Novel Perturbed b-Metric and Perturbed Extended b-Metric Spaces with Banach-Type Fixed Point Theorem

Abstract

In this paper, we introduce a new general framework, called \emph{perturbed extended $b$-metric spaces}, denoted by $(X,\mathcal{D}_{\zeta},\hbar)$, which extends the classical and extended $b$-metric structures through the inclusion of an explicit perturbation mapping $\hbar$. This formulation is motivated by the observation that distance measurements in many analytical and applied contexts are often affected by intrinsic or external deviations that cannot be captured by the usual metric-type geometries. We also identify a meaningful specialization arising when the control function $\zeta$ is constant, leading to the notion of a \emph{perturbed $b$-metric space}, introduced here as a natural restriction of the general framework. We establish several fundamental properties of spaces of the form $(X,\mathcal{D}_{\zeta},\hbar)$ and develop a Banach-type fixed point theorem in the perturbed extended $b$-metric setting. Conditions ensuring the existence and uniqueness of fixed points of a self-map $T:X\to X$ are derived, together with convergence of Picard iterations $T^{n}v \to \vartheta$. Illustrative examples are provided to show that the presence of the perturbation term $\hbar$, together with the influence of the control function $\zeta$, may cause $\mathcal{D}_{\zeta}$ to lose the usual extended $b$-metric behaviour and prevent $\mathcal{D}_{\zeta}$ from satisfying the standard extended $b$-metric axioms. We further discuss natural extensions of this perturbation philosophy to multi-point settings, encompassing $S$-metric spaces and their extended and $S_b$-metric variants. The results presented here open new directions for the study of contractive operators and fixed point theory in generalized metric environments.