{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:36:04Z","timestamp":1753886164647,"version":"3.41.2"},"reference-count":9,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2008,12,3]],"date-time":"2008-12-03T00:00:00Z","timestamp":1228262400000},"content-version":"vor","delay-in-days":337,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["10771173","2008B110012"],"award-info":[{"award-number":["10771173","2008B110012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation Project of Henan","award":["10771173","2008B110012"],"award-info":[{"award-number":["10771173","2008B110012"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Let <jats:italic>K<\/jats:italic> be a nonempty closed convex subset of a reflexive and strictly convex Banach\nspace <jats:italic>E<\/jats:italic> with a uniformly <jats:italic>G\u00e2teaux<\/jats:italic> differentiable norm, <jats:italic>\u2131<\/jats:italic> = {<jats:italic>T<\/jats:italic>(<jats:italic>h<\/jats:italic>) : <jats:italic>h<\/jats:italic> \u2265 0} a generalized asymptotically\nnonexpansive self\u2010mapping semigroup of <jats:italic>K<\/jats:italic>, and <jats:italic>f<\/jats:italic> : <jats:italic>K<\/jats:italic> \u2192 <jats:italic>K<\/jats:italic> a fixed contractive mapping\nwith contractive coefficient <jats:italic>\u03b2<\/jats:italic> \u2208 (0, 1). We prove that the following implicit and modified implicit\nviscosity iterative schemes {<jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>} defined by <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> = <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) + (1 \u2212 <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>)<jats:italic>T<\/jats:italic>(<jats:italic>t<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>)<jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> and <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> = <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:italic>y<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> + (1 \u2212 <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>)<jats:italic>T<\/jats:italic>(<jats:italic>t<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>)<jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:italic>,<\/jats:italic><jats:italic>y<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> = <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>) + (1 \u2212 <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>)<jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub> strongly converge to <jats:italic>p<\/jats:italic> \u2208 <jats:italic>F<\/jats:italic> as <jats:italic>n<\/jats:italic> \u2192 <jats:italic>\u221e<\/jats:italic> and <jats:italic>p<\/jats:italic> is the unique solution to the following variational\ninequality: \u2329<jats:italic>f<\/jats:italic>(<jats:italic>p<\/jats:italic>) \u2212 <jats:italic>p<\/jats:italic>, <jats:italic>j<\/jats:italic>(<jats:italic>y<\/jats:italic> \u2212 <jats:italic>p<\/jats:italic>)\u232a \u2264 0 for all <jats:italic>y<\/jats:italic> \u2208 <jats:italic>F<\/jats:italic>.<\/jats:p>","DOI":"10.1155\/2008\/687815","type":"journal-article","created":{"date-parts":[[2008,12,3]],"date-time":"2008-12-03T20:33:04Z","timestamp":1228336384000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces"],"prefix":"10.1155","volume":"2008","author":[{"given":"Yali","family":"Li","sequence":"first","affiliation":[]},{"given":"Jianjun","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Lei","family":"Deng","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2008,12,3]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00251595"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.04.059"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2007.05.021"},{"key":"e_1_2_6_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2006.06.009"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.09.023"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(81)90013-5"},{"volume-title":"Nonlinear Functional Analysis: Fixed Point Theory and Its Applications","year":"2000","author":"Takahashi 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