A Note On Fixed Point Theorem For Fuzzy Mappings

This paper explores Heilpern's notions of fuzzy mapping and the fixed point theorem for fuzzy mappings. The fixed point theorem for fuzzy mappings as introduced by Heilpern has been generalized and some characterizations are done in this context

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

Fuzzy games with a countable space of actions and applications to systems of generalized quasi-variational inequalities

In this paper, we introduce an abstract fuzzy economy (generalized fuzzy game) model with a countable space of actions and we study the existence of the fuzzy equilibrium. As applications, two types of results are obtained. The first ones concern the existence of the solutions for systems of generalized quasi-variational inequalities with random fuzzy mappings which we define. The last ones are new random fixed point theorems for correspondences with values in complete countable metric spaces.Comment: 18 page

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

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. 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