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

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

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.

Suggested Citation

  • Roy, Subhadip & Gdawiec, Krzysztof & Saha, Parbati & Choudhury, Binayak S., 2026. "Use of Ishikawa and Picard–Ishikawa iterations with s-convexity in the generation of Mandelbrot and Julia sets: A comparative analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PB), pages 739-757.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pb:p:739-757
    DOI: 10.1016/j.matcom.2025.11.007
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    References listed on IDEAS

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    1. Tanveer, Muhammad & Nazeer, Waqas & Gdawiec, Krzysztof, 2023. "On the Mandelbrot set of zp+logct via the Mann and Picard–Mann iterations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 184-204.
    2. Nawaz, Bashir & Ullah, Kifayat & Gdawiec, Krzysztof, 2024. "Generation of Mandelbrot and Julia sets by using M-iteration process," Chaos, Solitons & Fractals, Elsevier, vol. 188(C).
    3. Swati Antal & Nihal Özgür & Anita Tomar & Krzysztof Gdawiec, 2025. "Fractal generation via generalized Fibonacci–Mann iteration with s-convexity," Indian Journal of Pure and Applied Mathematics, Springer, vol. 56(4), pages 1593-1607, December.
    4. Wang, Yupin & Li, Xiaodi & Wang, Da & Liu, Shutang, 2022. "A brief note on fractal dynamics of fractional Mandelbrot sets," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    5. Rawat, Shivam & Prajapati, Darshana J. & Tomar, Anita & Gdawiec, Krzysztof, 2024. "Generation of Mandelbrot and Julia sets for generalized rational maps using SP-iteration process equipped with s-convexity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 148-169.
    6. Sharma, Saurabh & Tomar, Anita & Padaliya, Sanjay Kumar, 2025. "On the evolution and importance of the Fibonacci sequence in visualization of fractals," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    7. Tassaddiq, Asifa, 2022. "General escape criteria for the generation of fractals in extended Jungck–Noor orbit," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 1-14.
    8. Adhikari, Nabaraj & Sintunavarat, Wutiphol, 2024. "The Julia and Mandelbrot sets for the function zp−qz2+rz+sincw exhibit Mann and Picard–Mann orbits along with s-convexity," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
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