Abstract
The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.
Highlights
In 1975, Mandelbrot [1] introduced the concept of fractal theory, which studies patterns in the highly complex and unpredictable structures that exist in nature
We have investigated a generalization of the Banach-contraction principle through the novel generalized θ-contraction
For construction of new type selfsimilar sets, we have developed a new iterated function systems (IFSs) consisting of finite collection of generalized θ kn -contractions Tn : X m → X, n ∈ N N, and named it as generalized θ-contraction IFS
Summary
In 1975, Mandelbrot [1] introduced the concept of fractal theory, which studies patterns in the highly complex and unpredictable structures that exist in nature. Using Banach fixed point theorem, we can get an attractor or a fractal by iteration of a finite collection of contraction maps of an IFS. Jeli and Samet [27] proposed a new type of contractive mappings known as θ-contraction (or JS-contraction), and they proved a fixed point result in generalized metric spaces. They demonstrated that the Banach fixed point theorem remains as a particular case of θ-contraction. The new system is called generalized θ-contraction IFS, and it consists of a finite collection of θ kn -contraction functions on a complete metric product space.
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