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

54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
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