J. Fixed Point Theory Appl. (2020) 22:58 https://doi.org/10.1007/s11784-020-00790-9 c  Springer Nature Switzerland AG 2020 Journal of Fixed Point Theory and Applications Cyclic iterated function systems R. Pasupathi, A. K. B. Chand and M. A. Navascu´ es Abstract. In this paper, we consider some generalization of the Banach contraction principle, namely cyclic contraction and cyclic ϕ-contraction. For the application to the fractal, we develop new iterated function sys- tems (IFS) consisting of cyclic contractions and cyclic ϕ-contractions. Further, we discuss about some special properties of the Hutchinson operator associated with the cyclic (c)-comparison IFS. Mathematics Subject Classification. 28A80, 37C25, 47H04, 47H09, 47H10. Keywords. Fixed point, cyclic contraction, iterated function system, fractal, cyclic ϕ-contraction, cyclic (c)-comparison function. 1. Introduction In his famous book “The Fractal Geometry of Nature”, Mandelbrot [22] intro- duced the concept of fractal to capture non-linearity in nature and in various physical phenomena. Fractal geometry has been proved to be a very effective mean for modeling objects with infinite details in nature. The fixed point theory plays an important role in the theory of iterated function systems (IFS), introduced by Hutchinson [16] and studied in detail by Barnsley [4] for construction of deterministic fractals. IFSs have become powerful tools for construction of various types of fractals in applied sciences. The appli- cations of IFSs are image processing, stochastic growth models, and random dynamical systems, for instance, one can consult [35]. Fractal interpolation functions (FIFs) are developed based on the theory of IFS, where these func- tions are attractors of suitable IFS associated with any given data set. Many researchers proposed mathematical models of different types of fractal curves and surfaces, see for instance [7–11, 26, 27]. The existence of attractor or deter- ministic fractal of IFS with finite number of maps in a complete metric space follows from the famous Banach contraction principle. The second author acknowledges the financial support received from the project MTR/2017/000574-MATRICS of the Science and Engineering Research Board (SERB), Government of India. 0123456789().: V,-vol