A note on local bases and convergence in fuzzy metric spaces

In the context of fuzzy metrics in the sense of George and Veeramani, we study when certain families of open balls centered at a point are local bases at this point. This question is related to p-convergence and s-convergence. © 2013 Elsevier B.V. All rights reserved.Samuel Morillas acknowledges the support of Universitat Politenica de Valencia under Grant PAID-05-12 SP20120696.Gregori Gregori, V.; Miñana Prats, JJ.; Morillas Gómez, S. (2014). A note on local bases and convergence in fuzzy metric spaces. Topology and its Applications. 163:142-148. https://doi.org/10.1016/j.topol.2013.10.013S14214816

On Principal Fuzzy Metric Spaces

[EN] In this paper, we deal with the notion of fuzzy metric space (X, M, *), or simply X, due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable and also that there exist p-Cauchy sequences which are not Cauchy. We prove that if every p-Cauchy sequence in X is Cauchy, then X is principal, and we observe that the converse is false, in general. Hence, we introduce and study a stronger concept than principal, called strongly principal. Moreover, X is called weak p-complete if every p-Cauchy sequence is p-convergent. We prove that if X is strongly principal (or weak p-complete principal), then the family of p-Cauchy sequences agrees with the family of Cauchy sequences. Among other results related to completeness, we prove that every strongly principal fuzzy metric space where M is strong with respect to an integral (positive) t-norm * admits completion.Samuel Morillas acknowledges financial support from Ministerio de Ciencia e Innovacion of Spain under grant PID2019-107790RB-C22 funded by MCIN/AEI/10.13039/501100011033. JuanJose Minana acknowledges financial support from Proyecto PGC2018-095709-B-C21 financiado por MCIN/AEI/10.13039/501100011033 y FEDER "Una manera de hacer Europa" and from project BUGWRIGHT2. This last project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 871260. Also acknowledge support of Generalitat Valenciana under grant CIAICO/2021/137. This publication reflects only the authors' views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2022). On Principal Fuzzy Metric Spaces. Mathematics. 10(16):1-10. https://doi.org/10.3390/math10162860110101

Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric spaces with applications

The aim of this paper is to introduce a class of multivalued mappings satisfying a Suzuki type generalized contractive condition in the framework of fuzzy metric spaces and to present fixed point results for such mappings. Some examples are presented to support the results proved herein. As an application, a common fixed point result for a hybrid pair of single and multivalued mappings is obtained. We show the existence and uniqueness of a common bounded solution of functional equations arising in dynamic programming. Our results generalize and extend various results in the existing literature.http://link.springer.com/journal/11784hb201

Some questions in fuzzy metric spaces

The George and Veeramani's fuzzy metric defined by $M^*(x,y,t)=\frac{min\{x,y\}+t}{max\{x,y\}+t}$ on $[0,\infty[$ (the set of non-negative real numbers) has shown some advantages in front of classical metrics in the process of filtering images. In this paper we study from the mathematical point of view this fuzzy metric and other fuzzy metrics related to it. As a consequence of this study we introduce, throughout the paper, some questions relative to fuzzy metrics. Also, as another practical application, we show that this fuzzy metric is useful for measuring perceptual colour differences between colour samples.The authors wish to thank both the associated editors coordinating this submission and the reviewers for their insightful suggestions and comments which have been useful to increase the scientific quality and presentation of the paper. Also, the authors thank Dr. M. Melgosa, Dr. R. Huertas and Dr. L. Gomez-Robledo from the Department of Optics of University of Granada, for providing data, information and invaluable comments and suggestions. Valentin Gregori and Samuel Morillas acknowledge the support of Spanish Ministry of Education and Science under Grant MTM 2009-12872-C02-01. Samuel Morillas acknowledges the support of Research Project FIS2010-19839, Ministerio de Educacion y Ciencia (Espana) with European Regional Development Funds (ERDFs).Gregori Gregori, V.; Miñana Prats, JJ.; Morillas Gómez, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems. 204:71-85. https://doi.org/10.1016/j.fss.2011.12.008718520