Ad-Contractions in Dislocated Metric Spaces

• Ad-contractions are contractive self-maps in dislocated metric spaces that extend classical fixed-point results by incorporating nonzero self-distances via the A_d control function class.
• The framework employs a uniform contraction constant and Banach–Picard iteration to guarantee unique fixed points for single mappings, sequences, and two-metric settings.
• Applications include integral-type contractions and solving fractional differential equations, while open problems address extensions to best-proximity points and b-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 (Panthi et al., 21 Jul 2025).

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 $\mathcal{A}$0 such that $\mathcal{A}$1 for all $\mathcal{A}$2, and convergent to $\mathcal{A}$3 if $\mathcal{A}$4. Completeness means every Cauchy sequence converges in $\mathcal{A}$5 (Panthi et al., 21 Jul 2025).

The decisive difference from the ordinary metric case is that the axioms do not require $\mathcal{A}$6 for all $\mathcal{A}$7. This is visible in the worked example

$\mathcal{A}$8

on $\mathcal{A}$9, where $\mathcal{A}_d$0. 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$1. A common misconception is that the theory is simply the classical $\mathcal{A}_d$2-contraction framework transplanted to a broader space; the source explicitly identifies the positive self-distance issue as the reason axiom (A3) is needed (Panthi et al., 21 Jul 2025).

2. The control-function class $\mathcal{A}_d$3

The class $\mathcal{A}_d$4 consists of functions

$\mathcal{A}_d$5

satisfying three conditions. First, $\mathcal{A}_d$6 is continuous. Second, there is a continuous $\mathcal{A}_d$7 with

$\mathcal{A}_d$8

such that whenever

$\mathcal{A}_d$9

one has

$d$0

Third, for all $d$1 and $d$2,

$d$3

A self-map $d$4 is an $d$5-contraction if there exists $d$6 such that

$d$7

These are the defining data of the theory (Panthi et al., 21 Jul 2025).

The role of the modulus function $d$8 is to produce a uniform contraction constant

$d$9

which then governs every iterative step. The source emphasizes that this uniform bound is new relative to the classical $X$0-framework. The role of (A3) is different: it enforces strict inequality in the first variable and is used to exclude distinct fixed points even when terms such as $X$1 and $X$2 may be positive. This division of labor between (A2) and (A3) is structurally central to the theory (Panthi et al., 21 Jul 2025).

3. Fixed point theorems for $X$3-contractions

The basic standing hypothesis is that $X$4 is a complete dislocated metric space and $X$5. Under these assumptions, the paper proves four principal theorems (Panthi et al., 21 Jul 2025).

For a single mapping, Theorem 1 states that if $X$6 is an $X$7-contraction, then $X$8 has a unique fixed point $X$9. The iteration is the standard Picard sequence $d:X\times X\longrightarrow[0,\infty)$0, and the contractive condition yields

$d:X\times X\longrightarrow[0,\infty)$1

This gives $d:X\times X\longrightarrow[0,\infty)$2, the sequence is Cauchy, completeness provides $d:X\times X\longrightarrow[0,\infty)$3, and passage to the limit in the contractive condition shows $d:X\times X\longrightarrow[0,\infty)$4. Uniqueness follows from

$d:X\times X\longrightarrow[0,\infty)$5

which is impossible if $d:X\times X\longrightarrow[0,\infty)$6.

For a sequence of mappings, Theorem 2 considers self-maps $d:X\times X\longrightarrow[0,\infty)$7 satisfying

$d:X\times X\longrightarrow[0,\infty)$8

for all $d:X\times X\longrightarrow[0,\infty)$9 and $x,y,z\in X$0. Then $x,y,z\in X$1 admits a unique common fixed point $x,y,z\in X$2, meaning $x,y,z\in X$3 for all $x,y,z\in X$4. The iteration is the diagonal scheme $x,y,z\in X$5, and the proof repeats the Cauchy-convergence-uniqueness pattern from Theorem 1.

For integral-type contractions, Theorem 3 introduces the family $x,y,z\in X$6 of all Lebesgue-integrable $x,y,z\in X$7 with $x,y,z\in X$8 for $x,y,z\in X$9, and defines

$$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$0

If $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$1 satisfies

$$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$2

for all $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$3, then $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$4 has a unique fixed point. The proof consists in replacing each distance $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$5 by $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$6, after which the argument falls back into the proof pattern of Theorem 1.

For the two-metric setting, Theorem 4 assumes two dislocated metrics $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$7 and $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$8 with $$\text{(i) } d(x,y)=d(y,x), \qquad \text{(ii) } d(x,y)=0\implies x=y, \qquad \text{(iii) } d(x,y)\le d(x,z)+d(z,y).$$9 for all $\{x_n\}\subset X$0, $\{x_n\}\subset X$1 complete, $\{x_n\}\subset X$2 $\{x_n\}\subset X$3-continuous, and the existence of $\{x_n\}\subset X$4 such that for all $\{x_n\}\subset X$5,

$\{x_n\}\subset X$6

together with the same inequality under $\{x_n\}\subset X$7 and $\{x_n\}\subset X$8. Under these hypotheses, $\{x_n\}\subset X$9 and $\varepsilon>0$0 have a unique common fixed point (Panthi et al., 21 Jul 2025).

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 (Panthi et al., 21 Jul 2025).

For the single-map theorem, the iteration is $\varepsilon>0$1. For the sequence theorem, it is the diagonal iteration $\varepsilon>0$2. For the two-metric theorem, it is the alternating iteration

$\varepsilon>0$3

In the last case, the two contractive inequalities in $\varepsilon>0$4 give

$\varepsilon>0$5

so the sequence is Cauchy in $\varepsilon>0$6 and also in $\varepsilon>0$7. Completeness of $\varepsilon>0$8 then yields a limit $\varepsilon>0$9, continuity gives $\mathcal{A}$00, and the argument proceeds to show $\mathcal{A}$01 and finally $\mathcal{A}$02.

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}$03-contractions

The paper states that in a classical metric space one has $\mathcal{A}$04, and axioms (A2) and (A3) of $\mathcal{A}$05 reduce to those of $\mathcal{A}$06-contractions in [1, 10]. It further states that when $\mathcal{A}$07 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 $\mathcal{A}$08 in Theorem 3, one recovers classical integral-type $\mathcal{A}$09-contraction results (Panthi et al., 21 Jul 2025).

The conceptual comparison is precise. The source does not present $\mathcal{A}$10 as an unrelated contractive class, but as a generalization of the classical $\mathcal{A}$11-theory to the more challenging setting of dislocated metric spaces. The new axiom (A2) introduces a uniform bound $\mathcal{A}$12 so the same $\mathcal{A}$13 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

$\mathcal{A}$14

even when $\mathcal{A}$15 and $\mathcal{A}$16 may not vanish.

This comparison also clarifies the title’s use of “Ad-contractions.” The notation signals dependence on the dislocated metric context, not merely a renamed copy of the $\mathcal{A}$17-framework. The extension is therefore best understood as conservative in proof strategy and nontrivial in ambient geometry.

6. Examples, corollaries, applications, and open problems

The source supports the abstract theory with explicit examples. In Example 5.1, on $\mathcal{A}$18,

$\mathcal{A}$19

Then $\mathcal{A}$20 is a dislocated metric with $\mathcal{A}$21, and

$\mathcal{A}$22

Here $\mathcal{A}$23 with $\mathcal{A}$24, and Theorem 1 yields the unique fixed point $\mathcal{A}$25 from $\mathcal{A}$26 (Panthi et al., 21 Jul 2025).

In Example 5.2, again on $\mathcal{A}$27,

$\mathcal{A}$28

Then $\mathcal{A}$29, $\mathcal{A}$30, and

$\mathcal{A}$31

Theorem 4 then gives a unique common fixed point $\mathcal{A}$32, and one also verifies $\mathcal{A}$33.

The corollaries and forward-looking remarks delimit the present scope of the theory. By choosing $\mathcal{A}$34 in Theorem 3, one recovers classical integral-type $\mathcal{A}$35-contraction results. Potential applications include existence and uniqueness of solutions to fractional differential and integral equations in dislocated metric spaces. The open problems listed in the source are best-proximity point theorems for non-self maps under $\mathcal{A}$36, extensions to $\mathcal{A}$37-metric or partial metric spaces, and further integral-type generalizations (Panthi et al., 21 Jul 2025).

These directions indicate how the framework may propagate: not by altering its core control mechanism, but by transplanting it to adjacent generalized distance structures and to operator equations where fixed-point existence and uniqueness serve as the analytic backbone.