On Fuzzy Double Controlled Fisher Iterated Function System with Application to Economics via Numerical Algorithm

Abstract

In this research, we generalize the fixed-point theorem for fuzzy Fisher contraction in fuzzy double controlled metric space, a newly developed mathematical structure that expands ordinary metric spaces and by integrating control functions. The interplay of fuzzy Fisher contraction and fuzzy double controlled metric space provides a new paradigm for investigating contraction mappings and their applications in fractal theory. Using fuzzy Fisher contraction in this generalized fuzzy metric spaces framework, we define a novel class of fractal set known as fuzzy double controlled Fisher fractals and discuss the fuzzy double controlled Fisher iterated function system that is the generalization classical iterative function system in fuzzy double controlled metric space. We develop the Collage theorem for fuzzy double controlled Fisher fractals, which is a useful tool for approximation in fractal generation.  This theorem extends the usual collage theorem by incorporating the flexibility of fuzzy double controlled metric conditions, yielding more generalized approximation results. In addition, we focus on non-trivial cases including the graphical behavior of fuzzy double controlled fisher contraction.