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Zbl 1205.54040
Du, Wei-Shih
A note on cone metric fixed point theory and its equivalence. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 5, A, 2259-2261 (2010). ISSN 0362-546X

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 $\xi_e(y)=\inf \{r\in {\Bbb R} :y\in re-K \}$, $y\in Y$, and $K$ is the pointed convex cone. He proves an interesting theorem which is equivalent to the Banach contraction principle.
[Huseyin Cakalli (Istanbul)]

MSC 2000:: *54H25 Fixed-point theorems in topological spaces

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

Cited in: Zbl 1254.54048 Zbl 1245.54038

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Zentralblatt MATH Berlin [Germany]