A Counterexample to a Relational Fixed-Point Uniqueness Theorem and Corrections to Two Illustrative Examples
Year: 2026
Type: article
Abstract: This independent mathematical preprint examines Theorem 3 and Examples 1–2 of Filali et al., Relational Generalized Nonlinear Contractions of Pant Type with an Application to Nonlinear Integral Equations, Axioms 15 (2026), 226, DOI: 10.3390/axioms15030226. A complete three-point counterexample satisfies the printed hypotheses but has two distinct fixed points. An additional infinite complete-space counterexample uses the universal relation and verifies the contraction at nonfixed points. The note identifies failures of relation closure and directedness in Example 1 and failures of relation closure and the asserted contraction in Example 2. It supplies a sufficient strengthened uniqueness criterion and an independent global nonlinear-contraction proof preserving the integral-equation conclusions. A further proposition proves that lower and upper solutions bound the unique solution, with discussion of Bachar–Khamsi, Remark 3.3. Revised 28 October 2026, version 1.1. Includes the seven-page PDF and an editable LaTeX source package with revision notes. This is an independent preprint, not a journal-issued correction, and it has not been peer reviewed. No counterexample to the existence assertion of Theorem 2 is claimed. AI disclosure: AI tools assisted with mathematical analysis and development and checking of the additional counterexample and order-bound argument, proof-consistency checks, English editing, organization and typesetting. The author is responsible for the mathematical statements, counterexamples, proofs, references and final content.
Author: Rafik Zeraoulia
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222417335