A Brief Study on Fractals as Julia and Mandelbrot Sets for Generalized Rational Map Using the Generalized Viscosity Approximation-Type Iterative Methods

In this article, we study the use of the generalized viscosity approximation-type iterative methods in the generation of fractals as Julia and Mandelbrot sets for generalized rational map of the form + + sin(ε♭), where p ≥ 2, p, q , ϑ, φ , ε \{0} and [1, ∞). Utilizing the proposed iterative methods, we establish a novel escape criterion and implement it within the escape time algorithms to generate and visualize Julia and Mandelbrot sets. This criterion is essential for terminating the iterative process and is the key to producing the resulting captivating fractal patterns. Through graphical and numerical experiments, we analyze how the iteration parameters influence the shape, size and color of the fractals. This work aims to inspire the application of fractal patterns in textile design and printing.