Abstract
In 1981, Borsik and Dobos studied the aggregation problem for metric spaces. Thus, they characterized those functions that allow one to merge a collection of metrics providing a single metric as a result (Borsik and Dobos in Math. Slovaca 31:193-205, 1981). Later on, in 1994, the notion of partial metric space was introduced by Matthews with the aim of providing an appropriate mathematical tool for program verification (Matthews in Ann. N.Y. Acad. Sci. 728:183-197, 1994). In the aforesaid reference, an extension of the well-known Banach fixed point theorem to the partial metric framework was given and, in addition, an application of such a result to denotational semantics and program verification was provided. Inspired by the applicability of partial metric spaces to computer science and by the fact that there are partial metrics useful in such a field which can be induced through aggregation, in 2012 Massanet and Valero analyzed the aggregation problem in the partial metric framework (Massanet and Valero in Proc. of the 17th Spanish Conference on Fuzzy Technology and Fuzzy Logic (Estylf 2012), pp.558-563, 2012). In this paper, motivated by the fact that fixed point techniques are essential in order to apply partial metric spaces to computer science and that, as we have pointed out above, some of such partial metrics can be induced by aggregation, we introduce a new notion of contraction between partial metric spaces which involves aggregation functions. Besides, since fixed point theory in partial metric spaces from an aggregation viewpoint still is without exploring, we provide a fixed point theorem in the spirit of Matthews for the new type of contractions and, in addition, we give examples which illustrate that the assumptions in such a result cannot be weakened. Furthermore, we provide conditions that vouch the existence and uniqueness of fixed point for this new class of contractions. Finally, we discuss the well-posedness for this kind of fixed point problem and the limit shadowing property for the new sort of contractions.
Highlights
In many practical problems appears the need to process simultaneously a few numerical values which are provided by some sources with the aim of making a decision in order to elaborate a plan of action or to solve a problem
In this paper, motivated by the fact that fixed point techniques are essential in order to apply partial metric spaces to computer science and that, as we have pointed out above, some of such partial metrics can be induced by aggregation, we introduce a new notion of contraction between partial metric spaces which involves aggregation functions
Since fixed point theory in partial metric spaces from an aggregation viewpoint still is without exploring, we provide a fixed point theorem in the spirit of Matthews for the new type of contractions and, in addition, we give examples which illustrate that the assumptions in such a result cannot be weakened
Summary
In many practical problems appears the need to process simultaneously a few numerical values which are provided by some sources (possibly of different natures) with the aim of making a decision in order to elaborate a plan of action or to solve a problem. In part, by the aforesaid facts Borsík and Doboš began in a research line on the aggregation of metrics [ ] They studied the properties that a function must satisfy in order to merge a collection of metrics into a single one. In order to recall such a fixed point theorem, let us introduce the notion of a contraction in the partial metric framework (see [ , ]): A mapping from a partial metric space (X, p) into itself is said to be a contraction if there exists c ∈ [ , [ such that p f (x), f (y) ≤ cp(x, y).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
