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We prove that there exists $x \\in A\\cup B$ such that $\\|x - Tx\\| = \\emph{dist}(A, B)$ whenever $T(A) \\subseteq B$, $T(B) \\subseteq A$. Also, we establish that if $T(A) \\subseteq A$ and $T(B) \\subseteq B$, then there exist $x \\in A$ and $y\\in B$ such that $Tx = x$, $Ty = y$ and $\\|x - y\\| = \\emph{dist}(A, B)$. We prove the aforesaid results when the pair $(A, B)$ has the rectangle pr"},"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":"1611.02484","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-11-08T11:36:00Z","cross_cats_sorted":[],"title_canon_sha256":"684b2797d1908a8f013ba346822e56bfbe3bbdf5e30a9a0ef8e56f819bc7221c","abstract_canon_sha256":"26342dd18b217d00baa82beb30f5447b642d0fa6cf6f22084e709af72afe13f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:51.919890Z","signature_b64":"ZTXsZAtGsLFyvC8L1nToHLU8P1YaWR1yPFxZ0aDvVOUJEks762ewPipyF89BH27YTHbGCoVYJJFNACj9Un3/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"edaf6b4690ee6bd6dda24947e6383053cf1bbd65353777e680b59dbac2164872","last_reissued_at":"2026-05-18T00:59:51.919219Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:51.919219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"P. Veeramani, S. Rajesh","submitted_at":"2016-11-08T11:36:00Z","abstract_excerpt":"Let $(A, B)$ be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and $T: A\\cup B \\rightarrow A\\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x \\in A\\cup B$ such that $\\|x - Tx\\| = \\emph{dist}(A, B)$ whenever $T(A) \\subseteq B$, $T(B) \\subseteq A$. Also, we establish that if $T(A) \\subseteq A$ and $T(B) \\subseteq B$, then there exist $x \\in A$ and $y\\in B$ such that $Tx = x$, $Ty = y$ and $\\|x - y\\| = \\emph{dist}(A, B)$. We prove the aforesaid results when the pair $(A, B)$ has the rectangle pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.02484","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":""},"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":"1611.02484","created_at":"2026-05-18T00:59:51.919333+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.02484v1","created_at":"2026-05-18T00:59:51.919333+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.02484","created_at":"2026-05-18T00:59:51.919333+00:00"},{"alias_kind":"pith_short_12","alias_value":"5WXWWRUQ5ZV5","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5WXWWRUQ5ZV5NXNC","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5WXWWRUQ","created_at":"2026-05-18T12:30:01.593930+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP","json":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP.json","graph_json":"https://pith.science/api/pith-number/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/graph.json","events_json":"https://pith.science/api/pith-number/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/events.json","paper":"https://pith.science/paper/5WXWWRUQ"},"agent_actions":{"view_html":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP","download_json":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP.json","view_paper":"https://pith.science/paper/5WXWWRUQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.02484&json=true","fetch_graph":"https://pith.science/api/pith-number/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/graph.json","fetch_events":"https://pith.science/api/pith-number/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/action/storage_attestation","attest_author":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/action/author_attestation","sign_citation":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/action/citation_signature","submit_replication":"https://pith.science/pith/5WXWWRUQ5ZV5NXNCJFD6MOBQKP/action/replication_record"}},"created_at":"2026-05-18T00:59:51.919333+00:00","updated_at":"2026-05-18T00:59:51.919333+00:00"}