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

Du, Wei-Shih

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 5, 2259-2261 (2010).

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.

Reviewer: Huseyin Cakalli (Istanbul)

MSC:

54H25

Fixed-point and coincidence theorems (topological aspects)

Keywords:

TVS-cone metric space; Banach contraction principle; scalarisation method

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