Fuzzy ample spectrum contractions in (more general than) non-Archimedean fuzzy metric spaces

TL;DR

This paper extends the concept of ample spectrum contractions to fuzzy metric spaces, providing new fixed point results that are more general and applicable in non-Archimedean and incomplete fuzzy spaces.

Contribution

It introduces two approaches to fuzzy ample spectrum contractions, broadening fixed point theory in non-Archimedean and incomplete fuzzy metric spaces with novel properties.

Findings

Established existence and uniqueness of fixed points for the new fuzzy contractions.

Generalized well-known fuzzy contractive operators, including Mihet's fuzzy ψ-contractions.

Overcame technical issues in Kramosil and Michálek fuzzy spaces.

Abstract

Taking into account that Rold\'an et al.'s ample spectrum contractions have managed to extend and unify more than ten distinct families of contractive mappings in the setting of metric spaces, in this manuscript we present a first study on how such concept can be implemented in the more general framework of fuzzy metric spaces in the sense of Kramosil and Mich\'alek. We introduce two distinct approaches to the concept of fuzzy ample spectrum contractions and we prove general results about existence and uniqueness of fixed points. The proposed notions enjoys the following advantages with respect to previous approaches: (1) they permit to develop fixed point theory in a very general fuzzy framework (for instance, the underlying fuzzy space is not necessarily complete); (2) the procedures that we employ are able to overcome the technical drawbacks raising in fuzzy metric spaces in the sense of Kramosil and Mich\'alek (that do not appear in fuzzy metric spaces in the sense of George and Veeramani); (3) we introduce a novel property about sequences that are not Cauchy in a fuzzy space in order to consider a more general context than non-Archimedean fuzzy metric spaces; (4) the contractivity condition associated to fuzzy ample spectrum contractions does not have to be expressed in separate variables; (5) such fuzzy contractions generalize some well-known families of fuzzy contractive operators such that the class of all Mihet's fuzzy $\psi$-contractions. As a consequence of these properties, this study gives a positive partial answer to a question posed by this author some years ago.