Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results

[EN] With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see "Am. Math. Month. 1967, 74, 436-437") that a metric space is complete if and only if any Banach contraction on any of its closed subsets has a fixed point. We apply our main result to deduce that a well-known fixed point theorem due to D. Mihet (see "Fixed Point Theory 2005, 6, 71-78") also allows us to characterize the fuzzy metric completeness.This research was partially funded by Ministerio de Ciencia, Innovacion y Universidades, under grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds.Romaguera Bonilla, S.; Tirado Peláez, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics. 8(2):1-7. https://doi.org/10.3390/math8020273S1782Connell, E. H. (1959). Properties of fixed point spaces. Proceedings of the American Mathematical Society, 10(6), 974-979. doi:10.1090/s0002-9939-1959-0110093-3Hu, 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/bf01472580Kirk, W. A. (1976). Caristi’s fixed point theorem and metric convexity. Colloquium Mathematicum, 36(1), 81-86. doi:10.4064/cm-36-1-81-86Caristi, J. (1976). Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society, 215, 241-241. doi:10.1090/s0002-9947-1976-0394329-4Suzuki, 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. (2007). A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(05), 1861-1870. doi:10.1090/s0002-9939-07-09055-7Romaguera, S., & Tirado, P. (2019). A Characterization of Quasi-Metric Completeness in Terms of α–ψ-Contractive Mappings Having Fixed Points. Mathematics, 8(1), 16. doi:10.3390/math8010016Samet, 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.2011.10.014Abbas, 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/fil1506217aCastro-Company, F., Romaguera, S., & Tirado, 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(1). doi:10.1186/s13663-015-0476-1Radu, V. (1987). Some fixed point theorems probabilistic metric spaces. Lecture Notes in Mathematics, 125-133. doi:10.1007/bfb0072718Sehgal, V. M., & Bharucha-Reid, A. T. (1972). Fixed points of contraction mappings on probabilistic metric spaces. Mathematical Systems Theory, 6(1-2), 97-102. doi:10.1007/bf01706080Ćirić, L. (2010). Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(3-4), 2009-2018. doi:10.1016/j.na.2009.10.00

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

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

On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

[EN] A Meir-Keeler type fixed point theorem for a family of mappings is proved in Mengerprobabilistic metric space (Menger PM-space). We establish that completeness of the space isequivalent to fixed point property for a larger class of mappings that includes continuous as wellas discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ)type non-expansive mappings is established.Bisht, RK.; Rakocević, V. (2021). On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators. Applied General Topology. 22(2):435-446. https://doi.org/10.4995/agt.2021.15561OJS435446222R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl. 445 (2017), 1239-1242. https://doi.org/10.1016/j.jmaa.2016.02.053R. K. Bisht, A probabilistic Meir-Keeler type fixed point theorem which characterizes metric completeness, Carpathain J. Math. 36, no. 2 (2020), 215-222. https://doi.org/10.37193/CJM.2020.02.05R. K. Bisht and V. Rakočević, Generalized Meir-Keeler type contractions and discontinuity at fixed point, Fixed Point Theory 19, no. 1 (2018), 57-64. https://doi.org/10.24193/fpt-ro.2018.1.06R. K. Bisht and V. Rakočević, Discontinuity at fixed point and metric completeness, Appl. Gen. Topol. 21, no. 2 (2020), 349-362. https://doi.org/10.4995/agt.2020.13943Lj. B. Ćirić, On contraction type mappings, Math. Balkanica 1 (1971), 52-57.T. Hicks and B. E. Rhoades, Fixed points and continuity for multivalued mappings, International J. Math. Math. Sci. 15 (1992), 15-30. https://doi.org/10.1155/S0161171292000024D. S. Jaggi, Fixed point theorems for orbitally continuous functions, Indian J. Math. 19, no. 2 (1977), 113-119.G. F. Jungck, Generalizations of continuity in the context of proper orbits and fixed pont theory, Topol. Proc. 37 (2011), 1-15.A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329. https://doi.org/10.1016/0022-247X(69)90031-6K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31, no. 11 (2017), 3501-3506. https://doi.org/10.2298/FIL1711501PA. Pant, R. P. Pant and M. C. Joshi, Caristi type and Meir-Keeler type fixed point theorems, Filomat 33, no. 12 (2019), 3711-3721. https://doi.org/10.2298/FIL1912711PA. Pant, R. P. Pant and W. Sintunavarat, Analytical Meir-Keeler type contraction mappings and equivalent characterizations, RACSAM 37 (2021), 115. https://doi.org/10.1007/s13398-020-00939-8R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289. https://doi.org/10.1006/jmaa.1999.6560R. P. Pant, N. Y. Özgür and N. Tac s, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43, no. 1 (2020), 499-517. https://doi.org/10.1007/s40840-018-0698-6R. P. Pant, A. Pant, R. M. Nikolić and S. N. Ješić, A characterization of completeness of Menger PM-spaces, J. Fixed Point Theory Appl. 21, (2019) 90. https://doi.org/10.1007/s11784-019-0732-9R. P. Pant, N. Y. Özgür and N. Taş, Discontinuity at fixed points with applications, Bulletin of the Belgian Mathematical Society-Simon Stevin 25, no. 4 (2019), 571-589. https://doi.org/10.36045/bbms/1576206358O. Popescu, A new type of contractions that characterize metric completeness, Carpathian J. Math. 31, no. 3 (2015), 381-387. https://doi.org/10.37193/CJM.2015.03.15B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245. https://doi.org/10.1090/conm/072/956495S. Romaguera, w-distances on fuzzy metric spaces and fixed points, Mathematics 8, no. 11 (2020), 1909. https://doi.org/10.3390/math8111909B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 415-417. https://doi.org/10.2140/pjm.1960.10.313B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, Elsevier 1983.V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings in PM-spaces, Math. System Theory 6 (1972), 97-102. https://doi.org/10.1007/BF01706080P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325-330. https://doi.org/10.1007/BF01472580T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 1861-1869. https://doi.org/10.1090/S0002-9939-07-09055-7N. Taş and N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point Theory 20, no. 2 (2019), 715-728. https://doi.org/10.24193/fpt-ro.2019.2.4

Fixed Point Result in Menger Space with EA Property

This paper’s main objective is to define Menger space (PQM) and the concept of weakly compatible by using the notion of property (EA) & JSR maps to define new property to prove a common fixed point theorem for 4 self maps in Menger space (PQM). Key Words: Fixed Point, Probabilistic Metric Space, Menger space, JSR mappings, property EA Subject classification: 47H10, 54H2

Multivalued generalizations of fixed point results in fuzzy metric spaces

This paper attempts to prove fixed and coincidence point results in fuzzy metric space using multivalued mappings. Altering distance function and multivalued strong {bn}-fuzzy contraction are used in order to do that. Presented theorems are generalization of some well known single valued results. Two examples are given to support the theoretical results