{"paper":{"title":"The Gronwall inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Ralph Howard","submitted_at":"2025-03-31T00:53:32Z","abstract_excerpt":"We prove the following version generalization of the Gronwall inequality:\n  Let $\\mathbf X$ be a Banach space and $U\\subset \\mathbf X$ an open convex set in $\\mathbf X$. Let $f,g\\colon [a,b]\\times U\\to \\mathbf X$ be continuous functions and let $y,z\\colon [a,b]\\to U$ satisfy the initial value problems \\begin{align*} y'(t)&=f(t,y(t)),\\quad y(a)=y_0,\\\\ z'(t)&=g(t,z(t)),\\quad z(a)=z_0. \\end{align*} Also assume there is a constant $C\\ge 0$ so that $$ \\|g(t,x_2)-g(t,x_1)\\|\\le C\\|x_2-x_1\\| $$ and a continuous function $\\phi\\colon [a,b]\\to [0,\\infty)$ so that $$ \\|f(t,y(t))-g(t,y(t))\\|\\le \\phi(t). $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.23639","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.23639/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}