Abstract
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.
Highlights
In the 1970’s Benoit Mandelbrot introduced to the world new field of mathematics
One of the generalizations is the use of results from fixed point theory, namely the use of various iteration processes instead of the Picard one that is used in the generation of Mandelbrot and Julia sets
IV we present some graphical examples of Mandelbrot and Julia sets obtained with the derived criteria
Summary
In the 1970’s Benoit Mandelbrot introduced to the world new field of mathematics. He named this field fractal geometry (fractus -- from Latin divided, fractional). Relative superior Julia sets, Mandelbrot sets and tricorn, multicorns by using the S-iteration scheme were presented in [4] and [5]. Mishra et al [8], [9] developed fixed point results in relative superior Julia sets, tricorn and multicorns by using the Ishikawa iteration with s-convexity. Kang et al [10] introduced new fixed point results for fractal generation using the implicit Jungck-Noor orbit with s-convexity, whereas Nazeer et al [11] used the Jungck-Mann and Jungck-Ishikawa iterations with s-convexity. In this paper we study the use of Jungck-CR iteration with s-convexity in the generation of Mandelbrot and Julia sets. We present some graphical examples of Mandelbrot and Julia sets via the Jungck-CR iteration with s-convexity.
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