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The main assumption in this theorem is the separation among bounded solutions of homogeneous equations $$ x'=A(t)x\\ .\\ \\ \\ (**) $$ In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. In this case, the separation from zero of "},"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":"1907.02512","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-07-03T09:24:48Z","cross_cats_sorted":[],"title_canon_sha256":"35551904600b53de7ecaf278f6dac54955952effbd84876e5916fe92461d4d3d","abstract_canon_sha256":"f82931414a33ebb2cac52e478f3f7473e1055313dcd631816cae4efc2f8c803b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:28.355511Z","signature_b64":"pTwfoCxdWrtaHjy/AxxdtWanxe7sBkKXQIVmkqhtgidPje9UNCCsXhhLNpmP3nVk41di8ur5/sYZqb0NnODmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fa3897255aff2351a25b6266f26b2370add3ba4cf606135a36f2c57686ed3be","last_reissued_at":"2026-05-17T23:41:28.354763Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:28.354763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Levitan Almost Periodic Solutions of Linear Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"David Cheban","submitted_at":"2019-07-03T09:24:48Z","abstract_excerpt":"The known Levitan's Theorem states that the linear differential equation $$ x'=A(t)x+f(t) \\ \\ \\ (*) $$ with Bohr almost periodic coefficients $A(t)$ and $f(t)$ admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations $$ x'=A(t)x\\ .\\ \\ \\ (**) $$ In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. 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