On the Mandelbrot set of [formula omitted] via the Mann and Picard–Mann iterations

Abstract

Since its introduction, the Mandelbrot set has been studied and generalized in various directions. Some authors generalized it by using iterations from fixed point theory, whereas others characterized it by using different complex functions or polynomials. In this paper, we replace the constant c in the classical zp+c function with logct, where t∈R and t≥1. Moreover, we prove escape criteria for the Mann and Picard–Mann iterations in which we use the modified function. Then, we present graphical and numerical examples showing the behaviour of the generated sets depending on the parameters of the iterations and the parameter t. Using the proposed approach, we can generate a great variety of fascinating fractal patterns, and when t∈N the sets form rosette patterns.