{"corpus_id":37655547,"paper_sha":"f3478c833fa34f619782ca69bcf33e07c3ef6f13","doi":"10.1090/S0002-9939-1972-0298500-3","arxiv_id":null,"pmid":null,"pmcid":null,"mag_id":1969117912,"dblp_id":null,"acl_id":null,"title":"A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS","year":1972,"publication_date":null,"venue":"","journal":{"name":"","pages":"171-174","volume":"35"},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":6,"citation_count":1116,"influential_citation_count":182,"is_open_access":false,"arxiv_categories":null,"arxiv_license":null,"arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Let <italic>K</italic> be a subset of a Banach space <italic>X</italic> . A mapping <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F colon upper K right-arrow upper K\"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=\"false\"> → </mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">F:K \\to K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to be asymptotically nonexpansive if there exists a sequence <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace k Subscript i Baseline right-brace\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{ {k_i}\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of real numbers with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k Subscript i Baseline right-arrow 1\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=\"false\"> → </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{k_i} \\to 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i right-arrow normal infinity\"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo stretchy=\"false\"> → </mml:mo> <mml:mi mathvariant=\"normal\"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">i \\to \\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar upper F Superscript i Baseline x minus upper F Superscript i Baseline y double-vertical-bar less-than-over-equals k Subscript i Baseline double-vertical-bar x minus y double-vertical-bar comma x comma y element-of upper K\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo symmetric=\"true\">‖</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo symmetric=\"true\">‖</mml:mo> </mml:mrow> <mml:mo> ≦ </mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo symmetric=\"true\">‖</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo symmetric=\"true\">‖</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\left \\| {{F^i}x - {F^i}y} \\right \\| \\leqq {k_i}\\left \\| {x - y} \\right \\|,x,y \\in K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is proved that if <italic>K</italic> is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F colon upper K right-arrow upper K\"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=\"false\"> → </mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">F:K \\to K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is asymptotically nonexpansive, then <italic>F</italic> has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. 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