Exploring Iterated Function Systems of Generalized Contractions in the Realm of Generalized Metric Spaces

Abstract

In the 17th century, mathematics branched into the study of fractals, recognizing that many newlinenatural objects with irregular shapes possess complex geometries and non-integer dimensions. newlineThis led to the concept of fractal dimensions. The study of iterated function system (IFS) provides newlinea direct approach to generating deterministic fractals. Despite the widespread applications of newlinefractal theory, its non-integer dimension and self-similar properties remain central to its analysis. newlineMandelbrot and Hutchinson were pivotal in introducing fractals to mathematics, but Barnsley s newlineIFS methodology truly popularized them. newlineIFS serves as a sophisticated method for constructing fractals through a set of maps that newlinedelineate the shape s similarities. Over the years, IFS has emerged as a formidable tool in the newlinecreation and analysis of new fractal sets. Fixed point theory underpins many mathematical newlineproblems, such as optimization, variational inequalities, and the generation of fractals. The newlineBanach contraction principle, formulated by the Polish mathematician Stefan Banach in 1922, is newlinea fundamental concept in the theory of fixed points in metric spaces. It states that a self mapping newlinethat contracts on a metric space that is complete will have exactly one unique fixed point. newlineHutchinson-Barnsley theory leverages this principle to define and construct fractal attractors as newlinecompact invariant subsets of a complete metric space, generated by IFS of contractions. Michael newlineBarnsley s seminal work, Fractals Everywhere , discusses this process in detail newline