COMMON FIXED POINT OF WEAKLY COMPATIBLE MAPPINGS UNDER A NEW PROPERTY IN FUZZY METRIC SPACES

In this paper, we prove some common fixed point theorems for weakly compatible mappings under a new property in fuzzy metric spaces. We prove a new result under (S-B) property defined by Sharma and Bamboria. Keywords: Fixed point; Fuzzy metric space; (S-B) property

Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces

The purpose of this paper is to give some new fixed point theorems for contractive  type mappings in fuzzy metric spaces. The results presented improve and generalize some recent result. The result is a genuine generalization of the corresponding result of S.Sedghi et al. (2010).   Keywords: Fuzzy metric space, t-norm, g-convergent, coupled common fixed point. 2010 MSC: Primary 54E70; Secondary 54H25

COMMON FIXED POINT THEOREMS FOR HYBRID PAIRS OF MAPPINGS USING IMPLICIT RELATIONS IN FUZZY METRIC SPACE

The purpose of this paper is to introduce the notion of a new type of common limit range property for a hybrid pair of single and multivalued mappings in fuzzy metric space and establish some common fixed point theorems satisfying the same property using implicit relations. Some related results are also derived besides furnishing illustrative examples. Further, we also present an integral type common fixed pointtheorem in fuzzy metric space. Our results improve and extend some previously known results

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

Common Fixed Points of Compatible Maps in Fuzzy Metric Spaces

In this Paper a transposition of these notions is being made for 4-tuples A, B, C and D of self maps of fuzzy metric space (X, d). Then under suitable contractive conditions some common fixed point theorems involving such maps are stated and proved. These results open, in our opinion, a wider scope for the study of this topic in the framework of this new fuzzy metric space

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-

On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings

[EN] We present a procedure to construct a compatible metric from a given fuzzy metric space. We use this approach to obtain a characterization of a large class of complete fuzzy metric spaces by means of a fuzzy version of Caristi’s fixed point theorem, obtaining, in this way, partial solutions to a recent question posed in the literature. Some illustrative examples are also given.The authors thank the referees for several useful suggestions. Salvador Romaguera and Pedro Tirado acknowledge the support of the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.Castro Company, F.; Romaguera Bonilla, S.; Tirado Peláez, 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. 2015:226. https://doi.org/10.1186/s13663-015-0476-1S2015:226Kelley, JL: General Topology. Springer, New York (1955)Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math. 10, 314-334 (1960)Klement, E, Mesiar, R, Pap, E: Triangular Norms. Kluwer Academic, Dordrecht (2000)Hamacher, H: Über logische Verknüpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen. In: Progress in Cybernetics and Systems Research, pp. 276-287. Hemisphere, New York (1975)Kramosil, I, Michalek, J: Fuzzy metrics and statistical metric spaces. Kybernetika 11, 326-334 (1975)George, A, Veeramani, P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395-399 (1994)Gregori, V, Romaguera, S: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485-489 (2000)Radu, V: On the triangle inequality in PM-spaces. STPA, West University of Timişoara 39 (1978)Abbas, M, Ali, B, Romaguera, S: Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat 29(6), 1217-1222 (2015)Cho, YJ, Grabiec, M, Radu, V: On Nonsymmetric Topological and Probabilistic Structures. Nova Science Publishers, New York (2006)Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001)Mihet, D: A note on Hicks type contractions on generalized Menger spaces. STPA, West University of Timişoara 133 (2002)Mihet, D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431-439 (2004)Radu, V: Some fixed point theorems in PM spaces. In: Stability Problems for Stochastic Models. Lecture Notes in Mathematics, vol. 1233, pp. 125-133. Springer, Berlin (1985)Radu, V: Some remarks on the probabilistic contractions on fuzzy Menger spaces (The Eighth Intern. Conf. on Applied Mathematics and Computer Science, Cluj-Napoca, 2001). Autom. Comput. Appl. Math. 11(1), 125-131 (2002)Chauhan, S, Shatanawi, W, Kumar, S, Radenović, S: Existence and uniqueness of fixed points in modified intuitionistic fuzzy metric spaces. J. Nonlinear Sci. Appl. 7, 28-41 (2014)Hussain, N, Salimi, P, Parvaneh, V: Fixed point results for various contractions in parametric and fuzzy b-metric spaces. J. Nonlinear Sci. Appl. 8, 719-739 (2015)Mihet, D: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. J. Nonlinear Sci. Appl. 6, 35-40 (2013)Hicks, TL: Fixed point theory in probabilistic metric spaces. Zb. Rad. Prir.-Mat. Fak. (Novi Sad) 13, 63-72 (1983)Radu, V: Some suitable metrics on fuzzy metric spaces. Fixed Point Theory 5, 323-347 (2004)O’Regan, D, Saadati, R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195, 86-93 (2008)Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)Kirk, WA: Caristi’s fixed-point theorem and metric convexity. Colloq. Math. 36, 81-86 (1976)Ansari, QH: Metric Spaces: Including Fixed Point Theory and Set-Valued Maps. Alpha Science, Oxford (2010

A Characterization of Strong Completeness in Fuzzy Metric Spaces

[EN] Here, we deal with the concept of fuzzy metric space(X,M,*), due to George and Veeramani. Based on the fuzzy diameter for a subset ofX, we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.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 was 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 Nos. 779776 and 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.; Roig, B.; Sapena Piera, A. (2020). A Characterization of Strong Completeness in Fuzzy Metric Spaces. Mathematics. 8(6):1-11. https://doi.org/10.3390/math8060861S11186Menger, K. (1942). Statistical Metrics. Proceedings of the National Academy of Sciences, 28(12), 535-537. doi:10.1073/pnas.28.12.535George, 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)90162-7Gregori, 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. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xAtanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87-96. doi:10.1016/s0165-0114(86)80034-3Gregori, V., Romaguera, S., & Veeramani, P. (2006). A note on intuitionistic fuzzy metric spaces☆. Chaos, Solitons & Fractals, 28(4), 902-905. doi:10.1016/j.chaos.2005.08.113Gregori, V., & Sapena, A. (2018). Remarks to «on strong intuitionistic fuzzy metrics». Journal of Nonlinear Sciences and Applications, 11(02), 316-322. doi:10.22436/jnsa.011.02.12Abu-Donia, H. M., Atia, H. A., & Khater, O. M. A. (2020). Common fixed point theorems in intuitionistic fuzzy metric spaces and intuitionistic (ϕ,ψ)-contractive mappings. Journal of Nonlinear Sciences and Applications, 13(06), 323-329. doi:10.22436/jnsa.013.06.03Gregori, 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.2015.12.010Miheţ, D. (2007). On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 158(8), 915-921. doi:10.1016/j.fss.2006.11.012Wardowski, D. (2013). Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 222, 108-114. doi:10.1016/j.fss.2013.01.012Gregori, 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/s13398-015-0272-0Gregori, V., & Miñana, J.-J. (2017). Strong convergence in fuzzy metric spaces. Filomat, 31(6), 1619-1625. doi:10.2298/fil1706619gGrabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4George, A., & Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 90(3), 365-368. doi:10.1016/s0165-0114(96)00207-2Miheţ, D. (2008). Fuzzy -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), 739-744. doi:10.1016/j.fss.2007.07.006Vasuki, R., & Veeramani, P. (2003). Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 135(3), 415-417. doi:10.1016/s0165-0114(02)00132-xGregori, V., & Romaguera, S. (2004). Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems, 144(3), 411-420. doi:10.1016/s0165-0114(03)00161-1Gregori, 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.2011.12.008Ricarte, L. A., & Romaguera, S. (2014). A domain-theoretic approach to fuzzy metric spaces. Topology and its Applications, 163, 149-159. doi:10.1016/j.topol.2013.10.014Gregori, V., López-Crevillén, A., Morillas, S., & Sapena, A. (2009). On convergence in fuzzy metric spaces. Topology and its Applications, 156(18), 3002-3006. doi:10.1016/j.topol.2008.12.043Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Shukla, S., Gopal, D., & Sintunavarat, W. (2018). A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets and Systems, 350, 85-94. doi:10.1016/j.fss.2018.02.010Beg, 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.2018.2.37Zheng, D., & Wang, P. (2019). Meir–Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, 120-128. doi:10.1016/j.fss.2018.08.014Rakić, D., Došenović, T., Mitrović, Z. D., de la Sen, M., & Radenović, S. (2020). Some Fixed Point Theorems of Ćirić Type in Fuzzy Metric Spaces. Mathematics, 8(2), 297. doi:10.3390/math802029

Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions

[EN] In this paper, by using the concept of the α-Garaghty contraction, we introduce the new notion of the α-Θ-Garaghty type contraction and prove some fixed point results for this contraction in partial metric spaces. Also, we give some examples and applications to illustrate the main results.The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for the Master’s degree Program at KMUTT. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.Onsod, W.; Kumam, P.; Cho, YJ. (2017). Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions. Applied General Topology. 18(1):153-171. https://doi.org/10.4995/agt.2017.6694SWORD153171181T. Abdeljawad, Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed Point Theory Appl. 19 (2013). https://doi.org/10.1186/1687-1812-2013-19T. Abdeljawad and D. Gopal, Erratum to Meir-Keeler $alpha$-contractive fixed and common fixed point theorems, Fixed Point Theory Appl. 110 (2013). H. Alikhani, D. Gopal, M. A. Miandaragh, Sh. Rezapour and N. Shahzad, Some endpoint results for β-generalized weak contractive multifunctions, The Scientific World Journal (2013), Article ID 948472.Beg, I., Butt, A. R., & Radojević, S. (2010). The contraction principle for set valued mappings on a metric space with a graph. Computers & Mathematics with Applications, 60(5), 1214-1219. doi:10.1016/j.camwa.2010.06.003A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. (Debr.) 57 (2000), 31-37.Cho, S.-H., Bae, J.-S., & Karapınar, E. (2013). Fixed point theorems for α-Geraghty contraction type maps in metric spaces. Fixed Point Theory and Applications, 2013(1), 329. doi:10.1186/1687-1812-2013-329Chandok, S. (2015). Some fixed point theorems for (α, β)-admissible Geraghty type contractive mappings and related results. Mathematical Sciences, 9(3), 127-135. doi:10.1007/s40096-015-0159-4Geraghty, M. A. (1973). On contractive mappings. Proceedings of the American Mathematical Society, 40(2), 604-604. doi:10.1090/s0002-9939-1973-0334176-5GOPAL, D., ABBAS, M., PATEL, D. K., & VETRO, C. (2016). Fixed points of α -type F -contractive mappings with an application to nonlinear fractional differential equation. Acta Mathematica Scientia, 36(3), 957-970. doi:10.1016/s0252-9602(16)30052-2Gordji, M., Ramezani, M., Cho, Y., & Pirbavafa, S. (2012). A generalization of Geraghty’s theorem in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory and Applications, 2012(1), 74. doi:10.1186/1687-1812-2012-74Hussain, N., Karapinar, E., Salimi, P., & Akbar, F. (2013). alpha-Admissible mappings and related Fixed point Theorems. Journal of Inequalities and Applications, 2013(1), 114. doi:10.1186/1029-242x-2013-114Jachymski, J. (2007). The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 136(04), 1359-1373. doi:10.1090/s0002-9939-07-09110-1X. D. Liu, S. S. Chang, Y. Xiao and L. C. Zhao, Existence of fixed points for Θ-type contraction and Θ-type Suzuki contraction in complete metric spaces, Fixed Point Theory Appl. 8 (2016). J. Martinez-Moreno, W. Sintunavarat and Y. J. Cho, Common fixed point theorems for Geraghty's type contraction mappings using the monotone property with two metrics, Fixed Point Theory Appl. 174 (2015).S. G. Mathews, Partial metric topology, in Proceedings of the 11th Summer Conference on General Topology and Applications 728 (1995), 183-197, The New York Academy of Sci. C. Mongkolkehai, Y. J. Cho and P. Kumam, Best proximity points for Geraghty's proximal contraction mappings, Fixed Point Theory Appl. 180 (2013).W. Onsod and P. Kumam, Common fixed point results for φ-ψ-weak contraction mappings via f-α-admissible Mappings in intuitionistic fuzzy metric spaces, Communications in Mathematics and Applications 7 (2016), 167-178.V. L. Rosa and P. Vetro, Fixed point for Geraghty-contractions in partial metric spaces, J. Nonlinear Sci. Appl. 7 (2014), 1-10.Samet, 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.2011.10.01