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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.
##### Keywords:
contractive mappings; metric spaces; cone metric spaces
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