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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 $$ \\|f(t,y(t))-g(t,y(t))\\|\\le \\phi(t). $$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2503.23639","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2025-03-31T00:53:32Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"3be23bcfe5e2fb99d09f6479dd5ff31fc29f095f617ffa60810b1789d6dc5008","abstract_canon_sha256":"a9e71ec32554106127d25fdbec9fecf7a1ec8719c26d17e1383c8c47dcd5bfb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:41:56.987716Z","signature_b64":"kFqdKXVZK7CtAhZ4DrVdmc5WUt5bGhN/WQfBpwlIem1PMNufD4XJltCNpY/AaMv/+WbhHVCdmrXxAxaMZuwJCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cdc4d4c6fe679694069a6c8d7637c677a06b893bdc95ce10530c2e3762793ba","last_reissued_at":"2026-07-05T10:41:56.987265Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:41:56.987265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2503.23639","created_at":"2026-07-05T10:41:56.987322+00:00"},{"alias_kind":"arxiv_version","alias_value":"2503.23639v1","created_at":"2026-07-05T10:41:56.987322+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.23639","created_at":"2026-07-05T10:41:56.987322+00:00"},{"alias_kind":"pith_short_12","alias_value":"HTOE2TDP4Z4W","created_at":"2026-07-05T10:41:56.987322+00:00"},{"alias_kind":"pith_short_16","alias_value":"HTOE2TDP4Z4WSQDJ","created_at":"2026-07-05T10:41:56.987322+00:00"},{"alias_kind":"pith_short_8","alias_value":"HTOE2TDP","created_at":"2026-07-05T10:41:56.987322+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.02927","citing_title":"Generalization Bounds of Spiking Neural Networks via Rademacher Complexity","ref_index":15,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5","json":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5.json","graph_json":"https://pith.science/api/pith-number/HTOE2TDP4Z4WSQDJU3ENOY34M5/graph.json","events_json":"https://pith.science/api/pith-number/HTOE2TDP4Z4WSQDJU3ENOY34M5/events.json","paper":"https://pith.science/paper/HTOE2TDP"},"agent_actions":{"view_html":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5","download_json":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5.json","view_paper":"https://pith.science/paper/HTOE2TDP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2503.23639&json=true","fetch_graph":"https://pith.science/api/pith-number/HTOE2TDP4Z4WSQDJU3ENOY34M5/graph.json","fetch_events":"https://pith.science/api/pith-number/HTOE2TDP4Z4WSQDJU3ENOY34M5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5/action/storage_attestation","attest_author":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5/action/author_attestation","sign_citation":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5/action/citation_signature","submit_replication":"https://pith.science/pith/HTOE2TDP4Z4WSQDJU3ENOY34M5/action/replication_record"}},"created_at":"2026-07-05T10:41:56.987322+00:00","updated_at":"2026-07-05T10:41:56.987322+00:00"}