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

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

Coupled Coincidence and Common Fixed Point Theorems for Set-valued and Single-valued Mappings in fuzzy Metric Space

In this paper, we define the tangential property and the generalized coincidence property for a pair of set-valued and single-valued mappings and use it to prove some coupled coincidence and common fixed point theorems for a hybrid pair of mappings without appeal to the completeness of the underlying space

Fixed point theorems for fuzzy mappings and applications to ordinary fuzzy differential equations

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Some Coupled Fixed Point Theorems in Complete E−Fuzzy Metric Spaces

In this paper some coupled coincidence fixed point theorems are establishedfor the mappings under ϕ- contraction in complete generalizedE- fuzzy metric spaces. Also give an example to validate thetheorem and present an application of this theorem to integral equations

GENERALIZED WEAK CONTRACTION FOR HYBRID PAIR OF MAPPINGS WITH APPLICATION

We establish some common coupled fixed point theorems for hybrid pair of mappings under generalized weak contraction on a non complete metric space, which is not partially ordered. As an application, we study the existence and uniqueness of the solution to an integral equation and also give an example to show the fruitfulness of our results. The results we obtain generalize, extend and improve several classical results in the literature in metric spaces

COUPLED FIXED POINT THEOREMS IN ORDERED NON-ARCHIMEDEAN INTUITIONISTIC FUZZY METRIC SPACE USING k-MONOTONE PROPERTY

In this paper we define k-monotone property and proved coupled fixed point theorem in ordered non-Archimedean Intuitionistic fuzzy metric space. Key words: Non- Archimedean property, k-monotone property, mixed monotone mappings, coupled fixed point, Fuzzy metric space, Intuitionistic Fuzzy metric space, Cauchy sequence, complete fuzzy metric space.   Mathematics Subject Classification: 46S40, 47H10, 54H2