Hu's characterization of metric completeness revisited

[EN] In this note we show the somewhat surprising fact that the proof of the `if part' of the distinguished characterizations of metric completeness due to Kirk, and Suzuki and Takahashi, respectively, can be deduced in a straightforward manner from Hu's theorem that a metric space is complete if and only if any Banach contraction on bounded and closed subsets thereof has a xed point. We also take advantage of this approach to easily deduce a characterization of metric completeness via xed point theorems for ¿ ¿ ¿-contractive mappings.Romaguera Bonilla, S. (2022). Hu's characterization of metric completeness revisited. Advances in the Theory of Nonlinear Analysis and its Applications. 6:476-480. https://doi.org/10.31197/atnaa.1090077476480

A fixed point theorem of Kannan type that characterizes fuzzy metric completeness

[EN] We obtain a fixed point theorem for complete fuzzy metric spaces, in the sense of Kramosiland Michalek, that extends the classical Kannan fixed point theorem. We also show that, in fact, ourtheorem allows to characterize the fuzzy metric completeness, extending in this way the well-known Reich-Subrahmanyam theorem that a metric space is complete if and only if every Kannan contraction on it has afixed point.Romaguera Bonilla, S. (2020). A fixed point theorem of Kannan type that characterizes fuzzy metric completeness. Filomat (Online). 34(14):4811-4819. https://doi.org/10.2298/FIL2014811RS48114819341

w-Distances on Fuzzy Metric Spaces and Fixed Points

[EN] We propose a notion of w-distance for fuzzy metric spaces, in the sense of Kramosil and Michalek, which allows us to obtain a characterization of complete fuzzy metric spaces via a suitable fixed point theorem that is proved here. Our main result provides a fuzzy counterpart of a renowned characterization of complete metric spaces due to Suzuki and Takahashi.Romaguera Bonilla, S. (2020). w-Distances on Fuzzy Metric Spaces and Fixed Points. Mathematics. 8(11):1-9. https://doi.org/10.3390/math8111909S19811Suzuki, T., & Takahashi, W. (1996). Fixed point theorems and characterizations of metric completeness. Topological Methods in Nonlinear Analysis, 8(2), 371. doi:10.12775/tmna.1996.040Suzuki, T. (2001). Generalized Distance and Existence Theorems in Complete Metric Spaces. Journal of Mathematical Analysis and Applications, 253(2), 440-458. doi:10.1006/jmaa.2000.7151Al-Homidan, S., Ansari, Q. H., & Yao, J.-C. (2008). Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Analysis: Theory, Methods & Applications, 69(1), 126-139. doi:10.1016/j.na.2007.05.004Lakzian, H., & Lin, I.-J. (2012). The Existence of Fixed Points for Nonlinear Contractive Maps in Metric Spaces with -Distances. Journal of Applied Mathematics, 2012, 1-11. doi:10.1155/2012/161470Alegre, C., Marín, J., & Romaguera, S. (2014). A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory and Applications, 2014(1). doi:10.1186/1687-1812-2014-40Alegre, C., & Marín, J. (2016). Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces. Topology and its Applications, 203, 32-41. doi:10.1016/j.topol.2015.12.073Lakzian, H., Rakočević, V., & Aydi, H. (2019). Extensions of Kannan contraction via w-distances. Aequationes mathematicae, 93(6), 1231-1244. doi:10.1007/s00010-019-00673-6Alegre, C., Fulga, A., Karapinar, E., & Tirado, P. (2020). A Discussion on p-Geraghty Contraction on mw-Quasi-Metric Spaces. Mathematics, 8(9), 1437. doi:10.3390/math8091437Abbas, M., Ali, B., & Romaguera, S. (2015). Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat, 29(6), 1217-1222. doi:10.2298/fil1506217aRomaguera, S., & Tirado, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics, 8(2), 273. doi:10.3390/math8020273Kirk, W. A. (1976). Caristi’s fixed point theorem and metric convexity. Colloquium Mathematicum, 36(1), 81-86. doi:10.4064/cm-36-1-81-86Hu, T. K. (1967). On a Fixed-Point Theorem for Metric Spaces. The American Mathematical Monthly, 74(4), 436. doi:10.2307/2314587Subrahmanyam, P. V. (1975). Completeness and fixed-points. Monatshefte f�r Mathematik, 80(4), 325-330. doi:10.1007/bf01472580Grabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4Radu, V. (1987). Some fixed point theorems probabilistic metric spaces. Lecture Notes in Mathematics, 125-133. doi:10.1007/bfb0072718Riaz, M., & Hashmi, M. R. (2018). Fixed points of fuzzy neutrosophic soft mapping with decision-making. Fixed Point Theory and Applications, 2018(1). doi:10.1186/s13663-018-0632-5Riaz, M., & Hashmi, M. R. (2019). Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems. Journal of Intelligent & Fuzzy Systems, 37(4), 5417-5439. doi:10.3233/jifs-190550Hashmi, M. R., & Riaz, M. (2020). A novel approach to censuses process by using Pythagorean m-polar fuzzy Dombi’s aggregation operators. Journal of Intelligent & Fuzzy Systems, 38(2), 1977-1995. doi:10.3233/jifs-19061

On the weak form of Ekeland s Variational Principle in quasi-metric spaces

[EN] We show that a quasi-metric space is right K-sequentially complete if and only if it satisfies the property of the weak form of Eke land's Variational Principle. This result solves a question raised by S. Cobzas (2011) [3]. (C) 2015 Elsevier B.V. All rights reserved.The authors are very grateful to the referee for a careful reading of the paper and several useful suggestions and comments which allowed a substantial improvement of the first version. Salvador Romaguera acknowledges the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01Karapinar, E.; Romaguera Bonilla, S. (2015). On the weak form of Ekeland s Variational Principle in quasi-metric spaces. Topology and its Applications. 184:54-60. doi:10.1016/j.topol.2015.01.011S546018

The Meir-Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences

[EN] We obtain quasi-metric versions of the famous Meir¿Keeler fixed point theorem from which we deduce quasi-metric generalizations of Boyd¿Wong¿s fixed point theorem. In fact, one of these generalizations provides a solution for a question recently raised in the paper ¿On the fixed point theory in bicomplete quasi-metric spaces¿, J. Nonlinear Sci. Appl. 2016, 9, 5245¿5251. We also give an application to the study of existence of solution for a type of recurrence equations associated to certain nonlinear difference equationsPedro Tirado acknowledges the support of the Ministerio de Ciencia, Innovación y Universidades, under grant PGC2018-095709-B-C21Romaguera Bonilla, S.; Tirado Peláez, P. (2019). The Meir-Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences. Symmetry (Basel). 11(6):1-10. https://doi.org/10.3390/sym11060741S110116Alegre, C., Dağ, H., Romaguera, S., & Tirado, P. (2016). On the fixed point theory in bicomplete quasi-metric spaces. Journal of Nonlinear Sciences and Applications, 09(08), 5245-5251. doi:10.22436/jnsa.009.08.10Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Meir, A., & Keeler, E. (1969). A theorem on contraction mappings. Journal of Mathematical Analysis and Applications, 28(2), 326-329. doi:10.1016/0022-247x(69)90031-6Aydi, H., & Karapinar, E. (2012). A Meir-Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-26Chen, C.-M. (2012). Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-17Chen, C.-M. (2012). Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-41Chen, C.-M., & Karapınar, E. (2013). Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces. Journal of Inequalities and Applications, 2013(1). doi:10.1186/1029-242x-2013-410Choban, M. M., & Berinde, V. (2017). Multiple fixed point theorems for contractive and Meir-Keeler type mappings defined on partially ordered spaces with a distance. Applied General Topology, 18(2), 317. doi:10.4995/agt.2017.7067Di Bari, C., Suzuki, T., & Vetro, C. (2008). Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Analysis: Theory, Methods & Applications, 69(11), 3790-3794. doi:10.1016/j.na.2007.10.014Jachymski, J. (1995). Equivalent Conditions and the Meir-Keeler Type Theorems. Journal of Mathematical Analysis and Applications, 194(1), 293-303. doi:10.1006/jmaa.1995.1299Karapinar, E., Czerwik, S., & Aydi, H. (2018). (α,ψ)-Meir-Keeler Contraction Mappings in Generalized b-Metric Spaces. Journal of Function Spaces, 2018, 1-4. doi:10.1155/2018/3264620Mustafa, Z., Aydi, H., & Karapınar, E. (2013). Generalized Meir–Keeler type contractions on G-metric spaces. Applied Mathematics and Computation, 219(21), 10441-10447. doi:10.1016/j.amc.2013.04.032Nashine, H. K., & Romaguera, S. (2013). Fixed point theorems for cyclic self-maps involving weaker Meir-Keeler functions in complete metric spaces and applications. Fixed Point Theory and Applications, 2013(1). doi:10.1186/1687-1812-2013-224Park, S., & Bae, J. S. (1981). Extensions of a fixed point theorem of Meir and Keeler. Arkiv för Matematik, 19(1-2), 223-228. doi:10.1007/bf02384479Piątek, B. (2011). On cyclic Meir–Keeler contractions in metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 74(1), 35-40. doi:10.1016/j.na.2010.08.010Rhoades, B. ., Park, S., & Moon, K. B. (1990). On generalizations of the Meir-Keeler type contraction maps. Journal of Mathematical Analysis and Applications, 146(2), 482-494. doi:10.1016/0022-247x(90)90318-aSamet, B. (2010). Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(12), 4508-4517. doi:10.1016/j.na.2010.02.026Samet, B., Vetro, C., & Yazidi, H. (2013). A fixed point theorem for a Meir-Keeler type contraction through rational expression. Journal of Nonlinear Sciences and Applications, 06(03), 162-169. doi:10.22436/jnsa.006.03.02Schellekens, M. (1995). The Smyth Completion. Electronic Notes in Theoretical Computer Science, 1, 535-556. doi:10.1016/s1571-0661(04)00029-5Romaguera, S., & Schellekens, M. (1999). Quasi-metric properties of complexity spaces. Topology and its Applications, 98(1-3), 311-322. doi:10.1016/s0166-8641(98)00102-3García-Raffi, L. M., Romaguera, S., & Schellekens, M. P. (2008). Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms. Journal of Mathematical Analysis and Applications, 348(1), 346-355. doi:10.1016/j.jmaa.2008.07.026Mohammadi, Z., & Valero, O. (2016). A new contribution to the fixed point theory in partial quasi-metric spaces and its applications to asymptotic complexity analysis of algorithms. Topology and its Applications, 203, 42-56. doi:10.1016/j.topol.2015.12.074Romaguera, S., & Tirado, P. (2011). The complexity probabilistic quasi-metric space. Journal of Mathematical Analysis and Applications, 376(2), 732-740. doi:10.1016/j.jmaa.2010.11.056Romaguera, S., & Tirado, P. (2015). A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem. Fixed Point Theory and Applications, 2015(1). doi:10.1186/s13663-015-0431-1Stevo, S. (2002). The recursive sequence xn+1 = g(xn, xn−1)/(A + xn). Applied Mathematics Letters, 15(3), 305-308. doi:10.1016/s0893-9659(01)00135-

Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results

[EN] With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see "Am. Math. Month. 1967, 74, 436-437") that a metric space is complete if and only if any Banach contraction on any of its closed subsets has a fixed point. We apply our main result to deduce that a well-known fixed point theorem due to D. Mihet (see "Fixed Point Theory 2005, 6, 71-78") also allows us to characterize the fuzzy metric completeness.This research was partially funded by Ministerio de Ciencia, Innovacion y Universidades, under grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds.Romaguera Bonilla, S.; Tirado Peláez, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics. 8(2):1-7. https://doi.org/10.3390/math8020273S1782Connell, E. H. (1959). Properties of fixed point spaces. Proceedings of the American Mathematical Society, 10(6), 974-979. doi:10.1090/s0002-9939-1959-0110093-3Hu, T. K. (1967). On a Fixed-Point Theorem for Metric Spaces. The American Mathematical Monthly, 74(4), 436. doi:10.2307/2314587Subrahmanyam, P. V. (1975). Completeness and fixed-points. Monatshefte f�r Mathematik, 80(4), 325-330. doi:10.1007/bf01472580Kirk, W. A. (1976). Caristi’s fixed point theorem and metric convexity. Colloquium Mathematicum, 36(1), 81-86. doi:10.4064/cm-36-1-81-86Caristi, J. (1976). Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society, 215, 241-241. doi:10.1090/s0002-9947-1976-0394329-4Suzuki, T., & Takahashi, W. (1996). Fixed point theorems and characterizations of metric completeness. Topological Methods in Nonlinear Analysis, 8(2), 371. doi:10.12775/tmna.1996.040Suzuki, T. (2007). A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(05), 1861-1870. doi:10.1090/s0002-9939-07-09055-7Romaguera, S., & Tirado, P. (2019). A Characterization of Quasi-Metric Completeness in Terms of α–ψ-Contractive Mappings Having Fixed Points. Mathematics, 8(1), 16. doi:10.3390/math8010016Samet, B., Vetro, C., & Vetro, P. (2012). Fixed point theorems for -contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), 2154-2165. doi:10.1016/j.na.2011.10.014Abbas, M., Ali, B., & Romaguera, S. (2015). Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat, 29(6), 1217-1222. doi:10.2298/fil1506217aCastro-Company, F., Romaguera, S., & Tirado, P. (2015). On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings. Fixed Point Theory and Applications, 2015(1). doi:10.1186/s13663-015-0476-1Radu, V. (1987). Some fixed point theorems probabilistic metric spaces. Lecture Notes in Mathematics, 125-133. doi:10.1007/bfb0072718Sehgal, V. M., & Bharucha-Reid, A. T. (1972). Fixed points of contraction mappings on probabilistic metric spaces. Mathematical Systems Theory, 6(1-2), 97-102. doi:10.1007/bf01706080Ćirić, L. (2010). Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(3-4), 2009-2018. doi:10.1016/j.na.2009.10.00

A note on phi-contractions in probabilistic and fuzzy metric spaces

[EN] In a recent paper Fang (2015) [1], J.X. Fang generalized a crucial fixed point theorem for probabilistic phi-contractions on complete Menger spaces due to Jachymski (2010) [3]. In this note we show that actually Fang s theorem is an easy consequence of Jachymski s theorem. We also observe that the proof of a fixed point theorem for complete metric spaces deduced by Fang from his main result is not correct and present a new proof of it.Alegre Gil, MC.; Romaguera Bonilla, S. (2017). A note on phi-contractions in probabilistic and fuzzy metric spaces. Fuzzy Sets and Systems. 313:119-121. https://doi.org/10.1016/j.fss.2016.06.014S11912131

Weakly Cauchy filters and quasi-uniform completeness

[EN] It is well-known that the notion of a Smyth complete quasi-uniform space provides an appropriate notion of completeness to study many interesting quasi-metric spaces which appear in theoretical computer science. We observe that several of these spaces actually possess a stronger form of completeness based on the use of weakly Cauchy filters in the sense of H. H. Corson and we develop a theory of completion and completeness for this kind of filters. In parallel, we also study a more general notion of completeness based on the use of certain stable filters. Thus our results extend and generalize important theorems of Á. Császár, J. R. Isbell and N. R. Howes on uniform completeness.The second author ackowledges the support of the GDES, grant PB95-0737Pérez Peñalver, MJ.; Romaguera Bonilla, S. (1999). Weakly Cauchy filters and quasi-uniform completeness. Acta Mathematica Hungarica. 82(3):195-216. https://doi.org/10.1023/A:1026408831548S19521682

Cofinal bicompleteness and quasi-metrizability

[EN] We introduce the notions of a cofinally bicomplete quasi-uniformity and of a cofinally bicomplete quasi-pseudometric. The Sorgenfrey quasi-metric and the Kofner quasi-metric are interesting examples of cofinally bicomplete quasi-metrics. We observe that the finest quasi-uniformity of any quasi-pseudometrizable bitopological space is cofinally bicomplete and characterize those quasi-pseudometrizable bitopological spaces which admit a cofinally bicomplete quasi-pseudometric. Anecessary and sufficient condition for cofinal bicompleteness of quasi-pseudometrizable topological spaces is derived. Finally, bitopological quasi-metrizable spaces whose supremum topology is locally compact are characterized.The second author acknowledges the support of the DGES, under grant PB95-0737Pérez Peñalver, MJ.; Romaguera Bonilla, S. (1999). Cofinal bicompleteness and quasi-metrizability. RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITA DI TRIESTE. 30(Supl):165-172. http://hdl.handle.net/10251/107328S16517230Sup

Standard fuzzy uniform structures based on continuous t-norms

This paper deals with fuzzy uniform structures previously introduced by the authors [Fuzzy uniform structures and continuous t-norms, Fuzzy Sets Syst. 161 (2009) 1011-1021]. Our approach involves a covariant functor psi from the category of fuzzy uniform spaces and fuzzy uniformly continuous mappings (in our sense) to the category of uniform spaces and uniformly continuous mappings. We show that psi is well-behaved with respect to some significant fuzzy uniform concepts. and its behavior provides a method to introduce notions of tine fuzzy uniform structure and Stone-tech fuzzy compactification in this context. Our method also applies to obtain fuzzy versions of some classical results on topological algebra and hyperspaces. The case of quasi-uniform structures is also analyzed. (C) 2011 Elsevier B.V. All rights reserved.This research is supported by the Plan Nacional I+D+i, under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02.Gutiérrez, J.; Romaguera Bonilla, S.; Sanchis, M. (2012). Standard fuzzy uniform structures based on continuous t-norms. Fuzzy Sets and Systems. 195:75-89. doi:10.1016/j.fss.2011.10.008S758919