Use of Ishikawa and Picard–Ishikawa iterations with s-convexity in the generation of Mandelbrot and Julia sets: A comparative analysis

Subhadip Roy, Krzysztof Gdawiec, Parbati Saha and Binayak S. Choudhury

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 241, issue PB, 739-757

Abstract: In this paper, we consider a two-step Ishikawa iteration and a three-step Picard–Ishikawa iteration extended with s-convexity for complex polynomials of the form zp+1+c for the generation of sequences in the complex plane which lead to the formation of Mandelbrot and Julia sets. Various patterns that emerge from these processes of generation are displayed, analyzed and their variations with the change in the various parameters of the iteration processes are discussed. The outcomes highlight the notable differences in the images obtained by the two iterations. Additionally, we examine the relationship between the iteration parameters and two numerical measures, the average escape time, and the non-escaping area index. The numerical examples demonstrate that this dependency is nonlinear and often differs significantly between the two iterations.

Keywords: Escape criterion; Fractals; Ishikawa iteration; Julia set; Mandelbrot set (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:241:y:2026:i:pb:p:739-757

DOI: 10.1016/j.matcom.2025.11.007

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