w-Distances on Fuzzy Metric Spaces and Fixed Points

[EN] We propose a notion of w-distance for fuzzy metric spaces, in the sense of Kramosil and Michalek, which allows us to obtain a characterization of complete fuzzy metric spaces via a suitable fixed point theorem that is proved here. Our main result provides a fuzzy counterpart of a renowned characterization of complete metric spaces due to Suzuki and Takahashi.Romaguera Bonilla, S. (2020). w-Distances on Fuzzy Metric Spaces and Fixed Points. Mathematics. 8(11):1-9. https://doi.org/10.3390/math8111909S19811Suzuki, 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. (2001). Generalized Distance and Existence Theorems in Complete Metric Spaces. Journal of Mathematical Analysis and Applications, 253(2), 440-458. doi:10.1006/jmaa.2000.7151Al-Homidan, S., Ansari, Q. H., & Yao, J.-C. (2008). Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Analysis: Theory, Methods & Applications, 69(1), 126-139. doi:10.1016/j.na.2007.05.004Lakzian, H., & Lin, I.-J. (2012). The Existence of Fixed Points for Nonlinear Contractive Maps in Metric Spaces with -Distances. Journal of Applied Mathematics, 2012, 1-11. doi:10.1155/2012/161470Alegre, C., Marín, J., & Romaguera, S. (2014). A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory and Applications, 2014(1). doi:10.1186/1687-1812-2014-40Alegre, C., & Marín, J. (2016). Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces. Topology and its Applications, 203, 32-41. doi:10.1016/j.topol.2015.12.073Lakzian, H., Rakočević, V., & Aydi, H. (2019). Extensions of Kannan contraction via w-distances. Aequationes mathematicae, 93(6), 1231-1244. doi:10.1007/s00010-019-00673-6Alegre, C., Fulga, A., Karapinar, E., & Tirado, P. (2020). A Discussion on p-Geraghty Contraction on mw-Quasi-Metric Spaces. Mathematics, 8(9), 1437. doi:10.3390/math8091437Abbas, 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/fil1506217aRomaguera, S., & Tirado, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics, 8(2), 273. doi:10.3390/math8020273Kirk, W. A. (1976). Caristi’s fixed point theorem and metric convexity. Colloquium Mathematicum, 36(1), 81-86. doi:10.4064/cm-36-1-81-86Hu, 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/bf01472580Grabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4Radu, V. (1987). Some fixed point theorems probabilistic metric spaces. Lecture Notes in Mathematics, 125-133. doi:10.1007/bfb0072718Riaz, M., & Hashmi, M. R. (2018). Fixed points of fuzzy neutrosophic soft mapping with decision-making. Fixed Point Theory and Applications, 2018(1). doi:10.1186/s13663-018-0632-5Riaz, M., & Hashmi, M. R. (2019). Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems. Journal of Intelligent & Fuzzy Systems, 37(4), 5417-5439. doi:10.3233/jifs-190550Hashmi, M. R., & Riaz, M. (2020). A novel approach to censuses process by using Pythagorean m-polar fuzzy Dombi’s aggregation operators. Journal of Intelligent & Fuzzy Systems, 38(2), 1977-1995. doi:10.3233/jifs-19061

Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Research ArticleIn uniform spaces (...) with symmetric structures determined by the D-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the J-families of generalized pseudodistances on (...) are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by J-families. Results are new also in locally convex and metric spaces. Examples are provided

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