Set-Valued ?-Fractal Functions

In this paper, we introduce the concept of the a-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its a-fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an a-fractal function is an attractor of an iterated function system.

Automated summary

In this paper, the concept of a-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers is introduced. The properties of such fractal functions are studied, and the perturbation error between the given continuous function and its a-fractal function is estimated. A new graph of a set-valued function that differs from the standard graph in the literature is defined, and bounds on the fractal dimension of this newly defined graph for special classes of set-valued functions are established. Additionally, the necessity of defining this new graph is explained through examples. Finally, it is proven that the new graph of an a-fractal function is an attractor of an iterated function system.