Fractal dimensions of mixed Katugampola fractional integral associated with vector valued functions

Abstract

The aim of this article is to study the fundamental properties of the mixed Katugampola fractional integral (K-integral) of vector-valued functions and fractal dimensional results. We show that the mixed K-integral preserves the basic properties such as boundedness, continuity, and bounded variation of vector-valued functions. We also estimate the Hausdorff dimension of the graph of the vector-valued function and the graph of the mixed K-integral on the rectangular region. Moreover, we prove that the upper bound of the box dimension of the graph of each coordinate function of mixed K-integral of vector-valued functions is 3−min{μ1,μ2}, where μ1 and μ2 are order of the fractional integral with 0<μ1<1,0<μ2<1. Moreover, we give an example of unbounded variational vector-valued functions. In the end, we discuss some problems for future direction.