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Zbl 1213.54067
A note on the equivalence of some metric and cone metric fixed point results.
(English)
[J] Appl. Math. Lett. 24, No. 3, 370-374 (2011). ISSN 0893-9659

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 ($K=\Bbb R_+$).
[Mihai Turinici (Iaşi)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
54E40 Special maps on metric spaces

Keywords: Topological vector space; convex cone; Minkowski functional; conical metric; fixed point

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