A note on the equivalence of some metric and cone metric fixed point results. (English) Zbl 1213.54067

Kadelburg, Zoran; Radenović, Stojan; Rakočević, Vladimir

Appl. Math. Lett. 24, No. 3, 370-374 (2011).

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) \(d(x_1,y_1)\leq d(x_2,y_2)\) \(\Rightarrow\) \(d_q(x_1,y_1)\leq d_q(x_2,y_2)\). As a consequence, most of the fixed point results for \(K\)-metric spaces are deductible from their standard versions (\(K=\mathbb R_+\)).

Reviewer: Mihai Turinici (Iaşi)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
54E40	Special maps on metric spaces

Keywords:

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

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