Fixed Point Results in Metric Spaces Using Setbased Iteration Functions

In this paper, we establish several unique fixed point (UFP) results in a Real Metric (RM) space using a novel set-based iteration framework. Set-based iteration function provide a natural and robust extension of classical iteration by allowing mapping from element to subsets of the underlying space. The approach is based on the construction of an ordered pair G-set and its associated G₀-subset, generated through iterative sequences of a self-mapping. Unlike classical pointwise iterations, the proposed framework interprets the iteration process as a sequence of subsets governed by metric distances. We prove the existence and uniqueness of fixed-points for Banach, Kannan, Chatterjea, and generalized contractive mappings by analyzing the monotonic behaviour of an associated distance T- sequence. The completeness of the metric space ensures convergence, while the contraction conditions guarantee uniqueness. The results unify and extend several classical fixed point theorems and provide a flexible structure applicable to nonlinear analysis, optimization, coding theory, and control systems. The exposition is self-contained and written to meet contemporary research standards.