Fixed Point Theorems Related to Fuzzy Metric Spaces

In the present paper some fixed point and common fixed point theorems in complete Fuzzy 2-metric spaces and fuzzy 3- metric spaces are established which are motivated by Gahler [13-15], Sharma, Sharma and Isekey [30],Sharma , S.[31], Key Words: Fuzzy metric spaces, fuzzy 2- metric spaces, fuzzy 3- metric spaces fixed point, Common fixed point

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 Fixed Point Theorems for Countable Faintly Compatible Mappings in Fuzzy Metric Space

In the present paper, we prove some fixed point theorems for countable faintly compatible mappings in fuzzy metric space. Our results extend and generalized the result of S. Manro and A. Tomar [Faintly compatible maps and existence of common fixed points in fuzzy metric space, Anal. of Fuzzy Math. and Informatics, 10 (2014), 1-8]. Keywords: Fuzzy Metric Spaces, non-compatible mappings, faintly compatible mappings and reciprocally continuous mappings

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

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

Convexity and boundedness relaxation for fixed point theorems in modular spaces

[EN] Although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. A recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. In this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. To relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. Afterwards we present an application to a particular form of integral inclusions to support our generalized version of Nadler’s theorem in modular spaces.The authors gratefully acknowledge the reviewer and the editor for their useful observations and recommendations.Lael, F.; Shabanian, S. (2021). Convexity and boundedness relaxation for fixed point theorems in modular spaces. Applied General Topology. 22(1):91-108. https://doi.org/10.4995/agt.2021.13902OJS91108221M. Abbas, F. Lael and N. Saleem, Fuzzy b-metric spaces: Fixed point results for ψ-contraction correspondences and their application, Axioms 9, no. 2 (2020), 1-12. https://doi.org/10.3390/axioms9020036A. Ait Taleb and E. Hanebaly, A fixed point theorem and its application to integral equations in modular function spaces, Proceedings of the American Mathematical Society 128 (1999), 419-426. https://doi.org/10.1090/S0002-9939-99-05546-XM. R. Alfuraidan, Fixed points of multivalued mappings in modular function spaces with a graph, Fixed Point Theory and Applications 42 (2015), 1-14. https://doi.org/10.1186/s13663-015-0292-7A. H. Ansari, T. Dosenovic, S. Radenovic, N. Saleem, V. Sesum-Cavic and J. Vujakovic, C-class functions on some fixed point results in ordered partial metric spaces via admissible mappings, Novi Sad Journal of Mathematics 49, no. 1 (2019), 101-116. https://doi.org/10.30755/NSJOM.07794A. H. Ansari, J. M. Kumar and N. Saleem, Inverse-C-class function on weak semi compatibility and fixed point theorems for expansive mappings in G-metric spaces, Mathematica Moravica 24, no. 1 (2020), 93-108. https://doi.org/10.5937/MatMor2001093HA. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64, no. 4 (2014), 941-960. https://doi.org/10.2478/s12175-014-0250-6I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Unianowsk, Gos. Ped. Inst. 30 (1989), 26-37.S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181M. Berziga, I. Kédimb and A. Mannaic, Multivalued fixed point theorem in b-metric spaces and its application to differential inclusions, Filomat 32 no. 8 (2018), 2963-2976. https://doi.org/10.2298/FIL1808963BR. K. Bishta, A remark on asymptotic regularity and fixed point property, Filomat 33 no. 14 (2019), 4665-4671. https://doi.org/10.2298/FIL1914665BM. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Mod. Math. 4 (2009), 285-301.M. Bota, A. Molnar and C. Varga, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory 12, no. 2 (2011), 21-28.N. Bourbaki, Topologie Generale; Herman, Paris, France, 1974.M. S. Brodskii and D. P. Milman, On the center of a convex set, Doklady Acad. N. S. 59 (1948), 837-840.S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993), 5-11.S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 263-276.T. Dominguez-Benavides, M. A. Khamsi and S. Samadi, Asymptotically regular mappings in modular function spaces, Scientiae Mathematicae Japonicae 2 (2001), 295-304. https://doi.org/10.1016/S0362-546X(00)00117-6S. Dhompongsa, T. D. Benavides, A. Kaewcharoen and B. Panyanak, Fixed point theorems for multivalued mappings in modular function spaces, Sci. Math. Japon. (2006), 139-147.Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006), 103-112. https://doi.org/10.1016/j.jmaa.2005.12.004K. Fallahi, K. Nourouzi, Probabilistic modular spaces and linear operators. Acta Appl. Math. 105 (2009), 123-140. https://doi.org/10.1007/s10440-008-9267-6N. Hussain, V. Parvaneh, J. R. Roshan and Z. Kadelburg, Fixed points of cyclic weakly (ψ, φ , L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl. 2013 (2013), 256. https://doi.org/10.1186/1687-1812-2013-256M. A. Japon, Some geometric properties in modular spaces and application to fixed point theory, J. Math. Anal. Appl. 295 (2004), 576-594. https://doi.org/10.1016/j.jmaa.2004.02.047M. A. Japon, Applications of Musielak-Orlicz spaces in modern control systems, Teubner-Texte Math. 103 (1988), 34-36.W. W. Kassu, M. G. Sangago and H. Zegeye, Convergence theorems to common fixed points of multivalued ρ-quasi-nonexpansive mappings in modular function spaces, Adv. Fixed Point Theory 8 (2018), 21-36.M. A. Khamsi, A convexity property in modular function spaces, Math. Japonica 44, no. 2 (1996), 269-279.M. A. Khamsi, W. K. Kozlowski and C. Shutao, Some geometrical properties and fixed point theorems in Orlicz spaces, J. Math. Anal. Appl. 155 (1991), 393-412. https://doi.org/10.1016/0022-247X(91)90009-OM. A. Khamsi, W. M. Kozlowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Analysis, Theory, Methods and Applications 14 (1990), 935-953. https://doi.org/10.1016/0362-546X(90)90111-SM. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30, no. 1 (1984), 1-9. https://doi.org/10.1017/S0004972700001659S. H. Khan, Approximating fixed points of (λ, ρ)-firmly nonexpansive mappings in modular function spaces, arXiv:1802.00681v1, 2018. https://doi.org/10.1007/s40065-018-0204-xN. Kir and H. Kiziltunc, On some well known fixed point theorems in b-metric spaces, Turk. J. Anal. Number Theory 1, no. 1 (2013), 13-16. https://doi.org/10.12691/tjant-1-1-4D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007), 132-139. https://doi.org/10.1016/j.jmaa.2006.12.012W. M. Kozlowski, Modular Function Spaces, Marcel Dekker, 1988.P. Kumam and W. Sintunavarat, The existence of fixed point theorems for partial q-set-valued quasicontractions in b-metric spaces and related results, Fixed Point Theory Appl. 2014 (2014), 226. https://doi.org/10.1186/1687-1812-2014-226M. A. Kutbi and A. Latif, Fixed points of multivalued maps in modular function spaces, Fixed Point Theory and Applications 2009 (2009), 786357. https://doi.org/10.1155/2009/786357F. Lael and K. Nourouzi, On the fixed points of correspondences in modular spaces, International Scholarly Research Network ISRN Geometry 2011 (2011), 530254. https://doi.org/10.5402/2011/530254A. Lukács and S. Kajántó, Fixed point theorems for various types of F-contractions in complete b-metric spaces, Fixed Point Theory 19, no. 1 (2018), 321-334. https://doi.org/10.24193/fpt-ro.2018.1.25J. Markin, A fixed point theorem for set valued mappings, Bull. Am. Math. Soc. 74 (1968), 639-640. https://doi.org/10.1090/S0002-9904-1968-11971-8K. Mehmet and K. Hukmi, On some well known fixed point theorems in b-metric space, Turkish Journal of Analysis and Number Theory 1 (2013), 13-16. https://doi.org/10.12691/tjant-1-1-4R. Miculescu and A. Mihail, New fixed point theorems for set-valued contractions in $b-$metric spaces, J. Fixed Point Theory Appl. 19 (2017), 2153-2163. https://doi.org/10.1007/s11784-016-0400-2J. Musielak and W. Orlicz, On modular spaces, Studia Mathematica 18 (1959), 49-65. https://doi.org/10.4064/sm-18-1-49-65J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034, Lecture Notes in Mathematics, Springer-Verlag, 1983. https://doi.org/10.1007/BFb0072210S. B. Nadler, Multi-valued contraction mappings, Pacific Journal of Mathematics 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475H. Nakano, Modular Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.F. Nikbakht Sarvestani, S. M. Vaezpour and M. Asadi, A characterization of the generalization of the generalized KKM mapping via the measure of noncompactness in complete geodesic spaces, J. Nonlinear Funct. Anal. 2017 (2017), 8.K. Nourouzi and S. Shabanian, Operators defined on n-modular spaces, Mediterranean Journal of Mathematics 6 (2009), 431-446. https://doi.org/10.1007/s00009-009-0016-5W. Orlicz, Über eine gewisse klasse von Raumen vom Typus B, Bull. Acad. Polon. Sci. A (1932), 207-220.W. Orlicz, Über Raumen LM, Bull. Acad. Polon. Sci. A (1936), 93-107.M. O. Olatinwo, Some results on multi-valued weakly jungck mappings in b-metric space, Cent. Eur. J. Math. 6 (2008), 610-621. https://doi.org/10.2478/s11533-008-0047-3M. Pacurar, Sequences of almost contractions and fixed points in b-metric spaces, Analele Univ. Vest Timis. Ser. Mat. Inform. XLVIII 3 (2010), 125-137.S. Radenovic, T. Dosenovic, T. A. Lampert and Z. Golubovíc, A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations, Applied Mathematics and Computation 273 (2016), 155-164. https://doi.org/10.1016/j.amc.2015.09.089N.Saleem, I. Habib and M. Sen, Some new results on coincidence points for multivalued Suzuki-type mappings in fairly?? complete spaces, Computation 8, no. 1 (2020), 17. https://doi.org/10.3390/computation8010017N. Saleem, M. Abbas, B. Ali, and Z. Raza, Fixed points of Suzuki-type generalized multivalued (f, θ, L)-almost contractions with applications, Filomat 33, no. 2 (2019), 499-518. https://doi.org/10.2298/FIL1902499SN. Saleem, M. Abbas, B. Bin-Mohsin and S. Radenovic, Pata type best proximity point results in metric spaces,?? Miskolac Notes 21, no. 1 (2020), 367-386. https://doi.org/10.18514/MMN.2020.2764N. Saleem, I. Iqbal, B. Iqbal, and S. Radenovic, Coincidence and fixed points of multivalued F-contractions in generalized metric space with application, Journal of Fixed Point Theory and Applications 22 (2020), 81. https://doi.org/10.1007/s11784-020-00815-3S. Shabanian and K. Nourouzi, Modular Space and Fixed Point Theorems, thesis (in persian), 2007, K.N.Toosi University of Technology.W. Shan He, Generalization of a sharp Hölder's inequality and its application, J. Math. Anal. Appl. 332, no. 1 (2007), 741-750. https://doi.org/10.1016/j.jmaa.2006.10.019S. L. Singh and B. Prasad, Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl. 55, no. 11 (2008), 2512-2520. https://doi.org/10.1016/j.camwa.2007.10.026W. Sintunavarat, S. Plubtieng and P. Katchang, Fixed point result and applications on b-metric space endowed with an arbitrary binary relation, Fixed Point Theory Appl. 2013 (2013), 296. https://doi.org/10.1186/1687-1812-2013-296T. Van An, L. Quoc Tuyen and N. Van Dung, Stone-type theorem on b-metric spaces and applications, Topology and its Applications 185-186 (2015), 50-64. https://doi.org/10.1016/j.topol.2015.02.00

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

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