T Extended Weakly Contractive, Kannan, and Geraghty Mappings Fixed Points, Equivalences,

Abstract: We develop a unified T-extended framework for weakly contractive, weakly Kannan, and Geraghty classes of self-maps S on a metric space (X, d), where distances are measured on the auxiliary image via d(Tx, Ty), and the dynamics is governed by the composition of T and S. Under standard assumptions on the auxiliary map T (continuity, injectivity, subsequential convergence), fixed point theorems and Picard convergence are established for each class. The main contribution is twofold. First, it is shown that the T-extended weakly contractive class coincides with the T-extended Geraghty class, and that the T-extended weakly Kannan class coincides with the T-extended Kannan-Geraghty class. Second, the mechanism behind these equivalences is clarified by transporting the problem to an induced map F from T(X) to T(X), defined by F(Tx) = T(Sx), where the extended properties reduce exactly to the classical ones with the same control functions. A Delta-type ratio criterion on T(X) and quantitative Picard convergence rates are also provided. Examples, including Volterra smoothing operators, are presented to highlight the role of the auxiliary map. All results extend naturally to rectangular (Branciari) metric spaces.

• The paper introduces a T-extension framework that unifies fixed point results for weakly contractive, Kannan, and Geraghty mappings using an auxiliary map T.
• It proves the equivalence of T-extended classes by transferring classical contractive conditions to the T-image, ensuring fixed point existence and uniqueness.
• Quantitative Picard convergence rates are derived, enabling practical online estimation and diagnostics in iterative methods and nonstandard metric spaces.

Unified Fixed Point Analysis of $T$-Extended Weakly Contractive, Kannan, and Geraghty Mappings

Introduction

The paper "T Extended Weakly Contractive, Kannan, and Geraghty Mappings: Fixed Points, Equivalences, and Rates" (2604.23400) introduces a comprehensive $T$-extension framework for studying fixed points of various classes of contractive-type selfmaps on metric spaces. The authors systematically generalize classical fixed point theories—such as Banach contraction, Kannan, and Geraghty contractions—by evaluating the contractive properties via an auxiliary map $T$ and establishing fixed point existence, uniqueness, and convergence results in this extended context. The core contributions comprise identifying structural equivalences between classes, deriving explicit quantitative Picard convergence rates, and providing illustrative examples to clarify the role of the auxiliary map.

$T$-Extended Fixed Point Classes

The paper develops three $T$-extended classes of selfmaps $S : X \rightarrow X$ on a metric space $(X, d)$:

• $T$-Extended Weakly Contractive: $S$ satisfies $$d(TSx, TSy) \leq \overline{\alpha}(Tx, Ty) d(Tx, Ty)$$ for some $T$0 admitting uniform bounds over metric annuli.
• $T$1-Extended Weakly Kannan: $T$2 satisfies $T$3 with analogous annulus bounds.
• $T$4-Extended Geraghty: $T$5 satisfies $T$6 with $T$7, where $T$8 comprises functions $T$9 such that $T$0.

These contractive inequalities are defined relative to the $T$1-image metric, and results are predicated on $T$2 being continuous, injective, and (sub)sequentially convergent, ensuring that contractive behavior transported via $T$3 propagates fixed point properties back to $T$4.

Equivalences and Structural Mechanisms

A principal theoretical advance in the paper is the proof that, under the $T$5-extended framework, certain classes coincide:

• $T$6-Extended Weakly Contractive $T$7 $T$8-Extended Geraghty
• $T$9-Extended Weakly Kannan $T$0 $T$1-Extended Kannan-Geraghty

The mechanism underlying these equivalences is elucidated by considering the induced map $T$2, defined by $T$3. In this context, the contractive properties of $T$4 with respect to $T$5 translate naturally to classical contractive-type properties of $T$6 on the subspace $T$7. Consequently, the fixed point existence and uniqueness results for $T$8-extended classes follow directly from their classical counterparts on $T$9, contingent on $T$0's injectivity (which is shown to be essential).

Fixed Point Existence, Uniqueness, and Picard Convergence

The authors prove existence and uniqueness of fixed points for $T$1-extended weakly contractive, weakly Kannan, and Geraghty mappings. For each class:

• Existence/Uniqueness: If $T$2 satisfies the standing assumptions, $T$3 admits a unique fixed point.
• Convergence: If $T$4 is sequentially convergent, the Picard iteration $T$5 converges for any starting point.

Additionally, the paper introduces a $T$6-type criterion that characterizes Picard convergence in terms of observed contraction ratios, extending Geraghty's classical sequence-based fixed point criterion to the $T$7-image.

Quantitative Picard Convergence Rates

Explicit quantitative bounds are established for the Picard iterates. If a uniform bound $T$8 exists for contraction ratios along the Picard sequence—either for weakly contractive or Geraghty-type controls—then for iterations $T$9, the distance to the fixed point in $S : X \rightarrow X$0-image satisfies:

$S : X \rightarrow X$1

where $S : X \rightarrow X$2 and $S : X \rightarrow X$3 (with $S : X \rightarrow X$4 the fixed point). These results provide practical metrics for assessing convergence speed and enable diagnostic monitoring, including online adaptation using observed ratios.

Examples and Practical Implications

Several examples demonstrate the $S : X \rightarrow X$5-extended framework’s efficacy:

• Volterra Smoothing: The map $S : X \rightarrow X$6 (with $S : X \rightarrow X$7 the Volterra operator) is nonexpansive in the classical sense but becomes contractive after composition with $S : X \rightarrow X$8, illustrating how $S : X \rightarrow X$9 can induce fixed point formation where none exists classically.
• Linear Injection and Scalar Contractions: Linear and nonlinear examples show the applicability of the framework to both standard and non-standard mappings.

These constructions emphasize the auxiliary map's role as a structural device for generating contractive dynamics and expand the applicability of fixed point theorems to settings not amenable to classical techniques.

Extension to Rectangular (Branciari) Metric Spaces

The results, including fixed point existence, uniqueness, Picard convergence, and contractive criteria, extend without alteration to rectangular (Branciari) metric spaces. The reformulation applies the $(X, d)$0-image approach, leveraging the more general triangle inequality structure.

Observability, Online Estimation, and Diagnostics

The paper addresses practical estimation of convergence rates and fixed point distances along Picard iterates. It distinguishes between a priori bounds (uniform theoretical constants) and a posteriori estimates (computed from observed ratios). The analysis highlights that online monitoring of contraction ratios is necessary for validated numerical bounds, as finite historical data is insufficient for predictive guarantees without further conditions.

Theoretical and Practical Implications

The $(X, d)$1-extended framework provides a unifying lens for fixed point analysis across multiple contractive paradigms, simplifying equivalence proofs and extending applicability. The equivalence results clarify the relationship between classical and extended classes, potentially enabling new algorithmic approaches, especially for mappings where direct contractivity is unavailable but $(X, d)$2-induced contractivity is achievable. The Picard rate analysis and diagnostics provide computational tools for practitioners, relevant for iterative schemes, optimization, and approximate fixed point computation in numerical analysis.

From a theoretical standpoint, the transport of fixed point theorems through auxiliary maps suggests avenues for further abstraction—such as broader metric morphisms or more generalized contraction mechanisms—potentially impacting areas including nonlinear analysis, functional equations, and systems theory.

Conclusion

The paper develops a robust $(X, d)$3-extended framework for weakly contractive, Kannan, and Geraghty mappings, demonstrates pairwise equivalence of extended classes in the image metric, and establishes precise existence, uniqueness, and convergence results bolstered by explicit quantitative bounds. The auxiliary map framework enhances both theoretical rigor and practical application, with implications for general fixed point theory, operator analysis, and iterative computation. Extension to rectangular metric spaces and the analysis of observable diagnostic tools further augment the framework’s utility and generality.