Construction of Iterated Function System using Contraction Mapping Principle

Abstract

Fractals, briefly defined as self-similar structure or mathematically they are subsets of simple geometrical spaces such as $\mathbb{R, C}$ and $\mathbb{R}^{2}$. Fractals are viewed as significant on the grounds that they characterize pictures that are generally cannot be characterized by Euclidean geometry. Fractals are depicted utilizing calculations and manages objects that do not have whole number measurements. Some of the examples of fractals are the Cantor set, the Koch curve, the Sierpinski triangle, and the Julia set etc.

In Chapter 1, we give a brief introduction of fractals, their existence in nature or real life and some of their applications. Also we discuss the two vital properties of fractals that is self-similarity and fractional dimension with the help of Koch curve. We also give a brief introduction of Hutchinson operator, iterated function system and an attractor of iterated function system.

The study of Picard operator is similar to the study of contractive type mappings in the context of metric spaces. It is easy to see that almost all contractive type mappings on a complete metric space are Picard operators. In Chapter 2, we introduce weak $\theta_m$-contraction and give some results on the existence of Picard operator for such class of mappings in the setting of metric spaces.

In Chapter 3, we define weak $\theta_m$ iterated function system and present some results on the existence of a unique attractor for such an iterated function system. Also we define weak $\theta_m$ multifunction iterated system and prove some results on the existence of the attractor for such iterated multifunction system.