An Application of Viscosity Approximation Type Iterative Method in the Generation of Mandelbrot and Julia Fractals
Sudesh Kumari, Krzysztof Gdawiec, Ashish Nandal, Naresh Kumar, Renu Chugh
Abstract
Abstract In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type $$T(z) = z^{n} + p z + r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:math> , where $$p, r\in {\mathbb {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math> , and $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> . This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method.