Variants of Julia and Mandelbrot sets as fractals via Jungck-Ishikawa fixed point iteration system with s-convexity

Abstract

In this paper, we generate some non-classical variants of Julia and Mandelbrot sets, utilizing the Jungck-Ishikawa fixed point iteration system equipped with $ s $-convexity. We establish a novel escape criterion for complex polynomials of a higher degree of the form $ z^n + az^2 -bz + c $, where $ a, \; b $ and $ c $ are complex numbers and furnish some graphical illustrations of the generated complex fractals. In the sequel, we discuss the errors committed by the majority of researchers in developing the escape criterion utilizing distinct fixed point iterations equipped with $ s $-convexity. We conclude the paper by examining variation in images and the impact of parameters on the deviation of dynamics, color and appearance of fractals. It is fascinating to notice that some of our fractals represent the traditional Kachhi Thread Works found in the Kutch district of Gujarat (India) which is useful in the Textile Industry.

Keywords

• escape criterion
• fixed point
• jungck-ishikawa orbit
• s-convexity