Cyclic iterated function systems
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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 systems (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] introduced 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 applications 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 functions 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 deterministic 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
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The framework of IFS theory by Hutchinson has been extended to more general spaces, generalized contractions, and infinite IFSs or more generally multifunction systems. Hata [15] constructed IFS using condition φ functions. The concept of infinite IFSs was introduced by Fernau [13]. Other remarkable works in this direction have been done by Gw´ o´zd´z-Lukowska and Jachymski [14], Mauldin and Urba´ nski [23], Klimek and Kosek [20], Le´sniak [21], and Secelean [33]. Secelean [32] investigated countable iterated function systems on a compact metric space. The concept of constructing new IFS composed of different contractions called F -contractions was introduced by Secelean [34]. In the reference [2], the authors introduced the concept of topological IFS attractor, that generalizes the usual IFS attractor. That is to say, every IFS attractor is a topological IFS
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