The Effect of Fuzziness on the Stability of Fixed Points between Certainty and Probability

This paper is the research on how fuzziness may be incorporated in classical fixed-point theory. Fixed-point iteration is a basic numerical algorithm to solve equations of the form x= g(x). Nevertheless, in the real world, there is usually uncertainty, which deterministic models cannot represent. With the addition of the fuzzy parameters into the governing functions, fixed points that exist as a single exact value change to stability intervals. The study illustrates using analytic and quantitative examples of algebraic functions (linear, quadratic, trigonometric, and radical) the way in which the nature of fixed points changes through fuzziness. The results show that the linear systems maintain constant periods even in small perturbations of the system with fuzzy values, whereas nonlinear systems can have considerable topological variations, such as the disappearance of the fixed points. The work connects the theory of fixed-point and fuzzy logic, and provides a stronger mathematical framework on modeling uncertain dynamical systems, and has found use in control theory, economics, and artificial intelligence.