Fractals of two types of enriched [formula omitted]-Hutchinson–Barnsley operators

Abstract

The aim of this paper is to introduce and develop two novel classifications of enriched (q,θ)-contractions on Banach spaces. The paper includes illustrative examples to demonstrate these concepts and establishes the convergence of Krasnoselskii’s iteration method when applied to approximate the fixed point of such enriched (q,θ)-contractions. Additionally, the paper explores the application of these concepts in constructing the fractals of the corresponding Hutchinson–Barnsley operators. The above construction is illustrated by some examples. These discoveries provide new fixed-point solutions for iterated function systems under various contractive conditions. Finally, as an application of our main result, the existence of the solution to the problem of fourth order differential equation is presented. Furthermore, the findings not only validate but also enhance and expand upon multiple established conclusions in the existing literature.