Abstract
The aim of this article is to clearly formulate various possible assumptions for a comparison function in contractive conditions and to deduce respective (common) fixed point results in partial metric spaces. Since standard metric spaces are special cases, these results also apply for them. We will show by examples that there exist situations when a partial metric result can be applied, while the standard metric one cannot.
Highlights
In recent years many authors have worked on domain theory in order to equip semantics domain with a notion of distance
We will show by examples that there exist situations when a partial metric result can be applied, while the standard metric one cannot
It is easy to see that every closed subset of a complete partial metric space is complete
Summary
In recent years many authors have worked on domain theory in order to equip semantics domain with a notion of distance. Matthews 1 introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and obtained, among other results, a nice relationship between partial metric spaces and so-called weightable quasimetric spaces. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. Different assumptions for Abstract and Applied Analysis function φ were made Sometimes these assumptions were not clearly stated, and sometimes the assumptions were stronger than needed. We will show by examples that there exist situations when a partial metric result can be applied, while the standard metric one cannot
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