The periodic points of ε-contractive maps in fuzzy metric spaces

[EN] In this paper, we introduce the notion of ε-contractive maps in fuzzy metric space (X, M, ∗) and study the periodicities of ε-contractive maps. We show that if (X, M, ∗) is compact and f : X −→ X is ε-contractive, then P(f) = ∩ ∞n=1f n (X) and each connected component of X contains at most one periodic point of f, where P(f) is the set of periodic points of f. Furthermore, we present two examples to illustrate the applicability of the obtained results.Project supported by NNSF of China (11761011) and NSF of Guangxi (2020GXNSFAA297010) and PYMRBAP for Guangxi CU(2021KY0651)Sun, T.; Han, C.; Su, G.; Qin, B.; Li, L. (2021). The periodic points of ε-contractive maps in fuzzy metric spaces. Applied General Topology. 22(2):311-319. https://doi.org/10.4995/agt.2021.14449OJS311319222M. Abbas, M. Imdad and D. Gopal, ψ-weak contractions in fuzzy metric spaces, Iranian J. Fuzzy Syst. 8 (2011), 141-148.I. Beg, C. Vetro, D, Gopal and M. Imdad, (Φ, ψ)-weak contractions in intuitionistic fuzzy metric spaces, J. Intel. Fuzzy Syst. 26 (2014), 2497-2504. https://doi.org/10.3233/IFS-130920A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1989), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4V. Gregori and J. J. Miñana, Some remarks on fuzzy contractive mappings, Fuzzy Sets Syst. 251 (2014), 101-103. https://doi.org/10.1016/j.fss.2014.01.002V. Gregori and J. J. Miñana, On fuzzy $Psi$-contractive sequences and fixed point theorems, Fuzzy Sets Syst. 300 (2016), 93-101. https://doi.org/10.1016/j.fss.2015.12.010V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 125 (2002), 245-252. https://doi.org/10.1016/S0165-0114(00)00088-9J. Harjani, B. López and K. Sadarangani, Fixed point theorems for cyclic weak contractions in compact metric spaces, J. Nonl. Sci. Appl. 6 (2013), 279-284. https://doi.org/10.22436/jnsa.006.04.05X. Hu, Z. Mo and Y. Zhen, On compactnesses of fuzzy metric spaces (Chinese), J. Sichuan Norm. Univer. (Natur. Sei.) 32 (2009), 184-187.I. Kramosil and J. Michàlek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336-344.D. Mihet, Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Sys. 159 (2008), 739-744. https://doi.org/10.1016/j.fss.2007.07.006D. Mihet, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst. 251 (2014), 83-91. https://doi.org/10.1016/j.fss.2014.04.010B. Schweizer and A. Sklar, Statistical metrics paces, Pacif. J. Math. 10 (1960), 385-389. https://doi.org/10.2140/pjm.1960.10.313Y. Shen, D. Qiu and W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Letters 25 (2012), 138-141. https://doi.org/10.1016/j.aml.2011.08.002S. Shukla, D. Gopal and A. F. Roldán-López-de-Hierro, Some fixed point theorems in 1-M-complete fuzzy metric-like spaces, Inter. J. General Syst. 45 (2016), 815-829. https://doi.org/10.1080/03081079.2016.1153084S. Shukla, D. Gopal and W. Sintunavarat, A new class of fuzzy contractive mappings and fixed point theorems, Fuzzy Sets Syst. 359 (2018), 85-94. https://doi.org/10.1016/j.fss.2018.02.010D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 222 (2013), 108-114. https://doi.org/10.1016/j.fss.2013.01.012D. Zheng and P. Wang, On probabilistic Ψ-contractions in Menger probabilistic metric spaces, Fuzzy Sets Syst. 350 (2018), 107-110. https://doi.org/10.1016/j.fss.2018.02.011D. Zheng and P. Wang, Meir-Keeler theorems in fuzzy metric spaces, Fuzzy Sets Syst. 370 (2019), 120-128. https://doi.org/10.1016/j.fss.2018.08.01

On locally contractive fuzzy set-valued mappings

We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given.The third author thanks the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.Ahmad, J.; Azam, A.; Romaguera Bonilla, S. (2014). On locally contractive fuzzy set-valued mappings. Journal of Inequalities and Applications. 2014(74):1-10. doi:10.1186/1029-242X-2014-74S11020147

Common fixed points for generalized ψ-contractions in weak non-Archimedean fuzzy metric spaces

[EN] Fixed point theory in fuzzy metric spaces plays very important role in theory of nonlinear problems in applied science. In this paper, we prove an existence result of common fixed point of four nonlinear mappings satisfying a new type of contractive condition in a generalized fuzzy metric space, called weak non-Archimedean fuzzy metric space. Our main results can be applied to solve the existence of solutions of non-linear equations in fuzzy metric spaces. Some examples supporting our main theorem are also given. Our results improve and generalize some recent results contained in Vetro (2011)[16]to generalized contractive conditions under some suitable conditions and many known results in the literature.S. Suantai was partially supported by Chiang Mai University. Yeol Je Cho was supported by Basic Science Research Program through National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). J. Tiammee would like to thank the Thailand Research Fund and Office of the Higher Education Commission under Grant No. MRG6180050 for the financial support and Chiang Mai Rajabhat University.Suantai, S.; Cho, YJ.; Tiammee, J. (2019). Common fixed points for generalized ψ-contractions in weak non-Archimedean fuzzy metric spaces. Applied General Topology. 20(1):1-18. https://doi.org/10.4995/agt.2019.7638SWORD118201Y. J. Cho, Fixed points in fuzzy metric spaces, J. Fuzzy Math. 5 (1997), 949-962.A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems. 90 (1997), 365-368. https://doi.org/10.1016/S0165-0114(96)00207-2M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1989), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002), 245-252. https://doi.org/10.1016/S0165-0114(00)00088-9M. Jain, K. Tas, S. Kumar and N. Gupta, Coupled fixed point theorems for a pair of weakly compatible maps along with CLRg-property in fuzzy metric spaces, J. Appl. Math. 2012 (2012) Art. ID 961210, 13 pp. https://doi.org/10.7763/ijapm.2012.v2.130G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), 771-779. https://doi.org/10.1155/S0161171286000935G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4 (1996), 19-215.I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 336-344.S. Manro and C. Vetro, Common fixed point theorems in fuzzy metric spaces employing CLRg and JCLRst properties, Ser. Math. Inform. 29 (2014), 77-90.J. Martínez-Moreno, A. Roldán, C. Roldán and Y. J. Cho, Multi-dimensional coincidence point theorems for weakly compatible mappings with the CLRg-property in (fuzzy) metric spaces, Fixed Point Theory Appl. 2015, 2015:53. https://doi.org/10.1186/s13663-015-0297-2D. Mihet, Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems 159 (2008), 739-744. https://doi.org/10.1016/j.fss.2007.07.006A.-F. Roldán-López-de-Hierro and W. Sintunavarat, Common fixed point theorems in fuzzy metric spaces using the CLRg-property, Fuzzy Sets and Systems 282 (2016), 131-142. https://doi.org/10.1016/j.fss.2014.11.005A. Sapena, A contribution to the study of fuzzy metric spaces, Applied General Topology 2, no. 1 (2001), 63-75. https://doi.org/10.4995/agt.2001.3016B. Schweizer and A. Sklar, Statistical metric space, Pacific J. Math. 10 (1960), 314-334. https://doi.org/10.2140/pjm.1960.10.313C. Vetro, Fixed points in weak non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. 162 (2011), 84-90. https://doi.org/10.1016/j.fss.2010.09.018L. A. Zadeh, Fuzzy sets, Inform. Control. 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-

New generalized fuzzy metrics and fixed point theorem in fuzzy metric space

In this paper, in fuzzy metric spaces (in the sense of Kramosil and Michalek (Kibernetika 11:336-344, 1957)) we introduce the concept of a generalized fuzzy metric which is the extension of a fuzzy metric. First, inspired by the ideas of Grabiec (Fuzzy Sets Syst. 125:385-389, 1989), we define a new G-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by M Grabiec). Next, inspired by the ideas of Gregori and Sapena (Fuzzy Sets Syst. 125:245-252, 2002), we define a new GV-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by V Gregori and A Sapena). Moreover, we provide the condition guaranteeing the existence of a fixed point for these single-valued contractions. Next, we show that the generalized pseudodistance J:X×X→[0,∞) (introduced by Włodarczyk and Plebaniak (Appl. Math. Lett. 24:325-328, 2011)) may generate some generalized fuzzy metric NJ on X. The paper includes also the comparison of our results with those existing in the literature

On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.The first author 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 to the University of Basque Country by its support through Grant UFI 11/07

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

Cauchyness and convergence in fuzzy metric spaces

[EN] In this paper we survey some concepts of convergence and Cauchyness appeared separately in the context of fuzzy metric spaces in the sense of George and Veeramani. For each convergence (Cauchyness) concept we find a compatible Cauchyness (convergence) concept. We also study the relationship among them and the relationship with compactness and completeness (defined in a natural sense for each one of the Cauchy concepts). In particular, we prove that compactness implies p-completeness.Almanzor Sapena acknowledges the support of Ministry of Economy and Competitiveness of Spain under grant TEC2013-45492-R. Valentín Gregori acknowledges the support of Ministry of Economy and Competitiveness of Spain under grant MTM 2012-37894-C02-01.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2017). 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. https://doi.org/10.1007/s13398-015-0272-0S25371111Alaca, C., Turkoglu, D., Yildiz, C.: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 29, 1073–1078 (2006)Edalat, A., Heckmann, R.: A computational model for metric spaces. Theor. Comput. Sci. 193, 53–73 (1998)Engelking, R.: General topology. PWN-Polish Sci. Publ, Warsawa (1977)Fang, J.X.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 46(1), 107–113 (1992)George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395–399 (1994)George, A., Veeramani, P.: Some theorems in fuzzy metric spaces. J. Fuzzy Math. 3, 933–940 (1995)George, A., Veeramani, P.: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90, 365–368 (1997)Grabiec, M.: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385–389 (1989)Gregori, V., Romaguera, S.: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485–489 (2000)Gregori, V., Romaguera, S.: On completion of fuzzy metric spaces. Fuzzy Sets Syst. 130, 399–404 (2002)Gregori, V., Romaguera, S.: Characterizing completable fuzzy metric spaces. Fuzzy Sets Syst. 144, 411–420 (2004)Gregori, V., López-Crevillén, A., Morillas, S., Sapena, A.: On convergence in fuzzy metric spaces. Topol. Appl. 156, 3002–3006 (2009)Gregori, V., Miñana, J.J.: Some concepts realted to continuity in fuzzy metric spaces. In: Proceedings of the conference in applied topology WiAT’13, pp. 85–91 (2013)Gregori, V., Miñana, J.-J., Sapena, A.: On Banach contraction principles in fuzzy metric spaces (2015, submitted)Gregori, V., Miñana, J.-J.: std-Convergence in fuzzy metric spaces. Fuzzy Sets Syst. 267, 140–143 (2015)Gregori, V., Miñana, J.-J.: Strong convergence in fuzzy metric spaces Filomat (2015, accepted)Gregori, V., Miñana, J.-J., Morillas, S.: Some questions in fuzzy metric spaces. Fuzzy Sets Syst. 204, 71–85 (2012)Gregori, V., Miñana, J.-J., Morillas, S.: A note on convergence in fuzzy metric spaces. Iran. J. Fuzzy Syst. 11(4), 75–85 (2014)Gregori, V., Morillas, S., Sapena, A.: On a class of completable fuzzy metric spaces. Fuzzy Sets Syst. 161, 2193–2205 (2010)Gregori, V., Morillas, S., Sapena, A.: Examples of fuzzy metric spaces and applications. Fuzzy Sets Syst. 170, 95–111 (2011)Kramosil, I., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326–334 (1975)Mihet, D.: On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets Syst. 158, 915–921 (2007)Mihet, D.: Fuzzy $$\varphi$$ φ -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst. 159, 739–744 (2008)Mihet, D.: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431–439 (2004)Mishra, S.N., Sharma, N., Singh, S.L.: Common fixed points of maps on fuzzy metric spaces Internat. J. Math. Math. Sci. 17(2), 253–258 (1994)Morillas, S., Sapena, A.: On Cauchy sequences in fuzzy metric spaces. In: Proceedings of the conference in applied topology (WiAT’13), pp. 101–108 (2013)Ricarte, L.A., Romaguera, S.: A domain-theoretic approach to fuzzy metric spaces. Topol. Appl. 163, 149–159 (2014)Sherwood, H.: On the completion of probabilistic metric spaces. Z.Wahrschein-lichkeitstheorie verw. Geb. 6, 62–64 (1966)Sherwood, H.: Complete Probabilistic Metric Spaces. Z. Wahrschein-lichkeitstheorie verw. Geb. 20, 117–128 (1971)Tirado, P.: On compactness and G-completeness in fuzzy metric spaces. Iran. J. Fuzzy Syst. 9(4), 151–158 (2012)Tirado, P.: Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets. Fixed Point Theory 13(1), 273–283 (2012)Vasuki, R., Veeramani, P.: Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets Syst. 135(3), 415–417 (2003)Veeramani, P.: Best approximation in fuzzy metric spaces. J. Fuzzy Math. 9, 75–80 (2001

Common fixed points of self maps satisfying an integral type contractive condition in fuzzy metric spaces

In this paper, first we prove fixed point theorems for different variant of compatible maps, satisfying a contractive condition of integral type in fuzzy metric spaces, which improve the results of Branciari [2], Rhoades [33], Kumar et al. [23] Subramanyam [35] and results of various authors cited in the literature of "Fixed Point Theory and Applications". Secondly, we introduce the notion of any kind of weakly compatible maps and prove a fixed point theorem for weakly compatible maps along with the notion of any kind of weakly compatible. At the end, we prove a fixed point theorem using variants of R-Weakly commuting mappings in fuzzy metric spaces

On Convergences of contractive maps in metric spaces

In this paper, we introduce a new class of contraction maps, called A – contractions in fuzzy metric space. Under different sufficient conditions, existence of common fixed point for a pair of maps, four maps and also for a sequence of maps will be established here. Also it is shown that A – contractions is more generalized than TS – Contraction, B – Contraction in FM-space. If two fuzzy metrics are given on a set , which are related, a pair of self map can have common fixed point though the contractive condition with respect one fuzzy metric is given. Our result extends, generalized and fuzzifies several fixed point theorems with A – contractions on metric space.We give generalizations and convergences of these maps