New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates

Abstract

The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting field of research nowadays. We introduce in this paper the dynamics of polynomials z n -z + c = 0 for n  2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. In order to solve this function by Jungck -type iterative schemes, we write it in the form of Sz = Tz, where the function T, S are defined as Tz = z n + c and Sz = z.Only mathematical explanations are derived by applying Jungck Ishikawa Iteration for polynomials in the literature but in this paper we have generated Relative Mandelbrot sets and Relative Julia sets.

The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets

“…Other iterations from this group that were used in the study of Mandelbrot and Julia sets are the following: the Ishikawa iteration [6], the Noor iteration [2], the S-iteration [20,21] and the Abbas iteration [24]. The implicit iterations are the following: Jungck-Mann [39] and Jungck-Ishikawa [17]. Moreover, in both groups, several researchers proposed the use of s-convexity.…”

Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

“…[5] presented a construction of famous fractal images-Mandelbrot set and Julia set using 3D iterated function system which gives real look and feel of complex natural fractal images. Also [6] applied Jungck Ishikawa iteration to generate new relative superior Mandelbrot sets and relative superior Julia sets.…”