Multivariate Fractal Functions in Some Complete Function Spaces and Fractional Integral of Continuous Fractal Functions

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α-fractal function is a fractal function of similar kind.

On three dimensional fractal dynamics with fractional inputs and applications

<abstract><p>The environment around us naturally represents number of its components in fractal structures. Some fractal patterns are also artificially simulated using real life mathematical systems. In this paper, we use the fractal operator combined to the fractional operator with both exponential and Mittag-leffler laws to analyze and solve generalized three-dimensional systems related to real life phenomena. Numerical solutions are provided in each case and applications to some related systems are given. Numerical simulations show the existence of the models' initial three-dimensional structure followed by its self- replication in fractal structure mathematically produced. The whole dynamics are also impacted by the fractional part of the operator as the derivative order changes.</p></abstract>

Iterated function systems consisting of continuous functions satisfying Banach’s orbital condition

We introduce the concept of iterated function system consisting of continuous functions satisfying Banach’s orbital condition and prove that the fractal operator associated to such a system is weakly Picard. Some examples are provided.

A Fractal Operator on Some Standard Spaces of Functions

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as anα-fractal operator on, the space of all real-valued continuous functions defined on a compact intervalI. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the spaceof all bounded functions and the Lebesgue space, and in some standard spaces of smooth functions such as the spaceofk-times continuously differentiable functions, Hölder spacesand Sobolev spaces. Using properties of theα-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.