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The equivalence of cone metric spaces and metric spaces. (English) Zbl 1221.54055
The article deals with operators in cone metric (\(K\)-metric) spaces \((X,d)\) whose “metric” takes values in a cone \(P\) in a Banach space \(E\). The authors prove that then \(D(x,y) = \inf\{u \in P|\;u \geq d(x,y)\} \|u\|\) defines a usual metric in \(X\). Their main theorem says that \((X,d)\) is complete if and only if \((X,D)\) is complete. As an application, they consider fixed points of operators in \((X,d)\) satisfying the contractive condition \(d(Tx,Ty) \leq kd(x,y)\), \(x, y \in X\), with \(k < 1\). All results are derived without assuming \(P\) to be normal.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54E35 Metric spaces, metrizability
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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