A note on cone metric fixed point theory and its equivalence. (English) Zbl 1205.54040

A topological vector space valued cone metric space is a generalization of a cone metric space in the sense that the ordered Banach space in the definition is replaced by an ordered locally convex Hausdorff topological vector space \(Y\). The author obtains a metric \(d_{p}=\xi_e \circ p\) on a topological vector space valued cone metric space \((X,p),\) where \(\xi_e\) is a nonlinear scalarization function defined as \(\xi_e(y)=\inf \{r\in {\mathbb R} :y\in re-K \}\), \(y\in Y\), and \(K\) is the pointed convex cone. He proves an interesting theorem which is equivalent to the Banach contraction principle.


54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI


[1] Chen, G. Y.; Huang, X. X.; Yang, X. Q., Vector Optimization (2005), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, Germany · Zbl 1105.90079
[2] Du, W.-S., On some nonlinear problems induced by an abstract maximal element principle, J. Math. Anal. Appl., 347, 391-399 (2008) · Zbl 1148.49013
[3] Gerth (Tammer), Chr.; Weidner, P., Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67, 297-320 (1990) · Zbl 0692.90063
[4] Göpfert, A.; Tammer, Chr.; Zălinescu, C., On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal., 39, 909-922 (2000) · Zbl 0997.49019
[5] Göpfert, A.; Tammer, Chr.; Riahi, H.; Zălinescu, C., Variational Methods in Partially Ordered Spaces (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1140.90007
[6] Huang, L.-G.; Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332, 1468-1476 (2007) · Zbl 1118.54022
[7] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama, Japan
[8] Rezapour, Sh.; Hamlbarani, R., Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 345, 719-724 (2008) · Zbl 1145.54045
[9] Kannan, R., Some results on fixed point-II, Amer. Math. Monthly, 76, 405-408 (1969) · Zbl 0179.28203
[10] Chatterjea, S. K., Fixed-point theorems, C. R. Acad. Bulgare Sci., 25, 727-730 (1972) · Zbl 0274.54033
[11] Abbas, M.; Jungck, G., Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341, 416-420 (2008) · Zbl 1147.54022
[12] Abbas, M.; Rhoades, B. E., Fixed and periodic point results in cone metric spaces, Appl. Math. Lett., 22, 511-515 (2009) · Zbl 1167.54014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.