Ciències Matemàtiques i Informàtica
http://hdl.handle.net/11201/256
2021-06-24T00:00:13ZA Banach contraction principle in fuzzy metric spaces defined by means of t-conorms
http://hdl.handle.net/11201/155567
A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms
Gregori, Valentín; Miñana, Juan-José
[eng] Fixed point theory in fuzzy metric spaces has grown to become an intensive field of research. The difficulty of demonstrating a fixed point theorem in such kind of spaces makes the authors to demand extra conditions on the space other than completeness. In this paper, we introduce a new version of the celebrated Banach contracion principle in the context of fuzzy metric spaces. It is defined by means of t-conorms and constitutes an adaptation to the fuzzy context of the mentioned contracion principle more 'faithful' than the ones already defined in the literature. In addition, such a notion allows us to prove a fixed point theorem without requiring any additional condition on the space apart from completeness. Our main result (Theorem 1) generalizes another one proved by Castro-Company and Tirado. Besides, the celebrated Banach fixed point theorem is obtained as a corollary of Theorem 1.
A characterization of strong completeness in fuzzy metric spaces
http://hdl.handle.net/11201/152645
A characterization of strong completeness in fuzzy metric spaces
Gregori, Valentín; Miñana, Juan José; Roig, Bernardino; Sapena, Almanzor
[eng] Here, we deal with the concept of fuzzy metric space (X,M,*) , due to George and Veeramani. Based on the fuzzy diameter for a subset of X , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.
A duality relationship between fuzzy metrics and metrics
http://hdl.handle.net/11201/151930
A duality relationship between fuzzy metrics and metrics
Miñana, J.J.; Valero, O.
[eng] Based on the duality relationship between indistinguishability operators and (pseudo-)metrics, we address the problem of establishing whether there is a relationship between the last ones and fuzzy (pseudo-)metrics. We give a positive answer to the posed question. Concretely, we yield a method for generating fuzzy (pseudo-)metrics from (pseudo)-metrics and vice-versa. The aforementioned methods involve the use of the pseudo-inverse of the additive generator of a continuous Archimedean $t$-norm. As a consequence we get a method to generate non-strong fuzzy (pseudo-)metrics from (pseudo-)metrics. Examples that illustrate the exposed methods are also given. Finally, we show that the classical duality relationship between indistinguishability operators and (pseudo)-metrics can be retrieved as a particular case of our results when continuous Archimedean $t$-norms are under consideration.
A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics
http://hdl.handle.net/11201/153431
A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics
Gregori, Valentín; Miñana, Juan-José; Miravet, David
[eng] In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set X. In this paper, we adapt such a relationship to the fuzzy context, in the sense of George and Veeramani, by establishing a duality relationship between fuzzy quasi-metrics and fuzzy partial metrics on a set X, defined using the residuum operator of a continuous t-norm . Concretely, we provide a method to construct a fuzzy quasi-metric from a fuzzy partial one. Subsequently, we introduce the notion of fuzzy weighted quasi-metric and obtain a way to construct a fuzzy partial metric from a fuzzy weighted quasi-metric. Such constructions are restricted to the case in which the continuous t-norm is Archimedean and we show that such a restriction cannot be deleted. Moreover, in both cases, the topology is preserved, i.e., the topology of the fuzzy quasi-metric obtained coincides with the topology of the fuzzy partial metric from which it is constructed and vice versa. Besides, different examples to illustrate the exposed theory are provided, which, in addition, show the consistence of our constructions comparing it with the classical duality relationship.