Mathematical Modeling of Fractals via Proximal F-Iterated Function Systems

Abstract

We propose a novel approach to fractals by leveraging the approximation of fixed points, emphasizing the deep connections between fractal theory and fixed-point theory. We include a condition of isomorphism, which not only generates traditional fractals but also introduces the concept of generating two fractals simultaneously, using the framework of the best proximity point: one as the original and the other as its best proximity counterpart. We present a notion of the Proximal F−Iterated Function System (F−PIFS), which is constructed using a finite set of F*−weak proximal contractions. This extends the classical notions of Iterated Function Systems (IFSs) and Proximal Iterated Function Systems (PIFSs), which are commonly used to create fractals. Our findings show that under specific conditions in a metric space, the F−PIFS has a unique best attractor. In order to illustrate our findings, we provide an example showing how these fractals are generated together. Furthermore, we intend to investigate the possible domains in which our findings may be used.

Keywords

• fractals
• F − iterated function system%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> proximal <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm500"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">F</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>iterated function system
• metric space
• best proximity point