Projective contractions, generalized metrics, and fixed points

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.