New perturbed parametric metric spaces for distance modeling: fixed-point analysis with applications to the fixed-circle problem

In this study, we introduce a new generalization of metric spaces, called Perturbed Parametric Metric Spaces (PPMS). This framework extends the classical metric space by incorporating perturbation functions and the presence of a non-negative parameter τ in its distance function, rather than a standard two-variable metric $d(\mu ,\nu )$, providing a more flexible and robust treatment of distance measurement that reflects underlying variations and imperfections in complex systems. Specifically, the measurement of distance between two points is subject to errors, often arising from factors such as instrumental inaccuracies or environmental influences. We develop the foundational properties and initiate some topological notions of PPMS, provide illustrative examples, and establish several fixed-point theorems within this setting with application to fixed-circle problem. These results demonstrate the applicability of PPMS in nonlinear analysis and show how it unifies and generalizes various existing metric-type spaces. Our approach opens new perspectives for future research in functional analysis.