Fixed-Point Theorems in Fuzzy Metric Spaces and Their Applications in Uncertainty Modeling
DOI:
https://doi.org/10.31305/rrijm.2026.v11.n03.030Keywords:
Fixed-Point Theorems, Fuzzy Metric Spaces, Convergence, ContinuityAbstract
At the core of the thesis lies an extensive investigation of Banach’s Contraction Principle, a foundational result asserting that any contraction mapping on a complete metric space has a unique fixed point. The principle's implications for the existence and uniqueness of solutions to differential and integral equations serve as a recurring theme, revealing the theorem’s continued relevance in mathematical modeling and computational applications. The thesis meticulously outlines the formal proof structure, elucidating the contraction property, the iterative approach to constructing Cauchy sequences, and the convergence behavior ensured by completeness. This is followed by an analytical treatment of Brouwer’s Fixed Point Theorem, which guarantees the existence of a fixed point for any continuous function mapping a compact convex subset of Euclidean space into itself. The topological conditions underpinning this theorem are shown to be foundational not only for abstract mathematics but also for economics and game theory, wherein equilibrium concepts hinge upon the assurance of such fixed points.
References
[1] Arrow, K. J., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy. Econometrica, 22(3), 265-290.
[2] Milgrom, P., & Roberts, J. (1990). Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58(6), 1255-1277.
[3] Scarf, H. (1967). The approximation of fixed points of a continuous mapping. SIAM Journal on Applied Mathematics, 15(5), 1328-1343.
[4] Kreyszig, E. (1989). Introductory Functional Analysis with Applications. Wiley.
[5] Munkres, J. R. (2000). Topology. Prentice Hall.
[6] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[7] Dugundji, J., & Granas, A. (1982). Fixed Point Theory. Springer-Verlag.
[8] Kirk, W. A., & Khamsi, M. A. (2001). An Introduction to Metric Spaces and Fixed Point Theory. Wiley.
[9] Kirk, W. (1975). Caristi's fixed point theorem and metric convexity. Colloquium Mathematicum, 36, 81-86.
[10] Zeidler, E. (1986). Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems. Springer-Verlag.
[11] Burton, T. A. (1983). Fixed points and stability in the theory of differential equations. Kluwer Academic Publishers.
[12] Deimling, K. (1985). Nonlinear Functional Analysis. Springer-Verlag.
[13] Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1), 48-49.
[14] Czerwik, S. (1993). Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1), 5-11.
[15] Gähler, S. (1963). 2-metrische Räume und ihre topologische Struktur. Mathematische Nachrichten, 26, 115-148.
[16] Mustafa, Z., & Sims, B. (2006). A new approach to generalized metric spaces. Journal of Nonlinear and Convex Analysis, 7(2), 289-297.
[17] Khamsi, M. A., & Kirk, W. A. (2011). An Introduction to Metric Spaces and Fixed Point Theory. Wiley.
[18] Agarwal, R. P., & O'Regan, D. (2018). Recent Developments in Fixed Point Theory and Applications. Springer.
[19] Baleanu, D., & Mustafa, O. G. (2020). On new developments in fractional calculus and fixed point theory. Mathematical Modelling and Analysis, 25(3), 293-312.
[20] De La Sen, M. (2019). Multi-valued mappings and fixed points II. Mathematics and Computers in Simulation, 165, 27-53.
[21] Jachymski, J. (2021). The role of fixed point theory in machine learning models. Journal of Computational and Applied Mathematics, 380, 112987.
[22] Feinberg, M. (2019). Foundations of Chemical Reaction Network Theory. Springer.
[23] Hopcroft, J. E., & Ullman, J. D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.
[24] Smith, H. L., & Thieme, H. R. (2011). Dynamical Systems and Population Persistence. American Mathematical Society.
[25] Arrow, K. J., & Hahn, F. H. (1971). General Competitive Analysis. Holden-Day.
