--- title: Ad-Contractions in Dislocated Metric Spaces url: https://www.emergentmind.com/topics/ad-contractions type: topic --- # Ad-Contractions in Dislocated Metric Spaces Searching arXiv for the specified paper and closely related work on A-contractions / dislocated metric spaces. Ad-contractions, written in the source paper as \(\mathcal{A}_d\)-contractions, are a class of contractive self-maps introduced by Panthi & Panthi to extend the framework of \(\mathcal{A}\)-contractions from ordinary metric spaces to dislocated metric spaces, where self-distances need not vanish. The theory establishes fixed point results for a single mapping, a sequence of mappings, an integral-type contraction, and mappings defined relative to two dislocated metrics, while preserving a Banach–Picard style proof architecture. The central contribution is the replacement of the classical control-function class by a new class \(\mathcal{A}_d\), together with an axiom tailored to the nonzero self-distance phenomenon that is characteristic of dislocated metrics [2507.15635]. ## 1. Dislocated metric spaces as the ambient setting A dislocated metric space, or \(d\)-metric space, is a nonempty set \(X\) equipped with a function \[ d:X\times X\longrightarrow[0,\infty) \] such that for all \(x,y,z\in X\), \[ \text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y). \] In this setting, a sequence \(\{x_n\}\subset X\) is Cauchy if for every \(\varepsilon>0\) there exists \(N\) such that \(d(x_m,x_n)<\varepsilon\) for all \(m,n\ge N\), and convergent to \(x\in X\) if \(d(x_n,x)\to 0\). Completeness means every Cauchy sequence converges in \(d\) [2507.15635]. The decisive difference from the ordinary metric case is that the axioms do not require \(d(x,x)=0\) for all \(x\). This is visible in the worked example \[ d(x,y)=|x-y|+x+y \] on \(X=[0,1]\), where \(d(x,x)=2x\). This feature is not a minor notational variation: it changes the logic of uniqueness arguments and motivates the introduction of an additional control condition in the definition of \(\mathcal{A}_d\). A common misconception is that the theory is simply the classical \(\mathcal{A}\)-contraction framework transplanted to a broader space; the source explicitly identifies the positive self-distance issue as the reason axiom (A3) is needed [2507.15635]. ## 2. The control-function class \(\mathcal{A}_d\) The class \(\mathcal{A}_d\) consists of functions \[ \alpha:\mathbb{R}_+^3\longrightarrow\mathbb{R}_+ \] satisfying three conditions. First, \(\alpha\) is continuous. Second, there is a continuous \(\psi:[0,\infty)\to[0,1)\) with \[ k=\sup_{t\ge 0}\psi(t)<1 \] such that whenever \[ a\le \alpha(a,b,b) \quad\text{or}\quad a\le \alpha(b,a,b) \quad\text{or}\quad a\le \alpha(b,b,a), \] one has \[ a\le \psi(b)\,b. \] Third, for all \(t>0\) and \(s_1,s_2\ge 0\), \[ \alpha(t,s_1,s_2)0\) for \(\varepsilon>0\), and defines \[ \Psi(s)=\int_0^s\phi(t)\,dt. \] If \(T:X\to X\) satisfies \[ \int_0^{d(Tx,Ty)}\phi(t)\,dt \le \alpha\!\Bigl(\Psi(d(x,y)),\,\Psi(d(x,Tx)),\,\Psi(d(y,Ty))\Bigr) \] for all \(x,y\in X\), then \(T\) has a unique fixed point. The proof consists in replacing each distance \(d(\cdot,\cdot)\) by \(\Psi(d(\cdot,\cdot))\), after which the argument falls back into the proof pattern of Theorem 1. For the two-metric setting, Theorem 4 assumes two dislocated metrics \((X,d)\) and \((X,\delta)\) with \(d(x,y)\le \delta(x,y)\) for all \(x,y\), \((X,d)\) complete, \(T\) \(d\)-continuous, and the existence of \(\alpha\in\mathcal{A}_d\) such that for all \(x,y\), \[ \delta(Tx,Sy)\le \alpha(\delta(x,y),\delta(x,Tx),\delta(y,Sy)), \] together with the same inequality under \(T\leftrightarrow S\) and \(x\leftrightarrow y\). Under these hypotheses, \(S\) and \(T\) have a unique common fixed point [2507.15635]. ## 4. Iterative mechanism and proof architecture The source identifies the proof strategies as variations of the Banach–Picard iteration. The common template has four steps: show iterates approach each other geometrically; conclude Cauchy and invoke completeness; pass to the limit in the contractive inequality to get a fixed point; use axiom (A3) to rule out two distinct fixed points [2507.15635]. For the single-map theorem, the iteration is \(x_{n+1}=Tx_n\). For the sequence theorem, it is the diagonal iteration \(x_{n+1}=f_{n+1}x_n\). For the two-metric theorem, it is the alternating iteration \[ x_{2n+1}=T\,x_{2n},\qquad x_{2n+2}=S\,x_{2n+1}. \] In the last case, the two contractive inequalities in \(\delta\) give \[ \delta(x_n,x_{n+1})\le k\,\delta(x_{n-1},x_n), \] so the sequence is Cauchy in \(\delta\) and also in \(d\). Completeness of \((X,d)\) then yields a limit \(z\), continuity gives \(Tz=z\), and the argument proceeds to show \(\delta(z,z)=0\) and finally \(Sz=z\). The significance of this architecture is methodological rather than merely formal. The results do not abandon the standard fixed-point paradigm; instead, they show how the Picard method survives under a weakened ambient geometry, provided the control function is strengthened appropriately. This suggests that the main technical burden lies not in constructing a new iteration scheme, but in controlling the interaction between the contractive inequality and nonzero self-distances. ## 5. Relation to classical \(\mathcal{A}\)-contractions The paper states that in a classical metric space one has \(d(x,x)=0\), and axioms (A2) and (A3) of \(\mathcal{A}_d\) reduce to those of \(\mathcal{A}\)-contractions in [1, 10]. It further states that when \(d\) is an ordinary metric, the extra axiom (A3) is redundant and all theorems reduce to those in Akram et al. and Saha & Dey. By choosing \(\phi(t)=1\) in Theorem 3, one recovers classical integral-type \(\mathcal{A}\)-contraction results [2507.15635]. The conceptual comparison is precise. The source does not present \(\mathcal{A}_d\) as an unrelated contractive class, but as a generalization of the classical \(\mathcal{A}\)-theory to the more challenging setting of dislocated metric spaces. The new axiom (A2) introduces a uniform bound \(\sup\psi<1\) so the same \(k\) works in every iterative step, while (A3) is essential only when self-distances may be positive. A common misunderstanding would be to regard (A3) as a technical embellishment. In the source it is tied directly to uniqueness, since the contradiction argument for two distinct fixed points requires \[ \alpha(t,s_1,s_2)