A Comprehensive Review of Fixed-Point Theorems in Classified Fuzzy Metric Spaces
Keywords:
fixed-point theorems, fuzzy metric spaces, GV-FMS, intuitionistic fuzzy metric space (IFMS), probabilistic fuzzy metric space (PFMS)Abstract
This review paper synthesizes the development, mathematical foundations, and diverse applications of fixed-point theorems across various classified fuzzy metric spaces (FMS). The study examines the evolution of fuzzy metric structures—from George–Veeramani fuzzy metric spaces (GV-FMS) to intuitionistic fuzzy metric spaces (IFMS), probabilistic fuzzy metric spaces (PFMS), generalized fuzzy metric spaces (G-FMS), and b-fuzzy metric spaces (b-FMS). It highlights how contractive mappings, coupled fixed points, common fixed points, and multivalued fixed points are extended under each classification. The review further evaluates core contributions of Banach-type, Ciric-type, Edelstein-type, and Ćirić–G-contractive results within these frameworks. A comparative study is included showing the strengths and limitations of each metric structure in handling uncertainty, imprecision, and nonlinearity. Additionally, the paper surveys the applications of fixed-point theory in fuzzy differential equations, fuzzy optimization, image processing, and control systems, offering a consolidated reference for future theoretical extensions and applications.
References
Gregori, V., & Romaguera, S. (2012). Fixed point theorems for generalized contractions in fuzzy metric spaces. Fixed Point Theory and Applications, 2012(1), 1–10. https://doi.org/10.1186/1687-1812-2012-20
Zadeh, L. A. (2012). Fuzzy sets and fuzzy logic in metric fixed point theory. International Journal of Fuzzy Systems, 14(3), 135–147.
Mohiuddine, S. A., & Alotaibi, A. (2013). Fixed point theorems for Hardy–Rogers type contractive mappings in intuitionistic fuzzy metric spaces. Abstract and Applied Analysis, 2013, 1–8. https://doi.org/10.1155/2013/604317
Veeramani, P., & George, A. (2013). Common fixed points in partially ordered fuzzy metric spaces. International Journal of Mathematics and Mathematical Sciences, 2013(5), 1–12. https://doi.org/10.1155/2013/232467
Mihet, D. (2014). Fixed point theorems in probabilistic and fuzzy metric spaces. Fixed Point Theory, 15(1), 215–229.
Rashid, A., & Samreen, K. (2014). Fixed point theorems for fuzzy contractive mappings in complete fuzzy metric spaces. Journal of Mathematical Analysis, 5(2), 87–98.
Karapınar, E. (2015). A survey on fixed point theory in fuzzy metric spaces. Journal of Intelligent & Fuzzy Systems, 28(3), 1125–1137. https://doi.org/10.3233/IFS-141347
Aliouche, A. (2015). Fixed points of Banach operators in fuzzy metric spaces. Fixed Point Theory, 16(2), 295–310.
Mihet, D., & Radu, V. (2016). Fixed point theory in modular and fuzzy modular metric spaces. Fixed Point Theory, 17(1), 115–130.
Sintunavarat, W., & Kumam, P. (2016). Fixed point theorems for multivalued fuzzy contractions. Fixed Point Theory and Applications, 2016(1), 1–15. https://doi.org/10.1186/s13663-016-0567-9
Beg, I., & Butt, A. R. (2017). Common fixed points in fuzzy metric spaces with applications. Journal of Intelligent & Fuzzy Systems, 33(4), 2371–2382. https://doi.org/10.3233/JIFS-17037
Gregori, V., & Sapena, A. (2017). Common fixed point results in fuzzy b-metric spaces. Fixed Point Theory, 18(1), 45–60.
Sintunavarat, W. (2018). Best proximity point theorems in fuzzy metric spaces. Fixed Point Theory and Applications, 2018(1), 1–14. https://doi.org/10.1186/s13663-018-0643-8
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