Fixed Point Results via a Uniform Class of A_d-Contractions

Abstract

This paper studies fixed point arguments in dislocated metric spaces, where the self-distance of a point is not required to vanish. Starting from the class $\mathcal{A}$ of control functions introduced by Akram, Zafar, and Siddiqui for $\mathcal{A}$-contractions, we identify a uniformity issue in the contractive condition used in iterative fixed point arguments. The original condition provides a contraction factor only for an individual comparison, whereas a Picard iteration requires a single factor controlling every step of the orbit. To address this issue, we introduce a uniform subclass $\mathcal{A}_d$, in which each control function admits a global contraction constant $k_\alpha<1$. Within the sequential framework of dislocated metric spaces, we establish fixed point theorems for a single self-map, a countable family of self-maps, an integral-type contraction, and a pair of mappings associated with two compatible dislocated metrics. We prove that $\mathcal{A}_d$ is a proper subclass of $\mathcal{A}$ by constructing an explicit control function in $\mathcal{A}\setminus\mathcal{A}_d$ whose associated Picard orbit admits no uniform geometric decay. Additional examples illustrate the applicability of the resulting theory to interval models and to a function-space model with a nonzero fixed point.