The Picard\u2013Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets

Abdul Aziz Shahid,Krzysztof Gdawiec,Waqas Nazeer

doi:10.1007/s00605-021-01591-z

Abstract

In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the (k+1)st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

Highlights

The Mandelbrot and Julia sets represent some of the most beautiful examples of fractal structures generated by non-linear dynamic systems
To get an idea on this non-trivial dependency we present numerical examples showing the dependence of two measures—time and the average number of iterations (ANI)—on the parameters in the Picard–Mann iteration with s-convexity (α, s)
For the purpose of this study, the Picard–Mann iteration was extended with the use of the s-convex combination

Summary

Introduction

The Mandelbrot and Julia sets represent some of the most beautiful examples of fractal structures generated by non-linear dynamic systems. Working at IBM, Mandelbrot studied their work and plotted the Julia sets for z2 + c, where c ∈ C is the parameter During the generation, he used the following feedback iteration process: zn+1 = zn2 + c,. The Mann iteration belongs to the explicit group of iterations 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] In both groups, several researchers proposed the use of s-convexity. We study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets.

Preliminaries

Main results

Graphical examples

Numerical examples

Conclusions