{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:36Z","timestamp":1740134496515,"version":"3.37.3"},"reference-count":5,"publisher":"Wiley","license":[{"start":{"date-parts":[[2007,1,1]],"date-time":"2007-01-01T00:00:00Z","timestamp":1167609600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100004944","name":"Naresuan University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004944","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2007]]},"abstract":"<jats:p>Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E1\"><mml:mi>C<\/mml:mi><\/mml:math>be a nonempty closed convex subset of Hilbert space<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E2\"><mml:mi>H<\/mml:mi><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E3\"><mml:mi>P<\/mml:mi><\/mml:math>a metric projection of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E4\"><mml:mi>H<\/mml:mi><\/mml:math>onto<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E5\"><mml:mi>C<\/mml:mi><\/mml:math>and let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E6\"><mml:mi>T<\/mml:mi><\/mml:math>be a nonexpansive nonself-mapping from<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E7\"><mml:mi>C<\/mml:mi><\/mml:math>into<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E8\"><mml:mi>H<\/mml:mi><\/mml:math>. For a contraction<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E9\"><mml:mi>f<\/mml:mi><\/mml:math>on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E10\"><mml:mi>C<\/mml:mi><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E11\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>t<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><mml:mo>\u2286<\/mml:mo><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>, let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E12\"><mml:mrow><mml:msub><mml:mi>x<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math>be the unique fixed point of the contraction<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E13\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u21a6<\/mml:mo><mml:msub><mml:mi>t<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><mml:mi>f<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2212<\/mml:mo><mml:msub><mml:mi>t<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mn>(1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>n)<\/mml:mi><mml:mstyle displaystyle=\"true\"><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>j<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mi>n<\/mml:mi><\/mml:msubsup><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>P<\/mml:mi><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mi>j<\/mml:mi><\/mml:msup><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mstyle><\/mml:mrow><\/mml:math>. Consider also the iterative processes<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E14\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>y<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E15\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>z<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>generated by<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E16\"><mml:mrow><mml:msub><mml:mi>y<\/mml:mi><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><mml:mi>f<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:msub><mml:mi>y<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2212<\/mml:mo><mml:msub><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mn>(1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>(n<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1))<\/mml:mn><mml:msup><mml:mrow><mml:mstyle displaystyle=\"true\"><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>j<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mi>n<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>P<\/mml:mi><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mstyle><\/mml:mrow><mml:mi>j<\/mml:mi><\/mml:msup><mml:msub><mml:mi>y<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><mml:mo>,<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E17\"><mml:mrow><mml:msub><mml:mi>z<\/mml:mi><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mn>(1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>(n<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1))<\/mml:mn><mml:mstyle displaystyle=\"true\"><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>j<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mi>n<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mi>P<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:msub><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><mml:mi>f<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:msub><mml:mi>z<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2212<\/mml:mo><mml:msub><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>T<\/mml:mi><mml:mi>P<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mi>j<\/mml:mi><\/mml:msup><mml:msub><mml:mi>z<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>,<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><\/mml:mrow><\/mml:mstyle><\/mml:mrow><\/mml:math>where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E18\"><mml:mrow><mml:msub><mml:mi>y<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mi>z<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msub><mml:mo>\u2208<\/mml:mo><mml:mi>C<\/mml:mi><mml:mo>,<\/mml:mo><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>is a real sequence in an interval<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E19\"><mml:mrow><mml:mrow><mml:mo>[<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>. Strong convergence of the sequences<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E20\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>x<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><mml:mo>,<\/mml:mo><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>y<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E21\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>z<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>to a fixed point of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E22\"><mml:mi>T<\/mml:mi><\/mml:math>which solves some variational inequalities is obtained under certain appropriate conditions on the real sequences<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E23\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>\u03b1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E24\"><mml:mrow><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:msub><mml:mi>t<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2007\/48648","type":"journal-article","created":{"date-parts":[[2007,4,16]],"date-time":"2007-04-16T09:02:46Z","timestamp":1176714166000},"page":"1-10","source":"Crossref","is-referenced-by-count":0,"title":["Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces"],"prefix":"10.1155","volume":"2007","author":[{"given":"Rabian","family":"Wangkeeree","sequence":"first","affiliation":[]}],"member":"311","reference":[{"doi-asserted-by":"publisher","key":"1","DOI":"10.1016\/0362-546X(94)00278-P"},{"doi-asserted-by":"publisher","key":"2","DOI":"10.1006\/jmaa.1997.5398"},{"doi-asserted-by":"publisher","key":"3","DOI":"10.1016\/j.jmaa.2004.02.010"},{"issue":"1","key":"4","first-page":"171","volume":"57","year":"2003","journal-title":"Scientiae Mathematicae Japonicae"},{"doi-asserted-by":"publisher","key":"5","DOI":"10.1016\/j.jmaa.2004.04.059"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2007\/048648.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2007\/048648.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,8,11]],"date-time":"2016-08-11T10:15:43Z","timestamp":1470910543000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2007\/048648\/abs\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007]]},"references-count":5,"alternative-id":["048648","48648"],"URL":"https:\/\/doi.org\/10.1155\/2007\/48648","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2007]]}}}