Revisiting fractal through nonconventional iterated function systems

This paper is a pre-step in conducting a restudy for an emerging theory in applied sciences, namely Fractal interpolation. It is one of the best-fit models for capturing irregular data that arise in physical situations. On the other hand, it has fixed point theory as the staunch basis, so any inspection of it would get governed by the Hutchinson–Barnsley theory of fractals. In this regard, we classify an enormous collection of maps owned by the literature of fixed point theory into two — conventional and nonconventional. Suitably, every conventional iterated function system (IFS) has delivered fractal, but nonconventional IFSs are yet to make a mark. Therefore, the present work introduces a novel nonconventional map of the Ćirić–Reich–Rus genre to fulfill this gap. It incorporates a parameter δ∈(0,∞), in a Ćirić–Reich–Rus condition, for the first time in the literature. Consequently, we obtain extension, improvement, and generalization of the results produced in Sahu et al. (2010), Shaoyuan et al. (2015), Dung and Petruşel (2017) and Abbas et al. (2022). In addition, a rational map and a Suzuki-type Kannan map are considered to prove the point.