Fractal Construction Using Interpolative Kannan‐Type Operators in Controlled Settings

ABSTRACT This paper presents new developments in fixed point theory and fractal analysis within controlled metric spaces. We introduce a modified generalized controlled function for the Hausdorff controlled metric, addressing and correcting key assumptions in the work of Alamgir et al. [4]. This refinement facilitates the analysis of the Hutchinson operator and the existence of attractors, and enables the establishment of fixed point theorems for interpolative Kannan contractions, supported by illustrative examples. Additionally, we resolve gaps in mathematical rigor found in prior studies, notably those by Thangaraj et al. [13, 14], thereby reinforcing the theoretical framework. Our approach is extended to iterated function systems and multi‐function systems, proving both the existence and uniqueness of attractors and offering a refined Collage theorem. The paper concludes by identifying future research directions, including the study of weaker contractive conditions, such as ‐weak, enriched Ćirić‐type, and Hardy‐Rogers contractions, to enhance generalizability and potential applications.