In today's world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of s -convex combination. The Jungck-CR iteration process with s-convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria.

Fractal patterns generated in the complex plane by root finding methods are well known in the literature. In the generation methods of these fractals, only one root finding method is used. In this paper, we propose the use of a combination of root finding methods in the generation of fractal patterns. We use three approaches to combine the methods: (1) the use of different combinations, e.g. affine and s-convex combination, (2) the use of iteration processes from fixed point theory, (3) multistep polynomiography. All the proposed approaches allow us to obtain new and diverse fractal patterns that can be used, for instance, as textile or ceramics patterns. Moreover, we study the proposed methods using five different measures: average number of iterations, convergence area index, generation time, fractal dimension and Wada measure. The computational experiments show that the dependence of the measures on the parameters used in the methods is in most cases a non-trivial, complex and non-monotonic function.

The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomialz³− 1 on a complex plane.

In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton’s root-finding method the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives. These processes applied to polynomials on the complex plane produce images showing basins of attractions for polynomial zeros or images representing the number of iterations required to obtain polynomial roots. These latter images were called by Kalantari as polynomiographs. We use both: the colouring by roots to present basins of attractions, and the colouring by iterations that reveal the speed of convergence and dynamic properties of processes visualised by polynomiographs.