Weak Commuting Mappings and its Variants for Generalized φ-Weak Contraction in Perturbed Metric Spaces

In this paper, we establish several common fixed-point theorems in the setting of perturbed metric spaces, where the usual distance d(x, y) is replaced by the perturbed distance d_η(x, y) = d(x, y) + η(x, y), with η representing a non-negative perturbation kernel. Working in this generalized framework, we investigate weakly commuting mappings and their pointwise extensions, including R-weakly commuting and reciprocally continuous mappings. Further, we study R-weakly commuting mappings of type (P ) under a generalized ∗-weak contraction condition formulated with cubic and quadratic powers of the perturbed distance d_η(x, y). These results significantly extend the classical metric-space theorems to situations involving measurement errors and structural perturbations.