Complex-Valued Suprametric Spaces, Related Fixed Point Results, and Their Applications to Barnsley Fern Fractal Generation and Mixed Volterra–Fredholm Integral Equations

Abstract

The novelty of this work is that it is the first to introduce complex-valued suprametric spaces and apply it to Fractal Generation and mixed Volterra–Fredholm Integral Equations. In the realm of fuzzy logic, complex-valued suprametric spaces provide a robust framework for quantifying the similarity between fuzzy sets; for instance, utilizing a complex-valued suprametric approach, we compared the similarity between fuzzy sets represented by complex-valued feature vectors, yielding quantitative measures of their relationships. Thereafter, we establish related fixed point results and their applications in algorithmic and numerical contexts. The study then delves into the generation of fractals, exemplified by the Barnsley Fern fractal, utilizing sequences of affine transformations within complex-valued suprametric spaces. Moreover, this article presents two algorithms for soft computing and fractal generation. The first algorithm uses complex-valued suprametric similarity for fuzzy clustering, iteratively assigning fuzzy sets to clusters based on similarity and updating cluster centers until convergence. The distinctive pattern of the Barnsley Fern fractal is produced by the second algorithm’s repetitive affine transformations, which are chosen at random. These techniques demonstrate how well complex numbers cluster and how simple procedures can create complicated fractals. Moving beyond fractal generation, the paper addresses the solution of mixed Volterra–Fredholm integral equations in the complex plane using our results, demonstrating numerical illustrations of complex-valued integral equations.