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Remarks on Fixed Point Assertions in Digital Topology, 10

Laurence Boxer

TL;DR

This paper surveys the literature on fixed-point assertions in digital topology, revealing pervasive errors, miscitations, and trivialities. Through targeted critique of works by EgeKaraca-BanPriyanka, GopalEtal, MishTrip, ParvRaman, SalujaEtal, ShaheenEtAl, and others, it demonstrates that many claimed fixed-point results in digital metric spaces are unsupported or reducible to simple counterexamples. The authors provide corrected formulations (where possible) and highlight the structural issues—such as conflating metric and digital continuity, improper use of the Hausdorff metric, and ambiguous notation—that undermine the validity of the claimed theorems. The study underscores the need for rigorous, well-sourced, and clearly stated arguments in digital fixed-point theory, with practical implications for research quality and reproducibility in the field.

Abstract

The topic of fixed points in digital metric spaces continues to draw publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We continue the work of our earlier papers that discuss publications with bad assertions concerning fixed points of self-functions on digital images.

Remarks on Fixed Point Assertions in Digital Topology, 10

TL;DR

This paper surveys the literature on fixed-point assertions in digital topology, revealing pervasive errors, miscitations, and trivialities. Through targeted critique of works by EgeKaraca-BanPriyanka, GopalEtal, MishTrip, ParvRaman, SalujaEtal, ShaheenEtAl, and others, it demonstrates that many claimed fixed-point results in digital metric spaces are unsupported or reducible to simple counterexamples. The authors provide corrected formulations (where possible) and highlight the structural issues—such as conflating metric and digital continuity, improper use of the Hausdorff metric, and ambiguous notation—that undermine the validity of the claimed theorems. The study underscores the need for rigorous, well-sourced, and clearly stated arguments in digital fixed-point theory, with practical implications for research quality and reproducibility in the field.

Abstract

The topic of fixed points in digital metric spaces continues to draw publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We continue the work of our earlier papers that discuss publications with bad assertions concerning fixed points of self-functions on digital images.

Paper Structure

This paper contains 25 sections, 6 theorems, 39 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Adjacencies, continuity, fixed point
  4. Digital metric spaces
  5. A popular error
  6. Banach contraction principle of EgeKaraca-BanPriyanka
  7. GopalEtal's compressions
  8. Overall
  9. Improper citations in "Preliminaries" section
  10. "Preliminaries Proof 4"
  11. GopalEtal's "Results Proof 1"
  12. GopalEtal's "Results Proof 2"
  13. GopalEtal's "Results Proof 3.3"
  14. MishTrip's mappings
  15. Improper citations and related errors
  16. ...and 10 more sections

Key Result

Theorem 2.5

Bx99 A function $f: X \to Y$ between digital images $(X,\kappa)$ and $(Y,\lambda)$ is $(\kappa,\lambda)$-continuous if and only if for every $x,y \in X$, if $x \leftrightarrow_{\kappa} y$ then $f(x) \leftrightarroweq_{\lambda} f(y)$.

Figures (7)

  • Figure 1: "Proof 4" of the "Preliminaries" section of GopalEtal
  • Figure 2: "Proof 1" of the "Results" section of GopalEtal
  • Figure 3: In the argument for "Proof 1" of the "Results" section of GopalEtal, it seems likely that the symbols $\xi_n$ and $x_n$ are identified for all $n$.
  • Figure 4: The assertion "Proof 2" of the "Results" of GopalEtal
  • Figure 5: Unjustified claim in "proof" of "Proof 2" in "Results" of GopalEtal
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • ...and 8 more