A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms

[EN] 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.Juan-José Miñana acknowledges financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación/¿Proyecto PGC2018-095709-B-C21. This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direcció General d¿Innovació i Recerca, Govern de les Illes Balears) and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union¿s Horizon 2020 research and innovation programme under grant agreements No 779776 and No 871260, respectively. 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. Valentín Gregori acknowledges the support of Generalitat Valenciana under grant AICO-2020-136.Gregori Gregori, V.; Miñana, J. (2021). A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 115(3):1-11. https://doi.org/10.1007/s13398-021-01068-6S111115

On fuzzy phi-contractive sequences and fixed point theorems

In 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. (C) 2015 Elsevier B.V. All rights reserved.Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo of Generalitat Valenciana, Spain, by Programa Vali+d para investigadores en formacion under Grant ACIF/2012/040.Gregori Gregori, V.; Miñana, JJ. (2016). On fuzzy phi-contractive sequences and fixed point theorems. Fuzzy Sets and Systems. 300:93-101. doi:10.1016/j.fss.2015.12.010S9310130

Strong convergence in fuzzy metric spaces

[EN] In this paper we introduce and study the concept of strong convergence in fuzzy metric spaces (X, M,*) in the sense of George and Veeramani. This concept is related with the condition Lambda M-t > 0(x, y, t) > 0, which frequently is required or missing in this context. Among other results we characterize the class of s-fuzzy metrics by the strong convergence defined here and we solve partially the question of finding explicitly a compatible metric with a given fuzzy metric.Valentn Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P (MINECO/FEDER, UE).Gregori Gregori, V.; Miñana, J. (2017). Strong convergence in fuzzy metric spaces. Filomat. 31(6):1619-1625. https://doi.org/10.2298/FIL1706619GS1619162531

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 Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics

[EN] 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.Juan-Jose Minana acknowledges financial support from FEDER/Ministerio de Ciencia, Innovacion y Universidades-Agencia Estatal de Investigacion/Proyecto PGC2018-095709-B-C21, and by Spanish Ministry of Economy and Competitiveness under contract DPI2017-86372-C3-3-R (AEI, FEDER, UE). This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears) and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union's Horizon 2020 research and innovation program under grant agreements No 779776 and No 871260, respectively. 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.; Miravet, D. (2020). A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics. Mathematics. 8(9):1-16. https://doi.org/10.3390/math809157511689MATTHEWS, S. G. (1994). Partial Metric Topology. Annals of the New York Academy of Sciences, 728(1 General Topol), 183-197. doi:10.1111/j.1749-6632.1994.tb44144.xGeorge, 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)90162-7Roldán-López-de-Hierro, A.-F., Karapınar, E., & Manro, S. (2014). Some new fixed point theorems in fuzzy metric spaces. Journal of Intelligent & Fuzzy Systems, 27(5), 2257-2264. doi:10.3233/ifs-141189Gregori, V., & Miñana, J.-J. (2016). On fuzzy ψ -contractive sequences and fixed point theorems. Fuzzy Sets and Systems, 300, 93-101. doi:10.1016/j.fss.2015.12.010Gregori, V., Miñana, J.-J., Morillas, S., & Sapena, A. (2016). Cauchyness and convergence in fuzzy metric spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 111(1), 25-37. doi:10.1007/s13398-015-0272-0Gutiérrez García, J., Rodríguez-López, J., & Romaguera, S. (2018). On fuzzy uniformities induced by a fuzzy metric space. Fuzzy Sets and Systems, 330, 52-78. doi:10.1016/j.fss.2017.05.001Beg, I., Gopal, D., Došenović, T., … Rakić, D. (2018). α-type fuzzy H-contractive mappings in fuzzy metric spaces. Fixed Point Theory, 19(2), 463-474. doi:10.24193/fpt-ro.2018.2.37Gregori, V., Miñana, J.-J., & Miravet, D. (2018). Fuzzy partial metric spaces. International Journal of General Systems, 48(3), 260-279. doi:10.1080/03081079.2018.1552687Zheng, D., & Wang, P. (2019). Meir–Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, 120-128. doi:10.1016/j.fss.2018.08.014Romaguera, S., & Tirado, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics, 8(2), 273. doi:10.3390/math8020273Wu, X., & Chen, G. (2020). Answering an open question in fuzzy metric spaces. Fuzzy Sets and Systems, 390, 188-191. doi:10.1016/j.fss.2019.12.006Camarena, J.-G., Gregori, V., Morillas, S., & Sapena, A. (2008). Fast detection and removal of impulsive noise using peer groups and fuzzy metrics. Journal of Visual Communication and Image Representation, 19(1), 20-29. doi:10.1016/j.jvcir.2007.04.003Camarena, J.-G., Gregori, V., Morillas, S., & Sapena, A. (2010). Two-step fuzzy logic-based method for impulse noise detection in colour images. Pattern Recognition Letters, 31(13), 1842-1849. doi:10.1016/j.patrec.2010.01.008Gregori, V., Miñana, J.-J., & Morillas, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems, 204, 71-85. doi:10.1016/j.fss.2011.12.008Morillas, S., Gregori, V., Peris-Fajarnés, G., & Latorre, P. (2005). A fast impulsive noise color image filter using fuzzy metrics. Real-Time Imaging, 11(5-6), 417-428. doi:10.1016/j.rti.2005.06.007Gregori, V., & Romaguera, S. (2004). Fuzzy quasi-metric spaces. Applied General Topology, 5(1), 129. doi:10.4995/agt.2004.2001Park, J. H. (2004). Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals, 22(5), 1039-1046. doi:10.1016/j.chaos.2004.02.051Rodrı́guez-López, J., & Romaguera, S. (2004). The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems, 147(2), 273-283. doi:10.1016/j.fss.2003.09.007Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10(1), 313-334. doi:10.2140/pjm.1960.10.313Sapena Piera, A. (2001). A contribution to the study of fuzzy metric spaces. Applied General Topology, 2(1), 63. doi:10.4995/agt.2001.3016Miñana, J.-J., & Valero, O. (2020). On Matthews’ Relationship Between Quasi-Metrics and Partial Metrics: An Aggregation Perspective. Results in Mathematics, 75(2). doi:10.1007/s00025-020-1173-xKarapınar, E., Erhan, İ. M., & Öztürk, A. (2013). Fixed point theorems on quasi-partial metric spaces. Mathematical and Computer Modelling, 57(9-10), 2442-2448. doi:10.1016/j.mcm.2012.06.036Künzi, H.-P. A., Pajoohesh, H., & Schellekens, M. P. (2006). Partial quasi-metrics. Theoretical Computer Science, 365(3), 237-246. doi:10.1016/j.tcs.2006.07.05

A fixed point theorem for fuzzy contraction mappings

In this paper, we give a fixed point theorem for fuzzy contraction mappings in quasi-pseudo-metric spaces which is a generalization of the corresponding one for metric spaces given by S. Heilpern

Contractive sequences in fuzzy metric spaces

[EN] In this paper we present an example of a fuzzy psi-contractive sequence in the sense of D. Mihet, which is not Cauchy in a fuzzy metric space in the sense of George and Veeramani. To overcome this drawback we introduce and study a concept of strictly fuzzy contractive sequence. Then, we also make an appropriate correction to Lemma 3.2 of Gregori and Minana (2016) [5]. (C) 2019 Elsevier B.V. All rights reserved.Valentín Gregori acknowledges the support of Ministry of Economy and Competitiveness of Spain under Grant MTM 2015-64373-P (MINECO/FEDER, UE). Juan José Miñana acknowledges the support of Programa Operatiu FEDER 2014 2020 de les Illes Balears (50%), by project ref. PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and of project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776. 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.; Miravet-Fortuño, D. (2020). Contractive sequences in fuzzy metric spaces. Fuzzy Sets and Systems. 379:125-133. https://doi.org/10.1016/j.fss.2019.01.003S12513337

Fuzziness in Chang's fuzzy topological spaces

It is known that fuzziness within the concept of openness of a fuzzy set in a Chang's fuzzy topological space (fts) is absent. In this paper we introduce a gradation of openness for the open sets of a Chang jts (X, $\mathcal{T}$) by means of a map $\sigma\;:\; I^{x}\longrightarrow I\left(I=\left[0,1\right]\right)$, which is at the same time a fuzzy topology on X in Shostak 's sense. Then, we will be able to avoid the fuzzy point concept, and to introduce an adeguate theory for $\alpha$-neighbourhoods and $\alpha-T_{i}$ separation axioms which extend the usual ones in General Topology. In particular, our $\alpha$-Hausdorff fuzzy space agrees with $\alpha${*} -Rodabaugh Hausdorff fuzzy space when (X, $\mathcal{T}$) is interpreservative or $\alpha$-locally minimal

Fuzzy Partial Metric Spaces

"This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of General Systems on 01 Dec 2018, available online: https://doi.org/10.1080/03081079.2018.1552687"[EN] 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.Valentin Gregori acknowledges the support of the Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P (MINECO/Feder, UE). Juan Jose Minana acknowledges the partially support of the Ministry of Economy and Competitiveness of Spain under Grant TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds, by the Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project ref. PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and by project ROBINS. The latter has received research funding from the European Union framework under GA 779776. 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.; Miravet-Fortuño, D. (2018). Fuzzy Partial Metric Spaces. International Journal of General Systems. https://doi.org/10.1080/03081079.2018.1552687SBukatin, M., Kopperman, R., & Matthews, S. (2014). Some corollaries of the correspondence between partial metrics and multivalued equalities. Fuzzy Sets and Systems, 256, 57-72. doi:10.1016/j.fss.2013.08.016Camarena, J.-G., Gregori, V., Morillas, S., & Sapena, A. (2010). Two-step fuzzy logic-based method for impulse noise detection in colour images. Pattern Recognition Letters, 31(13), 1842-1849. doi:10.1016/j.patrec.2010.01.008Demirci, M. (2012). The order-theoretic duality and relations between partial metrics and local equalities. Fuzzy Sets and Systems, 192, 45-57. doi:10.1016/j.fss.2011.04.014George, 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)90162-7Grabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4Grečova, S., & Morillas, S. (2016). Perceptual similarity between color images using fuzzy metrics. Journal of Visual Communication and Image Representation, 34, 230-235. doi:10.1016/j.jvcir.2015.04.003Gregori, V., Miñana, J.-J., & Morillas, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems, 204, 71-85. doi:10.1016/j.fss.2011.12.008Gregori, 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.2010.03.013Gregori, V., & Romaguera, S. (2000). Some properties of fuzzy metric spaces. Fuzzy Sets and Systems, 115(3), 485-489. doi:10.1016/s0165-0114(98)00281-4Gregori, V., & Sapena, A. (2002). On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 125(2), 245-252. doi:10.1016/s0165-0114(00)00088-9Gutiérrez García, J., Rodríguez-López, J., & Romaguera, S. (2018). On fuzzy uniformities induced by a fuzzy metric space. Fuzzy Sets and Systems, 330, 52-78. doi:10.1016/j.fss.2017.05.001Höhle, U., & Klement, E. P. (Eds.). (1995). Non-Classical Logics and their Applications to Fuzzy Subsets. doi:10.1007/978-94-011-0215-5Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular Norms. Trends in Logic. doi:10.1007/978-94-015-9540-7MATTHEWS, S. G. (1994). Partial Metric Topology. Annals of the New York Academy of Sciences, 728(1 General Topol), 183-197. doi:10.1111/j.1749-6632.1994.tb44144.xMenger, K. (1942). Statistical Metrics. Proceedings of the National Academy of Sciences, 28(12), 535-537. doi:10.1073/pnas.28.12.535Miheţ, D. (2008). Fuzzy -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), 739-744. doi:10.1016/j.fss.2007.07.006Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10(1), 313-334. doi:10.2140/pjm.1960.10.313Shukla, S., Gopal, D., & Roldán-López-de-Hierro, A.-F. (2016). Some fixed point theorems in 1-M-complete fuzzy metric-like spaces. International Journal of General Systems, 45(7-8), 815-829. doi:10.1080/03081079.2016.1153084Ying, M. (1991). A new approach for fuzzy topology (I). Fuzzy Sets and Systems, 39(3), 303-321. doi:10.1016/0165-0114(91)90100-5Yue, Y. 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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