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

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
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##### References:
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