ElsevierGregori Gregori, V.; Miñana, JJ. (2016 AbstractIn this paper we give a fixed point theorem in the context of fuzzy metric spaces in the sense of George and Veeramani. As a consequence of our result we obtain a fixed point theorem due to D. Mihet and generalize a fixed point theorem due to D. Wardowski. Also, we answer in a positive way to a question posed by D. Wardowski, and solve partially an open question on Cauchyness and contractivity.

ElsevierGregori Gregori, V.; Miñana Prats, JJ.; Morillas Gómez, S. (2014)

In this paper we survey some concepts of convergence and Cauchyness appeared separately in the context of fuzzy metric spaces in the sense of George and Veeramani. For each convergence (Cauchyness) concept we find a compatible Cauchyness (convergence) concept. We also study the relationship among them and the relationship with compactness and completeness (defined in a natural sense for each one of the Cauchy concepts). In particular, we prove that compactness implies pcompleteness.

In this paper we provide a concept of fuzzy partial metric space (X, P, *) as an extension to fuzzy setting in the sense of Kramosil and Michalek, of the concept of partial metric due to Matthews. This extension has been defined using the residuum operator → * associated to a continuous t-norm * and without any extra condition on *. Similarly, it is defined the stronger concept of GV-fuzzy partial metric (fuzzy partial metric in the sense of George and Veeramani). After defining a concept of open ball in (X, P, *), a topology TP on X deduced from P is constructed, and it is showed that (X, TP) is a T0-space.

Abstract. In this paper we discuss the concept of Cauchy sequence due to Grabiec, that we call G-Cauchy, in the context of fuzzy metric spaces. It leads to introduce and study a concept of weak G-completeness in fuzzy and classical context. Then, we generalize the celebrated Grabiec's fuzzy Banach Contraction Principle. Also, we extend the Mihet's fixed point theorem given for weak B-contractive mappings.