Analytical properties and the box-counting dimension of nonlinear hidden variable recurrent fractal interpolation functions constructed by using Rakotch's fixed point theorem

Abstract

Rakotch contraction is a generalization of Banach contraction, which implies that in the case of using Rakotch's fixed point theorem, we can model more objects than using Banach's fixed point theorem. Moreover, hidden variable recurrent fractal interpolation functions (HVRFIFs) with Hölder function factors are more general than the fractal interpolation functions (FIFs), recurrent FIFs and hidden variable FIFs with Lipschitz function factors. We demonstrate that HVRFIFs can be constructed using the Rakotch's fixed point theorem, and then investigate the analytical and geometric properties of those HVRFIFs. Firstly, we construct a nonlinear hidden variable recurrent fractal interpolation functions with Hölder function factors on the basis of a given data set using Rakotch contractions. Next, we analyze the smoothness of the HVRFIFs and show that they are stable on the small perturbations of the given data. Finally, we get the lower and upper bounds for their box-counting dimensions.