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A note on the equivalence of some metric and cone metric fixed point results. (English) Zbl 1213.54067
Let $E$ be a Hausdorff topological vector space and $K$ be a proper closed convex cone of it, with nonempty interior. The following is the main result of the paper: Theorem. Let $(X,d)$ be a $K$-metric space. Take $e\in \text{int}(K)$ and let $q_e$ be the Minkowski functional of $[-e.e]$. Then i) $d_q:=q_e\circ d$ is a standard metric on $X$, ii) $d(x_1,y_1)\le d(x_2,y_2)$ $\Rightarrow$ $d_q(x_1,y_1)\le d_q(x_2,y_2)$. As a consequence, most of the fixed point results for $K$-metric spaces are deductible from their standard versions ($K=\Bbb R_+$).

54H25Fixed-point and coincidence theorems in topological spaces
54E40Special maps on metric spaces
Full Text: DOI
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