A Novel Approach to Fractal Generation through Strong Coupled Fixed Points in Intuitionistic Fuzzy Metric Spaces

In this manuscript, we explore the concept of strong-coupled fixed points in the context of intuitionistic fuzzy metric spaces (IFMS). Our approach is grounded in the idea of intuitionistic fuzzy contractive couplings (IFCCs), which provide a framework for understanding fixed points in fuzzy settings. We begin by introducing a novel formulation of coupling, which combines the principles of coupled fuzzy contractions with cyclic mappings. This combination leads to a more generalized and effective method of identifying strong-coupled fixed points, extending previous results in fuzzy metric spaces. A key contribution to this paper is the proof of the existence of a unique strong-coupled fixed point. We establish this result through rigorous theoretical analysis and provide a corollary that strengthens the foundation of our work. Several non-trivial examples are presented to demonstrate the applicability of the theory and the robustness of the strong-coupled fixed point in various scenarios. Additionally, we present a practical application of our findings: the construction of a strong-coupled fractal set within the framework of intuitionistic fuzzy metric spaces. This is achieved by applying an intuitionistic iterated function system (IIFS), which is based on a family of intuitionistic fuzzy contractive couplings. The fractal generation process is illustrated through several examples, demonstrating the theoretical results in action. To further solidify the applicability of our approach, we introduce an intuitionistic fuzzy version of the Hausdorff distance between compact sets, a crucial tool in measuring the "closeness" of sets within the intuitionistic fuzzy context. Several examples are provided to clarify the fractal generation process, showing how the intuitionistic fuzzy metrics and couplings contribute to the creation of self-similar fractals. This work not only enhances the understanding of fixed points in intuitionistic fuzzy spaces but also provides new insights into their application in fractal geometry, offering both theoretical advancements and practical tools for future research in this area.