2006) · Zbl 1098.41036
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