{"paper":{"title":"On fixed point results in metric spaces for large triangle-perimeter contractions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Fixed point theorem for large triangle-perimeter contractions holds only under an auxiliary condition.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ovidiu Popescu","submitted_at":"2026-05-13T08:14:32Z","abstract_excerpt":"In this paper we introduce a corrected extension of Burton's theory of large contractions in the context of triangle-perimeter contractions introduced by Petrov. Combining these two lines of research, we prove a fixed point result for large triangle-perimeter contractions with an auxiliary assumption, which is of utmost importance. Firstly, we give a counterexample to the main result related to large triangle-perimeter contractions that currently exists in literature. Then, we prove that given an additional condition, a fixed point result for large triangle-perimeter contractions holds. Lastly"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Given an additional condition, a fixed point result for large triangle-perimeter contractions holds.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The auxiliary condition imposed on the large triangle-perimeter contraction that is required for the fixed point theorem to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Large triangle-perimeter contractions admit fixed points when an auxiliary condition is added, as shown by a counterexample to the prior claim and a new proof that covers a strictly larger class than Burton's or Petrov's theories.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fixed point theorem for large triangle-perimeter contractions holds only under an auxiliary condition.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bb92128148aefe47ac1bbb597ac956fd507abbb8d9255c499e2622ae04d46278"},"source":{"id":"2605.13147","kind":"arxiv","version":1},"verdict":{"id":"bf4cc2d5-0a5a-4c5a-8866-24abba4b5476","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:42:14.424731Z","strongest_claim":"Given an additional condition, a fixed point result for large triangle-perimeter contractions holds.","one_line_summary":"Large triangle-perimeter contractions admit fixed points when an auxiliary condition is added, as shown by a counterexample to the prior claim and a new proof that covers a strictly larger class than Burton's or Petrov's theories.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The auxiliary condition imposed on the large triangle-perimeter contraction that is required for the fixed point theorem to hold.","pith_extraction_headline":"Fixed point theorem for large triangle-perimeter contractions holds only under an auxiliary condition."},"references":{"count":8,"sample":[{"doi":"","year":1922,"title":"Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922), 133-181","work_id":"1e5d2e01-0433-410f-a693-b14c5c9d53c4","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Burton, Integral equations, implicit functions, and fixed points","work_id":"4fd263de-4952-4d6e-8cb4-160a0e3d392a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1972,"title":"S. K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727–730","work_id":"962be2c1-b386-4f58-9f56-e5703c49491d","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"A. Dehici, M.B. Mesmouli, E. Karapinar, On the fixed points of large- Kannan contraction mappings and applications. Appl. Math.-E-Notes 2019, 19, 535–551. 9","work_id":"163b601e-55f5-438c-a59d-57cd884f8369","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"Kannan, Some results on fixed point - II, Amer","work_id":"a779d4e4-3e11-477a-9a33-9eaa353af43a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":8,"snapshot_sha256":"b8ede92dbdcb1d046a0cf9cbf544909c94d4fe2faa899bad4fc19a1bdba95e43","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}