Fixed point results of an implicit iterative scheme for fractal generations

Haixia Zhang; Muhammad Tanveer; Yi-Xia Li; Qingxiu Peng; Nehad Ali Shah; Haixia Zhang; Muhammad Tanveer; Yi-Xia Li; Qingxiu Peng; Nehad Ali Shah

doi:10.3934/math.2021761

AIMS Mathematics

2021, Volume 6, Issue 12: 13170-13186. doi: 10.3934/math.2021761

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Research article

Fixed point results of an implicit iterative scheme for fractal generations

1.
School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China
2.
Department of Mathematics and Statistics, The University of Lahore, Lahore Pakistan
3.
College of Mathematics and Finance, Xiangnan University, Chenzhou 423000, China
4.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea

Received: 06 May 2021 Accepted: 23 August 2021 Published: 15 September 2021
MSC : 37F45, 37F50, 47J25

In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.
- fixed point theory,
- fractals,
- Jungck-Ishikawa iteration
Citation: Haixia Zhang, Muhammad Tanveer, Yi-Xia Li, Qingxiu Peng, Nehad Ali Shah. Fixed point results of an implicit iterative scheme for fractal generations[J]. AIMS Mathematics, 2021, 6(12): 13170-13186. doi: 10.3934/math.2021761

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Abstract

In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.

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