Geraghty-Type Non-Self Mappings

• Geraghty-type non-self mappings are defined on distinct subsets of a metric space using a variable contraction controlled by a Geraghty function.
• They employ an auxiliary map S, which is continuous, injective, and subsequentially convergent, to transform non-self dynamics into a framework amenable to fixed point techniques.
• These mappings generalize classical fixed point theorems by establishing best proximity points and demonstrating uniqueness and convergence under proximal conditions.

Searching arXiv for the cited papers and closely related work on Geraghty-type non-self mappings. First, I’ll look up the core Geraghty/Suzuki-related fixed point paper and the non-self multivalued background paper. Searching arXiv for: (Abtahi, 2012) Geraghty Suzuki completeness Abtahi; (Khojasteh et al., 2011) generalized contractive non-self multi-valued mappings Geraghty-type non-self mappings are mappings for which the contractive mechanism is governed by a Geraghty control function, while the map itself is non-self, typically of the form $T:A\to B$ with $A,B\subseteq X$. In this setting, ordinary fixed points are generally unavailable when $A\cap B=\varnothing$, so the natural replacement is a best proximity point, namely a point $x^*\in A$ such that $d(x^*,Tx^*)=d(A,B)$. Recent work formulates this theory in a proximal form, often through an auxiliary map $S$, and proves existence and uniqueness under Geraghty-type and Kannan–Geraghty-type assumptions. Closely related literatures include Suzuki-inspired self-map generalizations of Geraghty’s theorem, generalized non-self multivalued endpoint theory controlled by Boyd–Wong/Amini-Harandi type functions, and auxiliary-image formulations in which self-map dynamics is transported to $T(X)$ (Fogh et al., 2 Oct 2025).

1. Core setting and definitions

The standard metric framework is a complete metric space $(X,d)$ together with two nonempty subsets $A,B\subseteq X$. A mapping is called non-self when it has the form

$T:A\to B,$

rather than $A,B\subseteq X$0. Because $A,B\subseteq X$1 and $A,B\subseteq X$2 may be disjoint, the quantity

$A,B\subseteq X$3

replaces the fixed-point equation as the fundamental geometric benchmark. The associated best proximity sets are

$A,B\subseteq X$4

$A,B\subseteq X$5

A point $A,B\subseteq X$6 is a best proximity point if

$A,B\subseteq X$7

The Geraghty mechanism is encoded by a control class. In the non-self proximal framework developed in 2025, the relevant class is

$A,B\subseteq X$8

This is the same logical feature that underlies classical Geraghty contractions: the contractive coefficient may vary with scale and may approach $A,B\subseteq X$9, but only near $A\cap B=\varnothing$0 (Fogh et al., 2 Oct 2025).

A distinctive feature of the recent non-self theory is the use of an auxiliary function

$A\cap B=\varnothing$1

satisfying

$A\cap B=\varnothing$2

In the principal theorems, $A\cap B=\varnothing$3 is assumed to be continuous on $A\cap B=\varnothing$4 and on $A\cap B=\varnothing$5, one-to-one, subsequentially convergent, and such that

$A\cap B=\varnothing$6

The paper uses “subsequentially convergent” exactly in the sense that if $A\cap B=\varnothing$7 converges, then $A\cap B=\varnothing$8 has a convergent subsequence (Fogh et al., 2 Oct 2025).

2. Geraghty ancestry and the self-map antecedents

The modern non-self theory is best understood against the background of Geraghty’s self-map fixed point principle and later Suzuki-type weakenings of global contraction. In the classical one-variable form recalled in the infinite-dimensional extension paper, a self-map $A\cap B=\varnothing$9 is a Geraghty contraction when

$x^*\in A$0

where $x^*\in A$1 belongs to a Geraghty class $x^*\in A$2 satisfying

$x^*\in A$3

This control condition is the prototype for later non-self and multivariable variants (Bardhana et al., 2021).

A distinct but closely related development appears in the Suzuki-inspired self-map theory of Abtahi. There the global contractive hypothesis is weakened to conditional forms involving $x^*\in A$4, while the convergence criterion is preserved through Geraghty-style subsequences and ratios

$x^*\in A$5

The central implication is that, under the Suzuki-type trigger

$x^*\in A$6

the subsequential criterion $x^*\in A$7 is equivalent to convergence of the Picard orbit to a unique fixed point in a complete metric space. The same paper introduces a test-function class

$x^*\in A$8

which is the closest analogue there to Geraghty’s variable-coefficient form (Abtahi, 2012).

That paper, however, is explicitly a self-map paper: all mappings are $x^*\in A$9, every theorem is stated for self-maps, and the proofs use ordinary Picard iteration $d(x^*,Tx^*)=d(A,B)$0. It therefore does not treat non-self mappings directly. Its significance for non-self work lies in what it calls the Suzuki-conditioned Geraghty mechanism: contraction is imposed only under a local trigger involving $d(x^*,Tx^*)=d(A,B)$1, while the convergence logic is carried by Geraghty-style subsequences (Abtahi, 2012).

This distinction corrects a frequent misconception. Not every Geraghty-type generalization is a non-self theorem. Some papers generalize the contractive side but remain strictly within self-map dynamics; others change the operator arity, as in $d(x^*,Tx^*)=d(A,B)$2 or $d(x^*,Tx^*)=d(A,B)$3, without entering standard non-self fixed point theory (Bardhana et al., 2021).

3. Proximal Geraghty-type non-self mappings

The 2025 best proximity framework gives a direct formulation of Geraghty-type non-self mappings. The principal condition is an $d(x^*,Tx^*)=d(A,B)$4-proximal Geraghty-type inequality: for all $d(x^*,Tx^*)=d(A,B)$5,

$d(x^*,Tx^*)=d(A,B)$6

where $d(x^*,Tx^*)=d(A,B)$7. The map is non-self because $d(x^*,Tx^*)=d(A,B)$8, and it is proximal because the inequality is imposed only on quadruples linked by the minimal-distance relations

$d(x^*,Tx^*)=d(A,B)$9

The corresponding best proximity conclusion is not $S$0, but

$S$1

Under the assumptions that $S$2 are nonempty subsets of a complete metric space, $S$3 and $S$4 are nonempty and closed, $S$5 is continuous on $S$6 and on $S$7, one-to-one, subsequentially convergent, and preserves $S$8, and

$S$9

the main theorem states that there exists a unique $T(X)$0 such that

$T(X)$1

Moreover, every sequence $T(X)$2 satisfying

$T(X)$3

converges to $T(X)$4 (Fogh et al., 2 Oct 2025).

The paper also introduces a proximal Kannan–Geraghty non-self condition. For all $T(X)$5,

$T(X)$6

implies

$T(X)$7

where

$T(X)$8

and $T(X)$9. Under the same structural assumptions, there again exists a unique $(X,d)$0 such that

$(X,d)$1

and every proximal sequence defined by

$(X,d)$2

converges to that point (Fogh et al., 2 Oct 2025).

These two formulas are the direct non-self Geraghty analogues developed in the paper. They recover constant-coefficient proximal contractions when $(X,d)$3, and recover constant-coefficient proximal Kannan conditions when $(X,d)$4 (Fogh et al., 2 Oct 2025).

4. Iterative mechanism and the role of the auxiliary map

The proof strategy proceeds in the $(X,d)$5-image rather than directly in $(X,d)$6. Starting from any $(X,d)$7, the assumptions

$(X,d)$8

allow the construction of a sequence $(X,d)$9 satisfying

$A,B\subseteq X$0

The Geraghty-type inequality then yields a recurrence of the same formal type as Geraghty’s classical fixed point theorem, but for the transformed sequence $A,B\subseteq X$1. The authors conclude by “standard arguments as in Geraghty’s fixed point theorem” that $A,B\subseteq X$2 is Cauchy, hence convergent in the complete ambient space (Fogh et al., 2 Oct 2025).

At that stage, the auxiliary hypotheses on $A,B\subseteq X$3 become decisive. Continuity is used to identify the limit of a convergent subsequence of $A,B\subseteq X$4 with the limit of $A,B\subseteq X$5. Injectivity is then used twice: first to show that the candidate limit is indeed a best proximity point, and second to prove uniqueness from an equality of $A,B\subseteq X$6-images. Subsequential convergence is the assumption that allows one to pass back from convergence of $A,B\subseteq X$7 to an actual convergent subsequence of $A,B\subseteq X$8 (Fogh et al., 2 Oct 2025).

The necessity of this condition is exhibited by a counterexample. Let

$A,B\subseteq X$9

with the Euclidean metric, and define

$T:A\to B,$0

Then $T:A\to B,$1 and $T:A\to B,$2. Define

$T:A\to B,$3

The transformed sequence may converge because

$T:A\to B,$4

but the original sequence $T:A\to B,$5 has no convergent subsequence in $T:A\to B,$6, since the second coordinate diverges to $T:A\to B,$7. The paper concludes that $T:A\to B,$8 is not subsequentially convergent and that, in this case, $T:A\to B,$9 has no best proximity point. The example isolates subsequential convergence of $A,B\subseteq X$00 as essential rather than cosmetic (Fogh et al., 2 Oct 2025).

A further misconception is thereby addressed. In auxiliary-image methods, convergence of the transformed orbit does not by itself imply convergence of the original orbit. The $A,B\subseteq X$01-space and the original domain need not have the same compactness or sequential behavior, and the transfer back to $A,B\subseteq X$02 is one of the genuinely nontrivial steps.

5. Neighboring frameworks and boundary cases

The literature surrounding Geraghty-type non-self mappings is heterogeneous. Some contributions are directly non-self but not Geraghty in the strict sense; others are Geraghty but not non-self; still others use auxiliary transport maps while remaining self-map theorems.

Framework	Mapping type	Relation to Geraghty-type non-self mappings
Best proximity with auxiliary $A,B\subseteq X$03	$A,B\subseteq X$04	Direct theory
Common strict fixed points with $A,B\subseteq X$05	$A,B\subseteq X$06	Indirect background
Infinite-input Geraghty operators	$A,B\subseteq X$07, $A,B\subseteq X$08	Indirect analogue
$A,B\subseteq X$09-extended auxiliary-image theory	$A,B\subseteq X$10	Methodological precursor

A prominent indirect precursor is the 2011 paper on generalized contractive non-self multivalued mappings. There the setting is a complete metric space $A,B\subseteq X$11, a closed subset $A,B\subseteq X$12, and non-self multivalued maps

$A,B\subseteq X$13

with contraction measured by the Hausdorff metric: $A,B\subseteq X$14 where

$A,B\subseteq X$15

The control function $A,B\subseteq X$16 is upper semicontinuous, satisfies $A,B\subseteq X$17 for all $A,B\subseteq X$18, and

$A,B\subseteq X$19

The main theorem states that $A,B\subseteq X$20 and $A,B\subseteq X$21 have a unique common strict fixed point in $A,B\subseteq X$22 if and only if they have the common approximate $A,B\subseteq X$23-boundary strict fixed point property, and moreover

$A,B\subseteq X$24

This is non-self, multivalued, and proximal in spirit, but it is not a Geraghty theorem in the classical $A,B\subseteq X$25 form (Khojasteh et al., 2011).

Another adjacent direction is the 2021 theory of infinite Geraghty-type extensions for operators

$A,B\subseteq X$26

Its fixed point notion is diagonal,

$A,B\subseteq X$27

and its contractive conditions involve $A,B\subseteq X$28. This paper extends Geraghty, Kannan–Geraghty, and Fisher–Geraghty ideas to multivariable and infinite-input operators, and applies them to infinite-dimensional Fredholm and Urysohn integral equations. It is not a non-self mapping paper in the standard sense $A,B\subseteq X$29, but it shows how Geraghty control can survive outside the classical self-map format (Bardhana et al., 2021).

A third neighboring strand is the 2026 $A,B\subseteq X$30-extended framework. There the principal object is still a self-map $A,B\subseteq X$31, but the contractive behavior is measured on the auxiliary image through

$A,B\subseteq X$32

with $A,B\subseteq X$33 continuous, injective, and subsequentially convergent. The paper proves that $A,B\subseteq X$34-extended weakly contractive and $A,B\subseteq X$35-extended Geraghty classes coincide, and likewise $A,B\subseteq X$36-extended weakly Kannan and $A,B\subseteq X$37-extended Kannan–Geraghty classes coincide. The key induced map is

$A,B\subseteq X$38

This is not a non-self theory, since $A,B\subseteq X$39 remains a self-map, but it offers an explicit transport principle from a generalized formulation to a classical self-map problem on $A,B\subseteq X$40 (Fogh et al., 25 Apr 2026).

6. Applications, examples, and methodological significance

The direct application developed in the 2025 non-self paper is a registration-inspired alignment model. For $A,B\subseteq X$41, let

$A,B\subseteq X$42

as compact subsets of $A,B\subseteq X$43 with the Euclidean metric. Then

$A,B\subseteq X$44

Take

$A,B\subseteq X$45

for $A,B\subseteq X$46. If

$A,B\subseteq X$47

and

$A,B\subseteq X$48

then

$A,B\subseteq X$49

and therefore

$A,B\subseteq X$50

Thus the Geraghty-type condition holds with the constant function $A,B\subseteq X$51. The theorem yields a unique $A,B\subseteq X$52 such that

$A,B\subseteq X$53

Since

$A,B\subseteq X$54

this equals $A,B\subseteq X$55 if and only if $A,B\subseteq X$56, so the unique best proximity point is

$A,B\subseteq X$57

The paper calls this the “unique and well-defined alignment anchor” (Fogh et al., 2 Oct 2025).

The same example also makes the iteration explicit. If $A,B\subseteq X$58, then the proximal relation

$A,B\subseteq X$59

forces

$A,B\subseteq X$60

hence

$A,B\subseteq X$61

The abstract existence theorem is therefore realized by a concrete geometric decay toward the unique best proximity point (Fogh et al., 2 Oct 2025).

From a broader methodological perspective, this suggests two major directions. First, Geraghty-type control can be transplanted from fixed point theory to best proximity theory provided the geometry is organized through proximal relations and an auxiliary transform. Second, auxiliary-image methods appear in both genuine non-self results and in self-map transport theorems, but the target statements differ sharply: non-self theories seek best proximity points, whereas auxiliary self-map theories reduce to fixed points of induced maps such as $A,B\subseteq X$62 (Fogh et al., 2 Oct 2025).

7. Conceptual boundaries and research directions

The current literature establishes a clear conceptual boundary. A direct theory of Geraghty-type non-self mappings now exists in best proximity form, but many papers adjacent to it remain outside that category. The 2012 Suzuki-inspired theorem is self-map only; the 2011 non-self multivalued paper is nonlinear and non-self but not Geraghty in the standard $A,B\subseteq X$63-class sense; the 2021 infinite-input paper is Geraghty but not non-self in the subset-to-subset sense; and the 2026 $A,B\subseteq X$64-extended theory is again a self-map framework transported to $A,B\subseteq X$65 (Abtahi, 2012).

Within the direct non-self theory, the strongest structural assumptions are also the most consequential. The requirements that $A,B\subseteq X$66 and $A,B\subseteq X$67 be nonempty and closed, that

$A,B\subseteq X$68

and that $A,B\subseteq X$69 be continuous, injective, subsequentially convergent, and preserve $A,B\subseteq X$70, are not ancillary hypotheses. They are the mechanisms that make proximal iteration possible, allow convergence to be proved in the transformed space, and permit that convergence to be lifted back to the original domain (Fogh et al., 2 Oct 2025).

A plausible implication is that future extensions will likely focus on weakening these auxiliary hypotheses rather than on altering the Geraghty control condition itself. The existing counterexample shows that subsequential convergence of the auxiliary map is tightly tied to the proof architecture. Likewise, the transport philosophy developed for self-maps on $A,B\subseteq X$71 suggests a route for further non-self generalizations: one may attempt to encode non-self dynamics into an induced self-map on an auxiliary image or proximity space, while preserving a Geraghty-type control law (Fogh et al., 25 Apr 2026).

In that sense, Geraghty-type non-self mappings now occupy a specific position within metric fixed point theory: they are not merely non-self analogues of classical contractions, but a proximal theory in which scale-dependent Geraghty control, auxiliary transforms, and best proximity geometry are combined to recover existence, uniqueness, and convergence when literal fixed points are unavailable (Fogh et al., 2 Oct 2025).