Examples of non-strong fuzzy metrics

Answering a recent question posed by Gregori et al. [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] we present two examples of non-strong fuzzy metrics (in the sense of George and Veeramani). © 2010 Elsevier B.V. All rights reserved.This research was supported by the Ministry of Science and Innovation of Spain under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02. J. Gutierrez Garcia also acknowledges financial support from the University of the Basque Country under Grant GIU07/27.Gutiérrez García, J.; Romaguera Bonilla, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems. 162(1):91-93. https://doi.org/10.1016/j.fss.2010.09.017S9193162

On Yager and Hamacher t-Norms and Fuzzy Metric Spaces

Recently, Gregori et al. have discussed (Fuzzy Sets Syst 2011;161:2193 2205) the so-called strong fuzzy metrics when looking for a class of completable fuzzy metric spaces in the sense of George and Veeramani and state the question of finding a non-strong fuzzy metric space for a continuous t-norm different from the minimum. Later on, Gutíerrez-García and Romaguera solved this question (Fuzzy Sets Syst 2011;162:91 93) by means of two examples for the product and the Lukasiewicz t-norm, respectively. In this direction, they posed to find further examples of nonstrong fuzzy metrics for continuous t-norms that are greater than the product but different from minimum. In this paper, we found an example of this kind. On the other hand, Tirado established (Fixed Point Theory 2012;13:273 283) a fixed-point theorem in fuzzy metric spaces, which was successfully used to prove the existence and uniqueness of solution for the recurrence equation associated with the probabilistic divide and conquer algorithms. Here, we generalize this result by using a class of continuous t-norms known as ω-Yager t-norms.The second author acknowledges the support of the Ministry of Economy and Competitiveness of Spain under grant MTM2012-37894-C02-01 and the support of Universitat Politecnica de Valencia under grant PAID-06-12-SP20120471.Castro Company, F.; Tirado Peláez, P. (2014). On Yager and Hamacher t-Norms and Fuzzy Metric Spaces. International Journal of Intelligent Systems. 29:1173-1180. https://doi.org/10.1002/int.21688S1173118029Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Gregori, V. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xGregori, V., Morillas, S., & Sapena, A. (2010). On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems, 161(16), 2193-2205. doi:10.1016/j.fss.2010.03.013Gutiérrez García, J., & Romaguera, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems, 162(1), 91-93. doi:10.1016/j.fss.2010.09.017Yager, R. R. (1980). On a general class of fuzzy connectives. Fuzzy Sets and Systems, 4(3), 235-242. doi:10.1016/0165-0114(80)90013-5Castro-Company, F., & Tirado, P. (2012). Some classes of t-norms and fuzzy metric spaces. doi:10.1063/1.4756272George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. doi:10.1016/0165-0114(94)90162-7Hadžić, O., & Pap, E. (2001). Fixed Point Theory in Probabilistic Metric Spaces. doi:10.1007/978-94-017-1560-7Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular Norms. Trends in Logic. doi:10.1007/978-94-015-9540-

On completable fuzzy metric spaces

In this paper we construct a non-completable fuzzy metric space in the sense of George and Veeramani which allows to answer an open question related to continuity on the real parameter t. In addition, the constructed space is not strong (non-Archimedean).Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo (Programa Vali+d para investigadores en formacion) of Generalitat Valenciana, Spain and the support of Universitat Politecnica de Valencia under Grant PAID-06-12 SP20120471.Gregori Gregori, V.; Miñana, J.; Morillas, S. (2015). On completable fuzzy metric spaces. Fuzzy Sets and Systems. 267:133-139. https://doi.org/10.1016/j.fss.2014.07.009S13313926

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

Optimal coincidence best approximation solution in non-Archimedean fuzzy metric spaces

In this paper, we introduce the concept of best proximal contraction theorems in non-Archimedean fuzzy metric space for two mappings and prove some proximal theorems. As a consequence, it provides the existence of an optimal approximate solution to some equations which contains no solution. The obtained results extend further the recently development proximal contractions in non-Archimedean fuzzy metric spaces and famous Banach contraction principle.http://ijfs.usb.ac.iram2016Mathematics and Applied Mathematic

Optimal coincidence point results in partially ordered non-Archimedean fuzzy metric spaces

In this paper, we introduce best proximal contractions in complete ordered non-Archimedean fuzzy metric space and obtain some proximal results. The obtained results unify, extend, and generalize some comparable results in the existing literature.M De la Sen thanks the Spanish Ministry of Economy and Competitiveness for partial support of this work through Grant DPI2012-30651. He also thanks the Basque Government for its support through Grant IT378-10, and the University of Basque Country for its support through Grant UFI 11/07.http://link.springer.com/journal/11784am2016Mathematics and Applied Mathematic

Optimal coincidence points of proximal quasi-contraction mappings in non-Archimedean fuzzy metric spaces

The aim of this paper is to present fuzzy optimal coincidence point results of fuzzy proximal quasi contraction and generalized fuzzy proximal quasi contraction of type1 in the framework of complete non- Archimedean fuzzy metric space. Some examples are presented to support the results which are obtained here. These results also hold in fuzzy metric spaces when some mild assumption is added to the set in the domain of mappings which are involved here. Our results unify, extend and generalize various existing results in literature.http://www.tjnsa.comam2016Mathematics and Applied Mathematic